PhD Thesis - Universität Augsburg

Phase Transitions and Real-Time Dynamics
of Spin and Charge Densities
in One-Dimensional Correlated Systems
Michael Sekania
Augsburg 2011

Phase Transitions and Real-Time Dynamics
of Spin and Charge Densities
in One-Dimensional Correlated Systems
Dissertation
zur Erlangung des Grades eines
Doktors der Naturwissenschaften
(Dr. rer. nat.)
eingereicht an
der Mathematisch-Naturwissenschaftlichen Fakultät
der Universität Augsburg
von M.Sc.
Michael Sekania
aus Tbilisi (Tiflis), Georgien
Augsburg 2011

Erstgutachter:
Zweitgutachter:
Prof. Dr. Arno P. Kampf
Prof. Dr. Thilo Kopp
Tag der Einreichung: 17. Januar 2011
Tag der mündlichen Prüfung: 03. Juni 2011

Contents
i
CONTENTS
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1. Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1 Model Hamiltonian for electrons in solids . . . . . . . . . . . . . . . . . . . 7
1.2 Hubbard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 t-J and Heisenberg models . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4 Extended Hubbard models . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4.1 Next-nearest neighbor interaction . . . . . . . . . . . . . . . . . . . 21
1.4.2 Peierls distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.4.3 Ionic potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2. Density-Matrix Renormalization Group . . . . . . . . . . . . . . . . . . . . . 23
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Matrix-Product State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.1 MPS, Blocks, and a Superblock . . . . . . . . . . . . . . . . . . . . 28
2.2.2 Operators in block effective basis . . . . . . . . . . . . . . . . . . . 30
2.3 Density-Matrix Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4 Infinite-System DMRG algorithm . . . . . . . . . . . . . . . . . . . . . . . 43
2.5 Finite-System DMRG algorithm . . . . . . . . . . . . . . . . . . . . . . . . 46
2.6 Implementation details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.6.1 Superblock Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 51
2.6.2 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.6.3 Wave-function transformations . . . . . . . . . . . . . . . . . . . . . 55
2.6.4 Additive quantum numbers . . . . . . . . . . . . . . . . . . . . . . 57
3. Real-time evolution using DMRG . . . . . . . . . . . . . . . . . . . . . . . . 61
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2 Early attempts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.3 Adaptive time-dependent DMRG . . . . . . . . . . . . . . . . . . . . . . . 68
3.4 Time-step targeting adaptive time-dependent DMRG . . . . . . . . . . . . 74
3.4.1 Performing time evolution . . . . . . . . . . . . . . . . . . . . . . . 75

ii
Contents
3.4.2 Hilbert space adaption strategy . . . . . . . . . . . . . . . . . . . . 78
3.4.3 Time-step targeting adaptive time-dependent DMRG based on the
Krylov-subspace methods for the time-evolution . . . . . . . . . . . 81
3.5 Accuracy of adaptive time-dependent DMRG . . . . . . . . . . . . . . . . . 85
3.5.1 t-DMRG based on the Suzuki-Trotter time-evolution scheme . . . . 89
3.5.1.1 Analytical estimates . . . . . . . . . . . . . . . . . . . . . 89
3.5.1.2 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . 92
3.5.2 t-DMRG based on the Arnoldi method for the time-evolution . . . 113
3.5.3 Comparison and summary . . . . . . . . . . . . . . . . . . . . . . . 118
3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4. Nature of the Band- to Mott-insulator transition in one-dimension . . . . 123
4.1 Ionic Hubbard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.1.2 Symmetry analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.1.3 Exact diagonalization results . . . . . . . . . . . . . . . . . . . . . . 128
4.1.4 DMRG results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.1.4.1 Excitation gaps . . . . . . . . . . . . . . . . . . . . . . . . 129
4.1.4.2 Correlation functions . . . . . . . . . . . . . . . . . . . . . 133
4.2 Adiabatic Holstein-Hubbard model . . . . . . . . . . . . . . . . . . . . . . 139
4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4.2.2 Theoretical models . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
4.2.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
4.2.3.1 Charge- and spin-structure factors . . . . . . . . . . . . . 140
4.2.3.2 Optical response . . . . . . . . . . . . . . . . . . . . . . . 142
4.2.4 Phase diagram in the adiabatic limit . . . . . . . . . . . . . . . . . 143
4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5. Real-time Dynamics of Spin and Charge Densities in the One-dimensional
Hubbard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.1 One-dimensional Hubbard model . . . . . . . . . . . . . . . . . . . . . . . 151
5.2 Computational setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5.2.1 Preparation of the initial state . . . . . . . . . . . . . . . . . . . . . 155
5.2.2 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
5.2.3 Simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.3 Real-time evolution in the tight-binding model . . . . . . . . . . . . . . . . 157
5.4 Real-time evolution in the Hubbard model . . . . . . . . . . . . . . . . . . 159
5.4.1 System at half-filling . . . . . . . . . . . . . . . . . . . . . . . . . . 159
5.4.2 System close to half-filling . . . . . . . . . . . . . . . . . . . . . . . 170

Contents
iii
5.5 Ballistic vs. subdiffusive dynamics . . . . . . . . . . . . . . . . . . . . . . . 177
5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
6. Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
A. Free Spinless Lattice Fermions in One-Dimension . . . . . . . . . . . . . . . 193
A.1 Time evolution of a wave packet . . . . . . . . . . . . . . . . . . . . . . . . 195
A.2 Time evolution of Fock states . . . . . . . . . . . . . . . . . . . . . . . . . 202
B. Ionic-Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

Introduction 1
INTRODUCTION
Theoretical understanding of strongly correlated electron systems is one of the most challenging
areas of modern solid state physics. The complex interplay between electronic,
spin and lattice degrees of freedom in such materials results in rich phase diagrams and
interesting transport phenomena, making these systems particularly interesting in view of
possible technological applications. High temperature superconductivity, metal-insulator
transitions, colossal magnetoresistance, multiferroic behavior, and colossal magnetocapacitive
effect — this is probably the most representative but, certainly, incomplete list of
exotic phenomena recently found in these systems. In addition, many correlated electron
systems exhibit a great sensitivity with respect to changes in internal or external parameters
such as doping, pressure, temperature, magnetic or electric fields, etc. All these
make correlated electron systems one of the basic playgrounds for future material design
applications.
A specific class of strongly correlated materials, the so-called quasi one-dimensional systems
such as nanowires, nanoneedles, nanorings, nanobelts, nanotubes etc., has recently
attracted considerable interest due to novel developments in the field of nanotechnology.
These artificial structures have rapidly become important model systems where parameters
can be more easily tuned to explore broader regions of the phase diagram of one-dimensional
correlated electron systems. Bulk quasi one-dimensional organic compounds, e.g. conjugated
polymers and charge transfer salts, had been known since the 1970’s. However,
recent progress in chemical synthesis and upheaval in organic electronics have breathed
new life into these materials. There are also bulk quasi one-dimensional inorganic materials
like halogen-bridged mixed-metal compounds and transition metal oxides which, as it
has been recently found, show gigantic optical nonlinearity. All these fascinating materials
exhibit novel physical properties owing to their unique geometry and strongly anisotropic
electronic structures, and provide potential building blocks for a wide range of nanoscale
electronics, optoelectronics, magnetoelectronics, and sensing devices.
For the effective design of “smart” materials with advanced properties that are required
for successful technological applications, it is crucial to have a fundamental understanding
of their characteristics under different conditions. Unfortunately, both experimental as
well as theoretical descriptions of such systems are quite problematic. Exact calculations
help in general to interpret and predict experimental data and create possibilities for more

2 Introduction
complex experimental investigations, revealing the properties of these materials in more
detail. However, even for a small cluster of atoms these calculations are very complicated,
since the number of parameters describing the physical states grows exponentially with
the number of particles. This strongly limits the size of investigated clusters. Many
problems in condensed matter physics can be described effectively within a single-particle
picture, but for many interesting and important quantum systems a description in terms
of weak effective interactions is not possible (e.g. Mott insulators). In systems where
interactions lead to strong correlation effects between the system components, mapping to
an effective single particle model can lead to an erroneous description of their behavior.
In low dimensions the situation is even more complicated since interactions are typically
enhanced due to spatial confinement of the electrons. In such cases the necessity of a full
many body description of the problem is unavoidable. To keep the complexity of these
problems manageable and to gain insight into the fundamental principles often substantial
approximations and far-going simplifications are required.
Strongly correlated systems are typically described by simplified models, such as the
Hubbard- or Heisenberg-type Hamiltonians, which are believed to capture some of their
essential physics. Despite the apparent simplicity of the Hamiltonians (see e.g. chapter 1),
few analytical exact solutions are available. Certain models in one-dimension can be solved
exactly by means of the Bethe ansatz, but in general, approximate methods like perturbation
theory must be applied. Some analytical approximations in certain limits do exist, but
the absence of a dominant exactly solvable contribution over the entire parameter range
has ultimately limited the applicability and controlled reliability of conventional perturbative
methods. Due to these reasons, numerical techniques have become an essential tool in
handling the inherent complexity of strongly correlated systems. The standard numerical
tools are well represented by exact diagonalization (ED), quantum Monte Carlo (QMC), numerical
renormalization group (NRG), and density matrix renormalization group (DMRG)
methods.
Recent progress in quantum state engineering in ultracold atomic gases has opened
unique opportunities to realize and manipulate a range of lattice models for strongly correlated
quantum systems. Subjected to an optical lattice, these gases are arguably the
purest realizations of the typical model Hamiltonians of strong correlation physics, such as
the Hubbard model [88] (for recent reviews see Refs. [23, 125, 152]). More important, the
interaction parameters, while being known precisely from microscopic calculations, can be
tuned experimentally over a huge range of values on quantum mechanically relevant timescales.
The unprecedented control over the parameters of the system additionally allows
realizations of various lattice geometries from one- to three-dimensional, and investigations
of non-equilibrium dynamics as well as excited many-body states. The developments in the
fields of nanoelectronics and spintronics also raise the questions how strongly correlated
quantum many-body systems react on external time dependent perturbations and how the

Introduction 3
transport can be calculated quantitatively far from equilibrium. These questions become
particularly important in the context of transport and decoherence as the size of devices
continues to shrink towards the atomic scale.
The theoretical description of time-dependent and out-of-equilibrium properties of
strongly correlated quantum systems is becoming one of the most challenging and fascinating
topics in condensed matter. In general, the level of the current knowledge in
this field is far from being complete, mainly due to the lack of controlled approaches as
well as well established theoretical concepts. Recently, there has been a progress in the
development of new approaches in this direction. It is worthwhile to mention that there is
the extension of dynamical mean field theory (DMFT), which allows the study of out-ofequilibrium
problems [46, 47, 71, 70, 236, 237]. In combination with the recently developed
continuous-time QMC techniques [176, 207, 254] based on stochastic samplings of diagrammatic
expansions of the time-evolution operator, nonequilibrium DMFT seems to become
a promising approach in high dimensions [48, 49, 253].
In the last decade considerable progress was made in the development of numerical tools
for simulations in low dimensions. In principle, all physical quantities can be accurately and
directly determined via exact diagonalization. However this approach fundamentally suffers
from the exponential increase of the computational effort with the number of quantum
subsystems composing the total system. While the use of iterative schemes, such as Lanczos,
can extend the usefulness of ED [12, 111, 161, 203], this intractable scaling ultimately
limits it to very small or restricted systems which are often not adequate to describe properties
of solids. In this respect QMC techniques are more attractive [69, 89, 185]. The ground
state QMC calculations scales cubically in the number of particles and the ground state
properties of a wide-ranging class of many-body Hamiltonians can be efficiently evaluated
for moderately sized systems in any dimensions. Extensions of QMC to dynamical time
evolution also exist [176, 207, 254]. However, despite the great potential of this method,
there are several restrictions and handicaps inherent to all QMC techniques, namely the
fermionic sign problem encountered in equilibrium (imaginary-time) simulations of strongly
correlated many-fermion systems, and the dynamical sign problem in real-time (dynamical)
simulations, which already shows up for a single particle.
For one-dimensional systems in particular, DMRG has enabled us to calculate the
static and dynamical properties of systems much larger than those possible with ED [129,
130, 196, 210]. Since the invention of DMRG by Steven White in 1992 [255, 256], it
has been under constant development. The algorithm was rapidly extended and adapted
to different situations, and has become the most reliable and versatile method for 1D
quantum systems. A multitude of quasi one-dimensional systems has been investigated
using this method [99, 100, 210]. Extensions to calculate dynamical correlation functions
and finite temperature properties have been developed in the course of the years [129, 130].
However, these extensions are mainly applicable to systems in equilibrium. Recently, the

4 Introduction
DMRG has been extended to treat time-dependent systems in situations out of equilibrium
[38, 67, 258], opening up new possibilities for the investigation of their properties. These
new techniques provide valuable insight into quantum systems assisting the progress of
several forefront areas of research, both in science — e.g., condensed matter, quantum
optics, atomic and nuclear physics, quantum chemistry — and technology — e.g., quantum
information processing, quantum computation, nanotechnology.
For this reasons it is of great interest to develop and improve unbiased numerical tools
that allow to study complex systems in different interesting setups. In the first part of
this thesis I describe a standard DMRG algorithm with all possible improvements that are
required for the development of the efficient computer code and provide the possibilities for
further extension of the method in order to investigate time-dependent problems. I devise
two different variants of the time-dependent DMRG algorithms and perform a comprehensive
analysis of their accuracy using exactly solvable models. These new methods are
flexible enough and allow to investigate — with high accuracy — diverse time-evolution
out-of-equilibrium problems in arbitrary quasi one-dimensional systems. Developed methods
help to understand the physical properties of complex systems and serve as a testing
playground for different new models or new modifications of already existing ones. They
also provide advanced tools for elaborate numerical experiments.
Besides methodical objectives of this thesis (illustration and further development of
mentioned techniques) my goal will be to use devised methods in order to analyze the
ground state properties of the different extended Hubbard models. In particular, in the
second part of this thesis I study the so-called ionic Hubbard model with an additional
alternating modulation of the on-site energies and discuss the possible existence of third
intermediate insulating phase between the band- and Mott-insulating phases. I also study
the real time dynamics of spin- and charge- densities induced by adding a single particle
in the ground state of the Hubbard model in the Mott-insulating phase (half band filling)
or close to it (one electron less). These investigations are particularly interesting, since
gigantic optical nonlinear properties, potentially useful for applications, e.g., terahertz
optical switches or even solar cells, have been reported in 1D Mott-Hubbard materials,
such as cuprates and halogen-bridged Ni-compounds [123, 137, 138, 163, 164].
Structure of this thesis:
• Chapter 1 presents the problem of strong electronic correlations in solids and discusses
typical models that are expected to capture some of the existing complexity in
interacting electronic system on a lattice. First, the Hubbard model is derived starting
from a general Hamiltonian describing the solid state matter. In addition, several
other models such as the t − J and the quantum Heisenberg models are introduced
as effective low-energy models of the Hubbard Hamiltonian in the limit of strong
interactions. These models are believed to capture some of the essential physics of

Introduction 5
strongly correlated electron systems. The chapter is closed with the presentation of
the extended Hubbard models that allow a more realistic treatment of the physical
systems.
• Chapter 2 discusses the methodology of the Density-Matrix Renormalization Group
based on Matrix Product States (MPS) and the density matrix projection. This
chapter provides the theoretical foundation that is necessary to understand how the
actual DMRG computations are performed and contains information crucial for the
method’s extensions. The standard DMRG algorithms for calculating ground state
properties of quantum lattice many-body system such as the infinite- and finitesystem
algorithms are presented. The chapter is also supplemented by additional
algorithmic improvements that are relevant for the development of an efficient computer
code, which are also included in the DMRG tool developed by us. The subsequent
Chapter 3 builds on the descriptions provided in this chapter as a background.
• The extensions to the standard DMRG algorithms that allow to study problems
with real-time evolution are presented in the Chapter 3. Two efficient time-dependent
DMRG algorithms with necessary technical and implementation details are discussed
in this chapter. A comprehensive analysis of the accuracy of the considered methods
based on the analytical solutions of exactly solvable models — studied in Appendix
A — are presented in Section 3.5. In the conclusion of this chapter I underline
advantages and differences of both devised t-DMRG methods and discuss the timeevolution
problems that can be studied by these methods.
• Applications of the developed DMRG method are presented in Chapter 4, where the
ground state phases of the ionic Hubbard as well as the adiabatic Holstein-Hubbard
model are studied. These models are also derived and motivated in Chapter 1. In the
case of the half-filled ionic Hubbard model a continuous transition from the bandto
the correlated-insulator phase is clearly identified. Shortly after the transition a
strong signal for a long range bond ordered density wave is identified in the correlatedinsulator
phase. With further increase of the interaction strength scaling behavior of
the staggered bond density and spin-spin correlation functions changes qualitatively
and approaches the scaling behavior of the Hubbard model (Mott insulator). In
the adiabatic Holstein-Hubbard model an additional alternating modulation in onsite
energy is caused by the lattice static distortion that leads to an extra lattice
elastic energy. Two scenarios emerge with a discontinuous band- to Mott-insulator
transition for strong coupling and two continuous transitions for weak coupling with
an intermediate phase where a long range charge-density wave order persists.
• In Chapter 5 real-time properties of strongly correlated systems are studied using
the time-dependent DMRG methods. I present results of the real-time dynamics

6 Introduction
of spin- and charge- densities induced by adding of a single particle to the ground
state of the Hubbard model in the Mott-insulating phase (half band filling) or in the
metallic phase close to it (one electron less). Even in this case of an extremely localized
initial perturbation, effects of spin-charge separation are clearly identified in
the space-time evolution of the spin and charge densities. In the first case, where the
Mott-insulating ground state serves as a “host” system, ballistic spreading of induced
spin and charge densities is observed. The speed of the charge-density propagation
is enhanced in comparison with the non-interacting case and weakly depends on the
on-site interaction strength. In contrast, the velocity of the spin-density spreading
is heavily influenced by the interaction strength. Particularly interesting is the second
case, where the induced charge density stops to propagate ballistically and the
spreading is rather subdiffusive.
• The summary and outlook close the thesis and the appendices provide examples
where an analytical solution for the time evolution of different initial states can be
still constructed.

7
1. MODELS
As it is well known, a macroscopic sample of any solid material consists of a very huge
number of atoms (O(10 23 ) cm −3 ) composed by a nucleus and electrons carrying positive
and negative charges, respectively. In general, a quantum-mechanical description of this
many-particle system requires a with the particle number exponentially growing number
of parameters. Therefore, in order to keep the complexity of the problem manageable
and to gain insight into the fundamental principles of collective behavior often bold
approximations and far-going simplifications are required. In this chapter we set up the
general Hamiltonian operator of solid-state physics and gradually approximate it to derive
the Hamiltonian for the purely electronic problem.
1.1 Model Hamiltonian for electrons in solids
The only relevant elementary interaction in solids is the electromagnetic interaction between
charged particles, i.e., nuclei and electrons. To treat the inter-atomic physics responsible
for the special properties of the many-body system making up a solid it is sufficient
to use a non-relativistic description in contrast to the intra-atomic physics of single atoms.
The relativistic effects, e.g., spin-orbit interaction, can also be neglected for most solids,
except those having constituents with large atomic masses, e.g. uranium. Thus, in the
following the quantum-mechanical description of a solid is considered to be given by the
non-relativistic Hamiltonian [77]
H tot = H (N)
kin + H(N−N) int + H (e)
kin + H(e−e) int + H (e−N)
int , (1.1a)
where the kinetic energy terms of the N N nuclei with atomic masses M m and of the N e
electrons with the electronic mass m e are given by
N N
H (N)
kin = ∑ P 2 m
,
2M m
m=1
N e
H (e)
kin = ∑ p 2 i
,
2m e
i=1
(1.1b)
where P m and p i are the momentum operators of the nuclei and electrons, respectively. The
electromagnetic interaction terms between the nuclei with atomic numbers Z m (N − N),

8 1. Models
the electrons (e − e), and the electrons and the nuclei (e − N) are given by
H (N−N)
int
= e2
2
N N
∑
n≠m=1
Z n Z m
|R n − R m | , H(e−e) int = e2
2
H (e−N)
int
= − e 2 N N
∑
∑N e
m=1 i=1
∑N e
i≠j=1
1
|r i − r j | ,
Z m
|R m − r i | , (1.1c)
where R m and r i are positions of the nuclei and electrons, respectively. The above Hamiltonian
H tot defines a quantum mechanical many-particle problem, which is impossible to
solve directly. Therefore further approximations and assumptions are necessary in order to
obtain models that are either solvable or at least tractable by some powerful approximate
methods. In the following, such simplified models, which are later studied in detail in the
following chapters of the present thesis, are derived and discussed. The derivation follows
the books by F. Gebhard [77] and A. Auerbach [11].
The binding energy of the condensed solid state material is about O(1 . . .10 eV) per
atom. The physics of electrons whose binding energy to the nucleus is much larger than
this energy scale is well described by the results obtained for an isolated atom. The nuclei
together with these electrons having a large binding energy can thus be treated as if one
had isolated ions. During crystallization, only the electron orbitals of the shells that are
not completely filled, and for which the binding energy of the electrons to the nucleus is
up to O(10 eV), will be substantially changed. For condensed states it is thus sufficient to
focus on the valence electrons, the interaction among them, their interaction with the ions,
and the ion-ion interaction.
The next step is to restrict the model to crystalline solids, in which the ions form
a regular lattice. Since the ion masses are much larger than the electron masses, the
motion of the ions in such a lattice at temperatures well below the condensation energy
is much slower than the motion of the electrons. Thus, mainly the electron dynamics is
responsible for many properties and collective phenomena in crystalline solids, such as
magnetic properties, or whether the material is conducting or insulating. It is therefore
justified to separate the motion of the electrons from the motion of the ions and the ion-ion
interaction, and consider the so-called adiabatic or Born-Oppenheimer approximation where
the ion positions are treated as fixed from the electronic point of view. The interaction
of the electrons with the ions can now be described by a periodic potential in which the
electrons move. This might sound like a bad approximation, since the periodic ion array
may respond to the presence of the electrons, but when it is required, it is also possible to
incorporate some aspects of the lattice dynamics in a purely electronic effective model. For
example, the electron-ion interaction can induce static lattice deformations that result in a
new potential with a different periodicity if the energy gain for the electrons moving in this
new potential is larger then the elastic energy necessary for the lattice deformation (e.g.

1.1. Model Hamiltonian for electrons in solids 9
Peierls effect) [193]. One ends up with a purely electronic model, in which static lattice
deformations are taken into account by this new potential. Therefore, the Hamiltonian
describing the electron dynamics in a crystalline solid takes the form
where H (e−I)
int
forming the lattice.
H (e)
tot = H (e)
kin + H(e−e) int + H (e−I)
int , (1.2)
= ∑ N e
i=1 V (e−I) (r i ) is the periodic potential resulting from the ions which are
If one neglects the electron-electron interaction H (e−e)
int in (1.2), the problem reduces
to a problem of many independent electrons, whose wave function is given by a Slater
determinant of single electron wave functions. The wave function of a single electron
moving in the periodic potential V (e−I) is obtained by solving the band structure equation
[ ] p
2
+ V (e−I) (r) φ kbσ (r) = ǫ kb φ kbσ (r) , (1.3)
2m e
where φ kbσ (r) and ǫ kb are the Bloch wave function and band energy, respectively [10, 22];
k is the electron’s quasi momentum, σ =↑, ↓ is the z-component of its spin, and b denotes
the band index. The Bloch wave functions form a basis of one-particle states and obey the
following relation [10, 22]
φ kbσ (r + R) = e ik·R φ kbσ (r) . (1.4)
Here the vector R describes any translation that maps the lattice onto itself. The eigenenergies
for the independent-electron problem are
∑N e
E(k tot ) = ǫ ki b i
, (1.5)
where k tot = ∑ i k i is the total momentum of the N e electrons. In the ground state the
lowest N e states are occupied, and the energy of the uppermost occupied state defines the
Fermi energy
E F = max ǫ ki b i
. (1.6)
i=1,...,N e
A complementary one-particle basis is given by the Wannier wave functions [10, 249, 250],
which are obtained from the Bloch wave functions by the following Fourier transformation
φ ibσ (r) = 1 √
L
∑
k∈BZ
i=1
e −ik·R i
φ kbσ (r − R i ) . (1.7)
Here, the sum over k runs over all momenta in the first Brillouin zone (BZ), and L denotes
the number of lattice sites. The Wannier wave functions are localized at a lattice site i with
position R i , and for narrow bands are almost identical with atomic orbitals. The index

10 1. Models
b for the Wannier functions denotes the orbital, and bands can be classified according to
their corresponding atomic orbitals.
It is convenient to use the second quantization formalism by introducing electron creation
(annihilation) operators c † α (c α ) operating on an occupation-number Fock space, in
which the many particle wave function can be written as a Fock state
|n α , n α ′, n α ′′, . . . 〉 , (1.8)
where n α are the eigenvalues of the number operator n α = c † αc α that measures the number
of particles in the state α. Due to Pauli’s principle, n α can only take the values 0 and 1,
and the creation/annihilation operators obey the anti-commutation relations
{c † α, c † β } = c† αc † β + c† β c† α = 0
{c α , c β
} = 0 (1.9)
{c † α , c β } = δ α,β .
Here, α and β denote triple indices that represent a spin index σ, an atomic orbital or
Bloch band index b, and a momentum or real space (lattice site) coordinate k or i, respectively.
The operators in momentum and real space are connected by the discrete Fourier
transformation
c † ibσ = √ 1 ∑
e −ik·R i
c † kbσ . (1.10)
L
k∈BZ
Using (1.7) and (1.10) the field operator ψ σ(r), † which creates an electron with spin σ
localized at r, can be expressed in terms of the above creation operators and wave functions
in two different ways
ψ σ(r) † = ∑
φ ∗ ibσ (r)c† ibσ . (1.11)
k∈BZ,b
φ ∗ kbσ (r)c† kbσ = ∑ i,b
Here “*” denotes complex conjugation. ψ † σ (r) and ψ σ ′ (r ′ ) obey the anti-commutation relations
{ψ † σ (r), ψ† σ ′ (r ′ )} = 0 ,
{ψ σ (r), ψ σ ′(r ′ )} = 0 , (1.12)
{ψ † σ (r), ψ σ ′ (r ′ )} = δ(r − r ′ )δ σ,σ ′ .
The local electron density operator is given by
ρ(r) = ∑ σ
ψ † σ (r)ψ σ (r) = ∑ i,b,σ
j,b ′ ,σ ′ φ ∗ ibσ (r)φ jb ′ σ ′ (r) c † ibσ c jb ′ σ ′ . (1.13)

1.1. Model Hamiltonian for electrons in solids 11
The electron-electron interaction part of the Hamiltonian (1.2) can be rewritten in terms
of local electron densities as
H (e−e)
int
= e2
2
= e2
2
= e2
2
∑N e
i≠j=1
∫
∫
1
|r i − r j |
d 3 r d 3 r ′ 1
|r i − r j | [ρ(r)ρ(r′ ) − δ(r − r ′ )ρ(r)]
d 3 r d 3 r ′ 1 ∑
ψ σ † |r i − r j |
(r)ψ† σ
(r ′ )ψ ′ σ ′(r ′ )ψ σ (r ′ ) , (1.14)
σ,σ ′
where in the last line the self-interaction term of the electrons is eliminated by normal
ordering of the field operators ψ † σ(r), making use of the anti-commutation relation (1.12).
Finally, this result is easily expressed in the site-local occupation-number space by
H (e−e)
int = ∑ [ e
2
∫
2
i,j,l,m
b,b ′ ,b ′′ ,b ′′′
σ,σ ′
= ∑
d 3 r d 3 r ′ 1
|r − r ′ | φ∗ ibσ (r)φ∗ jb ′ σ ′ (r ′ )φ lb ′′ σ ′ (r ′ )φ mb ′′′ σ (r) ]
× c † ibσ c† jb ′ σ ′ c lb ′′ σ ′ c mb ′′′ σ
V bb′ b ′′ b ′′′
i,j,l,m
b,b ′ ,b ′′ ,b ′′′
σ,σ ′
ijlmσσ ′ c† ibσ c† jb ′ σ ′ c lb ′′ σ ′ c mb ′′′ σ . (1.15)
This term is quartic in the creation/annihilation operators, and thus makes H (e)
tot a truly
correlated many-particle problem. It is the origin of all electron correlation phenomena
occurring in solids, such as magnetic ordering or the electronic properties of the normal
state of high-T C superconductors.
Similarly, the independent-electron part of the Hamiltonian (1.2) can be expressed as
H (e)
kin + H(e−I) int = ∑ ∫
σ,σ ′
= ∑ [∫
i,b,σ
j,b ′ ,σ ′
[ ] p
d 3 r ψ σ(r)
† 2
+ V (e−I) (r) ψ
2m σ ′(r)
e
[ ]
p
d 3 r φ ∗ 2
ibσ (r) + V (e−I) (r)
]φ jb
2m ′ σ ′(r) c † ibσ c jb ′ σ ′
e
= − ∑ t bb′
ij c† ibσ c jb ′ σ
(1.16)
′
i,b,σ
j,b ′ ,σ ′
where in the case of equal lattice sites after inserting relation (1.3) the hopping matrix

12 1. Models
elements t bb′
ij
are expressed as
1 ∑
ij = −δ bb ′ e −i(R j−R i )·k ǫ
L
kb
. (1.17)
t bb′
k∈BZ
Thus, the Hamiltonian (1.2) describing electrons on a lattice reads
H (e)
tot = − ∑
t b ij c† ibσ c jbσ +
∑
i,j,b,σ
i,j,l,m
b,b ′ ,b ′′ ,b
σσ ′ V bb′ b ′′ b ′′′
ijlmσσ ′ c† ibσ c† jb ′ σ
c ′ lb ′′ σ
c ′ mb ′′′ σ
(1.18)
′′′
Unfortunately, even after restricting the problem of a quantum-mechanical description of
a solid to a description of its electronic properties and making all these assumptions and
approximations, the obtained Hamiltonian describes a many-particle problem that is still
technically not tractable in most cases. Therefore further approximations are needed to
reduce its complexity, which will be described in the following section.
1.2 Hubbard model
It follows from Eq. (1.17) that the position of the bands relative to each other is determined
by the matrix elements t b ii . For bands that lie energetically far away from the Fermi
energy E F
, V bb′ b ′′ b ′′′
ijlmσσ /(E ′ F − tb ii) should be a small parameter, so that the electron-electron
interaction between these bands may be neglected. In case there is only one band b F
near
E F
, it may be justified to omit the band indices and focus only on the band that is closest
to the Fermi energy. This assumption leads to an effective one-band model.
The second approximation is to take into account only the maximum term in the
Coulomb interaction. Since the Coulomb interaction decreases as 1/r with distance r, the
maximum term is the local interaction of two electrons residing in the same orbital. For a
single band, the local Coulomb interaction matrix element (“Hubbard U”) reads
∫
U = 2V b Fb F b F b F
iiii↑↓
= e 2 d 3 r d 3 r ′ |ψ i↑b F
(r)| 2 |ψ i↓bF (r ′ )| 2
. (1.19)
|r − r ′ |
The omission of all other contributions of the electron-electron interaction is motivated
by strong screening of the electron-electron interaction, so that the effective interaction
between the electrons is not really long-range and decays stronger than 1/r. This is justified
if the band at the Fermi energy is only partially filled, i.e., if the Fermi energy lies in the
band, as it is the case for metals. However, since the Hubbard model [115, 116, 117,
118] is the conceptually simplest model incorporating the full many-body electron-electron
interaction, from a pragmatic point of view it is justified to use it also to study insulating
systems, if one does not expect a fundamental change of the physics by incorporating

1.2. Hubbard model 13
longer-ranged interactions. One always has to keep in mind that one cannot necessarily
expect quantitatively correct predictions then. Following this, in this thesis the Hubbard
model (partly with extensions, as described below) will be used to describe correlated
insulating systems.
Considering the kinetic part of the Hamiltonian (1.18), it follows from Eq. (1.17), that
as long as the site and band indices are independent, i.e., the atomic orbitals are identical
on all sites, the local matrix elements t b Fb F
ii are completely independent from the site index
i. Thus, the local contribution from the kinetic part reads
− ∑ i,σ
t b Fb F
ii c † ib F σ c ib F σ = −µ ∑ i
n i = −µ ˆN , (1.20)
where ˆN = ∑ i n i = ∑ i c† ib F σ c ib F σ denotes the total particle number operator and µ = tb Fb F
ii
can be treated as the chemical potential of the electrons. Very often the reference energy
is chosen in a way that µ = 0, and this term can be omitted. From Eq. (1.17) it follows,
that the non-local matrix elements obey t b Fb F
ij = (t b Fb F
ji ) ∗ . A further simplification of the
Hamiltonian which is compatible with the assumption that the Wannier wave functions
(1.11) are strongly localized around R i is the tight-binding approximation, were one retains
only the hopping matrix elements between nearest neighbors. This is justified in systems
where the atomic orbitals are quite localized in space, and thus do not overlap much with
the wave functions of other ions.
Finally, after omitting the band index b F , the Hamiltonian of the one-band Hubbard
model in second quantization is given by [90, 91, 92, 115, 116, 117, 118, 135]
H = − ∑
t ij c † iσ c jσ + U ∑ n i↑ n i↓ = ˆT + U ˆD (1.21)
i
〈i,j〉,σ
where 〈i, j〉 denotes site indices running over neighboring sites and ˆD = ∑ i n i↑n i↓ is the
operator for the number of double occupancies. In the absence of an electromagnetic
vector potential t ij can be chosen to be real. The obtained Hamiltonian (1.21) conserves
the particle number N e as well as the z-component of the total spin, since
[H, ˆN] = [H, S z ] = 0 and [ ˆN, S z ] = 0 . (1.22)
∑
Here S z = 1 2 i (n i↑ − n i↓ ) is z-component of the total spin operator. This implies that the
total numbers of particles with up- and down-spin, ˆN↑ = ∑ i n i↑ and ˆN ↓ = ∑ i n i↓ respectively,
are also separately conserved. The latter can be easily verified using the following
commutation relation
from which follows that
[ ˆN σ , c † jσ ′ ] = δ σ,σ ′ c † jσ ′ [ ˆN σ , c jσ ′] = −δ σ,σ ′ c jσ ′
[ ˆN σ , c † jσ ′ c mσ ′] = [ ˆN σ , c † jσ ′ c mσ ′]c mσ ′ + c † jσ ′ [ ˆN σ , c mσ ′] = 0

14 1. Models
and hence
[H, ˆN σ ] = 0, σ =↑, ↓ . (1.23)
Since ˆN = ˆN ↑ + ˆN ↓ and S z = 1 2 ( ˆN ↑ − ˆN ↓ ) the commutation relations (1.22) follow.
Symmetries
Because of the particle number conservation, a term −U/2 ˆN + U/4L, which sets the chemical
potential to zero at half band-filling, can be added to the Hamiltonian (1.21) without
affecting its eigenstates. The resulting Hamiltonian
H ′ = H − U ˆN/2 + UL/4 = − ∑
t ij c † iσ c jσ + U ∑ (n i↑ − 1/2)(n i↓ − 1/2) (1.24)
i
〈i,j〉,σ
is of higher symmetry than (1.21). It can be shown that the Hamiltonian H ′ (as well as
H) commutes with the generators of the global SU(2) spin algebra given by
S + = ∑ i
S − = ∑ i
c † i,↑ c i,↓ ,
c † i,↓ c i,↑ = (S+ ) † ,
(1.25a)
(1.25b)
S z = 1 ∑
n i,↑ − n i,↓ = 1 2
2 ( ˆN ↑ − ˆN ↓ ) ,
i
(1.25c)
where S + = S x + iS y and S − = S x − iS y holds. Therefore
[H ′ , S α ] = [H, S α ] = 0, α = x, y, z. (1.26)
and H ′ (as well as H) is invariant against the global spin rotations.
On a bipartite lattice — i.e., a lattice that can be split into two interpenetrating sublattices
(A and B) such that nearest neighbors of any site belong to the complementary
sublattice — with symmetric hopping matrix elements t ij = t ji only between A and B
sublattice sites the Hamiltonian H ′ displays another global SU(2) symmetry in the charge
sector known as η-pairing or pseudospin symmetry [107, 261]. The generators of the corresponding
charge SU(2) algebra are given by
η + = ∑ i
η − = ∑ i
(−1) R i
c † i,↑ c† i,↓ ,
(−1) R i
c i,↓
c i,↑
= (η + ) † ,
(1.27a)
(1.27b)
η z = 1 ∑
(n i,↑ + n i,↓ − 1) = 1 2
2 ( ˆN ↑ + ˆN ↓ − L) (1.27c)
i

1.2. Hubbard model 15
where (−1) R i
is set to +1 (−1) if the site i belongs to the A (B) sublattice. Defining
analogous to spin operators S x and S y pseudospin operators η x = 1 2 (η+ + η − ) and
η y = − i 2 (η+ − η − ) it follows that
[H ′ , η α ] = 0, α = x, y, z . (1.28)
It can be also shown that [η α , S β ] = 0 for α, β = x, y, z and since eigenvalues of S z and η z
are either both integer or both half odd-integer altogether the Hubbard Hamiltonian H ′
is SO(4) ≃ SU(2) × SU(2)/Z 2 invariant. Adding a magnetic field term BS z or a chemical
potential term µ ˆN to the Hamiltonian H ′ (1.24) breaks the spin- or pseudospin-rotational
symmetry, respectively, without altering the other.
On a bipartite lattice with symmetric hopping matrix elements t ij = t ji only between
A and B sublattice sites the particle-hole transformation for any spin projection (σ =↑ or
σ =↓) accompanied with the sign change on every B sublattice site
P σ : c iσ ↦→ (−1) R i
c † iσ ; c† iσ ↦→ (−1)R i
c iσ (1.29)
leaves the tight-binding part of the Hubbard Hamiltonian ˆT (1.24) unchanged while the
interaction part changes its sign
t ij
⎫⎪
U ⎬
⎧⎪ ⎨
P
↦−→
σ
ˆN σ
⎪ ⎭ ⎪ ⎩ ˆN¯σ
t ji
−U
L − ˆN σ
. (1.30)
ˆN¯σ
Here ¯σ denotes the opposite to σ projection of the spin. Thus, H ′ (U) is mapped to H ′ (−U).
The same transformation maps the generators of the spin and pseudospin (η-pairing) SU(2)
algebra onto each other
S +
S −
S z
η +
η −
η z
⎫⎪ ⎬
⎪ ⎭
P ↓
η
⎧⎪ +
η −
⎨
η z
←→
S + ,
⎪
S −
⎩
S z
S +
S −
S z
η +
η −
η z
−η
⎫⎪ ⎬
⎧⎪ −
−η +
P ⎨
↑ −η z
←→
−S − . (1.31)
⎪ ⎭ ⎪
−S +
⎩
−S z
Finally, a particle-hole transformation with the accompanied sign change on every B sublattice
site performed for both — up and down — spin projections (P ↑ P ↓ H ′ P ↓ P ↑ ) does not
alter the Hamiltonian H ′ (1.24), since the sign of the coupling is switched twice. However,
the empty lattice state is mapped onto the state with all sites doubly occupied. Thus the
eigenstates of the Hubbard Hamiltonian with N e electrons are mapped onto the eigenstates
with 2L − N e electrons, and hence it is enough to solve the Hubbard model in the case
N e L.

16 1. Models
Basic Properties
A complete solution of the one-band Hubbard model (1.21) is possible in the case of
vanishing interactions, U = 0. The remaining kinetic energy operator (the tight-binding
contribution) ˆT, is diagonal in momentum space (Bloch basis) and the model describes a
free Fermi gas which is an ideal metal. The model is also easily solved in the so-called
atomic limit t ij = 0, since U ˆD is diagonal in position space (Wannier basis). A lattice
site can be occupied with zero, one, or two electrons. For N e L only singly occupied
and empty sites are present in the ground state, while for N e > L only doubly and singly
occupied sites are encountered. Excited states can be classified according to the number
of doubly occupied sites. The ground- and all excited states are highly degenerate with
respect to the spin and charge degrees of freedom since neither the position of empty or
doubly occupied sites nor the position of singly occupied sites in the lattice has any influence
on the energy spectrum. Since the lattice sites are completely isolated the system is an
insulator.
The tight-binding ˆT and on-site interaction U ˆD terms (1.21) do not commute.
Therefore the Hubbard Hamiltonian can neither be diagonalized in the Bloch nor in
Wannier basis. The physics of the one-band Hubbard model may be understood as arising
from the competition between two contributions: the tight-binding contribution ˆT, that
prefers to delocalize the electrons and the on-site interaction U ˆD, that favors localization.
Depending on the relations between the magnitudes of the hopping matrix elements
t ij in different spatial directions, due to the lattice structure, the effective motion of the
electrons can be strongly anisotropic. If the hopping matrix elements in one direction are
much larger than in all the others, so the electrons move mainly in this direction, the
system is called a quasi one-dimensional (1D) system. Similarly, if the hopping matrix
elements in two spatial directions dominate, it is called quasi two-dimensional (2D). Of
course, all materials are three dimensional crystals, but with respect to the motion of the
electrons their dimensionality is reduced, and their electronic properties may be modeled
by Hubbard or related models on one- or two-dimensional lattices. Since in real 3D crystals
the hopping matrix elements perpendicular to the 1D chains (2D planes) are never exactly
zero, it is also an interesting question how this affects the 1D (2D) physics. The issue of
this crossover in the dimensionality is a research topic of its own (see, e.g., Ref. [9] and
references therein), and is not a subject of this thesis. The topic of this thesis is to study
certain cases of the half-filled Hubbard model (partly with extensions, see below) in 1D.
The Hubbard Hamiltonian (1.21) represents the simplest many-particle electron model
that can be deduced from the generic electronic Hamiltonian (1.18) incorporating true electronic
correlation effects beyond an effective one-particle description. Despite its apparent
simplicity, no full consistent treatment of the Hubbard model is available in general. Ex-

1.2. Hubbard model 17
ceptions are cases with two extremes of the lattice coordination numbers: two and infinity.
In the first case, which corresponds to a one-dimensional lattice, the Hubbard model is
integrable and many physical properties can be determined exactly. An exact solution is
obtained by means of the Bethe ansatz [54, 155, 156], but even there the structure of the
obtained solution is so complex that it is hard to calculate correlation functions [54]. In 1D,
field-theoretical methods such as bosonization [80, 85] could also be applied successfully. If
the crystal structure is such that it has a large coordination number z, i.e., every lattice site
has z direct neighbors, and the hopping matrix elements are comparable in all these directions,
it is justified to take the limit z → ∞, and the electronic motion may be described by
the Hubbard model on an “infinite-dimensional” lattice. In this limit, the Hubbard model
can be solved within dynamical mean-field theory (DMFT) [78, 128, 170], that maps it
to an effective self-consistent Anderson impurity model [5]. The latter can then be solved
numerically, e.g., by Wilson’s numerical renormalization group (NRG) [26, 205, 260] or
quantum Monte Carlo (QMC) [128]. A striking result obtained in the DMFT approach is
an understanding of the Mott transition between a paramagnetic metal and a correlated
insulator [77, 79].
In all the other cases one is forced to use approximate techniques, and a lot of theoretical
research has been done in the last decades to develop such techniques. Among
them are concepts like the Gutzwiller variational wave-function approach [90, 91, 92], diagrammatic
perturbation theory with different self-consistent and non-self-consistent summation
schemes, or mean-field solutions (e.g., Hartree-Fock theory or slave-boson techniques
[35, 144]) that reduce the Hamiltonian (1.21) to a single-particle problem. On
the other hand, there exist powerful numerical techniques, such as quantum Monte Carlo
(QMC) [21, 69, 89, 109], exact diagonalization (ED), like the Lanczos [12, 146] or Davidson
method [12, 40, 41], and the density-matrix renormalization group (DMRG) technique
[100, 196, 210, 255, 256], which allow to study the ground-state and low-energy properties
of the Hubbard model (with extensions). Each of these techniques has its own advantages
and disadvantages (see also the Introduction), but all of them are limited to relatively
small clusters with L ∼ 10 − 1000 sites, depending on the method. Therefore, if system
properties in the thermodynamic limit are required, extrapolation schemes have to be used
(i.e., for L → ∞). For 1D systems in particular — which we intend to study — DMRG has
become the most reliable and versatile method. In Chapter 2 and Chapter 3 we describe
the DMRG algorithms with its extentions and discuss in more details how they work. Later
in Chapter 4 and Chapter 5 DMRG will be employed in order to study the ground-state
properties and real-time dynamics of the 1D Hubbard model (with extensions). But before,
in this chapter we discuss the strong-interaction limit of the one-band Hubbard model
and briefly present some of the possible extensions of the Hamiltonian that allows a more
realistic treatment of the physical systems.

18 1. Models
1.3 t-J and Heisenberg models
If one is interested only in the low-energy physics of the Hubbard model at large U ≫ |t ij |
(i, j = 1, . . .,L), i.e., the physics on an energy scale much smaller than U, it is useful to derive
an effective Hamiltonian H eff which describes properly only the low-energy properties
of the model, but neglects the complexity of its high-energy physics. a
In the limit |t ij |/U ≪ 1 the kinetic term
ˆT = − ∑
〈i,j〉,σ
t ij c † iσ c jσ (1.32)
in the Hubbard Hamiltonian (1.21) can be considered as a small perturbation to the interaction
term
U ˆD = U ∑ n i↑ n i↓ = H [0] , (1.33)
i
which is taken as the unperturbed problem. H [0] gives the atomic limit of the Hubbard
model, i.e., it describes independent atomic orbitals (sites). Since ˆD is diagonal in position
space and counts the number of doubly occupied states (sites) the Hilbert space of
the problem decomposes into the subspaces each consisting of the states with the same
well-defined number of double occupied sites. For N e L the ground state of H [0] lies
completely in the subspace without double occupancies, which as discussed previously is
“infinitely” degenerate, and has the energy E 0 = 0. The perturbation ˆT lifts this large
degeneracy. The lowest energy level of U ˆD splits into many levels which are well separated
from the first exited level of U ˆD as long as the condition |t ij | ≪ U holds. Therefore,
the new effective Hamiltonian H eff must operate in the subspace spanned by these states,
without double occupancies, in order to describe the low-energy properties of the model
properly.
To make the effective Hamiltonian act only in the subspace with no double occupancies,
the following projection operator
P 0 = ∏ i
(1 − n i↑ n i↓ ) (1.34)
is introduced. Now, using the second-order (degenerate) perturbation theory [11, 54] the
effective Hamiltonian
H (2)
eff = P 0
(
H [0] + H [1] + H [2]) P 0 = P 0 ˆTP0 − 1 U P 0 ˆTP 1 ˆTP0 (1.35)
is obtained, where P 1 projects onto the subspace with one double occupancy. For the
construction of higher-order terms a more systematic approach has to be applied [133, 160,
a Here we only consider the case with N e L.

1.3. t-J and Heisenberg models 19
225]. The projection of H [0] is zero, the first order correction H [1] turns out to be the
hopping term ˆT, and the second order correlation H [2] consists of terms representing the
two-site spin exchange and a three-site term. The resulting effective Hamiltonian is the
so-called t − J model
[
H t−J = P 0 − ∑
t ij c † iσ c jσ
〈i,j〉,σ
+ ∑ (
J ij S i · S j − n )
in j
4
〈i,j〉
+ 1 ∑ (
t ij t jk c † iσ
U
σ σσ
c ′ kσ ′ · S j − 1 ∑
c † iσ
2
c kσ j) ]
n P 0 (1.36)
i≠j≠k≠i
with the exchange coupling constant J ij = 2|t ij | 2 /U, and the spin one half operators defined
as
S i = 1 ∑
c † iσ
2
σ σσ
c ′ iσ ′ , (1.37)
σσ ′
where the three components of the vector σ σσ ′ are the usual Pauli matrices.
Close to half-filling b the three-site term in (1.36) (the last term) may be considered to
be unimportant compared to the two-site spin exchange (the second term), and it is of
higher order in 1/U than the first term describing the projected hopping. Neglecting this
three-site term one obtains a simplified version of the t − J model Hamiltonian
H t−J = −P 0
∑
〈i,j〉,σ
σ
t ij c † iσ c jσ P 0 + ∑ (
J ij S i · S j − n )
in j
, (1.38)
4
〈i,j〉
For analyzing the ground-state properties of strongly-correlated electron systems, i.e.,
for systems with a large U ≫ |t ij | = t like, e.g., the 2D CuO 2 planes in the cuprates, the
t − J model (1.38) represents a reasonable approximation to the Hubbard model. Since
its Hilbert space consists only of three states per sites (instead of four in the case of the
Hubbard model), the reachable system sizes L in numerical treatments are larger than for
the Hubbard model, which leads to smaller finite size effects.
Heisenberg model
At half-filling, i.e., when the number of electrons N e equals the number of lattice sites L, all
eigenstates of H t−J (1.38) are “pure spin states” where every lattice site is occupied precisely
by one electron, n i = 1. Therefore, the projected hopping term as well as the three-site
term in the t − J model (1.36) do not contribute anymore and the model reduces to the
b At half-filling the number of electrons N e equals the number of lattice sites L.

20 1. Models
spin-1/2 quantum Heisenberg model
H eff = ∑ 〈i,j〉
(
J ij S i · S j − 1 )
. (1.39)
4
This model describes the physics of the magnetic interactions and hence the low-lying excitations
of the half-filled Hubbard model at U ≫ |t ij | are magnetic excitations; the model
is in a Mott insulator phase (an insulator resulting from electron-electron interactions).
For a bipartite lattice with only nearest-neighbor hopping and J ij = J, the ground state
of (1.39) is proven or expected to exhibit antiferromagnetic long-range order in all dimensions
d > 2 (J > 0). However, in 1D and 2D, antiferromagnetic long range order is not
possible at T > 0 [114, 169], and in 1D, even at T = 0 long-range antiferromagnetic order
is forbidden, so the ground state shows only quasi long-range order, i.e., the antiferromagnetic
correlations between two spins decay algebraically with increasing distance. The low
energy spin excitations in this model are collective spin wave excitations, which in the 1D
case cost zero energy, and the spectrum is gapless, i.e., it has a spin excitation gap ∆ S = 0
[154].
1.4 Extended Hubbard models
Since later we intend to study 1D strongly correlated systems, in this section we mainly
consider extentions to the 1D one-band Hubbard model given by the Hamiltonian
H = −t ∑ i,σ
(c † i,σ c i+1,σ + h.c.) + U ∑ i
n i↑ n i↓ . (1.40)
The one-dimensional Hubbard model has evolved from a toy model to a paradigm of
experimental relevance for strongly correlated electron systems, due to the synthesis of new
quasi one-dimensional materials and the refinement of experimental techniques. Although
it is not strictly a perfect model for any existing material, many of its qualitative features
seem to be realized in nature. At present there is a sizeable list of materials, for which the
electronic degrees of freedom are believed to be described by Hubbard-like Hamiltonians.
However, in all these cases the appropriate electronic Hamiltonians differ significantly from
a simple one-band Hubbard model.
In Chapter 4, we study the ground-state phase diagram of the half-filled 1D Hubbard
model with different extensions using numerical tools such as Lanczos exact diagonalization
and the density matrix renormalization group (DMRG) method. Since in 1D systems,
quantum fluctuations are especially important, due to the restricted dimensionality, and
could cause the system to be unstable against various small perturbations, the assumptions
that lead to the original Hubbard model (1.21) may be too severe. Additional terms

1.4. Extended Hubbard models 21
neglected so far could lead to ground-state phases of completely different nature. Thus,
various extensions to the Hubbard model have been proposed, which allow a more realistic
treatment of 1D systems. In the following we consider some of them.
1.4.1 Next-nearest neighbor interaction
In addition to the local on-site Coulomb interaction U, also the interaction
H int = V ∑ 〈i,j〉
n i n j (1.41)
of electrons between neighboring sites can be taken into account. For many systems where
the electron-electron interaction is not strongly screened [24, 136, 137], this is a much more
realistic choice than the reduction to the on-site interaction only. This model is called the
extended or U − V Hubbard model.
1.4.2 Peierls distortion
So far, the influence of the ion dynamics of the lattice has been neglected completely. As
mentioned above, the static response of the ions to the motion of the electrons can be
taken into account within a purely electronic model by an effective potential. The effective
potential describes a static alternating Peierls distortion δ of the lattice, resulting in a
modulation of the hopping amplitudes in the kinetic term of (1.21) [193]
H Peierls = −t ∑ i,σ
δ(−1) i (c † i,σ c i+1,σ + h.c.) (1.42)
The resulting Hamiltonian is called the Peierls-Hubbard model, which is discussed, e.g., as
a possible model for dimerized chain materials like polyacetylene [105].
1.4.3 Ionic potential
In the context of organic mixed-stack charge transfer (CT) crystals such as tetrathiafulvalene
(TTF)-p-chloranil [179, 178, 233, 234], the Hubbard model with an additional
alternating on-site energy modulation
H ion = ∆ 2
∑
(−1) i n i (1.43)
has been proposed thirty years ago [179, 178], which is usually called the ionic Hubbard
model. The organic CT crystals consist of quasi one-dimensional chains, formed by a
sequence of alternating donor (D) and acceptor (A) molecules (. . . D +ρ A −ρ D +ρ A −ρ . . .),
i

22 1. Models
where ρ denotes the amount of charge transfer from the donors to the acceptors. The
ionic Hubbard model serves as an appropriate effective model for the D − A chains, with
the sites with even i representing the LUMO (Lowest Unoccupied Molecular Orbital) of an
acceptor molecule, while sites with odd i represent the HOMO (Highest Occupied Molecular
Orbital) of a donor. In the neutral state, these orbitals would be occupied by either zero
or two electrons, respectively. The model parameters U and ∆ are used for an effective
description of the microscopic parameters, like, e.g., the electron affinity of the acceptor
molecules, the ionization potential of the donors, the Madelung energy gained by the pairs
ionized due to the charge transfer ρ, and the Coulomb interaction in the D +ρ A −ρ pairs [83].
Within this picture, ∆ is interpreted as the energy necessary to move an electron from the
donor to the acceptor. The ionic Hubbard model will be studied in detail in Section 4.1 of
Chapter 4.
In quasi 1D materials like halogen-bridged transition metal chain complexes, conjugate
polymers, or inorganic blue bronzes a similar on-site energy modulation can be effectively
generated by the strong competition among the itinerancy of the electrons, electronelectron
interactions and interactions of electrons with the motion of ions. In the adiabatic
limit, the latter interactions can be taken into account within a purely electronic model by
an effective potential
H ion = ∆ ∑
(−1) i n i + KL ( ) 2 ∆
(1.44)
2
2 2
i
which in addition includes the elastic energy of a harmonic lattice with a “stiffness constant”
K. This so-called adiabatic Holstein-Hubbard model will be studied in detail in Section 4.2
of Chapter 4.
The latter two extensions, the Peierls distortion and the ionic potential, obviously
change the translation symmetry of the Hubbard model (1.21), since the extended models
are only mapped onto themselves by a lattice translation by two sites. In momentum
space, this corresponds to a reduction of the Brillouin zone by one half (from [−π/a, π/a]
to [−π/2a; π/2a], with a being the lattice constant). For U = 0, a band gap opens at the
new Brillouin zone boundaries, so these models represent effective two-band models.

23
2. DENSITY-MATRIX RENORMALIZATION GROUP
Most of the studies presented in this thesis are devoted to numerical tools or are based on
numerical calculations. In this chapter we describe the standard Density-Matrix Renormalization
Group (DMRG) algorithms for calculating ground state properties of quantum
lattice many-body system. We also discuss algorithmic improvements that are relevant for
the development of an efficient computer code and are included in the DMRG program
developed by us. The subsequent chapter (Chapter 3) builds on this description as a background.
There we present two extensions of the basic DMRG algorithms that allow to
study the problems with time-evolution. In Chapter 4 the standard DMRG methods are
used to study the ground state phase diagram of the one-dimensional ionic Hubbard and
adiabatic Holstein-Hubbard models.
2.1 Introduction
The DMRG method was developed by Steven White [255, 256] in 1992 to overcome the
problems arising in the application of the real-space renormalization group approaches to
quantum lattice many-body systems in solid-state physics. Although initially DMRG was
intended for studies of only low-energy properties of one-dimensional strongly correlated
quantum systems, such as the Heisenberg and Hubbard models (see [196] and references
therein), since its invention it has been under constant development. The algorithm was
rapidly extended and adapted to different situations, becoming the most reliable and versatile
method for 1D systems. Its field of applicability has now extended beyond condensed
matter physics and is successfully used in statistical mechanics, nuclear and high energy
physics, quantum information theory, ab initio quantum chemistry, etc.; an incomplete
list can be found in recent reviews [99, 100, 210]. In the following chapter we consider
extensions to the basic DMRG algorithms that allow to study the problems with time
evolution. Numerous other extensions and applications of DMRG are discussed in various
books [64, 187, 196]. Additional information can be also found at Refs. [1, 186].
Like in other renormalization group techniques, the main idea of the DMRG algorithm
is to successively eliminate microscopic degrees of freedom in order to obtain a reduced
description of the system with many degrees of freedom, that is numerically manageable,
but nevertheless captures the essential physics of the original model. However, the key

24 2. Density-Matrix Renormalization Group
difference from most renormalization group approaches is to renormalize a system using the
information provided by a reduced density matrix rather than by an effective Hamiltonian,
hence the name density-matrix renormalization. A few years after the advent of DMRG,
it was understood that [43, 192, 201, 227] the approximate ground states produced by
DMRG have the form of matrix-product states (MPS) which can be explored and optimized
variationally with an efficient use of computational resources. Recently, this connection
between DMRG and matrix-product states has been emphasized (for recent reviews, see
[166, 241]) and has lead to significant extensions of the DMRG approach.
The outline of this chapter is as follows: First we introduce the matrix-product states
used in the DMRG framework and draw connection to the traditional DMRG blocks and
superblocks. Next we describe the density-matrix projection scheme and discuss conditions
where this scheme leads to a successful reduction of the problem’s complexity. In sections
2.4 and 2.5 we present the infinite-system DMRG and finite-system DMRG algorithms. In
section 2.6 we consider important improvements to the basic DMRG algorithms, e.g., wavefunction
transformations, additive quantum numbers, necessary for the effective DMRG
implementations.
2.2 Matrix-Product State
Consider a quantum system with L sites, each having a local Hilbert space H j
(j = 1, . . .,L). The Hilbert space of the whole system is H = ⊗ L
j=1 H j. Let
B(j) = {|s j 〉; s j = 1, . . .,d j } denote a complete basis for the site j. a The tensor product
of bases of sites yields a complete basis of the whole Hilbert space
{|s = (s 1 , . . ., s L )〉 = |s 1 〉 ⊗ · · · ⊗ |s L 〉; s j = 1, . . .,d j ; j = 1, . . .,L} . (2.1)
A general normalized state of the system can be expressed in this basis as
|ψ〉 =
∑
s 1 ,s 2 ...,s L
c(s 1 , s 2 , . . .,s L )|s 1 〉 ⊗ |s 2 〉 ⊗ · · · ⊗ |s L 〉 ≡ ∑ {s}
c(s)|s〉 , (2.2)
where c(s) ∈ C.
The total number of possible combinations in (2.1) and hence the dimension of the
whole Hilbert space is dim(H) = ∏ L
j d j; so it grows exponentially with the system size L.
For instance, for the Hubbard model d j = 4, B(j) = {|0〉, | ↑〉, | ↓〉, | ↑↓〉}, and dim(H) = 4 L .
Representing the system Hamiltonian in the basis (2.1), the complete eigensystem of the
Hamiltonian matrix can be found using any full diagonalization algorithm [87]. Unfortunately,
the exponentially growing Hilbert-space dimension strongly limits the system sizes
a all bases used here are orthonormal.

2.2. Matrix-Product State 25
accessible with this direct approach (typically few tens). Some other iterative diagonalization
schemes, like Lanczos or Davidson [12, 37, 40, 41, 146, 203], can also be used in order
to find the ground and a few low-energy states of the system. Additionally one can use
the symmetries of the studied problem to reduce the dimension of the Hilbert space. All
these typically double or even triple the system sizes accessible by the exact diagonalization
schemes, but finally all these methods suffer from the same problem of exponentially
growing Hilbert space size.
Since in most cases we are interested in the ground-state or low-energy properties of
the system, a natural question arises, whether effective representations for these states can
be found that allow us to study large size systems exactly or at least in some controllable
approximation. Let us first rewrite c(s) in a different form. Like in Wilson’s Renormalization
Group [260] let us start with a one-site system and grow it adding one site at
every iteration. We combine the first two sites and consider an a 2 -dimensional subspace of
the complete Hilbert space of these two sites H L(2) ⊂ H 1 ⊗ H 2 , a 2 d 1 d 2 . Here and later
L(n) will denote a set of the first left n contiguous sites starting from 1. Next we add
the third site and consider an a 3 a 2 d 3 dimensional subspace H L(3) of the Hilbert space
of the two+one system H L(2) ⊗ H 3 . Proceeding in the same way iteratively, one obtains
H L(L) ⊂ H L(L−1) ⊗ H L with dim(H L(L) ) = a L a L−1 d L . If one carries out this procedure
keeping all a j (j = 1, . . .,L) small enough, transforming all the necessary operators including
the Hamiltonian in this reduced basis, one can study the effective system obtained.
Therefore, we are looking for an optimal reduction procedure for the Hilbert space, obtained
after the addition of one site to the system, that keeps the space dimension small
and still represents the quantities of interest as accurately as possible. But before, let us
first study the structure of the basis vectors for the considered blocks of sites.
Consider any orthonormal set of vectors in H 1 , {|α 1 〉 ∈ H 1 ; α 1 = 1, . . ., a 1 },
〈α ′ 1 |α 1 〉 = δ α ′ 1 ,α 1 . (2.3)
One can express |α 1 〉 in the site basis B(1) = {|s 1 〉; s 1 = 1, . . .,d 1 } as
|α 1 〉 =
∑d 1
s 1 =1
A 1 (s 1 ) α1 |s 1 〉 , (2.4)
and due to (2.3)
∑d 1
s 1
A ∗ 1(s 1 ) α ′
1
A 1 (s 1 ) α1
= δ α ′
1 ,α 1
(2.5)
where the “∗” denotes the complex conjugate only. b
b For instance one can consider A 1 (s 1 ) α1 = δ s1,α 1
.

26 2. Density-Matrix Renormalization Group
Now let us add the second site and denote by |α 2 〉 the elements of the orthonormal
basis of H L(2) ⊂ H 1 ⊗ H 2 . One can write |α 2 〉 in terms of linear combinations of vectors
|α 1 〉 ⊗ |s 2 〉 as
|α 2 〉 =
=
∑a 1
∑d 2
α 1 =1 s 2 =1
∑a 1
∑d 1
α 1 =1 s 1 =1 s 2 =1
A 2 (s 2 ) α1 ,α 2
|α 1 〉 ⊗ |s 2 〉
∑d 2
A 1 (s 1 ) α1 A 2 (s 2 ) α1 ,α 2
|s 1 〉 ⊗ |s 2 〉 (2.6)
and since we demand orthonormality 〈α 2 ′ |α 2 〉 = δ α ′ 2 ,α , from (2.3) and (2.5) followsc
2
∑a 1
∑d 2
α 1 =1 s 2 =1
A ∗ 2(s 2 ) α1 ,α ′ 2 A 2(s 2 ) α1 ,α 2
= δ α ′
2 ,α 2
.
(2.7a)
Associating A 2 (s 2 ) with a (a 1 × a 2 )-dimensional matrix one can rewrite (2.7a) in a compact
form using a matrix notation
∑d 2
s 2 =1
(A 2 (s 2 )) † A 2 (s 2 ) = I , (2.7b)
where I is the (a 2 × a 2 )-dimensional identity matrix.
If one proceeds in the same way iteratively, after j steps the following relation for {|α j 〉}
is obtained
|α j 〉 =
a j−1
∑
d j
∑
α j−1 =1 s j =1
A j (s j ) αj−1 ,α j
|α j−1 〉 ⊗ |s j 〉 , (2.8)
and orthonormality conditions 〈α ′ j|α j 〉 = δ α ′
j ,α j
and 〈α ′ j−1|α j−1 〉 = δ α ′
j−1 ,α j−1
imply that
d j
∑
(A j (s j )) † A j (s j ) = I . (2.9)
s j =1
Here, once again A j (s j ) α ′
j ,α j
are associated with (a j−1 × a j )-dimensional matrices A j (s j )
and I is the (a j × a j )-dimensional identity matrix.
Actually starting with orthonormal sets {|s j 〉}, j = 1, . . .,L, and demanding (2.9) condition
for each j, the set {|α j 〉; α j = 1, . . .,a j } obtained at iteration j is also orthonormal.
Substituting recursively the definitions of previous |α j 〉 in Eq. (2.8)
|α j 〉 = ∑
s 1 ,s 2 ,...,s j
[A 1 (s 1 )A 2 (s 2 ) · · ·A j (s j )] αj |s 1 〉 ⊗ |s 2 〉 ⊗ · · · ⊗ |s j 〉 . (2.10)
c Remember that {|s j 〉} are orthonormal sets.

2.2. Matrix-Product State 27
Note, that since the A 1 (s 1 ) are row vectors and the rest of the A’s are matrices, the product
A 1 (s 1 )A 2 (s 2 ) · · ·A j (s j ) is a vector from which the α j -th component is taken (2.10). The
elements of the orthonormal basis in H L(j) , and consequently all the vectors therein, can
be expressed in the form (2.10). Their coefficients in the original basis |s 1 , s 2 , . . .,s j 〉 can
be then written as a product of matrices. Due to this structure, the vectors written in the
(2.10) form are called matrix-product states (MPS). Note also, that if a j = ∏ j
k=1 d k then
H L(j) = ⊗ j
k=1 H k and all vectors are represented exactly.
Coming back to the original problem, the vectors |ψ〉 belonging to H L(L) ⊂ H can be
written as the following MPS
|ψ〉 =
∑
A 1 (s 1 )A(s 2 ) · · ·A L (s L )|s 1 , s 2 . . .,s L 〉 . (2.11)
s 1 ,s 2 ,...,s L
Here, A L (s L ) are column vectors and since |ψ〉 was normalized (〈ψ|ψ〉 = 1) they fulfill the
orthonormality condition (2.9), i.e.
d L
∑
a∑
L−1
s L =1 α L =1
|A L (s L ) αL | 2 = 1 . (2.12)
In fact, MPS do not have a unique representation in terms of A’s. This follows from
the fact that a product of two matrices is left invariant when a nontrivial resolution of the
identity is inserted between them:
AB = AXX −1 B = A ′ B ′
A ′ = AX (2.13)
B ′ = X −1 B .
This freedom gives us the possibility of choosing a gauge, and thus imposing conditions on
the matrices A which simplify further calculations, or give a physical meaning. In what
follows we will choose (2.9) as one possible gauge, because it makes connection between
MPS and DMRG methods.
One can generalize (2.11) for a closed ring topology (e.g. periodic boundary conditions)
by substituting A 1 (s 1 ) and A L (s L ) vectors with (a 0 × a 1 )- and (a L × a 0 )-dimensional
(a 0 1) matrices, respectively, and taking the trace over the product of matrices
|ψ〉 =
∑
Tr (A 1 (s 1 )A(s 2 ) · · ·A L (s L )) |s 1 , s 2 . . .,s L 〉 (2.14)
s 1 ,s 2 ,...,s L
(for open chain a 0 = a L = 1 and the trace can be dropped). Although, one should admit
that it becomes hard to use the above considered orthonormality conditions (2.9) in case
of periodic boundary conditions (for more details see [213, 241, 242]).

28 2. Density-Matrix Renormalization Group
The states of the considered structure had appeared in the literature in many different
contexts and under different names before the invention of DMRG. The simplest case of
(2.14), corresponding to a homogeneous state with the same matrices for all sites, was first
considered in the eighties [60, 61] and occured as a ground state of certain spin chains with
competing interactions [141]. The best-known example is the spin-one chain with bilinear
and biquadratic interactions and a certain ratio of the couplings, where the valence-bond
ground state [2, 3] can be written in this form using (2 × 2) matrices.
In this thesis, we focus only on the case of open boundary condition (MPS (2.11)).
The systems with periodic boundaries can be studied within these states too, however the
effective dimensions a j tend to the square of that required for open boundary conditions
[150].
2.2.1 MPS, Blocks, and a Superblock
There is another way of expressing c(s) in MPS. Instead of starting from the site 1 and
growing the system by subsequent adding from the right the sites 2, 3, and so on (like
in the previous subsection), one can start from the site L and add from the left the sites
L − 1, L − 2, . . . , 1.
Similar to the procedure outlined in the previous subsection we start with some orthonormal
basis {|β L 〉} in H L and express it in terms of |s L 〉
where
|β L 〉 =
〈β ′ L|β L 〉 = δ β ′
L ,β L
⇐⇒
d L
∑
s L =1
B L (s L ) βL |s L 〉 , (2.15)
d L
∑
s L =1
B ∗ L(s L ) β ′
L
B L (s L ) βL
= δ β ′
L ,β L
. (2.16)
After adding from the left first the (L − 1)-th site, then the (L − 2)-th, and so on,
at step L − k = j we can write a basis of H R(j) ⊂ H j ⊗ H R(j+1) ⊂ ⊗ L
i=j H i in terms of
|s j 〉 ⊗ |β j+1 〉 as
with
b j+1
∑
d j
∑
β j+1 =1 s j =1
|β j 〉 =
d j
∑
b j+1
∑
s j =1 β j+1 =1
B j (s j ) βj ,β j+1
|s j 〉 ⊗ |β j+1 〉 , (2.17)
B ∗ j (s L−1) βj ,β j+1
B L−1 (s L−1 ) βj ,β j+1
= δ β ′
j ,β j
= 〈β ′ j |β j 〉 . (2.18)
Substituting |β j 〉 from the previous steps in (2.17)
∑
|β j 〉 = [B j (s j )B j+1 (s j+1 ) · · ·B L (s L )] βj |s j 〉 ⊗ |s j+1 〉 ⊗ · · · ⊗ |s L 〉 , (2.19)
s j ,s j+1 ,...,s L

2.2. Matrix-Product State 29
where the β j -th element of the vector B j (s j )B j+1 (s j+1 ) · · ·B L (s L ) is taken. Associating
Bj ∗ (s j ) βj ,β j+1
with (b j × b j+1 )-dimensional matrices B j (s j ) one can rewrite (2.18) in a matrix
form
d j
∑
s j =1
B j (s j )(B j (s j )) † = I . (2.20)
where I is the identity matrix. Here the difference between (2.20) and (2.9) becomes visible,
B j (s j ) matrices are right hand orthogonal while A j (s j ) are left hand orthogonal.
Using both A j (s j ) and B k (s k ) one can rewrite state |ψ〉 in yet another form, i.e.,
|ψ〉 = ∑ s
A 1 (s 1 )A 2 (s 2 ) · · ·A j (s j )C j B j+1 (s j+1 )B j+2 (s j+2 ) · · ·B L (s L )|s〉 (2.21)
where C j is a (a j × b j+1 )-dimensional matrix; A k (s k ) and B k (s k ) are (a k−1 × a k )- and
(b k × b k+1 )-dimensional matrices fullfilling the orthonormality conditions (2.9) and (2.20),
respectively.
One can draw some conclusions at this point. First of all, a trivial product state can
be written as MPS with a j , b j = 1 for j = 1, . . .,L. Since a 0 = b L+1 = 1, it follows that
a k a k−1 d k ∏ k
i=1 d i and b k d k b k+1 ∏ L
i=k d i. The quality of the optimal approximation
of any quantum state can be improved monotonically increasing a j and b j , j = 1, . . .,L:
if we take a ′ j−1 a j=1 and a ′ j a j than the optimal (a j−1 × a j )-dimensional matrix A j (s j )
can be taken as a submatrix of a larger (a ′ j−1 × a ′ j)-dimensional one, the quality of the
representation can thus only be improved (similarly for the rest of A’s and B’s).
MPS written in the (2.21) form splits lattice sites into two groups. The sites
k = 1, . . .,j make up a left block L(j) with a subspace H L(j) ⊂ ⊗ j
k=1 Hk of dimension
a j , spanned by an orthonormal basis B(L, j) = {|α j 〉; α j = 1, . . .,a j } defined
with (2.8). The sites k = j + 1, . . .,L build up a right block R(j + 1) with
a b j+1 -dimensional subspace H R(j+1) ⊂ ⊗ L
k=j+1 Hk spanned by an orthonormal basis
B(R, j + 1) = {|β j+1 〉; β j+1 = 1, . . ., b j+1 } given by (2.17). H L(j) (H R(j+1) ) and B(L, j)
(B(R, j + 1)) are also called block state space and effective block state-space basis, respectively.
In what follows, we will use α, a, and A to denote the left-block components and
β, b, and B for the right-block ones.
If we combine the left block L(j) with the right block R(j + 1), we obtain a socalled
superblock which contains all sites, from 1 to L. The tensor product basis
B(SB, j) = B(L, j) ⊗ B(R, j + 1) of the blocks bases is called superblock basis which spans
a (a j · b j+1 )-dimensional subspace of the whole Hilbert space H. The MPS given by (2.21)
can be expanded in this basis as
a j
b j +1
∑ ∑
|ψ〉 = [C j ] α,β |α j β j+1 〉 , (2.22)
α=1 β=1

30 2. Density-Matrix Renormalization Group
where [C j ] α,β denotes the matrix elements of C j , |α j β j+1 〉 = |α j 〉 ⊗ |β j+1 〉 ∈ B(SB, j), and
the square norm of |ψ〉 is given by 〈ψ|ψ〉 = TrC † j C j
Note, that if a j = ∏ j
i=1 d i and b j+1 = ∏ L
i=j+1 d i, the superblock basis B(SB, j) is a
complete basis of the whole Hilbert space H and any state |ψ〉 ∈ H can be written in the
form (2.22). For a large lattice these conditions imply that some matrix dimensions are still
exponentially large (at least 4 L/2 for a Hubbard model). However, a MPS is numerically
tractable, only if all matrix dimensions are kept small, for instance a i , b k m (1 i j,
j < k L) with m up to a few thousands. Therefore, matrix-product states with restricted
matrix size can be considered as an approximation for states in H. In section 2.3 we
determine the procedure that gives the optimal reduction of the single-block state-space
dimension and discuss conditions under which this reduction scheme leads to a successful
approximation. In particular, MPS can also be used as a variational ansatz for the ground
state of the system Hamiltonian H. For this, the system energy E = 〈ψ|H|ψ〉/〈ψ|ψ〉 as a
function of matrices A n (s n ), B n (s n ), and C j , has to be minimized with respect to these
variational parameters, subjected to the constraints (2.9) and (2.20). In the sections 2.4
and 2.5 we will present algorithms (the infinite-system DMRG and the finite-system DMRG
algorithms) for carrying out this minimization.
2.2.2 Operators in block effective basis
Before determining the optimal projection matrices A j (s j ) and B i (s i ), let us first construct
matrix representations of operators acting on the blocks L(j) (j = 1, . . ., L) in the above
considered effective basis B(L, j). For operators acting on the R(i), matrix representations
in the effective basis B(R, j) can be found in a similar way.
Any operator acting on the block of contiguous sites from 1 to l can be written as a
sum of products of local operators
Ô = ∑ ν
Ô 1 ν Ô2 ν · · ·Ôl ν , (2.23)
where each of Ôj ν is a local operator that acts only on the j-th site. Some, if not most of
them may just be trivial identity operators. A matrix representation of Ô, in the block
tensor-product basis ({|s 1 〉 ⊗ · · · ⊗ |s l 〉; s n = 1, . . .,d n ; n = 1, . . .,l}), which we denote as
O, can be written as a tensor product of matrix representations of the local operators in
their site basis
O = ∑ ν
O 1 ν ⊗ O2 ν ⊗ · · · ⊗ Ol ν , (2.24)
where Oν j is a matrix representation of the local operator Ôj ν in site basis |s j 〉. Now we wish
to find a matrix representation of Ô in the block effective basis B(L, l), which we simply

2.2. Matrix-Product State 31
call L(l)-representation of Ô and denote as OL(l) . d To do this we start with the simple case
of obtaining a L(j)-representation of the local operator Ôj . Its matrix representation in
the site local basis {|s j 〉} denoted as O j is given by O j = 〈s ′ s ′ j ,s j |Ôj |s j 〉. To find its matrix
j
representation in an effective basis B(L, j) (O L(j) ) we will use the procedure of obtaining
{|α j 〉} from {|α j−1 〉|s j 〉} (2.8)
O L(j)
α ′ j ,α j
= 〈α j ′ |Ôj |α j 〉 =
⎛
⎞ ⎛
⎞
= ⎝〈α j−1 ′ |〈s′ j | ∑
A ∗ j (s′ j ) α
⎠Ô j ⎝ ∑
′ A
j−1 ,α′ j
j (s j ) αj−1 ,α j
|α j−1 〉|s j 〉 ⎠
α ′ j−1 ,s′ j
α j−1 ,s j
⎛
⎞ ⎛
⎞
= ⎝〈α j−1 ′ | ∑
A ∗ j (s′ j ) ⎠
α ′ O j ⎝ ∑
A
j−1 ,α′ j s ′
α ′ j ,s j (s j ) αj−1 ,α
j
j
|α j−1 〉 ⎠
j−1 ,s′ j
α j−1 ,s j
= ∑
A ∗ j (s′ j ) α j−1 ,α ′ A j(s
j j ) αj−1 ,α j
O j . (2.25a)
s ′ j ,s j
α j−1 ,s ′ j ,s j
Here the only condition used is the orthonormality of the basis B(L, j − 1). We can rewrite
(2.25a) using the matrix notation as
O L(j) = ∑ s ′ j ,s j
A † j (s′ j )A j (s j )Oj . (2.25b)
s ′ j ,s j
Similarly, if we know the L(j − 1)-representation Q L(j−1) of some operator ˆQ that acts
only on sites of the block L(j − 1), we can obtain its L(j)-representation
Q L(j)
α ′ j ,α j
= 〈α j ′ | ˆQ|α j 〉 =
⎛
⎞ ⎛
⎞
∑
= ⎝〈α j−1|〈s ′ ′ j| A ∗ j(s ′ j) α ′
⎠
j−1 ,α
ˆQ ⎝ ∑
A ′
j
j (s j ) αj−1 ,α
|α
j
j−1 〉|s j 〉 ⎠
α ′ j−1 ,s′ j
α j−1 ,s j
⎛
⎞ ⎛
⎞
= ⎝〈s ′ j | ∑
A ∗ j (s′ j ) ⎠
α Q L(j−1) ⎝ ∑
A ′
j−1 ,α′ j α ′
α ′ j−1 ,α j (s j ) αj−1 ,α
|s
j−1
j j 〉 ⎠
j−1 ,s′ j
α j−1 ,s j
∑
= A ∗ j (s j ) α A ′ j−1 ,α′ j j (s j ) α j−1 ,α
Q L(j−1) , (2.26a)
j α ′
α ′ j−1 ,α j−1
j−1 ,α j−1,s j
or in matrix notation
Q L(j) = ∑ s j
A † j (s j )QL(j−1) A j (s j ) . (2.26b)
d Here and below superscript L(l) also shows that the operator acts on the reduced Hilbert space H L(l)
of the block L(l).

32 2. Density-Matrix Renormalization Group
Here we only used the orthonormality of the site local basis {|s j 〉}.
Equations (2.25) and (2.26) can be used to find a matrix representation of any local
operator, that acts on some site j < i in the basis B(L, i).
It remains to find a matrix representation of the product of two local operators ˆQ k and
Ô j , acting on the different sites k and j (k < j), respectively, e in the basis B(L, l). For this
we start with the L(k)-representation Q L(k) of ˆQ k (2.25) and using (2.26) iteratively obtain
its L(j − 1)-representation. Then using (2.8) we write the product of Q L(j−1) and matrix
representation of Ôj (O j ) in the basis B(L, j) and hence obtain the L(j)-representation of
ˆQ k Ô j ( ˆQ k Ô j ) L(j)
= 〈α
α j|( ′ ˆQ k Ô j )|α ′
j ,α j 〉
j
⎛
⎞ ⎛
⎞
∑
= ⎝〈α j−1|〈s ′ ′ j| A ∗ j(s ′ j) α ′ ⎠
j−1 ,α ′ ( ˆQ k Ô j ) ⎝ ∑
A
j
j (s j ) αj−1 ,α j
|α j−1 〉|s j 〉 ⎠
α ′ j−1 ,α′ j
α j−1 ,α j
⎛
⎞
⎛
⎞
= ⎝ ∑
A ∗ j (s′ j ) ⎠
α ′ Q L(j−1)
j−1 ,α′ j α ′
α ′ j−1 j−1O j ⎝ ∑
A ,α s ′ j ,s j (s j ) αj−1
⎠
,α
j
j
j−1 ,α′ j
α j−1 ,α j
∑
= A ∗ j (s′ j ) α ′ A j−1 ,α′ j j (s j ) α j−1 ,α j
Q L(j−1)
α ′ j−1 j−1O j , (2.27a)
,α s ′ j ,s j
α ′ j−1 ,α j−1,s ′ j ,s j
or in matrix notation
( ˆQ k Ô j ) L(j) = ∑ s ′ j ,s j
A † j (s′ j )QL(j−1) A j (s j ) O j s ′ j ,s j. (2.27b)
Finally, once more using (2.26) iteratively, from (2.27) we obtain its L(l)-representation.
This can be easily generalized for the product of more than two operators each acting on
a different site, e.g.,
Ô ν = Ô1 ν Ô2 ν · · ·Ôl ν . (2.28)
O L(l) (2.24) is then constructed by summing up L(l)-representations Oν
L(l) of Ô ν . Note
that we can take the sum but not the product of operators in the same L(l)-representation;
since B(L, l) is the basis of the block subspace and not the complete one,
( ˆQÔ)L(l) α ′ l ,α l
= 〈α ′ l |( ˆQÔ)|α l 〉 ≠ ∑ α ′′
l
〈α l ′ ′′
| ˆQ|α l 〉〈α′′ l |Ô|α l 〉 = [OL(l) Q L(l) ] α ′
l ,α l
. (2.29)
As mentioned at the beginning, typically most of Ôν j in Ôν are just identity operators Î,
corresponding to identity matrices in the site-local basis. In case of fermionic operators, due
to the anticommutation relations, one should take care about “-” signs where it is necessary.
e For k = j we take the product in the local basis of the site k and obtain a new local operator.

2.3. Density-Matrix Projection 33
This is done by substituting identity operators Î with a single site parity operator ˆP, where
it is necessary. Usually this procedure does not increase gradually the complexity of the
problem, since as we will see later in this chapter, in most cases the number of electrons is
a “good” quantum number and just counting them is enough.
To make everything more transparent, consider specifically the Hubbard model. The
local basis for each site is given by an orthonormal set {|0〉, | ↑〉, | ↓〉, | ↑↓〉}. The site
parity operator ˆP = (−1) ˆN (where ˆN is an electron number operator) and the matrix
representations of ˆP and the creation operator of the electron with ↑ spin projection (c † ↑ )
in the site basis are given by
and
⎛
P = ⎜
⎝
⎛
c † ↑ = ⎜
⎝
1 0 0 0
0 −1 0 0
0 0 −1 0
0 0 0 1
0 0 0 0
1 0 0 0
0 0 0 0
0 0 1 0
⎞
⎞
⎟
⎠ , (2.30)
⎟
⎠ . (2.31)
Choosing the simplest ordering of sites, from 1 to l left to right, the creation operator
acting on the site j is
C † j↑ = (−1)P j−1
i=1 N i
I 1 ⊗ · · · ⊗ I j−1 ⊗ c † ↑ ⊗ Ij+1 ⊗ · · · ⊗ I l (2.32)
where the sign is positive (negative) if there is an even (odd) number of electrons to the
left of site j. N i is the number of electrons on the i-th site. Using P one can rewrite it in
a different form
C † j↑ = P1 ⊗ · · · ⊗ P j−1 ⊗ c † ↑ ⊗ Ij+1 ⊗ · · · ⊗ I l . (2.33)
To summarize, we have determined all the formulas ((2.23), (2.25), (2.26), and (2.27))
necessary to obtain the L(l)-representation of any operator acting on the block L(l). R(l)-
representations of operators acting on the right block R(l) can be obtained in a similar
way.
2.3 Density-Matrix Projection
In this section we determine the procedure that gives the optimal reduction of the Hilbertspace
dimension for the left block of a given length l, and obtain the corresponding basismapping
matrices A l (s l ). The presented arguments can be generalized for the right block

34 2. Density-Matrix Renormalization Group
Figure 2.1: Sketch showing superblock consisting of the system and the environment
blocks, and the system block constructed with a block L(l) and a site l.
and the corresponding B i (s i ) matrices. Forthcoming sections will describe the construction
and the optimization of all A l (s l ) and B i (s i ) for the entire system (l, i = 1, . . .,L).
In the above outlined “growing” procedure the block size is increased iteratively, adding
one site at every step, until the desired length L is reached. f Consequently, the block at
some step becomes a part of another, bigger one at the following steps. To avoid strong
boundary effects during this procedure, White proposed [255, 256] to embed the block,
which we might call the system block, into another one called the environment block, and
search for the reduced representation of it in the environment given by the superblock
(system + environment block) state. g Later in the subsection we will see that embedding
into the environment for avoiding strong boundary effects has a deeper meaning, but at
this stage let us follow a historical development of this concept.
Let us look into this matter more precisely. Assume that at some point we reached
a block of length l with a state space of dimension a l spanned by a set of orthonormal
vectors {|α l 〉}. Now one site is added to it resulting in a system block (S) which has a
n S = a l · d l+1 dimensional Hilbert space H S with a basis {|α l 〉 ⊗ |s l+1 〉 ≡ |α l s l+1 〉} =: {|i〉}.
To avoid strong boundary effects we embed the system block into an “environment” (E)
consisting with remaining sites of the lattice. This is schematically depicted in Fig. 2.1.
At this point and for the following discussion it is not important whether the environment
contains all remaining sites or just a part of them. Let n E be the size and {|j〉}
the complete orthonormal basis of the Hilbert space H E of the environment block. Now
we wish to determine a procedure which finds a set of orthonormal system-block states
{|α l+1 〉} that spans an m := a l+1 n S dimensional subspace H L(l+1) of H S and gives the
f It is not necessary to have a fixed L from the beginning. It can be identified as the size of the system
for which the convergence in some quantity was reached.
g Sometimes the name system/environment block is missleading, because at the end we are interested in
features of the entire system (superblock). It is less confusing just to use the names left and right blocks,
but it still will be a good concept to give a name system to the block whose states are going to be optimized
and the rest name as an environment.

2.3. Density-Matrix Projection 35
optimal reduction of it. The elements of A l+1 (s l+1 ) matrices are then obtained using
A l+1 (s l+1 ) αl+1 α l
= 〈α l s l+1 |α l+1 〉 (see Eq. (2.8)). But before, we define what the “optimal
reduction” means.
Consider a normalized pure state |ψ〉 of the entire system, e.g., the ground state of the
superblock. In the superblock basis {|α l s l+1 〉} ⊗ {|j〉} this state reads
|ψ〉 =
∑a l
d l+1
∑
n E ∑
α l =1 s l+1 =1 j=1
ψ αl s l+1 ,j|α l s l+1 〉|j〉 =
∑n S
i=1
n E ∑
j=1
ψ ij |i〉|j〉; 〈ψ|ψ〉 = 1 . (2.34)
Let n be the smaller of n S and n E . Associating coefficients ψ ij with a (n S × n E )-dimensional
matrix ψ one can write a singular value decomposition (SVD) of it
ψ = UDV † , (2.35)
where U and V are (n S × n)- and (n E × n)-dimensional matrices with orthonormal columns
and D is a (n × n)-dimensional diagonal matrix with non-negative entries D µµ = λ µ also
called singular values. If n S = n (n E = n) then U (V ) is also a unitary matrix. Using a
SVD of ψ (2.35) |ψ〉 can be rewritten as
|ψ〉 =
∑n S
n∑
n E ∑
i=1 µ=1 j=1
⎛ ⎞ ⎛
n∑
U iµ λ µ V † µj |i〉|j〉 = ∑n S
λ µ
⎝ U iµ |i〉 ⎠ ⎝
µ=1
i=1
n E ∑
j=1
V †
µj |j〉 ⎞
⎠ (2.36)
∑
and since λ µ are the singular values of ψ and |ψ〉 is normalized:
µ λ2 µ = 1. In the
following, we assume without loss of generality, that λ 1 λ 2 · · · λ n . The matrices U
and V apply a basis transformation on the system and the environment blocks, respectively,
leading to the diagonalization of matrix ψ. Their orthonormality properties ensure that
|u µ 〉 = ∑ i U iµ|i〉 and |v µ 〉 = ∑ j V † µj |j〉 form the orthonormal basis of the system and the
environment, respectively, in which the Schmidt decomposition [119, 206]
|ψ〉 =
n∑ ∑n Sch
λ µ |u µ 〉|v µ 〉 = λ µ |u µ 〉|v µ 〉 (2.37)
µ=1
holds. After all these steps n S · n E coefficients ψ ij are reduced to n Sch = rank(ψ)
min(n S , n E ) non-zero coefficients λ µ , also known as Schmidt rank and Schmidt coefficients,
respectively, and |ψ〉 is still represented exactly. Upon tracing out the environment or the
system part of the state the reduced density matrices for the system or the environment
part are found to be
µ=1
∑
ˆρ S = Tr E |ψ〉〈ψ| = nS
∑
[ψψ † ] ii ′|i〉〈i ′ | = nS
[UD 2 U † ] ii ′|i〉〈i ′ |
ii ′ ii ′
∑
= nS
λ 2 µ|u µ 〉〈u µ | = n ∑Sch
λ 2 µ|u µ 〉〈u µ | ,
µ=1
µ=1
(2.38a)

36 2. Density-Matrix Renormalization Group
∑
ˆρ E = Tr S |ψ〉〈ψ| = nE ∑
[ψ † ψ] jj ′|j〉〈j ′ | = nE [V D 2 V † ] jj ′|j〉〈j ′ |
jj ′ jj ′
∑
= nE λ 2 µ|v µ 〉〈v µ | = n ∑Sch
λ 2 µ|v µ 〉〈v µ | .
µ=1
µ=1
(2.38b)
So even if system and environment are different, both reduced density matrices have the
same eigenvalue spectrum and hence the same number of nonzero eigenvalues bounded
by the smaller of the dimensions of system and environment. Their eigenvalues are given
by the squared singular values of the state coefficient matrix and the eigenvectors — |u µ 〉
for ˆρ S and |v µ 〉 for ˆρ E — corresponding to nonzero eigenvalues λ 2 µ are the orthonormal
vectors of the Schmidt decomposition (2.37). Therefore the analysis of the reduced density
matrices or the Schmidt decomposition yields exactly the same information. Note that the
columns of U (V ) are the vector representations of the eigenstates of the system-block
(environment-block) reduced density matrix in the basis {|i〉} ({|j〉}).
Now we come back to the original problem of finding a procedure for the optimal reduction
of the system-block state space from the n S -dimensional H S to the m-dimensional
H L(l+1) . If m n Sch , then from (2.37) and (2.38b) it follows that the projection to the subspace
spanned by the system-block reduced density-matrix eigenstates |u µ 〉 with nonzero
eigenvalues λ 2 µ is enough to reduce the Hilbert-space size and to leave the superblock state
|ψ〉 unchanged. What remains is the case when m < n Sch and hence it is not possible to
find a projection which does not alter |ψ〉. In what follows I will provide three different
criteria for determining the optimal reduction procedures, which, as we will see, lead to
the same space-reduction schemes.
(i) Optimization of expectation values [257]: If the superblock is in a pure state |ψ〉
(2.34), the physical state of the system block is fully described through a reduced
density matrix ˆρ S (2.38a).
Consider some bounded operator Ô acting on the system block (‖Ô‖ =
max φ |〈φ|Ô|φ〉/〈φ|φ〉| ≡ c Ô
< ∞). The expectation value of Ô is
〈ψ|Ô|ψ〉
〈Ô〉 = = Tr Sˆρ S Ô (2.39)
〈ψ|ψ〉
that can be expressed in the reduced density-matrix eigenbasis as
n S
〈Ô〉 = ∑
λ 2 µ 〈u µ |Ô|u µ 〉 . (2.40)
µ=1
Now, since we wish to reduce the system-block space size to the m, it is reasonable
to consider a projection onto the subspace spanned by the m dominant eigenvectors
(those with the largest eigenvalues) of the reduced density matrix in order to obtain

2.3. Density-Matrix Projection 37
an accurate approximation for the expectation value. The error produced for 〈Ô〉
would be bounded by
|〈Ô〉 appr − 〈Ô〉| c Ô
where 〈Ô〉 appr = ∑ m
µ=1 〈u µ|Ô|u µ〉 and
ǫ =
∑n S
µ=m+1
∑n S
µ=m+1
λ 2 µ = ǫ c Ô , (2.41)
m
λ 2 µ = 1 − ∑
λ 2 µ . (2.42)
ǫ is called the discarded weight. Note that in 〈Ô〉 appr we have neglected the trace in
the denominator (2.39) that gives a correction of (1 − ǫ) −1 and is irrelevant for the
arguments presented here, as ǫ → 0. Estimate (2.41) particularly holds for the energies.
For any specific operator it can be tightened since the pre-truncated 〈u a |Ô|u a〉
is known and other more efficient projections can be found. However, for arbitrary
bounded operators acting on the system block, the projection to the subspace spanned
by the dominant eigenstates of the reduced density matrix is optimal.
(ii) Optimization of the wave function [255, 256]: Since quantum-mechanical objects
are completely described by their wave functions, it is reasonable to search for an
optimally reduced representation | ˜ψ〉 of |ψ〉, that minimizes the square deviation
µ=1
S := ‖|ψ〉 − | ˜ψ〉‖ 2 , (2.43)
and for which the system-block state space is spanned by m-dimensional orthonormal
set of states.
As we have seen, the n Sch -dimensional system-block subspace, spanned by the orthonormal
set {|u µ 〉; µ = 1, . . .,n Sch }, represents |ψ〉 exactly (see Eq. (2.37)). Therefore
we only need to find an optimal reduction of this subspace to the m-dimensional
one.
Expanding | ˜ψ〉 in the basis {|u µ 〉; µ = 1, . . .,n Sch } ⊗ {|v ν 〉; ν = 1, . . ., n Sch }
| ˜ψ〉 =
∑n Sch
µ,ν
˜ψ µν |u µ 〉|v ν 〉 , (2.44)
S can be rewritten as
∑n Sch
S =
λ
∥ µ |u µ 〉|v µ 〉 −
µ=1
∑n Sch
µ,ν=1
˜ψ µν |u µ 〉|v ν 〉
∥
2
=
∑n Sch
µ,ν=1
∣
∣λ µ δ µν − ˜ψ µν
∣ ∣∣
2
(2.45)

38 2. Density-Matrix Renormalization Group
where δ µν is a Kronecker delta. In this form one can see that S becomes minimal if
˜ψ µν = ˜λ µ δ µν and
˜λ µ =
{
λµ , for µ = 1, . . .,m
0, else
. (2.46)
Recall that λ 1 λ 2 · · · λ nSch and we search for | ˜ψ〉 with an m-dimensional
system-block subspace. h Plugging this in (2.44), the optimally reduced representation
of |ψ〉 is found to be
for which the square deviation
| ˜ψ〉 =
‖|ψ〉 − | ˜ψ〉‖ 2 =
m∑
λ µ |u µ 〉|v µ 〉 , (2.47)
µ=1
∑n Sch
µ=m+1
m
λ 2 µ = 1 − ∑
λ 2 µ = ǫ , (2.48)
is minimal. Therefore, the projection of the system-block space to the subspace spanned
by the m eigenvectors of the reduced density matrix with the largest eigenvalues, is
the optimal reduction procedure for the system-block state space.
ǫ in (2.48) also measures the loss in norm of |ψ〉.
(iii) Optimization of the fidelity [13, 245]: This is closely related to the both optimization
criteria considered above. The concept of fidelity was originally formulated in
communication theory as a quantitative measure of the transmission accuracy. A
generalized quantum analog was later formulated by Richard Jozsa [132]. Fidelity
between two pure normalized quantum states on a finite dimensional Hilbert space
is given by the transition probability (also called overlap)
µ=1
F (|ψ 1 〉〈ψ 1 |, |ψ 2 〉〈ψ 2 |) = |〈ψ 1 |ψ 2 〉| 2 , (2.49)
which also corresponds to the closeness of states in Hilbert space. Eq. (2.49) can
be extended for the case of mixed quantum states ˆρ 1 and ˆρ 2 [132] using Uhlmann’s
theorem [238] that generalizes the transition probability for mixed states in terms of
h This is equivalent to expanding | ˜ψ〉 in the same way as |ψ〉 (2.34), | ˜ψ〉 ∑
= nS
∑n E
i=1 j=1
˜ψ ij |i〉|j〉, associating the
coefficients ˜ψ ij with the (n S × n E )-dimensional matrix ˜ψ, and reducing the problem to the well known one
from linear algebra, namely, approximating a matrix ψ of rank n < min(n S , n E ) with a rank m < n matrix
( ˜ψ), that minimizes Frobenius norm of difference S = ∑ |ψ ij − ˜ψ
(
ij | 2 = Tr(ψ − ˜ψ)(ψ − ˜ψ)
) 2 † = ‖ψ −
2 ˜ψ‖ F
ij
[45].

2.3. Density-Matrix Projection 39
their purifications. Detailed definition of the fidelity and its properties can be found
in [184].
Consider again a superblock in a normalized pure state |ψ〉 (2.37). We are looking
for a normalized pure state | ˜ψ〉 for which the fidelity
F(|ψ〉〈ψ|, | ˜ψ〉〈 ˜ψ|) = |〈ψ| ˜ψ〉| 2 (2.50)
is maximal and the corresponding system-block subspace has a dimension m. This
procedure is equivalent of maximizing the fidelity between the original systemblock
reduced density matrix ˆρ S and the one obtained after the optimal reduction/truncation
of the system-block state space ˆ˜ρ S (that we still have to determine).
According to Jozsa’s result [132] this maximization can be achieved by maximizing
the fidelity between respective purifications of ˆρ S (|ψ〉) and ˆ˜ρ S (| ˜ψ〉) in a larger Hilbert
space, in our case just the superblock.
The Schmidt decomposition of | ˜ψ〉 is
| ˜ψ〉 =
m∑
˜λ µ |ũ µ 〉|ṽ µ 〉 , (2.51)
where ˜λ µ 0, 〈 ˜ψ| ˜ψ〉 = ∑ m
µ=1 ˜λ 2 µ = 1, and ˜λ 1 ˜λ 2 · · · ˜λ m .
The expansion of | ˜ψ〉 in the basis of the superblock {|i〉} ⊗ {|j〉} reads
µ=1
| ˜ψ〉 =
∑n S
i=1
n E ∑
j=1
˜ψ ij |i〉|j〉 . (2.52)
Associating the coefficients ˜ψ ij with the (n S × n E )-dimensional matrix ˜ψ, F can be
written as
∣ F(|ψ〉〈ψ|, | ˜ψ〉〈 ˜ψ|) = |〈ψ| ˜ψ〉| ∑n S n E ∣∣∣∣∣
2
∑
2 =
ψij ∗ ˜ψ ij = |Trψ † ˜ψ| 2 . (2.53)
∣
i=1
Consider now Von Neumann’s trace inequality i
j=1
n∑
|Tr ψ † ˜ψ| s k (ψ)s k ( ˜ψ) (2.54)
k=1
where s k (ψ) (s k ( ˜ψ)) is the k-th largest singular value of ψ ( ˜ψ). The equality in
Eq. 2.54 holds, if both |ψ〉 and | ˜ψ〉 have the same local basis in the ordered Schmidt
i The prove of this inequality can be found in [171, 172]. Although the original trace inequality [247] is
formulated for square matrices, it can be easily generalized for rectangular ones.

40 2. Density-Matrix Renormalization Group
decomposition, i.e., the µ-th largest Schmidt coefficients correspond to the same local
basis element (see Eqs. (2.37) and (2.51)). j With Von Neumann’s trace inequality
(2.54), the maximum possible fidelity between |ψ〉 and | ˜ψ〉 is
max F(|ψ〉〈ψ|, | ˜ψ〉〈 ˜ψ|) =
{|ũ µ〉},{|ṽ µ〉}
(
∑ m
) 2
λ µ˜λµ (2.55)
where the sum runs only up to m since ˜λ µ>m = 0. Associating the singular values
λ µ and ˜λ µ with m-dimensional vectors λ := (λ 1 , λ 2 , . . .,λ m ) (|λ| =
√ ∑m
µ=1 λ2 µ ) and
˜λ := (˜λ 1 , ˜λ 2 , . . ., ˜λ m ) (|˜λ| = 1), respectively,
max
|˜λ|=1
( m
∑
µ=1
) 2
λ µ˜λµ = max(λ · ˜λ) 2 =
|˜λ|=1
µ
m∑
λ 2 µ = 1 − ǫ ⇐⇒ λ ‖ ˜λ . (2.56)
Therefore, | ˜ψ〉 that approximates |ψ〉 in a sense of maximal fidelity is given by
µ=1
| ˜ψ〉 =
1
√
m∑
λ 2 µ
µ=1
m∑
λ µ |u µ 〉|v µ 〉 . (2.57)
µ=1
Once more, the projection of the system-block state to the subspace spanned by the m
eigenvectors of the reduced density matrix with the largest eigenvalues, is the optimal
reduction procedure for the system-block state space.
As we have seen, all the considered optimization criteria lead to the same procedure for the
optimal reduction of the system-block state space. The new state space is spanned by a set
of m dominant eigenvectors {|u µ 〉 = ∑ i U iµ|i〉; µ = 1, . . .,m} of the reduced density matrix
ˆρ S and since the elements of this set are orthonormal it can be considered as an effective
basis {|α l+1 〉} = {|u µ 〉} of the block L(l). The matrix elements of the basis transformation
matrices then read
A l+1 (s l+1 ) αl ,α l+1
= 〈α l s l+1 |α l+1 〉 = U αl s l+1 ,α l+1
. (2.58)
The outlined procedure of the block state-space optimal reduction is successful and a
substantial reduction can be achieved if the density-matrix eigenvalues fall down quickly.
Typically this is the case for one-dimensional quantum lattice systems with a finite energy
gap. That is why DMRG algorithms, which we describe in the upcoming sections
(Sections 2.4 and 2.5) have been quite successful in studying such systems [99, 100, 196].
j
Consider SVD of ψ, ψ = UDV † , where D is a diagonal matrix with singular values of ψ ordered in
non-increasing sequence. TrU † DV ˜ψ = TrDV ˜ψU † and if V ˜ψU † = ˜D, where ˜D is a diagonal matrix with
singular values of ˜ψ ordered in non-increasing sequence, then Trψ † ˜ψ = ∑ n
k=1 s k(ψ)s k ( ˜ψ).

2.3. Density-Matrix Projection 41
Up to now, we have considered the superblock being in a pure state. One can also study
systems in mixed states. For example mixed states naturally arise when one considers the
system at finite temperature. It is also useful when one wishes to consider several states
simultaneously. In this case the mixed state can be represented by saying that the entire
system has a probability w k to be in state |ψ k 〉. In this case the superblock state is
completely described by the density matrix
ˆρ = ∑ k
w k |ψ k 〉〈ψ k | (2.59)
where ∑ k w k = 1. If the system is at a finite temperature, then the w k are normalized
Boltzmann weights. The above outlined arguments can be generalized for the mixed state
given by (2.59) and for the reduced density matrix of the system block
ˆρ S = ∑ k
∑n S
w k
ii ′
n E ∑
j
ψ k ij (ψk i ′ j )∗ |i〉〈i ′ | . (2.60)
It can be verified that the projection of the system-block Hilbert space to the subspace
spanned by the m dominant eigenstates of the system-block reduced density matrix (2.60)
is again the optimal procedure for the system-block space reduction.
The above outlined schemes of finding optimal space reduction for block L(l) and
consequently A l (s l ) matrices straightforwardly generalize for the block R(i) and B i (s i )
matrices (i = L − l + 1) by exchanging the roles of the system and the environment blocks.
Entanglement and when the scheme works
The essential feature of a non-classical state is its entanglement [20, 212]. Entanglement or
nonseparability refers to the fact, that a subsystem (e.g. system block) of a system (e.g.
superblock) cannot be adequately described without taking into account its counterparts
or in other words just one state of the subsystem is not enough to represent a global state of
the system. This leads to the existence of quantum correlations between two or more sets
of degrees of freedom of a physical system that are considered as subsystems. In order to
study entanglement between two subsystems (one speaks about bipartition entanglement,
too) a useful concept is the Schmidt decomposition [184, 206].
Schmidt decomposition of the state partitioned into the system and environment parts,
is given by equation (2.37). When n Sch = 1, corresponding to a reduced density matrix in
a pure state ˆρ = |u 1 〉〈u 1 |, the state |ψ〉 = |u 1 〉|v 1 〉 factorizes into the product of individual
states for the system block and for the rest of the entire system (environment block). If
n Sch > 1, the block and the rest of the lattice are entangled. Typically the ground state, as
well as low lying states, of interacting lattice systems are entangled and exhibit quantum
correlations.

42 2. Density-Matrix Renormalization Group
In order to represent the global (e.g. superblock) states accurately, it is necessary to
take into account entanglement between a portion of the lattice sites (system block) and the
rest of the lattice (which we “mimic” with an environment block). Therefore, the optimal
system-block space-reduction procedure should preserve system-environment entanglement
[73, 74, 75, 191]. Neglecting it completely and considering the states of the isolated system
block, causes the removal of the key parts necessary for reproducing the entanglement in the
final state. This is also equivalent to the application of extra strong boundary conditions
to the subsystem and is the reason why Wilson’s numerical RG [260], with those isolated
block states, fails to find the ground state of the regular lattice systems, which in general
is entangled [191, 255, 256]. Note however, that for an accurate representation of the true
global state, the appropriate choice of the environment is also very important [191]. A good
environment has to contain as much as possible degrees of freedom, that are maximally
entangled with the system-block states. In general, it is not possible to pick the best
environment from the beginning, due to the same reason of an exponentially large Hilbert
space of the subsystem with all remaining sites. Therefore, it can be only constructed
and improved iteratively. In the following sections we will see that the first part, namely
the construction of the environment blocks just like the system blocks is performed by the
so-called infinite-system DMRG algorithm and then the finite-system DMRG algorithm
improves them iteratively.
According to quantum information theory the amount of entanglement between two
parts can be measured with the entanglement entropy [184], namely the von Neumann
entropy of the reduced density matrix ˆρ S ,
∑n Sch
S vN (ˆρ S ) ≡ −Tr(ˆρ S log 2 (ˆρ S )) = − λ 2 µ log 2(λ 2 µ ) . (2.61)
µ=1
This measure vanishes for a product state and it is maximal for the flat probability distribution
λ 2 µ = 1/n Sch, where S vN = − ∑ µ 1/n Sch log 2 (1/n Sch ) = log 2 (n Sch ). Therefore we
always have n Sch 2 S vN
. This quantity imposes a useful bound on the minimal number
m of kept states during the reduction process. Therefore, the true global state can be approximated
accurately with the MPS, if the bipartition entanglements of the subsystems
constituting the entire system are bounded or maximally grow logarithmically with large
subsystem sizes (effecting in polynomial growth for the matrix dimension). In one spatial
dimension, the entanglement entropy of a block of l contiguous sites typically increases
with l until l becomes of the order of the correlation length ξ in the system, at this point
it saturates to some value S max , whereas it diverges logarithmically at a quantum critical
point [191, 246]. The question of representing a quantum state in terms of matrix products
has recently been investigated in more details [240, 241].

2.4. Infinite-System DMRG algorithm 43
2.4 Infinite-System DMRG algorithm
The infinite-system method is certainly the simplest DMRG algorithm and it is the starting
point of many other DMRG methods. It is mainly designed for computing the ground state
(or the low-energy spectrum) of a quantum chain in the thermodynamic limit (L → ∞).
In the following we call the superblock state or states used for finding the optimal block
states ((2.8) and (2.17)) target states. By targeting only one state, the block states are
more specialized for representing it, and fewer are needed for a given accuracy. If excited
states or matrix elements between different states are required, more than one target state
can be used. However, for a fixed number of states kept (a j , b j m, for j = 1, . . .,L), the
accuracy with which the properties of each individual state can be determined goes down
as more states are targeted.
For simplicity, we will assume that only the ground state is targeted in the following
and only present a rough sketch of the algorithm. Details of the efficient implementation
are discussed in subsequent sections.
The infinite-system DMRG algorithm for determining the ground state of the system
proceeds as follows:
1. Form a system block of size l (L(l)) with the Hilbert space H L(l) = ⊗ l
i=1 H i of
dimension a l = ∏ l
i=1 d i, that is small enough to be treated exactly.
The tensor product of constituent-sites bases gives the block basis
{|α l 〉} = {|s 1 , . . .,s l 〉}. The matrix representations (L(l)-representations) of
the required operators, including the Hamiltonian, are tensor products of matrix
representations in the sites bases.
In the same way construct the environment block R(L − l + 1).
2. Enlarge the system block by adding one site from the right. The Hilbert space of the
obtained system block has a dimension n S = a l · d l+1 and is spanned by the product
states {|α l 〉|s l+1 〉 ≡ |α l s l+1 〉}. The environment block is enlarged, similarly. The
added sites are often called “active” or “free” sites.
3. Join the system and environment blocks to form the superblock of length 2l + 2. The
superblock Hilbert-space dimension would be n S · n E .
4. Determine the target state |ψ〉. If the target state is the ground state, diagonalize
the superblock Hamiltonian numerically, obtaining only the ground state eigenvalue
and eigenvector ψ using the Lanczos or Davidson algorithms.
5. Perform the reduced density-matrix projection:
(a) form the reduced density matrix for the enlarged system block ˆρ S , Eq. (2.38a);

44 2. Density-Matrix Renormalization Group
Figure 2.2: Schematic representation of the infinite-system DMRG. Circles correspond
to sites and ovals to blocks.
(b) determine its eigenbasis {|u µ 〉} using a dense matrix diagonalization routine;
(c) form a new reduced basis {|α l+1 〉} for the enlarged system block taking a l
eigenstates of the reduced density matrix with the largest eigenvalues.
The elements of their vector representations in the product basis of the
enlarged system block give the elements of the space-reduction matrices
A l+1 (s l+1 ) αl ,α l+1
= 〈α l s l+1 |α l 〉.
(d) repeat all these steps to construct the reduced basis {|β L−l 〉} of dimension b L−l
and the projection matrices B L−l (s L−l ) for the environment block.
6. Construct the L(l + 1)-representations (see Section 2.2.2) of the Hamiltonian and
other operators acting on the system block that are required in the course of the
iteration (e.g. observables). Similarly, build R(L − l)-representations for the corresponding
operators acting on the environment block.
7. Calculate the desired ground state properties, e.g. correlations, from |ψ〉.
8. Repeat starting with step 2 — substituting l + 1 for l — until the desired length L
of the system is reached. k
This algorithm is schematically depicted in Fig. 2.2.
So far nothing was said how to determine a l and b L−l (l = 1, . . ., L/2), the numbers
to which the dimensions of system- and environment-block state spaces are reduced. The
above outlined algorithm can be performed in a couple of different ways. First we can
k For of odd L, at the final step either the system or the environment block is enlarged.

2.4. Infinite-System DMRG algorithm 45
1e-03
1e-04
(e(L)-e ex
)/e ex
1e-05
1e-06
m = 100
m = 200
m = 400
1e-07
0 500 1000 1500 2000
L
Figure 2.3: Convergence of the ground state energy per site (e(L)) obtained with the
infinite-system DMRG algorithm as a function of the superblock size L for three different
numbers of density-matrix eigenstates kept m. Results are for the one-dimensional
Hubbard model at half filling with U/t = 4.
use state-space dimensions which are (almost) constant a l , b L−l m. In this case, the
discarded weight (2.42) is variable. Second, the reduced density-matrix eigenbasis can be
truncated so that the discarded weight is approximately constant, ǫ l , ǫ L−l ε, where ε
denotes the discarded weight threshold. In this case, the number of kept states (and also
state-space dimensions), is variable. One can improve the latter scheme by limiting the
range of the number of density-matrix eigenstates kept. In this case both the number of
kept states and the discarded weight threshold become variable. In the considered cases,
physical quantities are calculated for several values of m or/and ε and their scaling is
analyzed for increasing m or decreasing ε. Some other, more elaborated reduction schemes
can be also used [151], but this strategy is not followed in the present thesis.
Different version of step 8 can also be considered. Like in the original formulation of
the infinite-system method by White [256, 257], instead of starting with the fixed L one
starts with L = 2l + 2 and at step 8 changes L to L + 2. Then everything is repeated until
convergence in some particular quantity (e.g. energy per site) is reached. In this case each
iteration pushes the ends of the chain farther from the two sites in the center. After the
convergence is reached each block represents one half of the infinite chain, so it converges
in two senses simultaneously: in the length of L(l) going to infinity and in the sense that
L(l) is adapted to respond to an infinite chain connected to it on the right.

46 2. Density-Matrix Renormalization Group
As an illustration, Fig. 2.3 shows the convergence of the ground state energy per site as
a function of the superblock size L in the one-dimensional Hubbard model at half filling for
U = 4t. The energy per site e(L) is calculated from the total energy E 0 for two consecutive
superblocks e(L) = (E 0 (L) − E 0 (L − 2))/2. The exact Bethe-ansatz results for an infinite
chain is e ex ≈ −1.573729367898451. l The block space dimensions a l , b l are chosen to be
not greater than a number m. As L increases, e(L) converges to a limiting value e(m).
This energy is always higher than the exact ground state energy e ex as expected for a
variational method. The error in e(m) is dominated by truncation errors, which decrease
rapidly as the number m increases.
Experience shows that the infinite-system DMRG algorithm yields accurate results for
local physical quantities such as spin density and short-range spin correlations in quantum
lattice models with “good” features (infinite homogeneous one-dimensional systems with
short-range interactions like the Heisenberg model on an open chain).
The present algorithm has been formulated to obtain the system’s ground state, but
it easily generalizes to the cases with several target states. These additional states are
obtained at step 4 and the reduced density matrix for the enlarged block is constructed
using Eq. (2.60) at step 5.a at each iteration of the infinite-system DMRG algorithm.
2.5 Finite-System DMRG algorithm
For many systems and physical quantities the infinite-system algorithm does not give satisfactory
results. Very often it does not generate the optimal block representation for
the ground state (i.e., does not find the best possible matrix-product state (2.21) for the
present matrix sizes). Problems arise, if the environment in the early stage of the chain
growth does not resemble the system of final length closely enough. Typically, if the wave
function changes qualitatively between the iterations, convergence may be poor or even
not reachable. This can happen in electronic systems when incommensurate fillings occur
for some lattice sizes. In systems close to a first-order transition one may be trapped in
a metastable state favored for small system sizes, e.g., by edge effects. In this case the
effective representations of the blocks, obtained in the early stage do not have to be optimal
for representing the final state. In these cases, one requires an algorithm in which the
same finite system is treated at each step and effective representations of the blocks are
improved iteratively. The idea is to optimize the chosen basis for a system of fixed length
L by shifting “active” sites through the system and instead of convergence to an infinitesystem
fixed point, have a variational convergence to the target-state (target-states) wave
function (wave functions).
∫ ∞
l
dω J 0 (ω)J 1 (ω)
e ex = −U − 4t
0 ω 1 + exp(Uω/2t) , where U = 4t and J 0/1 are Bessel functions [155, 156].

2.5. Finite-System DMRG algorithm 47
The finite-system DMRG algorithm for determining the ground state of the system
proceeds as follows:
0. Carry out the complete infinite-system algorithm storing L(l)- and R(l ′ )-
representations (l = 1, . . .,L/2, l ′ = L/2, . . .,L) of all required operators including
the block Hamiltonian and the operators needed to connect the blocks at each step.
1. Carry out steps 2-5 of the infinite-system DMRG algorithm to obtain left (system)
block L(l + 1), new L(l + 1)-representations of the Hamiltonian and the operators
needed to connect the blocks, and the space-reduction matrices A l+1 (s l+1 ). Store
them.
Right (environment) block R(l + 3) with R(l + 3)-representations of required operators
are retrieved form the storage.
2. Repeat step 2 until l = L − 3. This is the left-to-right phase of the algorithm.
3. Carry out steps 2-5 of the infinite-system algorithm to obtain now right (environment)
block R(L − l), new R(L − l)-representations of the Hamiltonian and the operators
needed to connect the blocks, and the space-reduction matrices B L−l (s L−l ). Store
them.
L(L − l − 2) block with L(L − l − 2)-representations of required operators are retrieved
form the storage.
4. Repeat step 3 until l = 1. This is the right-to-left phase of the algorithm.
5. Repeat step 1 until l = L/2 − 1.
6. Calculate the desired ground-state properties, e.g. correlation functions, from |ψ〉.
7. Repeat starting with step 1 until convergence in the ground state is reached.
This algorithm is schematically depicted in Fig. 2.4.
One iteration of the outermost loop (steps 1-5) is usually called a finite-system sweep;
complete left-to-right or right-to-left sweeps are also called complete half-sweeps. Typically,
when the left or right block reaches some minimum size and can be described exactly, the
sweep direction is switched.
We say that convergence is reached (step 7) when the energy at each iteration matches
closely the energy at the equivalent iteration of the previous sweep, to very high accuracy.
In this formulation of the finite-system algorithm, we have assumed that the infinitesystem
algorithm can be carried out to build up the lattice to the desired size and to
generate an initial set of environment blocks. If this is not the case, the finite-system

48 2. Density-Matrix Renormalization Group
Figure 2.4: Schematic representation of the complete sweep of the finite-system
DMRG algorithm. Circles correspond to sites, ovals to blocks.
method can still be applied, if a reasonable approximation for the environment block for
the first finite-system sweep can be found. This can be done using a few exactly treated
sites as the environment. As long as the initial procedure does not lead to a system-block
basis that is inadequate, convergence is reached after a relatively small number of finitesystem
sweeps for most systems, typically between 2 and 10 for one-dimensional systems.
Similar to the infinite-system DMRG algorithms, iterations of the finite-system sweeps
can be performed with the fixed number of kept reduced density-matrix eigenstates m, with
the fixed discarded weight threshold ε, or some combination of both; they will determine
the corresponding a l or b l at each iteration. For more elaborated schemes see Ref. [151].
For a given system size L, the finite-system algorithm almost always gives substantially
more accurate results than the infinite-system algorithm, and is therefore usually preferred
unless there is a specific reason to go to the thermodynamic limit. As an illustration in
Fig. 2.5 we show the behavior of the ground-state energy in the course of a DMRG run
for the one-dimensional Hubbard model. Note that the convergence is variational and the
representation of the system is qualitatively improved.
Note, that similar to the infinite-system algorithm, the finite-system algorithm can be
also generalized for the cases with several target states.
Before moving to the implementation details of the DMRG algorithms, it is worthwhile
to mention that the DMRG with the block-site-site-block configuration produces position

2.5. Finite-System DMRG algorithm 49
5e-03
4e-03
(E(i) - E ex
)/E ex
3e-03
2e-03
1e-03
0 50 100 150 200
i
Figure 2.5: Convergence of the ground-state energy (E(i)) obtained with the finitesystem
DMRG algorithm as a function of the superblocks’ bipartition position i for
m = 25 reduced density-matrix eigenstates kept. Results are for an L = 200 sites
one-dimensional Hubbard model with U/t = 4 and N ↑ = N ↓ = 63. Arrows show the
direction of the first three complete half-sweeps starting from top.
dependent MPS and the method is not truly variational, the energy might increase within
the finite-system sweep. This is due to the fact that when determining A l (s l ) or B l+1 (s l+1 )
the enlarged representations of both the system and the environment blocks are considered.
First the optimal representation within the a l d l+1 -dimensional (d l+1 b l+2 -dimensional) state
space is found by reobtaining the target state in the enlarged a l d l+1 d l+1 b l+2 -dimensional
space and then optimal reduction of this enlarged space to the subspace of dimension
a l+1 a l d l+1 (b l−1 d l−1 b l ) is obtained (reduced density-matrix projection). At the second
step some states with finite weights are usually discarded leading to an approximate
representation of the target state (see Section 2.3). The situation is completely different,
if one switches form the block-site-site-block configuration to the block-site-block configuration
(so-called two-site and one-site DMRG algorithms). Takasaki et al. [227] showed
that the quality of the results can be improved, if one switches to the block-site-block
configuration in the latter stages of the finite-system algorithm. When this is done, the
finite-system algorithm has some interesting properties. Namely, the only states discarded
by the space reduction procedure have zero weight in the wave function, so the wave function
is exactly represented after the truncation. Since no states that have nonzero weight
in the wave function are ever lost, the variational principle implies that the new states

50 2. Density-Matrix Renormalization Group
Figure 2.6: Schematic picture of the superblock consisting of the system and the
environment blocks. The system block contains the block L(l − 1) and the l-th site,
while the environment block is constructed with the (l + 1)-th site and the block R(l 2 ).
introduced at each iteration can only reduce the energy. Thus, the obtained ground state
energy can only monotonically decrease in the course of the DMRG sweep. Moreover, the
converged state is independent of the position of the added site in the lattice. This is
in sharp contrast to the behavior of the standard two-site DMRG algorithm, where there
is a large position dependence. However, the difference between properties calculated by
one-site and two-site variants of the finite-system algorithm goes to zero as the number of
kept states m is increased.
2.6 Implementation details
The typical superblock configuration encountered at the l-th step of the infinite- or finitesystem
DMRG algorithms includes the left block L(l − 1), the l-th and (l + 1)-th sites,
and the right block R(l + 2). The left block L(l − 1) with an a l−1 -dimensional effective
basis B(L, l − 1) and the l-th site with a d l -dimensional basis B(l) (see Section 2.2) form an
enlarged system block of length l with the Hilbert space H S of dimension n S = a l−1 · d l . Its
basis can be written as a tensor product of block and site bases {|α l−1 〉 ⊗ |s l 〉 ≡ |α l−1 s l 〉}.
The right block R(l + 2) with a b l+2 -dimensional effective basis B(l + 2) together with the
(l + 1)-th site with a basis B(l + 1) of dimension d l+1 form an n E = d l+1 · b l+2 -dimensional
enlarged environment block with a tensor product basis {|s l+1 〉 ⊗ |β l+2 〉 ≡ |s l+1 β l+2 〉}. To
simplify notation in the following we will use α and β instead of α l−1 and β l+2 , respectively.
The basis of the superblock which we denote as B(SB, l, l + 2) is made of the tensor product
of the bases of the system and the environment blocks
B(SB, l, l + 1) = {|αs l 〉 ⊗ |s l+1 β〉 ≡ |αs l s l+1 β〉} . (2.62)
This configuration is schematically illustrated in Fig. 2.6. Any superblock state and any operator
acting within the superblock can be written in its vector and matrix representations
in this basis.

2.6. Implementation details 51
2.6.1 Superblock Hamiltonian
At each step of infinite- and finite-system DMRG algorithms, we reobtain the ground
state for which the optimization of the block state-space basis is performed. Typically
one can diagonalize the system Hamiltonian for it, but since we are interested only in
particular eigenstates, we can use more elucidate algorithms which can obtain particular
eigenstates without a full diagonalization or even a full construction of the system Hamiltonian
matrix. These algorithms are iteratively calculating the desired eigenstate from some
(random) starting state through successive — typically costly — matrix-vector multiplications.
Therefore we only have to provide an efficient procedure for calculating the product
of the Hamiltonian matrix with the input vector. In DMRG one mainly uses iterative
diagonalization algorithms like Lanczos [37, 146] or Davidson [12, 40, 41, 203]. If diagonal
elements of the Hamiltonian matrix can be obtained cheaply (without constructing the
actual matrix), the latter method is preferable because of accelerated convergence to the
desired eigenstate caused by using a diagonal preconditioner. In addition, this method is
more stable in the case of degenerate eigenstates. Relatively new is a method introduced by
Sleijpen [219], that — for some extra moderate cost — gives an extra acceleration of convergence
compared to the typical Davidson one. More details about this method, known as
Jacobi-Davidson, can be found in the Refs. [12, 219]. In the DMRG program developed by
us, we mainly use the variant of the Davidson algorithm with the diagonal preconditioner
developed by Sadkane and Sidje, namely the variable-block Davidson method [204].
The most pleasant feature of these algorithms is the fact, that for an (N × N)-
dimensional matrix they require only a much smaller number Ñ ≪ N of iterations, so
that iterative approximations to eigenvalues converge very rapidly to the maximum and
minimum eigenvalues of Ĥ at machine precision. With slightly more effort other eigenvalues
at the edge of the spectrum can also be computed. Therefore, the complexity of
the problem reduces from O(N 3 ) in the case of full diagonalization to O(N 2 ), and the
explicit construction and storage of the Hamiltonian matrix is avoided. Typical values for
the number of iterations (matrix-vector multiplications) in DMRG calculations are of the
order of 100.
As discussed, the problem of obtaining the ground state (or some low-energy states)
reduces to finding an efficient procedure that calculates the product of the Hamiltonian
with a state
|ψ ′ 〉 = Ĥ|ψ〉 (2.63)
without an explicit construction of the Hamiltonian.
Usually the Hamiltonian is given as a sum of different terms that can be grouped taking
into account the structure of the superblock. Assuming nearest-neighbor interactions (close
to the end of this subsection it will become clear that a straightforward generalization to
long range interactions is also possible), the superblock Hamiltonian can be decomposed

52 2. Density-Matrix Renormalization Group
in the considered superblock configuration: left block (L), left site (•), right site (◦), and
right block (R),
Ĥ = ĤL + Ĥ• + ĤL• + Ĥ•◦ + Ĥ◦R + Ĥ◦ + ĤR (2.64)
where ĤL and ĤR contain all terms acting only within the left and right blocks, respectively;
Ĥ • and Ĥ◦ consist of terms acting only on the left and right sites, respectively; Ĥ L•
and Ĥ◦R enclose interactions between blocks and neighboring sites, and finally the term
Ĥ •◦ contains only the interaction terms between the sites.
Any superblock state |ψ〉 can be expressed in the basis B(SB, l, l + 1) (2.62)
and equation (2.63) transforms to
ψ ′ α ′ s ′ l s′ l+1 β′ =
|ψ〉 = ∑ α, s l
s l+1 , β
∑
α, s l , s l+1 , β
ψ αsl s l+1 β|αs l s l+1 β〉 (2.65)
H α ′ s ′ l s′ l+1 β′ ; α s l s l+1 β · ψ αsl s l+1 β , (2.66)
where ψ ′ and ψ are vector representations of |ψ ′ 〉 and |ψ〉 in the superblock basis, respectively,
and
H α ′ s ′ l s′ l+1 β′ ; α s l s l+1 β = 〈α ′ s ′ ls ′ l+1β ′ |Ĥ|αs ls l+1 β〉 (2.67)
is the matrix representation of Ĥ in the same basis. If H is constructed explicitly, then
one requires an extra memory storage for (a l−1 d l d l+1 b l+2 ) 2 elements and the same amount
of operations for evaluating (2.66). Considerig the Hamiltonian decomposition (2.64) one
can reduce these costs substantially. The first and the last three terms in (2.64) are only
acting within the system and environment blocks, respectively, hence
〈α ′ s ′ ls ′ l+1β ′ |ĤL/•/L•|αs l s l+1 β〉 = 〈α ′ s ′ l|ĤL/•/L•|αs l 〉〈s ′ l+1β ′ |s l+1 β〉
= [H L/•/L• ] α ′ s ′ l ; αs l δ s ′ l+1 ;s l+1 δ β ′ ;β (2.68)
〈α ′ s ′ l s′ l+1 β′ |Ĥ◦R/◦/R|αs l s l+1 β〉 = 〈α ′ s ′ l |αs l 〉〈s′ l+1 β′ |Ĥ◦R/◦/R|s l+1 β〉
= δ α ′ ;αδ s ′
l ;s l
[H ◦R/◦/R ] s ′
l+1 β ′ ; s l+1 β (2.69)
and only the matrix representations of Ĥ L/•/L• (Ĥ◦R/◦/R) in the system (environment)
block basis are required. What remains is the middle term Ĥ•◦ describing interactions
between the left and right sites and consequently the part of the interactions between the
system and the environment blocks. In general interactions between the system and the
environment blocks, which we denote as ĤSE, can be always written as a sum of products
of operators acting within the system and the environment blocks, respectively,
∑n ν
Ĥ SE = Ĉν S ˆD ν E (2.70)
ν=1

2.6. Implementation details 53
where each of ĈS ν ( ˆD ν E ) acts within the system (environment) block. The matrix representation
of (2.70) in B(SB, l, l + 1) will be
∑n ν
〈α ′ s ′ l s′ l+1 β′ |ĤSE|αs l s l+1 β〉 = 〈α ′ s ′ l |ĈS ν |αs l 〉〈s′ l+1 β′ | ˆD ν E |s l+1 β〉
ν=1
n ν
∑
= [Cν S ] α ′ s ′ l ; αs [DE l ν ] s ′ l+1 β′ ; s l+1 β (2.71)
ν=1
where C S ν (D E ν ) are the matrix representations of Ĉ S ν ( ˆD E ν ) in the system (environment)
block basis.
Equations (2.65)-(2.71) can be rewritten in a more compact and clear form by adopting
the following matrix notations: associating expansion coefficients ψ αsl s l+1 β and ψ′ α ′ s ′ l s′ l+1 β′
of |ψ〉 and |ψ ′ 〉 with (a l−1 · d l × d l+1 · b l+2 )-dimensional matrices ψ and ψ ′ , (αs l forms the
row index and the column index is s l+1 β) we can rewrite (2.63) using matrix notation as
n ν
ψ ′ = (H L + H • + H L• )ψ + ψ(H◦R T + HT ◦ + HT R ) + ∑
Cν S ψ(DE ν )T , (2.72)
where M T means the transpose of M even in the case when M is a matrix with complex
elements. At this stage one can conclude, that for evaluating (2.63) one requires the
matrix representation of the system- and environment-block Hamiltonians in the systemand
environment-block bases (see (2.68) and (2.69)) and matrix representations of all
operators participating in the interaction between the system and the environment blocks
in the corresponding system- and environment-block bases. This strategy was followed in
the early DMRG calculations like in the original papers [188, 255, 256].
Let us go one step further and try to use the composite structure of the system and
the environment blocks (block-site, site-block) to avoid the explicit construction of Hamiltonians,
similarly to what we did for the superblock. This saves the computer memory
required for storing matrix representations of all necessary operators in the system- and
environment-block bases, and at the same time reduces the number of operations needed
to evaluate it. m For this we consider each term of (2.72) separately. We start with H L• .
As for the Hamiltonian terms describing the interaction between the system and the environment
blocks, the term describing the interaction between the left block and the site
inside the system block can be written as
ν=1
n
∑ µ
Ĥ L• = Ĉµ L Ĉl µ , (2.73)
µ=1
m The operators acting on a bare site are usually very sparse.

54 2. Density-Matrix Renormalization Group
where ĈL µ are operators acting only within the left block (L(l − 1) in the considered case)
and Ĉl µ are local operators acting on the site l. Matrix representation of Ĥ L• in the
system-block basis H L• is
n
[H L• ] α ′ s ′ l ; αs = 〈α′ s ′ l l |ĤL•|αs ∑ µ
l 〉 = 〈α ′ |ĈL µ |α〉〈s′ l |Ĉl µ |s l 〉
µ=1
n µ
∑
=
µ=1
[C L(l−1)
µ ] α ′ ; α[C l µ ] s ′ l ; s l
(2.74)
and it is clear that only L(l − 1)-representations Cµ L(l−1) of Ĉµ L (see Section 2.2.2) and
the matrix representation Cµ l of Ĉl µ in its own site basis have to be known. From (2.74)
it becomes clear that n µ a 2 l−1 d2 l d l+1b l+2 operators are required in order to evaluate H L• ψ
directly, but this number can be reduced to n µ (a l−1 + d l )a l−1 d l d l+1 b l+2 by applying the
left block and the site parts separately
a
φ µ = ∑ l−1
α ′ s l ; s l+1 β
α=1
n µ
ψ ′ α ′ ,s ′ ; s l+1 β = ∑
[C L(l−1)
µ=1 s l =1
µ ] α ′ ; α ψ α,s l ; s l+1 β , (2.75a)
∑d l
[C l µ ] s ′ l ; s lφ µ α ′ ,s l ; s l+1 β . (2.75b)
The first requires a 2 l−1 d ld l+1 b l+2 operations and an extra storage for intermediate φ µ and
the second a l−1 d 2 l d l+1b l+2 operations — all together n µ (a l−1 + d l )a l−1 d l d l+1 b l+2 operations.
This can be easily generalized for simpler ĤL (H L ) and Ĥ• (H • ), for which the
corresponding counterparts are identity operators represented as Kronecker deltas in the
matrix element notation. Their action on ψ can be evaluated without an intermediate
result and requires only a 2 l−1 d ld l+1 b l+2 and a l−1 d 2 l d l+1b l+2 operations, respectively. Hence
the computational cost for evaluating the product of the system-block Hamiltonian with
the state matrix ψ is (n µ + 1)(a l−1 + d l )a l−1 d l d l+1 b l+2 instead of a 2 l−1 d2 l d l+1b l+2 which one
would have for an explicitly constructed system-block Hamiltonian. In case when d l = d
for l = 1, . . ., L and finite-system sweeps are performed with only m states kept, the number
of operations required for this part is (n µ + 1)(m + d)m 2 d 2 . Parts of the Hamiltonian
acting within the environment block (H ◦R , H ◦ and H R ) can be applied in a similar way.
There remains the term ∑ n ν
ν=1 CS ν ψ(Dν E ) T describing the interaction between the system
and the environment blocks. In the considered case where the interactions between the
system and the environment blocks are restricted to the ones between the free sites, Cν S and
Dν E correspond to matrix representations of the local operators (see Section 2.2.2 for local
operators) Ĉl ν and in the system- and environment-block bases, respectively. Hence
ˆD
l+1
ν
[Cν S ] α ′ s ′ l ; αs = 〈α′ s ′ l l |Ĉl ν |αs l 〉 = δ α ′ ;α[Cν l ] s ′ l ;s l
(2.76a)
[Dν S ] s ′ l+1 β′ ; s l+1 β = 〈s ′ l+1 β′ l+1
| ˆD ν |s l+1 β〉 = [Dν l+1 ] β ′ ;βδ s ′
l+1 ;s l+1
(2.76b)

2.6. Implementation details 55
where C l ν (Dl+1 ν ) are matrix representations of the local operators Ĉl ν
((l + 1)-th) site basis. Evaluating ∑ n ν
ν=1 Cl ν ψ (D l+1
l+1
( ˆD ν ) in the l-th
ν ) T in two steps like (2.74) and (2.75)
we will require only n ν a l−1 d 2 l d2 l+1 b l+2 operations instead of n ν (a l−1 d l d l+1 b l+2 ) 2 . Such a
decomposition is crucial when block-block interactions ĤLR appear for longer-ranged interactions.
In this case d l d l+1 times fewer operations will be required than in the case of
explicitly constructed interaction terms.
2.6.2 Measurements
Measurements in the DMRG framework correspond to the calculation of expectation values
of operators within a state or between states of the superblock, which are obtained in the
iterative diagonalization step: e.g., for any arbitrary operator Ô and states |ψ〉, |φ〉
o = 〈φ|Ô|ψ〉 . (2.77)
This procedure is straightforward provided that the necessary operators are available in
the appropriate basis.
Any operator Ô acting on a superblock state can be represented as a sum of products
of operators each acting on the left block, the left site, the right site, and the right block,
respectively
Ô = ∑ Ôν
L(l−1) Ô νÔl+1 l ν Ôν R(l+2) . (2.78)
ν
Therefore, if O L(l−1)
ν
Ô R(l+2)
ν
and Oν
R(l+2) , L(l − 1)- and R(l + 2)-representations of Ôν
L(l−1) and
, respectively, are known then o (2.77) can be calculated straightforwardly. These
representations can be constructed and stored in the course of the DMRG sweeps, or
obtained at the stage where the “measurements” are required using the space-reduction
matrices A k (s k ), k = 1, . . ., l − 1, and B k ′(s k ′), k ′ = l + 2, . . .,L (see Section 2.2.2).
Like in the previous subsection, the operator Ô can be decomposed similar to the system
Hamiltonian Ĥ (2.64) and the derived procedure of calculating |ψ′ 〉 = Ĥ|ψ〉 (2.66)-(2.76)
employed substituting Ĥ with Ô. Finally, expectation value o (2.77) is obtained as
o = Trφ † ψ ′ , (2.79)
where φ and ψ ′ are (a l−1 · d l × d l+1 · b l+2 )-dimensional expansion coefficients matrices of
the states |φ〉 and |ψ ′ 〉, respectively, and |ψ ′ 〉 = Ô|ψ〉.
2.6.3 Wave-function transformations
The most time-consuming part of the DMRG algorithm is the iterative diagonalization of
the superblock Hamiltonian. Here we discuss how to optimize this procedure significantly

56 2. Density-Matrix Renormalization Group
by reducing the number of steps in the iterative diagonalization, in some cases by up to
an order of magnitude.
The key operation and most time-consuming part of any iterative diagonalization procedure
is the calculation of the product of the Hamiltonian with an arbitrary wave function,
i.e., the operation Ĥψ. Typically, up to a few hundreds of such multiplications are required
to reach convergence. Therefore, reducing this number would directly lead to a proportional
speedup of the diagonalization part. Usually, if one starts with a good approximation
to the desired state the Lanczos or Davidson procedure will require much fewer iterations.
Since the same system is treated at each step of the finite-system algorithm, an obvious
starting point is the result |ψ l−1 〉 of a previous finite-system step (we assume left-to-right
sweep). However, this state vector is not in an appropriate basis for Ĥ at step l because
it was obtained using a different superblock configuration (step l − 1). In order to be able
to perform the multiplication Ĥ |ψl−1 〉, the wave function must be transformed from the
basis at step l − 1
to a basis
B(SB, l − 1, l) = {|α l−2 s l−1 〉 ⊗ |s l β l+1 〉 ≡ |α l−2 s l−1 s l β l+1 〉} (2.80)
B(SB, l, l + 1) = {|α l−1 s l 〉 ⊗ |s l+1 β l+2 〉 ≡ |α l−1 s l s l+1 β l+2 〉}. (2.81)
suitable to describe the system configuration at step l. This transformation is performed in
two steps. The left block is transformed from the original product basis {|α l−2 〉 ⊗ |s l−1 〉}
to the effective state-space basis obtained at the end of the previous DMRG step, {|α l−1 〉},
using
|α l−1 〉 = ∑
s l−1 ,α l−2
A l−1 (s l−1 ) αl−2 ,α l−1
|α l−2 〉 ⊗ |s l−1 〉 . (2.82)
Similarly, for the right basis
|β l+1 〉 = ∑
s l+1 ,β l+2
B l+1 (s l+1 ) βl+1 ,β l+2
|s l+1 〉 ⊗ |β l+2 〉 . (2.83)
To perform the wave-function transformation needed for the left-to-right part of the
DMRG, we expand the wave function at step l − 1 in B(SB, l − 1, l)
|ψ〉 =
∑
ψ αl−2 s l−1 s l β l+1
|α l−2 s l−1 s l β l+1 〉 (2.84)
α l−2 , s l−1
s l , β l+2
and insert ∑ α l−1
|α l−1 〉〈α l−1 |. Since a truncation is involved in the procedure of obtaining
{|α l−1 〉}, this is only an approximation,
∑
α l+1
|α l+1 〉〈α l+1 | ≈ 1 . (2.85)

2.6. Implementation details 57
Next we insert ∑ β l+1
|β l+1 〉〈β l+1 | which is exact for the considered sweep direction.
The coefficients of the wave function in the new basis B(SB, l, l + 1) therefore become
ψ αl−1 s l s l+1 β l+2
=
∑
A l−1 (s l−1 ) αl−2 ,α l−1
ψ αl−2 s l−1 s l β l+1
B l+1 (s l+1 ) βl+1 ,β l+2
.
α l−2 ,s l−1
β l+1
It is convenient to perform this procedure in two steps: first obtain the intermediate result
and then
ψ αl−1 s l β l+1
=
∑
α l−2 ,s l−1
A l−1 (s l−1 ) αl−2 ,α l−1
ψ αl−2 s l−1 s l β l+1
,
ψ αl−1 s l s l+1 β l+2
= ∑ β l+1
ψ αl−1 s l β l+1
B l+1 (s l+1 ) βl+1 ,β l+2
.
(2.86a)
(2.86b)
An analogous transformation is used for a step in the right to left sweep.
It is worthwhile to mention that in the case of finite-system sweeps performed with the
fixed number of kept states m and block-site-block configuration of the superblock (one-site
finite-system algorithm), discarded weight at each state-space reduction is zero, and the
considered wave-function transformation (2.86) becomes exact.
2.6.4 Additive quantum numbers
A considerable speedup of the DMRG algorithms and significant reduction of the required
computer resources can be achieved by using the symmetries of the studied problem. We
only consider the U(1) abelian symmetries leading to additive good (conserved) quantum
numbers e.g., total magnetization S z and total particle number N. These type of symmetries
are relatively easy to implement and are applicable for a wide class of models. In the
case of additive quantum number, the quantum number of the tensor product of two states
is given by the sum of the quantum numbers of both states. The use of other symmetries
and quantum numbers is described in [167, 166, 210].
Consider an operator ˆQ acting in H, which is the sum of Hermitian local site operators
ˆQ = ∑ L ˆQ j=1 j , where the operator ˆQ j acts only on the site j. If ˆQ commutes with the
system Hamiltonian Ĥ, eigenstates of Ĥ can be chosen to be eigenstates of ˆQ, too. If the
DMRG target state is an eigenstate of ˆQ (e.g., the ground state of Ĥ), then one can show
that the reduced density operators for the left and right blocks also commute with the
operators
j∑
L∑
ˆQ L(j) = ˆQ i and ˆQR(j+1) = ˆQ i , (2.87)
i=1
i=j+1
respectively. Therefore, the density-operator eigenstates (2.38a) and (2.38b) can be chosen
to be eigenstates of ˆQL(j) and ˆQ R(j+1) , respectively, and block basis states can be extra

58 2. Density-Matrix Renormalization Group
labeled by their quantum number corresponding to the eigenvalue of ˆQ L(j) or ˆQ R(j+1) . For
instance, the basis of the left block L(l − 1) becomes
B(L, l − 1) = {|r, α l−1 〉; r = 1, 2, . . .; α l−1 = 1, . . ., a r,l−1 } , (2.88)
where the index r numbers the possible quantum numbers qr
L(l−1) of ˆQ L(l−1) , α l−1 numbers
a r,l−1 basis states with the same quantum number, and ∑ r a r,l−1 = a l−1 .
In general, if we choose the site basis states in B(j) to be eigenstates of the site operator
ˆQ j and denote with |t, s j 〉 a basis state with quantum number q S(j)
t , then the tensor product
state of the enlarged system block basis
|r, α l−1 〉 ⊗ |r, s l 〉 = |r, α l−1 ; t, s l 〉 (2.89)
is an eigenstate of ˆQL(l) = ˆQ L(l−1) + ˆQ l with quantum number qp
L(l+1) = qr
L(l) + q S(l+1)
t .
Therefore, the elements of the corresponding state-space reduction matrices take the form
Similarly,
A l (t, s l ) r,αl−1 ;p,α l
= 〈p, α l |r, α l−1 ; t, s l 〉 = 0 if q L(l)
p
B l (r, s l ) p,βl ;r,β l+1
= 〈p, β l |t, s l ; r, β l+1 〉 = 0 if q R(l)
p
≠ q L(l−1)
r
≠ q S(l)
t
+ q S(l)
t . (2.90)
+ q L(l+1)
r . (2.91)
Using these rules computation time and computer memory can be saved (only the components
that do not vanish identically are taken into account).
Furthermore, since ˆQ = ˆQ L(l−1) + ˆQ l + ˆQ l+1 + ˆQ R(j+2) , a superblock basis state (2.62)
can be written as
|p, α l−1 ; r, s l 〉 ⊗ |t, s l+1 ; v, β l+2 〉 = |p, α l−1 ; r, s l ; t, s l+1 ; v, β l+2 〉 (2.92)
and its quantum number is given by q = q L(l−1) + q S(l) + q S(l+1) + q R(l+2) . The superblock
representation of a state |ψ〉 (2.65) then becomes
ψ p,α;r,sl ;t,s l+1 ;v,β = 0 if q ≠ q L(l−1)
p
+ q S(l)
r
+ q S(l+1)
t
+ q R(j+2)
v . (2.93)
Here again, computation time and computer memory can be saved.
The same sequence of arguments can be adopted for the operators that have a simple
commutation relation with ˆQ. For instance, consider Ô with [ ˆQ, Ô] = ∆q Ô, where ∆q is
a number. The matrix elements 〈k; κ|Ô|n; ν〉 of Ô in the eigenbasis of ˆQ vanish identically
if q k ≠ k n + ∆q. Similar constraints apply for the related operators ˆQ L(j) , ˆQR(j) , and ˆQ j .
For matrix representations of the local operators in the corresponding site basis, one finds
that
〈r ′ , s ′ j |Ôj |r, s j 〉 = 0 (2.94)

2.6. Implementation details 59
(a)
(b)
Figure 2.7: Sketch of matrix representations of the system block Hamiltonian (a) and
ground-state wave function (b), corresponding to the half-filled Hubbard model with
L = 512, N ↑ , N ↓ = 256, and U/t = 4, at step l = L/2 of the finite-system DMRG
algorithm with m = 1024 kept states.
for qp
S(j) ≠ q S(j)
p
+ ∆q, if [ ˆQ j , ′
Ôj ] = ∆q Ôj : e.g., for the particle creation c † ↑
and number
ˆN operators with [ ˆN, c † ↑ ] = c† ↑ only 〈↑ |c† ↑ |0〉 and 〈↑↓ |c† ↑
| ↑〉 do not vanish. For the L(l)-
representations of the left block operators one finds that
for q L(l−1)
r ′
≠ q L(l−1)
r
〈r ′ , α ′ l|Ô|r, α l−1〉 = 0 (2.95)
+ ∆q and [ ˆQ L (l), Ô] = ∆q Ô. All these straightforwardly generalizes
for R(j)-representations of operators acting on the right block. Therefore, if only matrix
elements that do not vanish identically are considered, computation time and computer
memory is saved.
In the case of n different conserved additive quantum numbers with the corresponding
operators ˆQ ν , ν = 1, . . .,n, that also commute among each other [ ˆQ ν , ˆQ ν ′] = 0, further
constraints can be applied to the basis states and matrix representations of the operators
with simple commutation relations with each ˆQ ν . Associating ˆQ ν with the n-dimensional
vector of operators ˆQ = ( ˆQ 1 , . . ., ˆQ n ) and the corresponding additive quantum numbers
with n-dimensional vector q = (q 1 . . .,q n ), all considered relations including Eqs. (2.87)-
(2.95) can be generalized for ˆQ. For instance, in the case of the Hubbard model the total
number of particles N and the total magnetization S z (equivalently numbers of particles
with up- and down-spins, N ↑ and N ↓ ) are conserved.
As an illustration, in Fig. 2.7 we schematically show matrix representations of the
system block Hamiltonian and ground-state wave function, corresponding to the half-filled

60 2. Density-Matrix Renormalization Group
Hubbard model with L = 512 sites and U/t = 4, at step l = L/2 of the finite-system DMRG
algorithm with m = 1024 kept states. Using both conserved quantities, the ground state
is searched in a 915084-dimensional subspace with (N = 512, S z = 0) quantum numbers
instead of 1024 × 4 × 4 × 1024 = 16777216-dimensional Hilbert space of the superblock.
Moreover, for the Hamiltonian acting in the system block a similar reduction is achieved,
if Hamiltonian matrices are constructed explicitly, but since the system block is treated
as a composite object of the left block and a site, further reduction is accomplished. The
largest subspace of the system block for the considered case corresponds to the subspace
with quantum numbers (N = 256, S z = 0) and has dimension 180625.
In summary, using constraints generated by conserved additive quantum numbers, matrix
representations of operators and target states can be considered as a sparse matrix with
quantum-numbers vectors as indices and dense rather small matrices as elements. Although
extra bookkeeping increases the complexity of the DMRG program, all these constraints
allow to exclude a large number of coefficients from the calculation, leading to a significant
speedup of the computation and substantial reduction of the required computer memory.

61
3. REAL-TIME EVOLUTION USING DMRG
In this chapter we consider extentions to DMRG which allow to study particular classes
of the time dependent problems. After an introduction to the subject, first we briefly
overview the early attempts made in this direction (Section 3.2). In Setion 3.3 we formulate
a unique time-evolution algorithm, the so called adaptive time-dependent DMRG,
which boosted developments in this field. As we shall see, although being a very efficient
scheme the adaptive time dependent DMRG has its own shortcomings. In Section 3.4, we
describe strategies which help to overcome these drawbacks and devise an algorithm which
circumvents these limitations without considerable losses in efficiency. Within the largest
Section 3.5 of this chapter, we analyze the accuracy of both considered time dependent
DMRG methods using two nontrivial time-evolution examples and draw some conclusions
concerning the abilities of these algorithms. Conclusions including some short outlook close
the present chapter. In this chapter we follow the classification of the algorithms proposed
by Schollwöck and White in Ref. [211].
3.1 Introduction
The time evolution of a state in quantum mechanics is governed by the time-dependent
Schrödinger-equation
i ∂ |ψ(t)〉 = H(t)|ψ(t)〉 , (3.1)
∂t
which is a partial differential equation (PDE) of first order in time. The formal solution of
(3.1) can be given as
|ψ(t)〉 = Û(t, t 0)|ψ(t 0 )〉 ,
(3.2a)
where the unitary time evolution operator
Û(t, t 0 ) = T
{
exp
(
− i ∫ t
)}
Ĥ(t ′ )dt ′
t 0
(3.2b)
contains now the complete dynamics of the system. In Eq. (3.2a) |ψ(t 0 )〉 is a system state
at initial time t 0 (t 0 can be arbitrary) and T is the time ordering operator. In the case of
time-independent Hamiltonians (3.2) simplifies to
Û(t, t 0 ) = e −iĤ(t−t 0)/
(3.3a)

62 3. Real-time evolution using DMRG
and
|ψ(t)〉 = e −iĤ(t−t 0)/ |ψ(t 0 )〉 .
(3.3b)
In the following we simply take = 1.
Although the formal solution (3.3), corresponding to the case of time-independent
Hamiltonians, looks quite simple, it is in general a highly nontrivial problem to determine
Û(t, t 0 ) (3.3a). It gets even more complicated in the case of time-dependent Hamiltonians.
For the single-particle Schrödinger equation, an approximate solution can be found
numerically using the finite-difference methods with an appropriately chosen discretization
in time and space; the most well-known variants are the Crank-Nicolson and Runge-Kutta
methods. For interacting many-particle systems, however, it is less evident how to formulate
a well-behaved and efficient algorithm. Even if we consider the systems modeled
with minimal-model Hubbard- or Heisenberg-type Hamiltonians, we still face the problems
which were partially discussed in the introduction of the previous chapter (see Section 2.2),
e.g., the fundamental problem of an exponentially growing Hilbert space with the system
size.
Taking into account the great success of the DMRG method in studying the static
and dynamic equilibrium properties of one-dimensional systems, it is natural to look for
the extension of this method to the time-dependent domain. In the following we discuss
early attempts in this direction and devise two time-dependent methods which have been
successfully employed to study time-dependent problems.
In order to see one of the advantages of time-evolution simulations, let us consider the
following setup. Typically all physical quantities of interest evolving in time can be reduced
to the evaluation of either equal-time n-point correlators, for example the density (1-point)
〈n i (t)〉 = 〈ψ(t)|n i |ψ(t)〉 (3.4)
or unequal-time n-point correlators like the (2-point) real-time Green’s function
G > ij(t) = −i 〈ψ|c i (t)c † j (0)|ψ〉 , (3.5)
where c † j (t) and c i (t) are Heisenberg representations of particle creation c† j and annihilation
c i operators, respectively, and n i (t) is the Heisenberg representation of the particle number
operator n i . Eq. (3.5) can be brought to the form of Eq. (3.4) introducing |φ〉 = c † j |ψ〉. The
desired correlator is then simply given as an equal-time matrix element between two timeevolved
states,
G > ij (t) = −i 〈ψ(t)|c i |φ(t)〉 . (3.6)
If both |ψ(t)〉 and |φ(t)〉 can be obtained efficiently, G > ij (t) can be evaluated in a single
calculation for all i and t as time proceeds. The frequency- and wave vector-dependent
Green’s function G > (k, ω) is then obtained by a double Fourier transformation. Obviously,

3.1. Introduction 63
the geometry of the problems (finite system-sizes and boundary effects) as well as algorithmic
limitations will impose physical constraints on the largest times and distances |i − j|
or correspondingly on the minimal frequency and the accessible wave vector resolutions.
Nevertheless, this approach is a very attractive alternative (see Ref. [67, 258]) to the current
very time-consuming calculations of G(k, ω) using the dynamical DMRG [98, 129, 145]. Although
the dynamical DMRG yields extremely accurate spectra, it is limited to only one
momentum and one narrow frequency range at a time. Therefore, constructing an entire
spectrum for a reasonable grid in momentum and frequency space can involve hundreds of
runs. Moreover, the dynamical DMRG method is restricted to equilibrium problems and
can not be used in truly out-of-equilibrium situations or in the cases with time-dependent
Hamiltonians.
The fundamental difficulty in obtaining the above correlators becomes obvious, if one
examines the time-evolution of the quantum state |ψ(t = 0)〉 under the action of some
(for simplicity) time-independent Hamiltonian Ĥ. If the eigensystem of the Hamiltonian
Ĥ|n〉 = E n |n〉 is known, then the time evolution of the state can be expressed as
|ψ(t)〉 = ∑ n
e −iEnt |n〉〈n|ψ(t = 0)〉 = ∑ n
e −iEnt y n |n〉 , (3.7)
where the magnitude of each expansion coefficient of |ψ(t)〉 is time-independent. Thus, a
projection onto the eigenstates corresponding to the large-magnitude expansion coefficients
provides a reasonable approximation to the time-evolving state and an optimal Hilbertspace
reduction procedure. In strongly correlated systems, however, one usually has no
good knowledge of the eigensystem. Instead, one uses some orthonormal basis with an
unknown eigenbasis expansion, |k〉 = ∑ n u kn|n〉 (|n〉 = ∑ k u′ nk |k〉). The time evolution of
the state then reads
|ψ(t)〉 = ∑ ( )
∑
e −iEnt y n u ′ nk |k〉 ≡ ∑ z k (t)|k〉 , (3.8)
k n
k
and the magnitude of each expansion coefficient z k (t) is time-dependent. For a general
orthonormal basis, the optimal Hilbert-space reduction at one fixed time (i.e. t = 0) will
therefore not ensure a reliable approximation of the time evolution. Note also, that the energy
differences now matter in the time evolution because of the phase factors e −i(En−E n ′)t
in |z k (t)| 2 . Therefore, in order to compute time-evolved states, the DMRG algorithm must
be extended in two ways: states beside extremal eigenstates must be generated, and the
basis must be adapted to the time-evolving state even in the case of time-independent
Hamiltonians. These extentions can be carried out differently and depending on the approach
used the time-dependent DMRG algorithms can be classified as static, dynamic,
and adaptive [211]. Later we will consider each of them.
Since both experimental as well as theoretical investigations of the time-evolution of
systems are difficult and include the vast varieties of possible setups, in the following we

64 3. Real-time evolution using DMRG
focus on some simpler set of problems. We consider isolated systems and simulate the time
evolution of a pure initial state governed by the time-dependent Schrödinger equation (3.1),
(i.e., at all instants of time the system remains in a pure state.) Furthermore, we focus on
cases where the time evolution is driven by the time-independent Hamiltonian, although
the developed methods can be used to study the problems where the time evolution is
driven by the time-dependent Hamiltonian. The possible setups can then be divided into
the following classes:
• One can construct an initial state by applying a given operator Ĉ to the ground state
|ψ GS 〉 (or some low-lying eigenstate) of the given Hamiltonian
|ψ(t = 0)〉 = Ĉ|ψ GS〉 , Ĥ 0 |ψ GS 〉 = E GS |ψ GS 〉 , (3.9a)
and study the time evolution of the obtained state
|ψ(t)〉 = e −iĤ0t |ψ(0)〉 .
(3.9b)
For example one can investigate the time evolution of an electron-hole pair created by
an incident photon. One can also create an extra particle or a Gaussian wave packet
in the ground state of the system and study the time evolution of the resulting state.
The former will also include the computation of the 2-point real-time Green’s function
(3.6).
• One can take as an initial state the ground state |ψ GS 〉 of a given Hamiltonian Ĥ0
|ψ(t = 0)〉 = |ψ GS 〉 , Ĥ 0 |ψ GS 〉 = E GS |ψ GS 〉 , (3.10a)
and study how it evolves in the course of time under the different Hamiltonian Ĥ1
|ψ(t)〉 = e −iĤ1t |ψ(0)〉 .
(3.10b)
Since |ψ GS 〉 is typically not an eigenstate of Ĥ 1 one obtains a nonequilibrium state
at t = 0.
This scenario includes the cases where:
– the system Hamiltonian contains some fields ( ˆf) which are switched off at the
beginning of the time evolution
Ĥ 0 = Ĥ + ˆf,
Ĥ 1 = Ĥ.
This can be used to prepare the system in the state with the desired properties,
e.g., particles in a trap or magnetic domain walls.

3.2. Early attempts 65
– the system Hamiltonian contains some fields ( ˆF) which are switched on at the
beginning of time evolution
Ĥ 0 = Ĥ,
Ĥ 1 = Ĥ + ˆF.
This allows to apply a bias voltage to the initial state in order to study the
conductivity of a small part of the entire system, e.g., quantum dot systems
coupled to leads at different voltages.
– the intrinsic parameters of the system e.g., the interaction strength, undergo a
sudden change in the entire system or only in a small/tiny part of it, the so
called global and local quantum quenches.
The former scenario can be realized in experiments on optical lattices by suddenly
changing the properties of the optical lattice whereas the latter can be
obtained in the system with a point contact between its two parts.
Using the methods developed in this thesis, it is possible to study problems of each of
these categories including arbitrary mixtures of them.
In the present work, we consider the time evolution of the particle density in cases
where the initial state is a fermionic domain wall or is created by adding a single particle
to the ground state of the half-filled system of spinless fermions. The former can also be
interpreted as the time evolution of the system obtained by connecting two subsystems
(at time t = 0), one completely filled with the spinless fermions and another completely
empty. These two nontrivial examples are used in the accuracy analysis of the developed
t-DMRG methods (see Section 3.5). In Chapter 5, we study the dynamics of the chargeand
spin- densities after adding a fermion to the ground state of the Hubbard model at or
in the vicinity of the half-filling.
3.2 Early attempts
Static time-dependent DMRG: The first so called time-dependent DMRG (t-DMRG)
method was developed by Cazalilla and Marston and it was employed to study the time
evolution of a one-dimensional system under an applied bias [32]. In their approach, an
infinite-system DMRG algorithm is used to grow the system up to a certain size, after
which either a quantum dot (paired with an ordinary site) or two sites linked by a tunneling
junction are inserted at the center of the chain. Finally, a time-dependent perturbation,
Ĥ ′ (t) is terned on, and the time-dependent Schrödinger equation
i ∂ ∂t |ψ(t)〉 = [(Ĥeff − E 0 ) + Ĥ′ (t)]|ψ(t)〉 (3.11)

66 3. Real-time evolution using DMRG
is numerically integrated forward in time. The initial state |ψ(0)〉 is chosen to be the
ground state of the superblock Hamiltonian (Ĥeff) of the unperturbed system, obtained
at the final step of the infinite-system DMRG algorithm. The ground state energy E 0 is
introduced to reduce the amplitude of the oscillations by making diagonal elements of Ĥ
smaller.
In this approach one works within a static effective basis which is optimal for the
state at t = 0 and is obtained using an infinite-system DMRG algorithm. Both |ψ(0)〉 and
the Hamiltonian Ĥ(t) = Ĥeff + Ĥ′ (t) driving the time evolution are represented in this
restricted basis. Consequently, one expects to lose accuracy when the wave function starts
to differ significantly from the initial state. In the system studied by Cazalilla and Marston
[32], the time evolution could be carried out for a reasonably long times. This relatively
good performance was possible because of the specific setup of the time-dependent coupling
located in the center of the system and hence modeled exactly by explicit sites in DMRG.
For different setups one would rather expect the method to break down for relatively short
time scales.
One can try to extend the accurate solution forward in time by increasing the number
of retained density-matrix eigenstates m. However, as pointed out by Luo et al. [158], this
type of enlargement of the effective state space is not efficient and even error prone. Despite
the fact that the DMRG truncation error becomes negligibly small and systematic convergence
with increasing m can be achieved by inclusion of the excited states (those which
are necessary for the long time evolution) into the set of basis states is not granted. This
scenario is schematically illustrated in Fig. 3.1a. This explains why the results obtained
by Cazalilla and Marston [32], deviate from the exact results at large time scales, even
though convergence with increasing m was reached [158] (see also Ref. [33]). Therefore,
enlarging the effective state space by only increasing m is not sufficient for the substantial
improvement of the quality of the results and the convergence is misleading.
Dynamic time-dependent DMRG: The first improvement of the above method was proposed
by Luo, Xiang, and Wang [158]. Having in mind the ability of DMRG to represent
a small set of states (“target states”) very well, they targeted several time evolved states
at a sequence of times spanning the whole studied time interval |ψ(t = 0)〉, |ψ(t = δt)〉,
|ψ(t = 2δt)〉, . . ., |ψ(t = N t δt)〉 simultaneously. The superblock density matrix is then constructed
as a superposition of these states
∑N t
ˆρ = w i |ψ(t i )〉〈ψ(t i )| (3.12)
i=0
where t i = iδt and the target states |ψ(t i )〉 are weighted with a factor w i , with ∑ i w i = 1.
Since the |ψ(t i )〉 are not known initially, it was suggested, within the framework of infinitesystem
DMRG to start with a system which is small enough in order to carry out the entire

3.2. Early attempts 67
(a)
(b)
Figure 3.1: Sketch showing the effective state spaces constructed within the “static”
(a) and “dynamic” (b) variants of t-DMRG during the simulation of the time-evolution
of the state |ψ(0)〉. The time evolution is carried out in: (a) fixed H eff , optimal for
the initial state |ψ(0)〉 only, (b) fixed enlarged H eff , optimal for all time-evolved states
|ψ(t i )〉 simultaneously. The solid line represents the “true” time-evolution when the
whole Hilbert space H is taken into account.
time evolution exactly. The obtained states are then used to construct an effective basis
for the next infinite-system DMRG iteration and the whole procedure is repeated until the
desired system length is reached.
In this approach one speculates that a better effective state space can be obtained for
the final system, taking into account how the time-evolution explores the Hilbert space for
intermediate chain lengths. Indeed, it was shown in Ref. [158] that this approach is much
more accurate than the one of Cazalilla and Marston. However, it is not very efficient, since
a quite large state space is required for a long interval of time, and the whole evolution has
to be performed at every DMRG step. The dynamic and static time-dependent DMRG
approaches are schematically compared in Fig. 3.1.
Yet another approach, falling into the considered class of dynamic t-DMRG, was proposed
by Schmitteckert in Ref. [208], who studied the transport through small interacting
nanostructures. In order to build an effective state space, equally well representing the
time-evolved system over the whole time interval, he targets two different sets of states.
The first set is composed of several low-lying eigenstates; they can be obtained within the
finite-system DMRG precision and the time evolution of them can be casted exactly. In
the subspace orthogonal to these eigenstates, he computes the time-evolved states |ψ(t i )〉
using one of the Krylov-subspace approximations (Arnoldi approximation) of the product
of matrix exponential with a vector, |ψ(t i+1 )〉 = e −iĤ(t i+1 −t i ) |ψ(t i )〉. These time-evolved
states |ψ(t i )〉 at fixed times t i constitute the second set of targeted states. In Schmitteckert’s
approach both infinite- and finite-system DMRG algorithms are used to simulate the

68 3. Real-time evolution using DMRG
time evolution of a system. The infinite-system DMRG algorithm is employed to construct
preliminary low-lying eigenstates, whereas the actual time evolution is carried out using
the finite-system DMRG algorithm. Performing finite-system DMRG sweeps a state space
suitable to describe all targeted states at good precision should be obtained. To increase
numerical efficiency, Schmitteckert starts with some small time and when convergence is
reached, time is increased bringing more and more |ψ(t i )〉 into the set of target states of
the finite-system DMRG.
Schmitteckert’s approach considerably improves the accuracy of the results and larger
time scales become feasible. Nevertheless, it is very expensive with regard to the usage
of memory and CPU-time. The number of DMRG kept states m grows with the desired
time interval as more and more different states have to be represented accurately. Since
the calculation time scales as m 3 , this type of approach will meet its limitations somewhat
later in time.
In the following sections we consider several more efficient DMRG methods for the time
evolution, which are now widely accepted and used in the physics community.
3.3 Adaptive time-dependent DMRG
The breakthrough came from the field of quantum information theory when Vidal developed
an algorithm for the efficient classical simulation of slightly entangled quantum
computations [243]. Later on, he used it to simulate the time evolution of one-dimensional
quantum many-body systems [244] in order to show that any quantum computation involving
only a sufficiently restricted amount of entanglement can be efficiently simulated on a
classical computer. Therefore, from an algorithmical point of view, this type of quantum
computation does not give any significant advantage over the classical one. This unique algorithm,
later called time-evolving block decimation (TEBD), for simulating near-neighbor
one-dimensional systems overlaps strongly with DMRG. Shortly after its introduction, two
papers appeared simultaneously which mapped ideas of TEBD on the powerful DMRG
“machinery” [38, 258]; the adaptive time-dependent DMRG was born. In this section we
consider the latter one in more detail. How TEBD and DMRG are connected can be found
in Ref. [38].
The crucial idea of the new method is to use the Suzuki-Trotter decomposition [221, 235]
for the time-evolution operator e −iĤδt for a small time step δt (|ψ(t + δt)〉 = e −iĤδt |ψ(t)〉). If
the system Hamiltonian (possibly time-dependent) contains nearest-neighbor interactions
only, then it can be split into local Hamiltonians acting on a single bond only:
Ĥ = ∑ i
ĥ 2i−1,2i + ∑ i
ĥ 2i,2i+1 =: Ĥodd + Ĥeven . (3.13)
All Ĥ odd and Ĥeven terms commute among each other, whereas ĥ2i−1,2i and ĥ2j,2j+1 in

3.3. Adaptive time-dependent DMRG 69
general do not commute, if they share the same site (i.e., i = j or i + 1 = j). The second
order Suzuki-Trotter (S-T(2)) approximation [221, 235] then can be written as a
e −iĤδt = e −iĤ odd δt/2 e −iĤ even δt e −iĤ odd δt/2 + O((δt) 3 ) , (3.14)
and a given state can be time evolved by repeated applications of the two-site (single bond)
time evolution operators e −iĥi,i+1δt .
Recall, that the standard DMRG representation of the wave function (see Sec. 2.5) at
a particular step j of a finite-system sweep is
|ψ〉 =
∑
ψ αj−1 s j s j+1 β j+2
|α j−1 〉|s j 〉|s j+1 〉|β j+2 〉 , (3.15)
α j−1 , s j
s j+1 , β j+2
where {|α j−1 〉} and {|β j+2 〉} constitute the effective bases of the left L(j − 1) and right
R(j + 2) blocks, respectively (not complete ones, but optimal for representing |ψ〉 accurately),
and {|s j 〉} and {|s j+1 〉} build the complete basis of the j-th and (j + 1)-th sites,
accordingly. Any arbitrary operator Ô acting only on sites j and j + 1 can be applied
exactly to |ψ〉 within the same bases
[Oψ] αj−1 s j s j+1 β j+2
= ∑
O sj s j+1 ;s ′ ψ j s′ j+1 α j−1 s ′ j s′ j+1 β , (3.16)
j+2
s ′ j s′ j+1
where O denotes the matrix representation of Ô in the two-site tensor-product basis. In
general, if Ô includes terms acting on other sites, then one cannot write this simple exact
relation; new bases have to be adapted in order to describe both |ψ〉 and Ô|ψ〉, which
might require several finite-system sweeps through the lattice.
Following Eq. (3.16) the two site time-evolution operator e −iĥj,j+1δt can be applied exactly
on DMRG step j. However, for some other term of the Suzuki-Trotter decomposition
(3.14) (e.g. e −iĥk,k+1δt ) one first has to transform the state to the appropriate two-block
two-site configuration, where this term precisely acts on the two free sites. This can be
accomplished using the step-to-step wave-function transformation, commonly used in the
finite-system DMRG algorithm to provide a good guess for the Lanczos or Davidson iterative
diagonalization (see Sec. 2.6.3). Note however, that such a transformation is not
always exact; when some effective states with finite weights are discarded, a truncation
error is additionally introduced into the representation of the time-evolving state and this
error will accumulate from step to step.
a For time-dependent Hamiltonian Ĥ(t) = Ĥodd(t) + Ĥeven(t)
T
[
exp
(
−i
∫ t+δt
t
See Suzuki [223] or Hatano and Suzuki [104].
Ĥ(t ′ )dt ′ )]
= e −iĤodd(t+δt/2)δt/2 e −iĤeven(t+δt/2)δt e −iĤodd(t+δt/2)δt/2 + O((δt) 3 ).

70 3. Real-time evolution using DMRG
Using the above mentioned ingredients one can now formulate the adaptive t-DMRG
algorithm for the time-evolution simulation based on the second order Suzuki-Trotter decomposition
of the time-evolution operator:
0. Using the conventional DMRG algorithm (infinite- plus finite-system algorithm), find
the ground state of the initial system using a Hamiltonian Ĥ0. Stop at one of the
chain edges, not in the middle of the chain.
At the end of this step, for every l = 2, . . .,L − 1, we have the effective bases B(L, l)
and B(R, L − l + 1), for the left and right blocks, respectively, and the corresponding
basis transformations A l (s l ) and B L−l+1 (s L−l+1 ). b
Some other methods, suited to obtain bases with corresponding basis transformations,
can also be used in place of the conventional DMRG.
1. When it is applicable (i) change the Hamiltonian Ĥ0 → Ĥ(t) or (ii) construct an
initial state by applying the given operator Ĉ to the ground state |ψ(0)〉 = Ĉ|ψ GS〉.
The latter can be constructed and targeted together with the ground state during the
finite-system DMRG sweeps in step 0. Alternatively an extra complete half-sweep
can be used, especially when Ĉ is the sum of terms acting on a single site (bond); in
this case, each term of Ĉ is applied when the site (bond) is one of the two central,
untruncated sites (the central untruncated bond).
2. In order to apply e −iĤ odd δt/2 to the targeted state, use a complete half-sweep of the
finite-system DMRG algorithm with the following modifications:
a. Instead of diagonalization, for each odd bond apply the local time-evolution
operator e −iĥ2i−1,2iδt/2 to the targeted state. This operation is carried out exactly
(see Eq. (3.16)) when the (2i − 1)-th and 2i-th sites are entirely represented in
the block-site-site-block configuration of the state.
Since the terms in H odd commute among each other, one can always rearrange
them to match the order of free sites during the finite-system half-sweep (e.g.,
for a left-to-right half-sweep one has e −iĥ1,2δt e −iĥ3,4δt · · ·, while for a right-toleft
half-sweep · · ·e −iĥ3,4δt e −iĥ1,2δt ). After each local time update, a new wave
function is obtained.
b. As always, perform the DMRG truncation (reduced density-matrix projection)
at each step obtaining a new effective basis for the block state space, with the
corresponding transformation matrices.
b Actually, one needs bases for all left or all right blocks only, with the corresponding basis transformations
A or B, depending on the chain edge from which the time-step simulation starts (for the left edge
all right blocks with corresponding B, and for the right edge all left blocks with corresponding A). Others
will be constructed during the first left-to-right or right-to-left complete half-sweep.

3.3. Adaptive time-dependent DMRG 71
Figure 3.2: Sketch of the effective state-space development during the adaptive t-
DMRG simulation of the time-evolution of the state |ψ(0)〉. The effective state space
H eff (t ′ ) is continuously adjusted in order to represent the time-evolved state |ψ(t)〉
accurately. The solid line represents the “true” time-evolution when the whole Hilbert
space H is taken into account.
c. Use the wave-function transformation (see Sec. 2.6.3) to shift the free sites by
two.
3. To apply e −iĤevenδt , carry out step 2 switching the half-sweep direction and applying
e −iĥ2i,2i+1δt to all even bonds in (step a).
4. Repeat step 2 without modification in order to apply the second e −iĤoddδt/2 in
Eq. (3.14).
5. This completes one time step (3.14) and the state |ψ(t + δt)〉 is obtained. As in
finite-system DMRG evaluate operators (DMRG “measurements”) when desired.
6. Repeat the complete Suzuki-Trotter time step, consisting of steps 2–5, until the
desired time is reached.
In this method, a new wave function is obtained after each application of the local timeevolution
operator (step 2.a). This operation is always performed on the two free sites in
the state’s two-block two-site configuration; new degrees of freedom are introduced in the
corresponding system block and a new effective state space basis, best describing the new
state is constructed with the consecutive DMRG truncation procedure (step 2.b). Hence,
the DMRG effective state spaces are continuously adapted to optimally represent the state
at any given point in the time evolution (accordingly the name adaptive t-DMRG). This
procedure is schematically depicted in Fig. 3.2.

72 3. Real-time evolution using DMRG
Strictly speaking, the above described algorithm, performes the block state-space reduction
on every step where the local time-evolution is carried out. Hence, the state
obtained after the time evolution is not necessarily the globally-optimal one representing
the “true” time-evolved state. Another closely related time-evolution method, introduced
by Verstraete et al. [242], solves this problem by performing the state-space reductions
only after all local time-evolution terms of the Suzuki-Trotter approximation of the global
time-evolution operator (3.14) are applied to the state, consequently making this alternative
very time consuming. On the other hand, the state update via the local-operators is
likely to be small and the global optimum to be rather well approximated with the current
algorithm in the case of small time steps δt.
The present method was shown (see Ref. [38, 258]) to perform better than the static
t-DMRG and produced more accurate results with a relatively small number of kept states
than the dynamic t-DMRG. Since only one state is targeted the present method is not only
less resource demanding, but also considerably faster.
For simplicity, above we have formulated the algorithm for the time-evolution of a single
state. Nevertheless, it can be straightforwardly generalized for the case of several target
states. In order to do this one has only to modify substeps (a) and (b) of step 2: at substep
(a) one applies the local time-evolution operators to each targeted state; at substep (b)
one optimizes the effective block-state basis for all target states simultaneously, similar to
the finite-system DMRG algorithm for several target states.
Several modifications can be made in order to improve the performance of the present
algorithm. The following rearranged form of Eq. (3.14) was suggested from the beginning
by White:
e −iĤδt =
L−1
∏
i=1
e −iĥi,i+1δt/2
1∏
i=L−1
e −iĥi,i+1δt/2 + O((δt) 3 ) . (3.17)
The new reordered approximant remains symmetric and is of the same order in δt. Importantly,
it reduces the number of the complete half-sweeps, required for the Suzuki-Trotter
time step, from three to two. Unfortunately, at the same time the Suzuki-Trotter error is
increased although only moderately. Therefore we mainly use the former procedure and
mention explicitly when the latter variant is considered.
The evaluation of the desired operators (DMRG measurements) can be carried out
during an extra complete half-sweep. Especially, when operators can be written as a sum
of terms acting on a single site or bond, each term can be evaluated when this site or bond
is one of the two central, untruncated sites or is the central untruncated bond, respectively.
This avoids storing and transforming these operators. This type of evaluation can also be
performed during the last complete half-sweep of the Suzuki-Trotter time step. In this case,
the obtained quantities might be slightly inaccurate (since the time-step is not finished,
and the state is not properly time evolved), but the extra half-sweep is also avoided. Note

3.3. Adaptive time-dependent DMRG 73
also, that there is no need to generate these operators at all those time steps where no
operator evaluation is desired (observables are required for relatively large time steps in
comparison to the typically very small δt in the Suzuki-Trotter time step). In this case
the last and the initial half-sweeps of the two consecutive intermediate (measurement free)
Suzuki-Trotter time steps can be combined and carried out in a single half-sweep (one
e −iĤoddδt instead of two e −iĤoddδt/2 ).
Let us now reconsider the example of calculating 〈ψ(t)|c i |φ(t)〉 (see Section 2.2). In
the case of systems with only nearest-neighbor terms in the Hamiltonian, 〈ψ(t)|c i |φ(t)〉
can be computed using the present method. Targeting |ψ(t = 0)〉 = |ψ GS 〉 and
|φ(t = 0)〉 = c † j |ψ GS〉 during the time-evolution simultaneously, 〈ψ(t)|c i |φ(t)〉 can be evaluated
for all i in one complete half-sweep at each of the given times. Although the time
evolution of |ψ GS 〉 can be performed exactly, targeting it together with |φ(t)〉 will keep
both states in the same effective bases.
So far nothing was said about the blocks’ effective state-space dimensions: a l and b l .
Similar to the finite-system DMRG, sweeps of the adaptive t-DMRG can be performed with
the fixed number of retained reduced density-matrix eigenstates m, with the fixed discarded
weight threshold ε, or some combination of both; they will determine the corresponding a l
or b l at each iteration. For the first case, where a l , b l m, some rough estimations of the
complexity of the method can be made. If the site Hilbert-space dimension is d, then the
number of operations required to prepare the local time-evolution operator scales as (d 2 ) 3 ; c
carrying out the local time evolution requires d 4 m 2 operations; the construction and the
diagonalization of the reduced density matrix scales as d 3 m 3 and the basis transformation
requires d 2 m 3 operations. Since typically m ≫ d, and the most expensive operations are
performed at each iteration, the computer simulation time for the algorithm will scale as
O(Ld 3 m 3 ), thus linear in L. One should note that since each of the e −iĥi,i+1δt possesses the
same symmetries as ĥi,i+1, using the additive good quantum numbers specific to the studied
problem, the number of required operations can be reduced significantly and thereby the
speed of the simulations substantially increased. The most significant reduction of the
required computer resources and speedup up of the algorithm is achieved by this.
In most cases an extra speeding up (or reduction of the Suzuki-Trotter decomposition
error, see Section 3.5.1) can be achieved by increasing the time-step size and using higher
order Suzuki-Trotter decompositions [104, 168, 222]. The former reduces the number of
Suzuki-Trotter time steps required for a given t and the latter keeps the error of the Suzuki-
Trotter approximation small. The detailed analysis of different types of high order Suzuki-
Trotter approximants is given in Ref. [168]. A nice review, how to construct the higher
c For the Hubbard model, d = 4, and ∼ (4 2 ) 3 operations are required. However, if the algorithm makes
use of conserved additive quantum numbers, the complete Hilbert space of two sites factorizes in one
subspace of dimension 4, 4 subspaces of dimension 2, and 4 of dimension 1, hence ∼ 4 3 + 4 · 2 3 + 4 · 1 3
operations are required.

74 3. Real-time evolution using DMRG
order approximants using the so called fractal decomposition can be found in Ref. [104].
The most efficient is the following symmetric fourth order Suzuki-Trotter decomposition
(S-T(4)) [168]
( 5∏
e −iĤδt = e −iaiĤoddδt e
)e −ibiĤevenδt −ia6Ĥoddδt + O((δt) 5 ) , (3.18)
i=1
where a 1 = a 6 = (14 − √ 19)/108, a 2 = a 5 = (20 − 7 √ 19)/108, a 3 = a 4 =(1 − 2a 1 − 2a 2 )/2,
b 1 = b 5 = 2/5, b 2 = b 4 = −1/10, and b 3 = 1 − 2b 1 − 2b 2 . d In this decomposition one requires
eleven complete half-sweeps per Suzuki-Trotter time step, but since the accuracy
grows with the fourth power and not quadratically as for the second order approximant
(requiring only three half-sweeps), the number of time steps, needed to reach a given time
with the demanded accuracy, decreases considerably. In the applications presented in this
work we only use the second- and fourth-order Suzuki-Trotter decompositions.
The algorithm devised here is efficient and fast, but it is limited to the Hamiltonians
with nearest-neighbor terms. A classification of the error sources appearing during the
time-evolution simulations with t-DMRG and the detailed error analysis for the current
method can be found in Section 3.5 and Section 3.5.1, respectively. How to deal with more
general Hamiltonians is discussed in the following section.
3.4 Time-step targeting adaptive time-dependent
DMRG
Although being efficient and fast, the above considered method is restricted to the systems
with nearest neighbor interactions on a single chain. In the case of short-range interactions
or narrow ladders with nearest-neighbor interactions one can combine sites into supersites
to obtain a new Hamiltonian with at most nearest-neighbor interactions between the supersites
(e.g., for the ladder, all sites in a rung constitute a single supersite). Unfortunately,
this approach becomes very inefficient as soon as several sites have to be included into
the super-one (e.g. wider ladders), and is not applicable to general long-range interaction
terms.
Note that the static and dynamic t-DMRG methods, do not have this particular limitation,
but are very inefficient (see Section 3.2). Summarizing the abilities of the static
and the dynamic t-DMRG methods we conclude that: the time evolution of a given state
|ψ(t 0 )〉 can be studied, only when the time-evolved state |ψ(t)〉 and the time-evolution
d Another fourth order approximation is constructed by the so called fractal decomposition
(symmetric decomposition with symmetric steps) [222], which reduces to (3.18) with coefficients
2a 1 = 2a 6 = a 2 = a 5 = 1/(4 − 3√ 4), a 3 = a 4 = (1 − 2a 1 − 2a 2 )/2, b 1 = b 2 = b 4 = b 5 = 1/(4 − 3√ 4), and
b 3 = 1 − 2b 1 − 2b 2 .

3.4. Time-step targeting adaptive time-dependent DMRG 75
driving Hamiltonian have a proper support on the effective state-space basis optimally
representing this state. Typically, this is the case for short enough times. In order to be
able to access the large times, an appropriate enlargement of the effective state space has
to be performed. The expanded state space should accommodate not only the time-evolved
state, but also the parts of the Hamiltonian relevant for the time evolution. So far, the
successful strategy was to include time-evolved states at a sequence of times covering the
whole time interval into the set of DMRG target states (dynamic t-DMRG). However,
since the number of required intermediate states grows with increasing time, the number
of DMRG kept states has to be increased accordingly in order to maintain the accuracy of
the targeted states on the same level. Therefore, this strategy is very inefficient for long
times, becoming computationally expensive and time consuming.
The time-step targeting (TST) method by Feiguin and White [67], which we consider
here, copes with the mentioned limitations of the adaptive t-DMRG and is nearly as
efficient. This new approach considers the actual time evolution and the adjustment of the
effective state-space basis separately. The time evolution of the whole state is carried out
for small but finite time-steps, as this was the case in the static and the dynamic t-DMRG
algorithms. However, the important new concept is to produce the basis which takes into
account only the states relevant for one, namely the current, time step instead of targeting
the entire studied time interval (see the sketch in Fig. 3.3). In this way the dimension of the
effective state-space basis becomes as small as possible and the efficiency of the calculation
is maximized. This new concept is a mixture of two previous schemes targeting precisely
one instant in time at any DMRG step in the adaptive t-DMRG and the entire range of
time to be studied in the dynamic t-DMRG.
In the following subsections: first we consider the time-evolution methods which can
operate within the fixed DMRG effective state-space basis and are free of particular limitations;
then we formulate how this state-space basis can be re-adapted during the timeevolution
simulation, and in the final subsection, we devise the time-step targeting adaptive
t-DMRG algorithm based on Krylov-subspace methods for the time-evolution.
3.4.1 Performing time evolution
In this subsection we consider the methods which can be used to efficiently obtain the
state |ψ(t + δt)〉 from a given |ψ(t)〉 and which are free of particular restrictions due to the
structure of the Hamiltonian.
The time-dependent Schrödinger equation (3.1), is a first-order partial differential equation
(PDE) which can be numerically solved by approximate explicit or implicit integration
schemes.
i) Runge-Kutta method: An example of an explicit scheme is the Runge-Kutta (R-K)
method [198]. The standard fourth-order R-K method is one of the most popular

76 3. Real-time evolution using DMRG
approaches. For time-independent Hamiltonians it is equivalent to a fourth order
Taylor series expansion of the time-evolution operator Û(δt) = exp(−iĤδt).
To obtain the state |ψ(t + δt)〉 from |ψ(t)〉, one first calculates the following four R-K
vectors:
|k 1 〉 = −i δt Ĥ(t)|ψ(t)〉, (3.19a)
|k 2 〉 = −i δt
(|ψ(t)〉 Ĥ(t + δt/2) + 1 )
2 |k 1〉 , (3.19b)
|k 3 〉 = −i δt
(|ψ(t)〉 Ĥ(t + δt/2) + 1 )
2 |k 2〉 , (3.19c)
|k 4 〉 = −i δt Ĥ(t + δt) (|ψ(t)〉 + |k 3〉), (3.19d)
from which the state at t + δt is obtained via
|ψ(t + δt)〉 = |ψ(t)〉 + 1 6 (|k 1〉 + 2|k 2 〉 + 2|k 3 〉 + |k 4 〉) + O((δt) 5 ) . (3.20)
One can see that the efficiency of this method strongly depends on the efficiency of
the matrix-vector products Ĥ(t′ )|ψ(t)〉.
It is important to note that the R-K and in general all polynomial approximations to
the exponentials do not conserve unitarity and are numerically unstable [175].
There are multiple implicit integration schemes that eliminate the lack of unitarity
and produce stable approximations. They received the name “implicit” because the state
at a later time step is obtained by solving an equation or inverting an operator. The
simplest second-order implicit integration scheme conserving unitarity, the Crank-Nicolson
method, was already employed by Daley et al. in Ref. [38]. They found that changing
the R-K method with the Crank-Nicolson one, improved the quality of the results
and the occurrence of artificial asymmetries with respect to reflection was decreased (see
Ref. [38]). Nonetheless, the main drawback of this method is the necessity of inverting off
a non-Hermitian denominator; a nontrivial problem which also determines the error in this
approach.
If the Hamiltonian is time independent, it appears advantageous to exploit the fact
that the formal solution is known,
|ψ(t + δt)〉 = e −iĤδt |ψ(t)〉 . (3.21)
In this case one only needs to find good means of evaluating matrix-vector products with
e −iHδt . Time evolutions with time-dependent Hamiltonians can be reduced to ones with
“time-independent” Hamiltonians by breaking the propagation time interval into properly
chosen time slices and considering a constant Hamiltonian within each slice. In general

3.4. Time-step targeting adaptive time-dependent DMRG 77
the numerical evaluation of the matrix exponentials is known to be non trivial. A nice
review covering most well known methods of computing matrix exponentials can be found
in Ref. [175]. Here, we only concentrate on a scheme which can be optimally implemented
within the DMRG framework and is among the most efficient approaches.
ii) Krylov-subspace methods: Krylov-subspace approximations to the matrix exponential
operator appear to be among the most efficient approaches when computing product
of matrix exponentials with vectors, e.g. e −iĤδt |ψ(t)〉. It is also part of a well known
matrix exponential software package Expokit [218]. In this approach, approximations
to the solution are obtained using the projection of the operator (i.e. Ĥ) onto an
n-dimensional Krylov subspace, generated by applying the operator (n − 1)-times to
an arbitrary initial vector, |u 0 〉,
K n (Ĥ, |u 0〉) = span{|u 0 〉, Ĥ|u 0〉, Ĥ2 |u 0 〉, . . ., Ĥn−1 |u 0 〉} . (3.22)
Efficient implementations of this operation are also available in DMRG. The Lanczos
and related Arnoldi method involve projecting an operator onto an orthonormalized
version of this Krylov subspace (3.22), where n is typically chosen to be much smaller
than the total dimension of the Hilbert space. In the present work, we only use the
Arnoldi algorithm, since it explicitly generates the orthonormal basis for K n (Ĥ, |u 0〉)
(typically using a modified Gram-Schmidt orthonormalization [87]) and is therefore
numerically more stable.
It is clear that Ĥ and iδtĤ span the same Krylov subspace. By projecting the time
evolution operator through one interval [t, t + δt] onto the orthonormal basis of n
Arnoldi vectors, one obtains [111, 202]
|ψ(t + δt)〉 = e −iĤδt |ψ(t)〉 ≈ V n e −iTnδt V † n |ψ(t)〉 , (3.23)
where the columns of the matrix V n are Arnoldi basis vectors,
V † n|u 0 〉 = ‖|u 0 〉‖[10 . . .0] T , and the initial vector |u 0 〉 = |ψ(t)〉; T n = V † nĤV n
represents the orthogonal projection of Ĥ onto the subspace K n (Ĥ, |ψ(t)〉) with
respect to the basis V n . In general, T n is a Hessenberg matrix, which becomes a
Hermitian tridiagonal one in the case of Hermitian operators.
Exact error bounds and convergence criteria for this approximation scheme for different
matrix types were derived by Hochbruck and Lubich [111]. For Hermitian
matrices one has:
∥
ε n := ∥e −iĤδt |u 0 〉 − V n e −iTnδt V † n |u 0〉 ∥
]( ) n
(̺ δt)2 e̺ δt
12 exp
[− for n 1 ̺ δt , (3.24)
16n 4n
2

78 3. Real-time evolution using DMRG
where ‖ · ‖ is the Euclidean norm and ̺ = |E max − E min | is the width of the Hamiltonian
spectrum. This bound is sharp for the worst possible case when the eigenvalues
of Ĥ are densely distributed and the initial vector |u 0〉 has no preferred “eigendirections”,
otherwise the convergence is considerably faster.
Unfortunately, this error bound is hardly used in practice. In general, ̺ is not known
and thus the error (3.24) cannot be computed. Nonetheless, one can assess the quality
of the result using the aposteriory error estimations [202]. Those can be evaluated
numerically cheaply and the following stopping criterion can be formulated for the
Krylov-subspace buildup [112, 113]
γ n δt ∣ [ e −iTnδt] ∣ ‖|u
n,1 0 〉‖ < tol,
(3.25a)
with
γ n =
( ) ∥
∥
∥Ĥ|v n〉 − V † ∥∥ nĤ|v n〉 V n .
(3.25b)
Here [ · ] n,1 denotes the (n, 1)-entry of a matrix, |v n 〉 is the n-th Arnoldi basis vector
(the n-th column of V n ), and “tol” defines the desired accuracy. The computation of
γ n is not numerically expensive, since both terms of (3.25b) are already evaluated
during the construction of T n .
Very good approximations to the large-matrix exponentials are often obtained with
relatively small n. Thus the original large-dimensional problem (3.21) is converted
to a smaller one (3.23). The explicit computation of exp(−iT n δt) is then performed
using some dense-matrix algorithm. e
Both considered methods strongly rely upon the efficient Ĥ|ψ〉 procedure, which is available
within DMRG, and hence both these methods can be easily included in the existing code.
3.4.2 Hilbert space adaption strategy
In the previous subsection we described how to obtain the state |ψ(t + δt)〉 from a given
|ψ(t)〉. The main operation used there, was the product of the system Hamiltonian with a
vector (e.g. Ĥ|ψ(t)〉). Recall that within DMRG one only operates with the projections of
the original Hamiltonian and the states onto the effective state-space basis. In this section
we consider how to readapt this basis in order to obtain an accurate representation of
|ψ(t + δt)〉.
We can perform the finite-system DMRG sweeps at each iteration reobtaining
|ψ(t + δt)〉 and constructing the effective state-space basis optimally representing this state
e For a Hermitian matrix one can use the eigendecomposition [87], while for non-Hermitian matrices the
scaling and squaring method [148] with irreducible Padé approximants of degree 13 is preferable [108].

3.4. Time-step targeting adaptive time-dependent DMRG 79
at the current DMRG bipartition of the system. However, to calculate |ψ(t + δt)〉 accurately,
an accurate representation of |ψ(t)〉 has to be also available at each iteration. Therefore,
the basis must be simultaneously re-adapted for |ψ(t)〉, too. If only one time step is
targeted during the simulation, which is our intent, then for an arbitrary time step |ψ(t)〉
cannot be reobtained in the finite-system DMRG sweeps. It cannot be explicitly recalculated
either, because it is not an extremal state of a particular functional, as this was the
case for the ground state of the system. For an arbitrary time step, the only information
available on |ψ(t)〉 is encoded as a series of basis transformations for successive DMRG partitions
of the system (2.21). Having all these in mind, we can only use the wave-function
transformation procedure (White’s vector prediction scheme; see Section 2.6.3), to rewrite
|ψ(t)〉 in the basis of the current iteration of the ongoing finite-system sweep. Note however,
that such a transformation (shifting sites) is not exact; it additionally introduces the
truncation error of the particular finite-system step into the representation of |ψ(t)〉. For
this reason, we should avoid performing superfluous finite-system sweeps when readjusting
the basis to the current time step.
Additional care has to be taken to assure that the re-adapting effective state-space basis
accurately represents the parts of the Hamiltonian relevant for the time evolution in the
current time interval [t, t + δt]. This was not necessary in the adaptive t-DMRG with the
Suzuki-Trotter decomposition, where each term of the time-evolution operator is exactly
represented in the tensor product basis of the corresponding two sites.
Several possibilities exist to improve the representation of the relevant parts of the
Hamiltonian within the time step. For example, one can directly target R-K vectors (3.19a)
in addition to the initial |ψ(t)〉 and final |ψ(t + δt)〉 states when the time evolution is carried
out using the R-K method. Similarly, one can directly target Krylov vectors (Ĥν |ψ(t)〉) or
Krylov-subspace Arnoldi orthonormal basis states when the time evolution is performed
using the Krylov-subspace methods. In fact, both are not the best choices. R-K vectors
represent derivatives and not the actual states. Vectors of the Krylov subspace do not
contain any information concerning the time-step size (δt), because both Ĥ and iδtĤ
generate the same Krylov subspace. Therefore, these directly targeted vectors only poorly
represent the states inside the given time interval [t, t + δt] and consequently, capturing
the parts of the Hamiltonian relevant for the current time-step is not granted.
When the state of the system evolves in time, its density matrix samples a region of the
Hilbert space that changes continuously. Representing accurately the region of the Hilbert
space, sampled within one time step (e.g. [t, t + δt]), one can guarantee that the parts of
the Hamiltonian relevant for the time evolution within this time step are captured correctly.
This can be achieved by targeting several intermediate time-evolved states, similar to the
approach by Luo et al. [158]. The weighted average density matrix, used to determine the
effective state-space basis, can be constructed from the time-evolved states at a sequence

80 3. Real-time evolution using DMRG
of times spanning the considered time interval as
∑N t
ˆρ = w i |ψ(t + δt i )〉〈ψ(t + δt i )| with 0 = δt 0 < δt 1 < · · · < δt Nt = δt . (3.26)
i=0
Here N t is the number of time-evolved states including the final one. The target state
|ψ(t + δt i )〉 is weighted with a factor w i , and ∑ N t
i=0 w i = 1. Typically one considers equal
time intervals δt i = iδt/N t . In this case, each of the intermediate states is weighted with
the same factor w i = w = (1 − w 0 − w Nt )/N t , i = 1, . . ., (N t − 1). The initial and final
states are also weighted equally with a factor which can be equal or larger than those for
the intermediate states (w 0 = w Nt w). A larger factor is usually taken, when a more
accurate representation of the initial and final states is required.
Increasing the number of intermediate states the quality of the time-step representation
can be improved, but at the same time computational costs grow and the method
becomes slower. Note also, that when the difference between two consecutive time-evolved
states decreases, at some point, the improvement, due to the targeting of these two states,
becomes irrelevant while the computational costs still increase uniformly. Therefore some
balance should be found between the time-step size and the number of intermediate states.
Some intermediate states can be computed relatively cheaply as compared to the product
of the Hamiltonian with the state. When the actual time evolution is carried out using
the R-K method, the R-K vectors (3.19a) can also be used to generate the states at other
times. The states at times t + δt/3 and t + 2δt/3 can be approximated (see Feiguin and
White [67]), with an error O((δt) 4 ), as
|ψ(t + δt/3)〉 ≈ |ψ(t)〉 + 1
162 [31|k 1〉 + 14|k 2 〉 + 14|k 3 〉 − 5|k 4 〉] (3.27a)
|ψ(t + 2δt/3)〉 ≈ |ψ(t)〉 + 1
81 [16|k 1〉 + 20|k 2 〉 + 20|k 3 〉 − 2|k 4 〉] . (3.27b)
When the Krylov-subspace time-evolution method is employed, one can use the same
Krylov subspace K n (Ĥ, |ψ(t)〉) and the same orthogonal projection T n of the Hamiltonian,
already constructed to obtain |ψ(t + δt)〉, in order to approximate the states at times
t + δt i , 0 < δt i < δt, as
|ψ(t + δt i )〉 = e −iĤδt i
|ψ(t)〉 ≈ V n e −iTnδt i
V † n|ψ(t)〉 . (3.28)
Typically, each of these states is at least as accurate as the final one |ψ(t + δt)〉, but for
additional safety, one can involve each of the intermediate times in the stopping criterion
for the Krylov-subspace buildup (3.25).
We can now formulate the time-step targeting scheme. To readjust the effective statespace
basis to the current time step, one performs one or several finite-system DMRG

3.4. Time-step targeting adaptive time-dependent DMRG 81
sweeps simultaneously targeting the initial |ψ(t)〉, the final |ψ(t + δt)〉, and several intermediate
|ψ(t + δt i )〉 time-evolved states; at each iteration one reobtains the final and all
intermediate states from the initial one and constructs a new re-adapted effective state
basis using the weighted average density matrix formed from the target states. Once this
basis is optimal enough, the final state is taken and one proceeds to the next time step.
This final state corresponds to the initial state of the upcoming time step. This process is
schematically depicted in Fig. 3.3.
Typically, some sweeps are required to achieve self-consistency between the targeted
states and the basis produced by the density matrix. However, since the accuracy of the
initial vector |ψ(t)〉 becomes worse from iteration to iteration and all intermediate states
including the final one are reobtained from it, it is advisable to keep the number of sweeps
as small as possible.
In the above considered scheme only the final state is being kept when progressing in
time. Hence, two consecutive time steps overlap with only one state (the final state in
the first step is the initial one for the following step). Different schemes are also possible.
For example, one can advance in time by δt i < δt, retaining the intermediate time-evolved
state |ψ(t + δt i )〉. In this case two consecutive time steps overlap not only with one state,
but with several ones. This may improve the reliability of the method, because parts of the
following time step are also included in the re-adaptation of the effective state-space basis.
Moreover, it is also not crucial to use the same time step for the time evolution methods
and the state-space readjustment. One can perform the time evolution with the time step
δt, computing ν consecutive time-evolved states — each of the states |ψ(t + µδt)〉 (µ ν,
µ ∈ N) is obtained from its preceding one |ψ(t + (µ − 1)δt)〉 — and readjust the statespace
basis for the time intervals ∆t = νδt. Nevertheless, for all considered alternatives,
one should keep in mind that the large number of intermediate time steps reduces the
speed of the simulations; if large time steps are taken, then the targeted states differ
considerably, reducing the quality of the results and requiring larger effective state-space
dimensions; additional sweeping may also become necessary, increasing the error due to
continuous deterioration of the initial state of the time step .
These additional alternatives require further investigations, which are out of the scope
of the present thesis. Here, we stick to the described time-step targeting scheme, using
the final state of the time step to advance in time and the same time-steps for the time
evolution and the state-space readjustment.
3.4.3 Time-step targeting adaptive time-dependent DMRG based
on the Krylov-subspace methods for the time-evolution
Using the components discussed in the above subsections, the time-step targeting adaptive
time-dependent DMRG based on Krylov-subspace methods for the time-evolution can be

82 3. Real-time evolution using DMRG
formulated as follows:
0. Using the conventional DMRG algorithm (infinite- plus finite-system algorithm), find
the ground state of the initial system.
At the end of this step, for every l = 2, . . .,L − 1, we have the effective bases B(L, l)
and B(R, L − l + 1), for the left and right blocks, respectively, and the corresponding
basis transformations A l (s l ) and B L−l+1 (s L−l+1 ).
1. When it is applicable (i) change the Hamiltonian or (ii) construct an initial state by
applying the given operator Ĉ to the ground state |ψ(0)〉 = Ĉ|ψ GS〉.
The latter can be constructed and targeted during the finite-system DMRG sweeps
in step 0.
If the system Hamiltonian is changed at t = 0, then additional care has to be taken
that the effective bases obtained during the conventional DMRG run (step 0.) accurately
represent this modified Hamiltonian, too.
2. To evolve the targeted state in time, use the finite-system DMRG sweep with the
following modifications:
a. Instead of diagonalization, at each iteration obtain N t time-evolved states at
a given sequence of times t + δt i , δt 1 < . . . < δt Nt = δt (typically δt i = iδt/N t ),
from the initial state |ψ(t)〉 using the Krylov-subspace methods, for instance the
Arnoldi method Eqs. (3.23) and (3.28).
The obtained states — the final |ψ(t + δt)〉 and the intermediates |ψ(t + δt i )〉
— together with the initial one (|ψ(t)〉) constitute the set of target states.
b. Construct the weighted average density matrix from the target states, using
given weight factors w i , ∑ N t
i=0 w i = 1, as
∑N t
ˆρ = w i |ψ(t + δt i )〉〈ψ(t + δt i )|
i=0
(typically w i = w for i = 1, . . .,N t − 1 and w 0 = w Nt w).
c. As always, perform the DMRG truncation (reduced density-matrix projection)
at each step obtaining a new effective basis for the block state space, with
corresponding transformation matrices.
d. Use wave-function transformation (see Sec. 2.6.3) only for the initial state |ψ(t)〉
to shift the free sites by one.

3.4. Time-step targeting adaptive time-dependent DMRG 83
Figure 3.3: Sketch of the effective state-space development during the time-step
targeting adaptive t-DMRG simulation of the time-evolution of the state |ψ(0)〉. The
effective state space H eff (t i ) is re-adapted to accurately represent two consecutive timeevolved
states |ψ(t i−1 )〉 and |ψ(t i )〉, and two intermediate states (one time step interval
is [t i−1 , t i ]). The solid line represents the “true” time-evolution when the whole Hilbert
space H is taken into account.
3. After performing one or several finite-system DMRG sweeps with the listed modifications
(see step 2.), advance in time (t + δt → t; |ψ(t + δt)〉 → |ψ(t)〉).
Evaluate operators (DMRG “measurements”) when desired.
4. Repeat steps 2–3 until the desired time is reached.
This algorithm does not differ much from the finite-system DMRG algorithm with the
state prediction (basis transformation). All operations are the same, except that now
instead of diagonalizing the orthogonal projection of the Hamiltonian we construct the
matrix exponential out of it. Since the time-evolution operator possesses the same symmetries
as the system Hamiltonian, we use the “advanced” DMRG machinery taking into
account (see Section 2.6.4) these symmetries during the time-evolution simulation, too;
these symmetries significantly reduce the computer resources required for the algorithm
and considerably increase the speed of the runs. Similar to the finite-system DMRG algorithm
finite-system sweeps can be performed with the fixed number of retained reduced
density-matrix eigenstates, with the fixed discarded-weight threshold, or with some combination
of both. The most expensive part still remains the matrix-vector multiplications
(i.e. Ĥ|ψ〉).
The time-step targeting adaptive t-DMRG algorithm by Feiguin and White [67] is
obtained from this algorithm by exchanging the time-evolution scheme from the Arnoldi
method to the R-K one in 2.a. The algorithm devised by Manmana et al. [161] is recovered,
if the Lanczos basis of the Krylov subspace is used instead of the Arnoldi basis. Time-step

84 3. Real-time evolution using DMRG
targeting adaptive t-DMRG algorithms are schematically depicted in Fig. 3.3. Note, that
the present algorithm employing the Arnoldi method allows to study the time evolutions
with non-Hermitian operators, too.
A single iteration of the R-K based time evolution is typically faster than the Krylovsubspace
one; it requires only 4 Hamiltonian-vector multiplications. However, Krylovsubspace
methods allow larger time steps with higher accuracy making them faster and
hence more attractive. For instance, in the example considered during the error analysis
(see Section 3.5.2) accurate results were obtained using the time steps δt = 0.5 or even
δt = 1.0 with up to 20 Arnoldi iterations (requires up to 20 Hamiltonian-vector multiplications).
The error due to the R-K approximation alone would be 0.5 5 ≈ 0.03 for δt = 0.5
(3.20).
The presented algorithm can be straightforwardly generalized for the cases with the
time-evolution of several states. In this case the final and all the intermediate states are
generated for each time-evolving state. The resulting sets are targeted simultaneously and
reweighted according to the weight factors of the original states.
Several modifications can also be made to improve the performance of the algorithm.
One can relax the accuracy requirements for the time evolution methods (3.25) for tentative
iterations of the finite-system sweeps. At the final iteration, when we advance in time, the
maximal accuracy can be demanded in order to obtain a more accurate final state which
then will be used as the initial one in the following time step. In fact, the computation
time involved in the last iteration of a sweep is typically miniscule as compared to the
time required for the entire sweep. The relative weights of the targeted states can also
be optimized. Feiguin and White [67] empirically found that targeting two intermediate
states at times t + δt/3 and t + 2δt/3 with the weight factors 1/6 and the initial and final
states with weights 1/3 each, produced excellent results. In our calculations we will follow
this parameterization of the time-step targeting scheme.
To summarize, the devised method is not as fast or as efficient as adaptive t-DMRG:
the whole time-evolution operator has to be applied to states at every iteration of the
finite-system sweeps. Additionally, several time-evolved states have to be simultaneously
targeted during the time-evolution simulation, requiring larger effective state-space dimensions.
Nevertheless, the potential to time-evolve with larger time steps partly compensates
these disadvantages and more importantly the restriction to the nearest-neighbor Hamiltonians
is lifted. Eventually, in exchange for the small loss of efficiency, we gain the ability
to treat the longer-range interactions, ladder systems, and narrow two-dimensional strips.
In addition, as we will see in the following section, the accuracy is much improved over the
lowest order Trotter methods.
In the following section we investigate and compare the accuracy of the considered
adaptive t-DMRG algorithms.

3.5. Accuracy of adaptive time-dependent DMRG 85
3.5 Accuracy of adaptive time-dependent DMRG
In this section, we will analyze the accuracy of the real-time evolution methods outlined
above. There are several method specific error sources:
• the algorithm employed to perform the time-evolution; e.g., the approximation of the
time evolution operator using the n-th order Suzuki-Trotter decomposition or the
Krylov-subspace projection of matrix exponentials;
• the representation of the Hamiltonian or the time-evolution operator, which are driving
the actual time-evolution, in the effective state-space basis;
• the truncation error in the targeted states (e.g., initial state |ψ(t 0 )〉 or the time-evolved
states |ψ(t)〉, |ψ(t + δt)〉) due to the cut-off of the block state-space basis. When the
target states cannot be reobtained or explicitly constructed (e.g. as an extremum of
some functional) at every iteration, the truncation error starts to accumulate from
step to step.
• the non-optimal block state-space basis; when the effective basis does not cover the
“relevant” parts of the Hilbert space, expectation values can be qualitatively wrong in
the worst case, even for a small discarded weight.
The last case is also relevant for conventional DMRG calculations, e.g., for obtaining the
ground state of the system, where one can get stuck in some metastable state and does
not converge to the actually wanted state. This scenario is usually avoided by performing
several finite-system sweeps with different parameter sets, which becomes non-trivial for
the time evolution.
There are some other errors generated by sources not listed above. These are errors
of standard numerical algorithms [12, 87], for example the full diagonalization used for
obtaining the optimal bases for the system and environment blocks (see Section 2.3), and
pure numerical ones, present in all calculations, caused by the floating-point arithmetic in
computers. f All these errors are assumed to be negligibly small.
The above listed error sources play different roles on different time scales, and have
an individual impact on the efficiency of the time-evolution method. Several simulation
parameters and strategies are available to control the inaccuracies caused by each of these
sources. One can already guess that the reduction of the truncation error, without loosing
the ability of the basis to represent the real states, will be the most demanding in computer
resources at large time scales. This possibly will set the limit to the time length accessible
with the method. On the other hand inaccuracies due to other error sources will play an
[86].
f ANSI/IEEE Standard 754-1985, Standard for Binary Floating Point Arithmetic. See else reference

86 3. Real-time evolution using DMRG
important role at small time scales. In the following we will investigate how each type of
error behaves in the course of time and how to control them simultaneously. In order to
do this, where it is possible we will use some analytical estimations and in addition we
consider a couple of nontrivial time-evolution examples for which the exact solutions are
available.
To estimate the errors made during the time-evolution simulation (mainly analytical
estimations) we use the following distinguishability measures for two quantum-mechanical
states ˆρ and ˆσ:
• Trace distance, also called variational or Kolmogorov distance, defined as
where ‖ · ‖ 1 denotes a trace norm.
D(ˆρ, ˆσ) := 1 2 Tr (|ˆρ − ˆσ|) = 1 2 ‖ˆρ − ˆσ‖ 1 . (3.29)
• Fidelity:
F(ˆρ, ˆσ) :=
( ) ( ) 2 2
Tr|
√ˆρ
√ˆσ| = Tr√ √ˆρˆσ
√ˆρ . (3.30)
Fidelity g is a popular measure of distance between density operators. It is not a
metric, but has some useful properties. One can also define metric distance measures
out of it, e.g.,
Bures distance:
B(ˆρ, ˆσ) 2 := 2(1 − √ F(ˆρ, ˆσ)) , (3.31)
and sine distance [82]:
C(ˆρ, ˆσ) := √ 1 − F(ˆρ, ˆσ). (3.32)
For detailed definitions of quantum distance measures and the relationships between them
see [72, 183]. The following bounds between the trace distance and the fidelity,
1 − √ F(ˆρ, ˆσ) D(ˆρ, ˆσ) √ 1 − F(ˆρ, ˆσ), (3.33)
first shown by Fuchs and van de Graaf in [72], prove to be useful in the following analysis.
For pure states ˆρ = |ψ〉〈ψ| and ˆσ := |φ〉〈φ|
F (|ψ〉〈ψ|, |φ〉〈φ|) = |〈ψ|φ〉| 2 , (3.34)
and all considered distinguishability measures are completely equivalent to one another,
namely
D (|ψ〉〈ψ|, |φ〉〈φ|) = √ 1 − F (|ψ〉〈ψ|, |φ〉〈φ|) = C (|ψ〉〈ψ|, |φ〉〈φ|). (3.35)
g Note, that an alternative definition of the fidelity F ′ = √ F is also used [72].

3.5. Accuracy of adaptive time-dependent DMRG 87
(a)
(b)
Figure 3.4: Schematic pictures of the initial states: (a) fermionic domain wall residing
in the left half of the chain, (b) particle added to the ground state of the system.
Trace distance is a good quantity when it is needed to estimate the error made in an
observable
|Tr(Ôˆρ) − Tr(Ôˆρ′ )| ‖Ô‖ op‖ˆρ − ˆρ ′ ‖ 1 = 2‖Ô‖ opD(ˆρ − ˆρ ′ ) , (3.36)
where ‖ · ‖ op denotes an operator norm.
Numerical analysis of the errors
To get an idea how errors develop during the time evolution, we will investigate two different
setups. We consider spinless fermions on an open chain. The model is equivalent to the
spin- 1 XX chain. The exact solution was first demonstrated by Lieb et al. in [154] (see also
2
Refs. [14, 15, 16]), almost half a century ago. Within this model we study the real-time
dynamics of the fermion domain wall (DE) (see Fig. 3.4a) and the time evolution of a single
particle (SP) added to the ground state of the N e fermion system at position i at time
t = 0 (see Fig. 3.4b). Since non-trivial exact solutions for both setups, also in the case of
open boundary conditions (conditions preferable for DMRG), exist (see Appendix A), they
provide a nice playground for the analysis of the accuracy. The former case was already
exploited by Gobert et al. (see Ref. [84]) in their analysis of the accuracy of adaptive
t-DMRG. However, several questions remained unanswered, which will be discussed below.
The model Hamiltonian of spinless lattice fermions reads
H = −J ∑ i
(c † i c i+1 + h.c.) , (3.37)
where J is the nearest-neighbor hopping amplitude, and c † i (c i ) creates (annihilates) a
fermion on lattice site i.
The initial state of the first case — the fermion domain wall — is essentially a Fock

88 3. Real-time evolution using DMRG
state in the real space
L/2
∏
|ψ(0)〉 dw =
i=1
c † i |0〉 , (3.38)
where |0〉 is a vacuum. This example (actually the time evolution of any real space Fock
state) is particularly interesting, since the initial state can be exactly written in the DMRG
MPS basis (2.21) with the restricted dimensions a j , b j = 1. h
For the second case we will consider the ground state (|ψ GS 〉) of the system of N e
spinless fermions and create an extra particle on the i-th site
|ψ(0)〉 sp = c † i |ψ GS〉 . (3.39)
Since it is not possible to directly compare the numerically obtained time-evolved state
with the exact one — in view of the exponentially large Hilbert space i — we introduce
some indirect error measures. As a measure of the overall error, we consider two kinds of
density deviations:
1. The maximum deviation of the on-site density (n i = c † i c i ) found by DMRG from the
exact result (for exact results see Appendix A.2), is given as [84]
Err max (n i , t) = max ∣ 〈n
DMRG
i (t)〉 − 〈n exact
i (t)〉 ∣ . (3.40a)
i=1,...,L
2. Another error measure can be defined by associating the expectation
value 〈n i (t)〉 with the i-th component of the L-dimensional vector
n(t) = (〈n 1 (t)〉, 〈n 2 (t)〉, . . ., 〈n L (t)〉), as follows
Err vec (n i , t) = √ 1 L∑
(〈n DMRG
i (t)〉 − 〈n exact
i (t)〉) 2
L
i=1
where ‖ · ‖ denotes the Euclidean norm.
= 1 √
L
∥ ∥n DMRG (t) − n exact (t) ∥ ∥ ,
(3.40b)
The second error measure in comparison with the first one gives a rough information
whether the error is concentrated on a few sites or is distributed over the entire system.
The remainder of this section is organized as follows: we start with the accuracy analysis
of the adaptive t-DMRG based on the Suzuki-Trotter time evolution scheme (see Section
3.5.1), then move to the time-step targeting adaptive t-DMRG based on the Arnoldi
method for the time-evolution (see Section 3.5.2) and conclude with comparisons and a
summary.
h In DMRG “language” this means, that the truncation error free state can be obtained by keeping only
one density-matrix eigenstate in every DMRG iteration.
i at least for reasonable chain sizes

3.5. Accuracy of adaptive time-dependent DMRG 89
3.5.1 t-DMRG based on the Suzuki-Trotter time-evolution
scheme
Error sources appearing in this method are the Suzuki-Trotter approximation of the time
evolution operator and the DMRG truncation error. The latter is introduced at every
iteration of the finite-system sweeps when the block state-space basis is truncated and it
accumulates from iteration to iteration.
Non-optimality of the block state-space basis might becomes an issue when the time
steps are really large, but since errors due to the Suzuki-Trotter approximation will also
be huge for this case, these runs are automatically excluded. There are examples where
the local optimality of the method causes problems (see the discussion in Section 3.3),
whereas algorithms with the globally optimal representation of the time-evolved states are
free of them. Typically these scenarios are encountered when the “true” time-evolving state
does not have an appropriate support on the DMRG effective state space. Nevertheless,
targeting of smartly-chosen auxiliary states helps to avoid these complications.
Before performing any numerical error analysis, let us first make some analytical estimates.
3.5.1.1 Analytical estimates
We consider both error sources independently. When discussing a particular error source
we assume that the errors produced by the other are negligibly small.
(i) The Suzuki-Trotter error: for an n-th order Suzuki-Trotter decomposition [68, 104, 221],
the error made in one time step δt is of order δt n+1 . Since t/δt time steps are necessary to
reach the time t, in the worst case the error will grow linearly in time t, resulting in the
error
ǫ(t) = c(n) · t
δt δtn+1 = c(n) · t δt n , (3.41)
where c(n) is some δt and t independent coefficient. The unapproximated time evolved
state |ψ(t)〉 can be written as
|ψ(t)〉 =
√
1 − ǫ(t) 2 |ψ S-T(t)〉 + ǫ(t)|ψ ⊥ S-T(t)〉 (3.42)
where |ψ S-T(t)〉 is the state kept after the Trotter expansion and |ψS-T ⊥ (t)〉 accounts for the
part that is neglected when doing the expansion. The fidelity of the state at time t is
F(|ψ(t)〉〈ψ(t)|, |ψ S-T(t)〉〈ψ S-T(t)|) = |〈ψ(t)|ψ S-T(t)〉| 2 = 1 − ǫ(t) 2 (3.43)
and the corresponding error ERR S-T(t), measured using the trace norm is
ERR S-T(t) = 2 √ 1 − |〈ψ(t)|ψ S-T(t)〉| 2 = 2ǫ(t) ∝ tδt n . (3.44)

90 3. Real-time evolution using DMRG
(ii) The DMRG truncation error: During the finite-system sweeps a truncation error is
introduced at every iteration when a tensor-product basis of dimension a j−1 d j for the left
block L(j) is reduced to a basis of dimension a j during a left-to-right sweep and, similarly,
when a tensor-product basis of dimension b j+1 d j for the right block R(j) is reduced to a
basis of dimension b j during the right-to-left sweep. Each state |ψ〉, which is defined using
the original tensor-product basis, is replaced by an approximate state | ˜ψ〉, which is defined
using the truncated basis. j While the truncation error ǫ that sets the scale of the error of
the wave function and the operators (see Section 2.3) is typically very small, it will strongly
accumulate in the course of time since O(Lt/δt) truncations are carried out up to time t.
To estimate the total error due to truncations, let us consider the j-th iteration of the
left-to-right sweep. The target state, denoted as |ψ [j] 〉, can be written using the Schmidt
decomposition as follows
n j
∑
|ψ [j] 〉 =
µ=1
λ [j]
µ |u [j]
µ 〉|v [j]
µ 〉 (3.45)
where n j = min(a j−1 d j , d j+1 b j+2 ) and λ [j]
1 λ [j]
2 · · · λ [j]
n j
0. Let ǫ j = ∑ n j
µ=a j +1(λ [j]
µ ) 2
denote the sum of discarded eigenvalues of the left block reduced density matrix when the
state-space size is reduced to the dimension a j . |ψ [j] 〉 can be rewritten as
where
|ψ [j]
tr 〉 =
|ψ [j] 〉 = √ 1 − ǫ j |ψ [j]
tr 〉 + √ ǫ j |ψ [j],⊥
tr 〉 (3.46a)
m
1 ∑ j
√ 1 − ǫj
µ=1
λ [j]
µ |u [j]
µ 〉|v [j]
µ 〉, 〈ψ [j]
tr |ψ [j]
tr 〉 = 1
(3.46b)
is the Schmidt decomposition of the state kept after the left block Hilbert space truncation
and
|ψ [j],⊥
tr 〉 = √ 1
n
∑ j
λ µ [j] |u µ [j] 〉|v µ [j] 〉, 〈ψ [j],⊥
tr |ψ [j],⊥
tr 〉 = 1 (3.46c)
ǫj
µ=m j +1
is the neglected part formed by the eigenfunctions corresponding to the smallest, “irrelevant”
Schmidt coefficients. Since 〈ψ [j],⊥
tr |ψ [j]
tr 〉 = 0 — they are formed by vectors lying in
the orthogonal subspaces — similar to the Trotter expansion the fidelity of the obtained
state, which we denote as ζ j (1), is
ζ j (1) = |〈ψ [j] |ψ [j]
tr 〉| 2 = 1 − ǫ j . (3.47)
After moving to the next bond (next iteration of the left-to-right sweep), the obtained
state |ψ [j+1] 〉, can be written similarly as
|ψ [j+1] 〉 = √ 1 − ǫ j+1 |ψ [j+1]
tr 〉 + √ ǫ j+1 |ψ [j+1],⊥
tr 〉 , (3.48)
j Usually a projection of a state to the truncated basis is performed with a consecutive normalization
of the obtained state.

3.5. Accuracy of adaptive time-dependent DMRG 91
and it is faithfully represented by |ψ [j+1]
tr 〉 up to
∣
ζ j+1 (1) = ∣〈ψ [j+1] |ψ [j+1]
tr 〉
∣ 2 = 1 − ǫ j+1 . (3.49)
The fidelity of the state obtained after the two consecutive block state-space truncations
(iterations j and j + 1), denoted as ζ j (2), is
ζ j (2) = |〈ψ [j] |ψ [j+1]
tr 〉| 2 = (1 − ǫ j+1 )|〈ψ [j] |ψ [j]
tr 〉| 2 = (1 − ǫ j )(1 − ǫ j+1 ) . (3.50)
This results can be easily generalized for the k consecutive iterations, including those from
the right-to-left sweeps, with the corresponding block state-space truncations, giving the
fidelity of the targeted state
k−1
∏
ζ j (k) = (1 − ǫ j+i ) . (3.51)
i=0
The upper bound for the total error made in the state, due to the blocks’ state-space
truncations, using (3.51) and (3.35) is
ERR tr (N(t)) = 2 √ N(t)
∏
1 − ζ(N) = 2
√ 1 − (1 − ǫ i ) , (3.52)
where N(t) is the number of DMRG iterations required to reach the time t. One should
note here that (3.50), (3.51) and consequently (3.52) hold only, if iterations are parts of the
same, left to right or right to left, DMRG half-sweeps. In this case each of the |ψ [q],⊥
tr 〉 is
orthogonal to any of |ψ [r]
tr 〉 (q, r = 1, . . .,N(t)). For iterations from different half-sweeps this
may not be the case any more, but as we will see in the following subsections, Eq. (3.52)
still gives a good estimate for the upper bound of the error caused by the state-space
truncations. k
If ν(n, L) is the number of DMRG iterations performed during the δt time step of the
n-th order Suzuki-Trotter decomposition, l then N(t) = ν(n, L) t/δt and
ν(n,L)t/δt
∏
ERR tr (t) ≡ ERR tr (N(t)) = 2
√ 1 − (1 − ǫ i ). (3.53)
k One can solve the mentioned problem — appearing during the change of the sweep direction — for
Eq. (3.52). Using the equivalence of the trace and sine distances for the pure states (3.35) and the triangular
inequality for the metric distance measures
i=1
i=1
C(|ρ〉〈ρ|, |ψ〉〈ψ| ′ ) C(|ψ〉〈ψ|, |φ〉〈φ|) + C(|φ〉〈φ|, |ψ〉〈ψ| ′ ),
the upper bound of the error within the half-sweep can be estimated using Eq. (3.51) and between the
half-sweeps using the triangular inequality.
l For the symmetric Suzuki-Trotter decomposition of 2nd order ν(2, L) = 3(L − 3), where L is the system
size.

92 3. Real-time evolution using DMRG
L is the system size. The maximum acquired value for ERR tr (t) is 2, corresponding to
two orthogonal states. Beside this, for ERR tr (t) ≈ 1 the error in the quantity and its
value are of the same order. Since ν(n, L) ∝ L, from (3.53) follows that the truncation
error should accumulate roughly exponentially with an exponent of Lt/δt. Therefore the
adaptive t-DMRG will eventually break down at long times.
Several remarks are in place: during the time-step sweeps, parts of the time evolution
operator are applied to the bonds of the same parity with the consecutive state-space
truncation procedure. In case of the 2nd order symmetric Suzuki-Trotter decomposition,
in the required three half-sweeps the local time evolutions are performed either on each
even or each odd bond during the single left-to-right or right-to-left half-sweep, respectively
(see Section 3.3). On the other hand, the discarded weight ǫ i must not be the same for
different iterations; it might be even 0, especially when the runs are performed keeping the
dimensions of the blocks’ state spaces fixed.
Finally, the total error in the worst cases will be given as a sum of errors due to
the Suzuki-Trotter approximation of the time evolution operator (3.44) and the DMRG
truncation error (3.53)
ν(n,L)t/δt
∏
ERR(t) = ERR S-T(t) + ERR tr (t) = c(n)tδt n + 2
√ 1 − (1 − ǫ i ). (3.54)
The first part of (3.54) can be reduced by decreasing the time step δt or by using a higher
order Suzuki-Trotter approximants. Unfortunately, both proposals will increase the number
of DMRG iterations required to reach the time t, with the possible consequent growth
of the accumulated truncation error (the second term in Eq. (3.54)). The second term in
(3.54) should decrease considerably with an increasing number of the kept states or with
a reduced discarded-weight threshold. Both proposals require more computer resources,
computation time as well as computer memory. Therefore, one needs to find a balance
between the accuracy and the price for it. Usually, in practice a partial compensation of
errors in observables may slow down the error growth and improve the situation. Since it
is hard to judge how the second term will develop in time (the ǫ i are not apriori known),
in the following subsection we will investigate the accuracy of the time evolution method
on a couple of concrete examples.
3.5.1.2 Numerical analysis
Domain wall dynamics: We begin with the time evolution of the domain wall of fermions.
The initial state (domain wall) can be written exactly as a DMRG MPS (2.21) with the
unit a j , b j = 1 (j = 1, . . ., L) dimensions. This example allows us to analyze the accuracy
of the considered time evolution method very explicitly. Recall that each term of the
Suzuki-Trotter approximation of the time evolution operator is applied exactly to the
i=1

3.5. Accuracy of adaptive time-dependent DMRG 93
1e+00
L=100, δt=0.02
Err max
(n i
,t), ERR tr
(t)
1e-02
1e-04
Err max
(n i
,t)
ERR tr
(t)
1e-06
m=10 5
m=20 5
m=40 5
1e-08
m=60 5
m=100 5
m=200 5
runaway time, t R
1e-10
0 10 20
time (t)
30 40
Figure 3.5: Density deviation Err max (n i , t) (bold lines) and the corresponding upper
bound of the DMRG truncation error ERR tr (t) (thin lines) as a function of time t
for different numbers of kept states m. The Suzuki-Trotter time step is performed
with δt = 0.02. A “runaway” time t R (circles) separates two regimes of Err max (n i , t).
Results are for the time evolution of a fermionic domain wall of size 50 residing in the
left half of the open L = 100 chain and were obtained by adaptive t-DMRG based on
S-T(2). The hopping amplitude is J = 0.5.
targeted state during the time-step sweeps of the adaptive t-DMRG. We partly repeat
the investigations performed by Gobert et al. presented in [84], but show that at each
iteration the discarded weight can be used to estimate the error due to the DMRG statespace
truncation procedure. In addition, we show that the form of the error progression
observed in [84] strongly depends on the way how the time-step sweeps are performed.
We consider the open chain of length L = 100. The domain wall is formed with the 50
spinless fermions residing on the first 50 sites of the chain (3.38). The hopping amplitude
is J = 0.5. As everywhere in this thesis, we assume the lattice constant a = 1, = 1, and
the time is measured in the inverse energy unit 1/J.
We mainly use the density deviation Err max (n i , t) (3.40a) to measure the error in the
obtained state |ψ(t)〉. Additionally we consider the upper bound of the DMRG truncation
error ERR tr (t) (3.53) to analyze the time development of the accumulated truncation error
separately. ERR tr (t) has a complex form, but since at each iteration the discarded weight
is known, one can easily obtain it numerically. The upper bound of the Suzuki-Trotter

94 3. Real-time evolution using DMRG
error ERR S-T(t) is rather simple and is therefore not considered independently.
We begin with the analysis of the adaptive t-DMRG accuracy for the case when the
time-step sweeps are performed with a fixed number m of the kept states. Two parameters,
the number of the kept states m and the time step size δt, can be used to control the
accuracy of the method during this type of simulations. First we investigate what varying
of m does. In Fig. 3.5 we plot Err max (n i , t) (bold lines) and the corresponding ERR tr (t)
(thin lines) vs. time t for the different numbers of kept states m. The Suzuki-Trotter
time step is performed with fixed δt = 0.02. Two regimes can be clearly identified in
the time evolution of the density deviation. These regimes are separated by a well defined
“runaway time” t R [84], which is increasing almost linearly in m. In the first regime, t < t R ,
the density deviation grows essentially linear in t, is independent of m, and obviously is
dominated by the Suzuki-Trotter error. For later times, t > t R , the density deviation is
m-dependent and growing faster than any power law, up to some saturation. The error in
this regime is entirely given by the accumulated truncation error. Note, that the density
deviation corresponding to the largest considered m (dotted bold line in Fig. 3.5) does
not exhibit any sudden deviation from the “linear” in t behavior, at least in the plotted
interval t ∈ [0, 40]. Since m → ∞ corresponds to the complete absence of the truncation
error, the same curve can be used as a measure of the deviation due to the Suzuki-Trotter
error alone and the runaway time can be read off very precisely as the moment in time
when the accumulated truncation error starts to dominate.
The considered density deviation is only useful for identifying typical scenarios of the
error development, since it will not be available for a general case. Therefore, we would
like to know what features of the accumulated truncation error are contained in ERR tr (t).
ERR tr (t) overestimates the density deviation caused by the accumulated truncation error,
but it still captures its development well (compare thin and bold lines in Fig. 3.5). At
the beginning of time t, in Fig. 3.5 one sees the interval in time where ERR tr (t) is less
than machine precision; the corresponding density deviation is free of any accumulated
truncation error in the same time interval. The size of this region is increasing with
growing m. For small m, ERR tr (t) starts to grow almost exponentially and is already
large when the moderate speed of growth — on a logarithmic scale — sets in. This is an
indicator of the rapid accumulation of the truncation error in the corresponding on-site
density. For large m, ERR tr (t) emerges later and grows considerably slower, supporting
the observed increase of the runaway time with the growing number of kept states m.
To study how the time step δt influences the accuracy of the adaptive t-DMRG we show
in Fig. 3.6 the density deviation Err max (n i , t) (bold lines) together with the corresponding
upper bound of the DMRG truncation error ERR tr (t) (thin lines) for m = 60 fixed number
of kept states and different time steps δt. In agreement with the above made observation,
two regimes separated by the runaway time t R can be identified in the time evolution
of the density deviation. At small times, t < t R , the density deviations corresponding to

3.5. Accuracy of adaptive time-dependent DMRG 95
1e+00
L=100, m=60
Err max
(n i
,t), ERR tr
(t)
1e-02
1e-04
Err max
(n i
,t)
ERR tr
(t)
1e-06
δt=0.2 δ
δt=0.1 δ
δt=0.05 δ
1e-08
δt=0.02 δ
δt=0.01 δ
δt=0.005 δ
runaway time, t R
1e-10
0 10 20
time (t)
30 40
Figure 3.6: Density deviation Err max (n i , t) (bold lines) and the corresponding upper
bound of the DMRG truncation error ERR tr (t) (thin lines) as a function of time t, for
different time steps δt and m = 60 fixed number of kept states. A “runaway” time t R
(circles) separates two regimes of Err max (n i , t).
different δt are parallel, increasing almost linearly in t, and decreasing with δt — as one
would expect for the Suzuki-Trotter error dominated density deviation. Indeed, as shown
in the left panel of Fig. 3.7 the density deviation for the fixed t = 4 is quadratic in δt and
the Suzuki-Trotter error dominates over the accumulated truncation one. At large times,
t > t R , the density deviation is no longer linear in t, it starts to grow almost exponentially in
t (see for example Err max (n i , t) for δt = 0.01 and t 16 in Fig. 3.6) and it does not exhibit
the simple monotonic behavior in δt (see also the right panel of Fig. 3.7 corresponding to
t = 40). The density deviation in this regime is obviously dominated by the accumulated
truncation error. The runaway time t R increases roughly linear in δt.
Surprisingly, the upper bound of the DMRG truncation error, ERR tr (t), (thin lines in
Fig. 3.6) is practically the same for all considered time steps δt. At small times, t < 6,
ERR tr (t) is smaller than the machine precision, indicating the absence of the truncation
error in all considered density deviations. At t ≈ 6 it starts to grow faster than any power
law and at the beginning is even bigger for larger δt. The latter is in contradiction with
the anticipated behavior, because the number of DMRG iterations performed to reach the
time t is smaller for larger δt. The unexpected pattern can be explained with the gradual
adjustment of the effective Hilbert space connected with several but small discarded weights

96 3. Real-time evolution using DMRG
1e-02
1e-03
1e-04
1e-05
m=60, t=4.0
m=60, t=40.0
1e-06
1e-07
Err max
(n i
,t)
Err vec
(n i
,t)
0.005 0.01 0.02 0.05 0.1 0.2 0.5
δt
0.01 0.02 0.05 0.1 0.2 0.5
δt
Figure 3.7: Density deviations Err max (n i , t) (squares) and Err vec (n i , t) (circles)
vs. time step δt, at times t = 4 (left panel) and t = 40 (right panel). Lines correspond
to the quadratic fit.
for small δt in contrast to a few but larger ones for large δt. For later times, the growth
rate — on a logarithmic scale — becomes smaller and starting from t ≈ 19 the anticipated
sequence is recovered — ERR tr (t) becomes smaller for larger δt. Although, the accumulated
truncation error is almost the same for different δt, the Suzuki-Trotter error will dominate
longer over the accumulated truncation error for a larger δt, due to the increase of the
Suzuki-Trotter error threshold (∝ δt 2 ) with δt.
The δt-dependence of the runaway time can also be identified in Fig. 3.7, where we
show the density deviations, Err max (n i , t) and Err vec (n i , t), vs. δt for the fixed number of
kept states m = 60. In contrast to the quadratic behavior of the density deviation in δt
at early time t = 4 (the left panel of Fig. 3.7), the clear deviation from this behavior is
identified at large time t = 40 for δt 0.1 (the right panel of Fig. 3.7), indicating that
t = 40 is larger than the runaway time corresponding to these time steps. Comparison
between Err max (n i , t) and Err vec (n i , t) reveals that the error is localized on a few sites at
t = 4, while it is spread over several ones at t = 40.
To shortly summarize, during the simulation of the time evolution of the fermionic domain
wall, like in Ref. [84] we have identified two distinct regimes in the time development
of the error in the on-site density. These regimes are clearly separated by the so called
runaway time t R . For early times t < t R , the error in the on-site density is dominated by
the Suzuki-Trotter error and is linear in t. For t > t R , it starts to grow rapidly, almost
exponentially in t and is dominated by the accumulated truncation error. The on-site den-

3.5. Accuracy of adaptive time-dependent DMRG 97
sity obtained in this regime can become unreliable in a few Suzuki-Trotter time steps. The
runaway time t R increases when the number of kept states m or the time step δt increases.
On the other hand, growing δt increases the error threshold of the Suzuki-Trotter approximation
and consequently the error in the measured quantities. Additionally, we have found
that the upper bound of the DMRG truncation error, defined with (3.53), overestimates
the error in the on-site density caused by the DMRG state-space truncations. Nonetheless,
it still captures the tendency of the time evolution of the accumulated truncation error and
can be used as a measure for the error caused by the DMRG state-space truncations.
Since for most interesting examples of the time evolution the exact solutions are not
known or hard or even impossible to obtain, in what follows we would like to find out:
• Is the existence of the runaway time separating two clearly distinguishable regimes
in the development of the error a generic feature of a simulation?
• Can one improve the quality of the results in the regime where the accumulated
truncation error is dominating?
• Does the upper bound of the DMRG truncation error (ERR tr (t)) always capture the
form of the time development of the accumulated truncation error?
The last question is the most important, since the only quantity available in a general case
is ERR tr (t).
To answer these questions, first we study what happens when the time-step sweeps are
performed with the fixed discarded weight threshold instead of the fixed number of kept
states and then perform similar investigations for the second exactly solvable example. We
close this subsection with the accuracy analysis of the adaptive t-DMRG based on the
fourth order Suzuki-Trotter approximation of the time evolution operator.
Up to now, all adaptive t-DMRG runs were performed with the fixed number of kept
states m. What if we repeat the simulations, but keep fixed the discarded-weight threshold,
denoted as ε, instead of m. Now ε can be used to control the accuracy of the time evolution
method.
In order to find out how ε influences the method precision, in Fig. 3.8 we show the density
deviation Err max (n i , t) (bold lines) and the corresponding upper bound of the DMRG
truncation error ERR tr (t) (thin lines), for different discarded-weight thresholds ε and two
different time steps δt = 0.02 and δt = 0.005. As one can see all ERR tr (t) behave similarly.
As expected ERR tr (t) increases with growing ε and when δt is reduced. It starts to grow
from the initial Suzuki-Trotter time step, but the speed of growth is significantly lower
than anticipated from Eq. (3.53). Namely, if the truncation of order ε is made during each
DMRG iteration, then ERR tr (t) has to grow like √ 1 − (1 − ε) γLt/δt , with γ ∼ 3. m However,
in most DMRG iterations the actual discarded weights are several orders of magnitude
m Actually 1 γ < 3, because in three DMRG half-sweeps, required for the second order Suzuki-Trotter

98 3. Real-time evolution using DMRG
1e+00
L=100
Err max
(n i
,t), ERR tr
(t)
1e-02
1e-04
1e-06
1e-08
Err max
(n i
,t)
ε=1e-10, δt=0.02
ε=1e-10, δt=0.005
ε=1e-12, δt=0.02
ε=1e-12, δt=0.005
ε=1e-14, δt=0.02
ε=1e-14, δt=0.005
ERR tr
(t)
5,δ
5,δ
5,δ
5,δ
5,δ
t,5δ
1e-10
0 10 20 30 40
time (t)
Figure 3.8: Density deviation Err max (n i , t) (bold lines) and corresponding upper
bound of the DMRG truncation error ERR tr (t) (thin lines) vs. time for different
discarded-weight thresholds ε and two different time steps δt = 0.02 and δt = 0.005.
Results correspond to the time evolution of a fermionic domain wall of size 50 residing
in the left half of the open L = 100 chain and are obtained by adaptive t-DMRG based
on S-T(2). The hopping amplitude is J = 0.5.
lower than the truncation weight threshold ε and in the plotted time interval observed form
is close to √ 1 − (1 − ε) 2γt2 /δt
. Reasons why for t 40 the exponent is proportional to t 2 ,
rather than to Lt, we will see later, when we discuss the time development of the bipartition
entanglement.
Let us now analyze what happens with the corresponding density deviations, plotted
with bold lines in Fig. 3.8. As one can see the accumulated truncation error is significant
from the beginning. It dominates the density deviation across the whole plotted region
and for most of the considered parameter sets. Exceptions are only the ε = 10 −14 and
δt = 0.02 case, for which the density deviation is dominated by the Suzuki-Trotter error
in the whole plotted time interval, and more interesting the ε = 10 −12 and δt = 0.02 case,
for which both the Suzuki-Trotter and the accumulated truncation errors tend to be of the
same order. For the latter case, the density deviation at small times, t < 1, is dominated
by the Suzuki-Trotter error. For 1 < t < 20 one sees a significant onset due to the accuapproximant,
state-space updates are only performed either on each even or on each odd bond during the
single left-to-right or right-to-left sweep, respectively.

3.5. Accuracy of adaptive time-dependent DMRG 99
mulated truncation error; starting from t ≈ 20, it becomes similar to the mentioned case
of the Suzuki-Trotter error dominated density deviation (ε = 10 −14 , δt = 0.02). Another
important observation is that the obtained on-site density remains reliable for long times,
even when the density deviations are dominated by the accumulated truncation error. See
for example Err max (n i , t) for ε = 10 −14 and δt = 0.005. Although dominated by the accumulated
truncation error, it remains below 10 −5 for t ∈ [0, 40] and gives more accurate
results than the Suzuki-Trotter error dominated case of ε = 10 −14 and δt = 0.02.
The upper bound of the DMRG truncation error (ERR tr (t)) overestimates the density
deviation due to the accumulated truncation error (compare bold and thin lines in Fig. 3.8),
but it remains a good measure for the error caused by the DMRG state-space truncations.
From the above observations one can conclude, that the concept of the runaway
time loses its significance when the adaptive t-DMRG runs are performed with the fixed
discarded-weight threshold. For the studied example it is clear that the sharp separation
between the regimes dominated by the different error sources strongly depends on the way
the simulations are performed: with the fixed number of kept states or with the fixed
discarded-weight threshold. Additionally, the results obtained with the simulations of the
latter type remain reliable even when the error is dominated by the accumulated truncation
one. It remains to find the “price” for the improved accuracy.
When the adaptive t-DMRG runs are performed with the fixed discarded-weight threshold
ε, the number of kept states m has to be adjusted at each DMRG iteration and a larger
m is needed for a smaller ε. Computer resources — time and the amount of memory —
required per DMRG iteration increase with m (as ∼ m 2 and ∼ m 3 , respectively). Because
m is a function of the iteration number i, we introduce the parameter M(t), corresponding
to the maximum number of kept states within a time-step sweep. M(t) is defined as
M(t) = max[m(i)], for i ∈]N(t − δt), N(t)] (3.55)
and represents the cost for the increased precision of the results. N(t) is a number of
interactions required to reach the time t. In Fig. 3.9 we show M(t) corresponding to
the simulations plotted in Fig. 3.8. One can see, that a few states are enough to keep the
discarded weights below the given threshold at the beginning, but as time grows, one needs
to increase M(t) permanently. For example, for ε = 10 −14 and δt = 0.02 (the top curve
in Fig. 3.9) less than 60 states are enough to reach the time t = 10, while for t = 40 one
requires almost 400 states. The price to keep the discarded weights below ε is the almost
exponential growth of M(t). Therefore the time-evolution simulation for the fermionic
domain wall, for fixed ε will break down at some point in time.
The second interesting observation based on Fig. 3.9 is that the speed of the growth of
M(t) is considerably slower for the smaller time step δt. This indicates that the adjustment
of the effective Hilbert space, connected to the local state-space updates, works more
efficiently for decreasing δt.

100 3. Real-time evolution using DMRG
M(t) = max[m(i)] for i∈]N(t-δt),N(t)]
450
400
350
300
250
200
150
100
50
ε=1e-14, δt=0.02
ε=1e-14, δt=0.005
ε=1e-12, δt=0.02
ε=1e-12, δt=0.005
ε=1e-10, δt=0.02
ε=1e-10, δt=0.005
L=100
0
0 10 20 30 40
time (t)
Figure 3.9: The maximum number of kept states within a time-step sweep, M(t),
as a function of time t, for different discarded-weight thresholds ε and two different
time steps δt = 0.02 (bold lines) and δt = 0.005 (thin lines). From top to bottom ε
increases. Results corespond to the simulations plotted in Fig. 3.8.
To get an idea about the discarded weight per iteration, we study how the von Neumann
entropy of the bipartition entanglement S vN (i, t) (2.61) behaves in time and space. In
Fig. 3.10 S vN (i, t) is shown in a false color plot as a function of the bipartition bond index i
and time t. For t = 0, S vN (i, t = 0) is zero at each bond, as is expected for a plain product
state. When the time evolution starts, entanglement is produced resulting in the rise of
S vN (i, t) on the border of the domain wall (the center of the chain in the present case).
It increases, but the growth rate progressively slows down. Simultaneously, the fermions
(holes) start to penetrate the empty (filled) part of the chain, occupying a region that
grows linearly in time until reaching the boundaries (see also Fig. A.8). Every bipartition
remains disentangled until the arrival of the wave front (see also Fig. 3.21). This explains
the observed exponent proportional to t 2 in the upper bound of the DMRG truncation
error, ERR tr (t), for t 40. Entanglement propagates through the system with maximal
group velocity, being 1 for the considered example and model parameters (see the sides
of the cone in Fig. 3.10). In time, more and more fermions are getting involved in the
evolution process, causing an extra growth of the entanglement entropy, which is maximal
in the middle of the chain. This supports the observed polynomial growth of the maximum
number of kept states (M(t)) in the adaptive t-DMRG runs with the fixed discarded-weight

3.5. Accuracy of adaptive time-dependent DMRG 101
Figure 3.10: Von Neumann entropy of the bipartition entanglement S vN (i, t) as a
function of time t and cutting bond index i. Time increases from top to bottom.
Results are for the time evolution of a fermionic domain wall of size 50 residing in the
left half of the open L = 100 chain and were obtained by adaptive t-DMRG based on
S-T(2) with the discarded-weight threshold ε = 10 −14 and the time step δt = 0.005.
The hopping amplitude is J = 0.5.
threshold.
Single-particle propagation: Let us now consider the second exactly solvable example:
the time evolution of the on-site density after an extra particle has been created on the i-th
site at time t = 0 in the ground state |ψ GS 〉 of N e free spinless fermions. We consider an
open chain of length L = 100, the hopping amplitude is J = 0.5, N e = 50 and i = 51. All
other assumptions are the same as in the previous example. Although the exact solution for
this model exists (see Appendix: A.1), the initial state |ψ(0)〉 = c † i |ψ GS〉 can not be written
as a trivial product state in real space, as this was the case for the fermionic domain wall.
It is also not always possible to write it as a DMRG MPS for any predefined m without
loss of accuracy. This also holds true for the ground state of N e fermions. n
In order to determine the ground state of the N e = 50 fermion system and obtain the
initial state |ψ(0)〉, we use the finite-lattice DMRG algorithm. For this we target both of
the states with equal weights 0.5. o In the following we will perform time evolution runs
n The exception is the trivial case N e = 0.
o One can also target only the ground state and use a preliminary “empty” sweep to build the initial

102 3. Real-time evolution using DMRG
Err max
(n i
,GS), Err max
(n i
,t=0)
1e-04
1e-05
1e-06
1e-07
1e-08
1e-09
1e-10
L=150, N e
=50 (+1)
100 200 300 400
m
Err max
(n i
,GS)
Err max
(n i
,t=0)
1e-14 1e-12 1e-10 1e-08
ε
Figure 3.11: Density deviation Err max (n i , ·) for the ground |ψ GS 〉 (circles) and the
initial |ψ(0)〉 (stars) states vs. the number of kept states m (left panel) and vs. the
discarded-weight threshold ε (right panel). Dashed lines are only guide for eyes.
similar to the previous example, but in the discussion we will mostly concentrate on the
differences in the results.
Since the initial state is known with a finite accuracy, we show in Fig. 3.11 the
density deviation for the ground state Err max (n i , G) (circles) and for the initial state
Err max (n i , t = 0) (stars) vs. the number of kept states m (left panel) and the discardedweight
threshold ε (right panel). As expected the accuracy of the states increases considerably
with growing m or decreasing ε.
We proceed with the analysis of the accuracy of the adaptive t-DMRG method for the
case when the time sweeps are performed with the fixed number of kept states (m). In
Fig. 3.12 we present the density deviation Err max (n i , t) (3.40a) (bold lines) and the upper
bound of the DMRG truncation error ERR tr (t) (3.53) (thin lines) as a function of time t.
The upper plot in Fig. 3.12 contains the results for different numbers m and fixed time
step δt = 0.02, while the bottom one deals with the fixed m = 150 and different δt. Similar
to the first example (fermionic domain wall), here two regimes in the time evolution of
the density deviation can also be identified. These regimes are separated by the runaway
time t R , which increases with growing m or δt. There are several differences, too: the
truncation error starts to accumulate from the beginning (actually it is already present
in the initial state) for t < t R (Suzuki-Trotter error dominated regime) one sees the weak
m-dependence in the density deviation (upper plot of Fig. 3.12), which might complicate
state.

3.5. Accuracy of adaptive time-dependent DMRG 103
1e-02
L=100
Err max
(n i
,t), ERR tr
(t)
1e-04
1e-06
Err max
(n i
,t)
ERR tr
(t)
m=50 5
m=75 5
1e-08
m=100
m=150
5
5
δt=0.02
m=200 5
m=300 5
runaway time, t R
1e-10
0 1e-02
10 20 30 40
Err max
(n i
,t), ERR tr
(t)
1e-04
1e-06
Err max
(n i
,t)
ERR tr
(t)
δt=0.2 δ
δt=0.1 δ
1e-08
δt=0.05
δt=0.02
δ
δ
m=150
δt=0.01 δ
δt=0.005 δ
runaway time, t R
1e-10
0 10 20
time (t)
30 40
Figure 3.12: Density deviation Err max (n i , t) (bold lines) and the corresponding upper
bound of the DMRG truncation error ERR tr (t) (thin lines) vs. time t. The upper
plot corresponds to different numbers of kept states m and a fixed time step
δt = 0.02, whereas the bottom one is for fixed m = 150 and different δt. Two regimes
in Err max (n i , t) are separated by a “runaway” time t R (circles). Results are for the time
evolution of a single particle added at position i = 51 at time t = 0 to the ground state
of the system of N e = 50 fermions on an open L = 100 chain. The time-evolution simulations
were performed by adaptive t-DMRG based on S-T(2). The hopping amplitude
is J = 0.5.

104 3. Real-time evolution using DMRG
the t R detection; for t > t R (accumulated truncation error dominated regime) the density
deviation saturates to a rather small number, therefore the results remain reliable even
after the runaway time (compare the density deviations for δt < 0.05 with δt = 0.05 on the
bottom plot of Fig. 3.12). Actually it is hard to segregate regimes for δt 0.02, because
for t > t R the accumulated truncation error is of the same order as the Suzuki-Trotter one.
Note also, that Err max (n i , t) does not increase linearly in t as it was the case for the timeevolution
of the domain wall, but it remains roughly unchanged in time. Its counterpart
Err vec (n i , t), not shown on the plot, still grows almost linearly in t.
The upper bound of the DMRG truncation error (ERR tr (t), thin lines in Fig. 3.12)
is already large at t = δt. As expected it increases when m is reduced. However, it has
approximately the same power law in t for different m and equal δt (see the upper plot in
Fig. 3.12). Surprisingly, ERR tr (t) and its growth rate increase with δt, even though the
number of Suzuki-Trotter time steps and consequently the number of DMRG iterations
required to reach the time t decreases linearly in δt. This is a manifestation of a better
performance of the local Hilbert space updates for decreasing δt and it is the reason for
more accurate results for a smaller δt, even in the accumulated truncation error dominated
regimes. Finally, the overall growth rate of ERR tr (t) is considerably lower in comparison
to the example of the domain wall (see Fig. 3.5). The upper bound of the DMRG truncation
error overestimates heavily the truncation error present in the on-site density, but
it captures the essence of the accumulation of the truncation errors and can be used as a
supplementarary indicator.
The above observations are supported with the data shown in Fig. 3.13. In Fig. 3.13
we plot the density deviations, Err max (n i , t) and Err vec (n i , t), vs. the time step δt for times
t = 4 (left panel) and t = 40 (right panel). The number of kept states is fixed at m = 150.
For δt > 0.02 the density deviations are quadratic in δt for both considered times and the
error in the on-site density is dominated by the Suzuki-Trotter error. For δt < 0.02, they
deviate from quadratic in δt even at small time t = 4 (left panel of Fig. 3.13) and the
error is dominated by the accumulated truncation one. For δt = 0.02 and δt = 0.05, for
t = 40 both errors, the Suzuki-Trotter and the accumulated truncation error, are of the
same order (right panel of Fig. 3.13). Furthermore, the comparison of Err max (n i , t) with
Err vec (n i , t) reveals that the error in the on-site density is concentrated on a few sites for
small times t = 4 (left panel of Fig. 3.13), while it starts to spread over the system for large
times t = 40 (right panel of Fig. 3.13), especially for those δt for which the accumulated
truncation error dominates over the Suzuki-Trotter one.
In the following we analyze the accuracy of the method for the case when the runs are
performed with fixed discarded-weight threshold ε. Fig. 3.14 shows the density deviation
Err max (n i , t) (bold lines) and the corresponding upper bound of the DMRG truncation
error ERR tr (t) (thin lines), for different discarded-weight thresholds ε and two different
Suzuki-Trotter time steps δt = 0.02 and δt = 0.005. ERR tr (t) becomes significant at the

3.5. Accuracy of adaptive time-dependent DMRG 105
1e-03
1e-04
m=150, t=4.0
m=150, t=40.0
1e-05
1e-06
1e-07
Err max
(n i
,t)
Err vec
(n i
,t)
0.005 0.01 0.02 0.05 0.1 0.2 0.5
δt
0.01 0.02 0.05 0.1 0.2 0.5
δt
Figure 3.13: Density deviations Err max (n i , t) (squares) and Err vec (n i , t) (circles)
vs. time step δt, at times t = 4 (left panel) and t = 40 (right panel). Lines correspond
to the quadratic fit.
first Suzuki-Trotter time step and grows considerably in the following few tens. Later,
starting from t ≈ 5, it grows approximately as √ 1 − (1 − ε) 2γ′ t 2 /δt
, with ε-dependent γ ′ .
This kind of behavior is supported by the time evolution of the bipartition entanglement
entropy (S vN (i, t)), presented in Fig. 3.16. At t = 0 there is a small dip in S vN (i, 0) at
i = (50, 51), which fills up in a few ten time steps. Clearly a cone-like onset on the initial
values of the bipartition entanglement entropy can also be identified, which explains the
exponent proportional to t 2 . Similar to the simulations with the fixed number of kept states,
two regimes in density deviation can still be identified (bold lines in Fig. 3.14), although
the separation between the regimes is less pronounced. In the first regime Err max (n i , t) is
dominated by the Suzuki-Trotter error, while in the second one the accumulated truncation
error is larger. The obtained on-site density remains reliable even in the accumulated
truncation error dominated regimes.
For the current example (particle propagation in the background of the free spinless
fermions), the upper bound of the DMRG truncation error (ERR tr (t)), overestimates heavily
the error in the on-site density caused by the truncations of the state spaces. This
massive overestimation is present in both type of adaptive t-DMRG runs: with the fixed
number of kept states (see Fig. 3.12) or with the fixed discarded-weight threshold (see
Fig. 3.14). Nonetheless, ERR tr (t) captures qualitatively the progression of the accumulated
truncation error. It also gives an adequate alignment of the density deviations obtained
with different control parameter sets as well as with different type of adaptive t-DMRG

106 3. Real-time evolution using DMRG
1e-02
L=100
Err max
(n i
,t), ERR tr
(t)
1e-04
1e-06
1e-08
Err max
(n i
,t)
ε=1e-10, δt=0.02
ε=1e-10, δt=0.005
ε=1e-12, δt=0.02
ε=1e-12, δt=0.005
ε=1e-14, δt=0.02
ε=1e-14, δt=0.005
ERR tr
(t)
5,δ
5,δ
5,δ
5,δ
5,δ
t,5δ
1e-10
0 10 20 30 40
time (t)
Figure 3.14: Density deviation Err max (n i , t) (bold lines) and the corresponding upper
bound of the DMRG truncation error ERR tr (t) (thin lines) vs. time for different
discarded-weight thresholds ε and two different time steps δt = 0.02 and δt = 0.005.
Results are for the time evolution of a single particle added at position i = 51 at time
t = 0 to the ground state of the system of N e = 50 fermions on an open L = 100
chain. The time-evolution simulations were performed by adaptive t-DMRG based on
S-T(2). The hopping amplitude is J = 0.5.
simulations. Further analysis involving non local observables are required in order to check
whether the error in the whole state is well estimated with ERR tr (t).
What is more interesting is the behavior of the maximum number of kept states M(t)
(3.55). Fig. 3.15 shows M(t) corresponding to the above discussed simulations. As one can
see there is a region at the beginning of the time evolution for which M(t) first drops quickly
and then continues to decrease slowly. The reason for this is the scheme of targeting both
the ground state |ψ GS 〉 and the initial states |ψ(t = 0)〉 during the finite DMRG sweeps.
During the time evolution sweeps only the latter is maintained, for which smaller numbers
of kept states are required for the same ε. In the mentioned time interval, the M(t)
corresponding to the same ε and different δt remain equal. Later, M(t) corresponding
to a larger δt = 0.02 starts to grow, while for δt = 0.005 it remains almost unchanged.
Nevertheless, the on-site densities obtained for δt = 0.005 are not worse than for δt = 0.02
and the same ε. They are even better when the counterpart is dominated by the Suzuki-
Trotter error (compare δt = 0.005 and δ = 0.02 for ε = 10 −14 in Fig. 3.14). This manifests

3.5. Accuracy of adaptive time-dependent DMRG 107
M(t) = max[m(i)] for i∈]N(t-δt),N(t)]
450
400
350
300
250
200
150
100
50
ε=1e-14, δt=0.02
ε=1e-14, δt=0.005
ε=1e-12, δt=0.02
ε=1e-12, δt=0.005
ε=1e-10, δt=0.02
ε=1e-10, δt=0.005
L=100
0
0 10 20 30 40
time (t)
Figure 3.15: The maximum number of kept states within a time-step sweep, M(t),
as a function of time t, for different discarded-weight thresholds ε and two different
time steps δt = 0.02 (bold lines) and δt = 0.005 (thin lines). From top to bottom ε
increases. Results corespond to the simulations plotted in Fig. 3.14.
the efficiency of the state-space adjustment process for decreasing time step δt. Another
important observation based on Fig. 3.15 is the saturation of M(t) as opposed to the
exponential growth in the example of the time evolution of the fermionic domain wall.
The von Neumann entropy of the bipartition entanglement, displayed in Fig. 3.10,
shows that the maximum of S vN (i, t) in the middle of the chain grows quickly from the
beginning and at t ≈ 8 starts to saturate. The perturbation (particle added to the ground
state of the system at the i-th site at t = 0) propagates trough the chain occupying a
region that increases linearly in t before reaching the boundaries (see also Fig. A.4d).
The entanglement produced during this evolution is adding to the entanglement that was
already present in the ground state. Every bipartition entanglement remains unchanged
until the arrival of the wave front (the cone-like onset in Fig. 3.16, see also Fig. 3.21). The
saturation of S vN (i, t), coinciding with the fact that the fronts of the density wave are far
from the partition place, and the production of bipartition entanglement at that place,
is stopped. This saturation of the entanglement entropy as well as the saturation of the
maximum number of kept states indicates, that the time evolution of the state obtained
from the ground state by a local perturbation at time t = 0 can be accurately carried out
for large times (see also Ref. [194]).

108 3. Real-time evolution using DMRG
Figure 3.16: Von Neumann entropy of the bipartition entanglement S vN (i, t) as a
function of time t and cutting bond index i. Time increases from top to bottom.
Results are for the time evolution of a single particle added at position i = 51 at time
t = 0 to the ground state of the system of N e = 50 fermions on an open L = 100
chain and were obtained by adaptive t-DMRG based on S-T(2) with the discardedweight
threshold ε = 10 −16 and the time step δt = 0.005. The hopping amplitude is
J = 0.5.
The 4th order Suzuki-Trotter decomposition
We close the present subsection with the accuracy analysis of the adaptive t-DMRG based
on 4th order Suzuki-Trotter approximation of the time-evolution operator. In order to do
this we consider the same examples: the domain wall dynamics and the time evolution of
a single particle. Recall, that (see Section 3.5.1.1) the error produced by the p-th order
Suzuki-Trotter decomposition alone, is proportional to the p-th power of the time-step (see
Eq. (3.44)). Therefore, here we compare the results obtained by the adaptive t-DMRG
using the 4th order approximant (3.18) and time step δt = 0.1 with the results obtained
using the 2nd order approximant (3.14) and δt = 0.01.
We start with the first example and show in Fig. 3.17 the density deviation Err max (n i , t)
(bold lines) with the corresponding upper bound of the DMRG truncation error ERR tr (t)
vs. time t, for two different numbers of kept states m = 60 and m = 200. As one can see,
there is no big change in the behavior of ERR tr (t) when the 4th order method is used in
place of the 2nd order one. From the analysis performed for the 2nd order method, we

3.5. Accuracy of adaptive time-dependent DMRG 109
Err max
(n i
,t), ERR tr
(t)
1e+00
1e-02
1e-04
1e-06
1e-08
Err max
(n i
,t)
(STr 2, δt=0.01, m=60)
( " , m=200)
(STr 4, δt=0.1, m=60)
( " , m=200)
L=100
ERR tr
(t)
()
()
()
()
1e-10
0 10 20 30 40
time (t)
Figure 3.17: Density deviation Err max (n i , t) (bold lines) and the corresponding upper
bound of the DMRG truncation error ERR tr (t) (thin lines) as a function of time t,
for two different numbers of kept states m = 60 and m = 200. Results are for the
time evolution of a fermionic domain wall of size 50 residing in the left half of the
open L = 100 chain. The time-evolution simulations where performed by adaptive t-
DMRG based on S-T(2) with δt = 0.01 and by adaptive t-DMRG based on S-T(4) with
δt = 0.1. The hopping amplitude is J = 0.5.
know that the state-space adjustment works less efficient for larger time steps. The same
holds for the 4th order method. Initially, the upper bound of the DMRG truncation error
is bigger for the 4th order method, due to the larger time step. At larger times, this order
is switched (at t ≈ 18 for m = 60 and at t ≈ 27 for m = 200) and as anticipated ERR tr (t)
becomes smaller in the case of the 4th order method as compared to the 2nd order one
(the total number of iterations and consequently the number of state-space truncations
required to reach a given time t for the 4th order method is 11t/0.1 and hence smaller than
3t/0.01 needed for the 2nd order one). This behavior is reflected in the time evolution
of the density deviations (Err max (n i , t)), when the accumulated truncation error starts to
dominate the error in the on-site density.
Interestingly, there is an additional two orders of magnitude reduction of the density
deviation in the Suzuki-Trotter error “dominated regime”, achieved using the 4th order
approximant instead of the 2nd order one (compare solid and dashed thick-lines with
dash-dot-dot and dash-dash-dot lines for t < 10). This implies that the proportionality

110 3. Real-time evolution using DMRG
coefficient c(p) (see Eq. (3.44)) for the considered 4th order approximant (3.18) is two
orders of magnitude smaller than for the 2nd order one, because δt p is the same (δt = 0.1
for p = 4 and δt = 0.01 for p = 2). Unfortunately, this extra gain in accuracy disappears as
soon as one enters the regime where the density deviation is dominated by the accumulated
truncation error. Nonetheless, Err max (n i , t) remains smaller for the 4th order method in
comparison with the 2nd order one for the considered time steps and equal numbers of
kept states m.
A similar behavior is encountered in the second example, the time-evolution of a single
particle added to the ground state of the free fermion system. The density deviation
Err max (n i , t) and the corresponding upper bound of the DMRG truncation error ERR tr (t)
are shown in Fig. 3.18. The top plot in Fig. 3.18 coresponds to the adaptive t-DMRG
runs with a fixed number of kept states m. In the regime where the Suzuki-Trotter error
dominates the density deviation, an additional reduction of two orders of magnitude is
achieved using the 4th order approximant instead of the 2nd order one. However, for
m = 150, both methods deliver results of similar quality, when both simulations enter
the regime where the DMRG truncation error becomes dominant in the density deviation
(t > 14). This behavior is reflected in the time evolution of the upper bound of the DMRG
truncation error (check dash and dash-dot-dot thin lines in the top plot of Fig. 3.18).
The situation does not change substantially when the adaptive t-DMRG runs are performed
with a fixed discarded-weight threshold ε, see the bottom plot of Fig. 3.18. The
growth of the upper bound of the DMRG truncation error is more moderate as compared
to the runs with a fixed number of kept states (compare thin lines of the top and bottom
plots in Fig. 3.18). As expected, the time-evolution of the density deviation reveals that
the accumulated truncation error is larger and grows faster for the 2nd order method than
for the 4th order one. The error due to the 4th order Suzuki-Trotter approximation of
the time-evolution operator is so small that we had to use the discarded-weight threshold
ε = 10 −16 in order to obtain a time interval of length ≈ 6 where the density deviation is
dominated by the Suzuki-Trotter error. The maximum number of kept states within a
time-step sweep (not shown here) increases faster for the 4th order method due to a larger
time step and the less efficient state-space adaption caused by it (see the accuracy analysis
for the 2nd order method). This space adjustment, however, works still better in the case
of the 4th order method as compared to the 2nd order one with the same time step size
(δt = 0.1 in this case).
To summarize, the efficiency of the adaptive t-DMRG method can be improved, without
reduction of its accuracy, substituting the 2nd order Suzuki-Trotter decomposition (3.14)
of the time-evolution operator with the 4th order one (3.18). This can be achieved, if
the Suzuki-Trotter error threshold as well as the number of half-sweeps required to reach
time t are smaller for the 4th order method than for the 2nd order one. This implies
that 0.01(δt 4 ) 4 < (δt 2 ) 2 and 11t/δt 4 > 3t/δt 2 , where δt 4 and δt 2 are the time steps for

3.5. Accuracy of adaptive time-dependent DMRG 111
Err max
(n i
,t), ERR tr
(t)
1e+00
1e-02
1e-04
1e-06
1e-08
L=100
Err max
(n i
,t)
(S-T(2), δt=0.01, m=150)
( " , m=300)
(S-T(4), δt=0.1, m=150)
( " , m=300)
ERR tr
(t)
()
()
()
()
Err max
(n i
,t), ERR tr
(t)
1e-10
0 10 20 30 40
1e+00
Err max
(n i
,t)
ERR tr
(t)
(S-T(2), δt=0.01, ε=1e-12) ()
( " , ε=1e-16) ()
1e-02
(S-T(4), δt=0.1, ε=1e-12) ()
( " , ε=1e-16) ()
1e-04
1e-06
1e-08
1e-10
0 10 20 30 40
time (t)
Figure 3.18: Density deviation Err max (n i , t) (bold lines) and the corresponding upper
bound of the DMRG truncation error ERR tr (t) (thin lines) vs. time t for two different
numbers of kept states m = 150 and m = 300 (top plot) and two different discardedweight
thresholds ε = 10 −12 and ε = 10 −16 (bottom plot). Results are for the time
evolution of a single particle added at position i = 51 at time t = 0 to the ground state
of the system of N e = 50 fermions on an open L = 100 chain. The time-evolution
simulations were performed by adaptive t-DMRG based on S-T(2) with δt = 0.01 and
by adaptive t-DMRG based on S-T(4) with δt = 0.1. The hopping amplitude is J = 0.5.

112 3. Real-time evolution using DMRG
the 4th and the 2nd order methods, respectively. When these criteria are fullfilled an
improvement is achieved for the time-evolution simulations with a fixed number of kept
states, but the situation is more subtle for the time-evolution runs with a fixed discardedweight
threshold. For the latter case the maximum number of kept states within a time-step
sweep can increase substantially for larger time steps, causing a strong slow down or even
crash of the simulations already at earlier times.
We would also like to point out that the second parameter set for the 4th order Suzuki-
Trotter decomposition of the time-evolution operator (see page 74, Section 3.3) gave essentially
the same results.
Conclusions
To summarize, during the time evolution simulations using the adaptive t-DMRG method
based on the Suzuki-Trotter approximation of the time evolution operator there are two
major error sources: the truncation error due to the errors in the representations of the
targeted states occurring when the block states with small but finite weights are discarded
(DMRG state-space reduction) and the algorithmic error due to the Suzuki-Trotter approximation
of the time-evolution operator. The truncation error is accumulating during
the time-step sweeps — from iteration to iteration — but can be directly controlled using
the number of kept states m or the discarded-weight threshold ε. This error can be
reduced increasing m or decreasing ε, but since the required computational resources —
computer memory (∝ m 2 ) as well as CPU time (∝ m 3 ) — increase some balance has to
be found (which depends on the problem studied). The Suzuki-Trotter error, which does
not depend on m and ε, can be directly controlled by the time step size δt and the order
of the used Suzuki-Trotter approximant. Decreasing δt and increasing the order of the
approximation reduces this error, but the computational time required to reach a given t
grows. Since both error sources are independent and have different dynamics in time (see
Section 3.5.1.1) there can be regimes where one is “dominating” the other.
One can now answer the questions asked at the beginning of this subsection on page 97:
• The concept of a runaway time separating regimes (dominated by one or another error
source) in the error development seems to be only useful when the time-evolution
runs are carried out using a fixed number of kept states. Furthermore, the second
considered example indicated that even in this case one can have situations when
both sources produce errors of the same order within the studied time interval.
• Reliable results can also be obtained in the regimes were the accumulated truncation
error is larger than the Suzuki-Trotter error, especially when time-evolution sweeps
are performed with fixed discarded weight threshold.

3.5. Accuracy of adaptive time-dependent DMRG 113
• The upper bound of the DMRG truncation error (ERR tr (t)) overestimates substantially
the accumulated truncation error in the local observables, but it captures well
its time development. Therefore, it can be used as a good indicator for the error
caused by the truncations of blocks-state spaces.
The studied examples reveals that a better strategy is to perform the time-evolution
simulations with a fixed discarded weight threshold. In general, in this type of simulations
one has a better control over the accumulated truncation error and the simulations are
faster, since a small m is needed to reach a small error in the observables and the maximum
value M(t) is only acquired at a few iterations of the time-step sweeps at the beginning
of the time evolution. However, the maximum number of kept states M(t) can grow
polynomial in t (the first example, see Fig. 3.9) or even exponentially in t (see Section 3.6)
causing the crash of the simulation at some point. Therefore, the best approach will be
to perform runs with fixed ε in a fixed range for m. Furthermore, since the upper error
bounds (see Section 3.5.1.1) can only be used as indicators for the time development of the
errors of one or another type, a precise convergence analysis in δt and m or ε (depending on
the way the simulations are performed) seems to be more suitable for obtaining accurate
results. In this work all time-evolution simulations using the adaptive t-DMRG based on
the Suzuki-Trotter approximation are performed with the above mentioned strategy for
two different time steps δt = 0.02 and δt = 0.01 in units of the inverse hopping amplitude.
3.5.2 t-DMRG based on the Arnoldi method for the
time-evolution
In this subsection we analyze the accuracy of the time-step targeting adaptive t-DMRG
based on the Arnoldi approximation of the time-evolution operator. Within this method
all four error sources (see page 85) are present in the simulations. Two of them can be
controlled directly, while for the other two, only implicit control parameters are available.
Let us briefly specify these error sources together with their control parameters following
the order on page 85.
The so called algorithmic error, caused by the Arnoldi approximation of the timeevolution
operator, can be controlled directly at every DMRG iteration. This error can
be reduced and made irrelevant in comparison with the other errors by increasing the
dimension of the Krylov subspace and making use of a relatively cheap runtime errorestimation,
available for this approximation (3.25). By adjusting the time-step size one
can keep the dimension of the Krylov subspace manageable.
Despite the fact that the error due to the Arnoldi approximation of the time-evolution
operator can be made negligible small, it is not quite clear how accurately the DMRG
effective basis represents the “true” time-evolution operator. This representation can be

114 3. Real-time evolution using DMRG
improved by increasing the number of targeted intermediate states or by reducing the time
step size. Unfortunately, there is no direct feedback upon the improvements made and it
is hard or even impossible to estimate the error produced by this source. Moreover, as
discussed below, both improvement strategies will also increase the truncation error in the
target states.
As already discussed, the truncation error will appear in all DMRG calculations, as soon
as parts of the targeted state are discarded. It can be directly controlled by adjusting the
number of retained density-matrix eigenstates in each iteration. In contrast to finite-system
DMRG, where the target states are reobtained at every iteration, here the truncation error
starts to accumulate from step to step. It can be controlled by reducing the overall number
of sweeps required to reach the time t. This is achieved by reducing the number of sweeps
inside a single time step or by increasing the time-step length.
More subtle is the question about the optimality of the block state-space basis. In
finite-system DMRG usually several sweeps are performed and optimal effective bases are
obtained. However, extra sweeps deteriorate the initial state within the time-step sweeps
(in this method it is the only state which is not reobtained during the time-step sweeps).
As one can see, varying each control parameter has different, sometimes opposite, effects
on errors produced by different error sources. It has also a different impact on the
computing resources, time and the amount of memory used.
We directly move to the numerical analysis of the accuracy of the method, since an
analytical estimate of all errors (e.g., errors caused by an inadequate representation of the
time-evolution operator and a non-optimal block state basis) is not possible.
We consider an example of the time evolution of the particle added to the ground state
of the system. We use the same parameters as for the previous method: chain length
L = 100, hopping amplitude J = 0.5; a particle is created in the ground state of N e = 50
spinless fermions on site i = 51 at time t = 0. The time is measured in units of 1/J.
Additionally, we restrict ourselves to only two intermediate states, |ψ(t + δt/3)〉 and
|ψ(t + 2δt/3)〉, targeted together with the initial |ψ(t)〉 and the final |ψ(t + δt)〉 states
during a δt time-step sweep. Intermediate states are weighted with 1/6 each, whereas the
initial and the final states with 1/3 each. To avoid substantial deterioration of the initial
state, only one complete finite-system sweep is performed before advancing in time.
We start the analysis of the accuracy of the present method with the case when timestep
sweeps are performed with a fixed number of kept states m. In Fig. 3.19 we plot
the density deviation Err max (n i , t) vs. time t for the three different numbers of kept states
m = 200, 400, 800 and the three different time steps δt = 0.5, 1.0, 2.0. As one can see, for
m = 200 (black lines), density deviations corresponding to different time steps δt behave
similarly. They are dominated by the accumulated truncation error, which, as expected, is
larger for a smaller time step. Increasing the number of kept states to m = 400 (cyan lines),
the density deviation decreases substantially. Results corresponding to different time steps

3.5. Accuracy of adaptive time-dependent DMRG 115
1e-03
L=100
1e-05
Err max
(n i
,t)
1e-07
1e-09
1e-11
m=200, δt=0.5
" , δt=1
" , δt=2
m=400, δt=0.5
" , δt=1
" , δt=2
m=800, δt=0.5
" , δt=1
" , δt=2
0 20 40 60 80
time (t)
Figure 3.19: Density deviation Err max (n i , t) as a function of time t for different numbers
of kept states m and different time steps δt. Results are for the time evolution of a
single particle added at position i = 51 at time t = 0 to the ground state of the system
of N e = 50 fermions on an open L = 100 chain. The time-evolution simulations were
performed by time-step targeting adaptive t-DMRG with the Arnoldi approximation of
the time evolution operator.
behave more or less similarly and they are analogous to the m = 200 case. However, at
t ≈ 28, a hump suddenly appears in the density deviation for the largest (among the considered)
time step δt = 2.0 and the on-site density becomes slightly inaccurate in comparison
with δt = 0.5. Retaining m = 800 reduced density-matrix eigenstates (red lines), we actually
create conditions where the accumulated truncation error is not the only major player.
The density deviations at the beginning remain of the same order (even decrease) as for
the initial state. The most accurate results now correspond to the δt = 0.5 case, where the
error in the on-site density remains below 10 −7 in the entire plotted time interval t ∈ [0, 80].
A hump in the density deviation also appears for δt = 1.0 at t ≈ 24; at t ≈ 30 the error
in the on-site density becomes larger than for δt = 0.5. Particularly interesting is the case
when δt = 2.0. At t ≈ 20, Err max (n i , t) suddenly starts to grow appreciably and around
t ≈ 42 it becomes of the same order as for the m = 400 case. Performing two finite-system
DMRG sweeps instead of one has only a minor influence on the density deviation in this
case. Therefore, a non-optimal representation of the time-evolved states is ruled out. This
rapid loss of accuracy is connected to an inadequate representation of the time-evolution

116 3. Real-time evolution using DMRG
operator in the effective DMRG basis. This case clearly illustrates that by increasing the
time step δt one reduces the accumulated truncation error, but at the same time one runs
into the danger that the relevant parts of the time-evolution operator are not captured by
the DMRG effective basis. Targeting more intermediate states can fix the latter problem,
but simultaneously one increases the truncation error, since more targeted states have to
be accommodated by the effective state-space having the same size.
We proceed with the analysis of the method’s accuracy when the time-step sweeps
are performed with a fixed discarded-weight threshold. In Fig. 3.20 the density deviation
Err max (n i , t) (top plot) and the corresponding maximum number of retained density-matrix
eigenstates M(t) within the time-step sweep (bottom plot) are shown as a function of
time t for three different time steps δt = 0.5, 1.0, 2.0 and three different discarded-weight
thresholds ε = 10 −10 , 10 −12 , 10 −14 . One can clearly identify that decreasing the discardedweight
threshold considerably reduces the density deviation for the same time step δt.
Before one looses the ability of the DMRG effective basis to adequately represent the
time-evolution operator, the results obtained for the largest considered time step δt = 2.0
(dot-dashed lines) are significantly more accurate than those for δt = 0.5 (solid lines). This
indicates the larger accumulated truncation error for a smaller time step. The relatively
large difference between the density deviations for the same ε and different δt at relatively
small time scales can be explained by the larger maximum number of kept states M(t)
within the time-step sweep (see the bottom plot in Fig. 3.20) — e.g., 500 for δt = 2.0
vs. 350 for δt = 0.5 for ε = 10 −12 at t = 2.0. The time-evolved on-site densities are as
accurate as the initial ones, for all the considered values of ε. At later times, the errors due
to the inadequate representation of the time-evolution operator become significant (earlier
for smaller ε), and the density deviations for δt = 2.0 and ε = 10 −14 and ε = 10 −12 become
of the same order at t ≈ 30; at t ≈ 60 the error is of the same order as for ε = 10 −10 .
M(t) for δt = 2.0 grows slowly and saturates at t ≈ 15 (dot-dashed line on the bottom
plot in Fig. 3.20). For ε = 10 −14 it starts to grow once again at a considerably higher
rate, but the accuracy of the representation of the time-evolution operator is not recovered
and Err max (n i , t) keeps increasing faster. This indicates that the plain enlargement of the
effective state space does not capture parts of the Hamiltonian which are relevant for the
time evolution. For δt = 0.5, M(t) increases most rapidly causing the breakdown of the
simulation. However, the accumulated truncation error remains largest for this time step in
comparison with the other time steps considered. Despite the errors due to an inadequate
representation of the time-evolution operator appearing for δt = 1 (dashed lines), M(t)
remains moderately large and the total error remains acceptable for δt = 1.0.
It remains to verify, if targeting more intermediate states solves the problem of the
accurate representation of the time-evolution operator, without a substantial increase of
M(t) or truncation errors. However, this analysis is beyond the scope of this thesis.
To summarize, by increasing the number of kept states m or by decreasing the discarded-

3.5. Accuracy of adaptive time-dependent DMRG 117
1e-03
L=100
1e-05
Err max
(n i
,t)
M(t) = max[m(i)] for i∈]N(t-δt),N(t)]
1e-07
ε=1e-10, δt=0.5
" , δt=1
" , δt=2
1e-09
ε=1e-12, δt=0.5
" , δt=1
" , δt=2
ε=1e-14, δt=0.5
" , δt=1
" , δt=2
1e-11
0 20 40 60 80
3000
2500
2000
1500
1000
500
ε=1e-10, δt=0.5
" , δt=1
" , δt=2
ε=1e-12, δt=0.5
" , δt=1
" , δt=2
ε=1e-14, δt=0.5
" , δt=1
" , δt=2
0
0 20 40 60 80
time (t)
Figure 3.20: Density deviation Err max (n i , t) (upper plot) and corresponding maximum
number of kept states within a time-step sweep, M(t) (bottom plot) as a function of
time t for different discarded-weight thresholds ε and different time steps δt. Results
are for the time evolution of a single particle added at position i = 51 at time t = 0
to the ground of the system of N e = 50 fermions on an open L = 100 chain. The
time-evolution simulations were performed by time-step targeting adaptive t-DMRG
with the Arnoldi approximation of the time evolution operator.

118 3. Real-time evolution using DMRG
weight threshold ε, the accuracy of the results can be increased. For an optimal representation
of the time-evolution operator and the targeted states, a systematic convergence of
all quantities with an increasing m or a decreasing ε can be realized. Enlarging the time
step in order to reduce the accumulated truncation error is not always a good idea. For
large time steps with only several intermediate target states, one starts to lose the ability
of the effective state space to capture the parts of the Hamiltonian, which are relevant for
the time evolution. This phenomenon is relatively hard to control, but it can be identified
as a sudden appreciable deviation of the values for the same observable, obtained with
relatively large but different numbers of kept states (or discarded-weight thresholds).
3.5.3 Comparison and summary
Comparing two different types of adaptive t-DMRG algorithms the following conclusions
can be made. At a first glance, the adaptive t-DMRG algorithm based on the Suzuki-
Trotter decomposition of the time-evolution operator (shortly Suzuki-Trotter t-DMRG)
seems to be preferable since it contains only two sources of error. But due to the large
error threshold of the 2nd order approximation of the time-evolution operator it turns out
that it compares poorly with the algorithm based on the Arnoldi method for the time
evolution (shortly Arnoldi t-DMRG). For example, the Suzuki-Trotter error for δt = 0.1 is
roughly two to four orders of magnitude larger (four at small times) than the accumulated
truncation error in the results obtained with Arnoldi t-DMRG using even larger time
steps δt = 0.5, 1.0, 2.0 (compare Fig. 3.12 with Fig. 3.19). Larger δt on the other hand
lead to a smaller time needed for the simulation. Using the fourth order Suzuki-Trotter
decomposition the error can be decreased by more than two orders of magnitude (see
Fig. 3.18), but it requires 11 DMRG half-sweeps against only 2 in the Arnoldi t-DMRG
(although one should admit that one half-sweep of the former algorithm is faster than one
half-sweep of the latter one). The truncation error alone is much smaller in the simulations
performed by Suzuki-Trotter t-DMRG than the one in the simulations performed with the
Arnoldi t-DMRG with the same δt and m. This is because the former targets only one
time-evolving state during the time-evolution simulation while the latter four of them.
However this disadvantage is compensated by a larger δt possible in the Arnoldi t-DMRG
making the truncation errors comparable.
An important advantage of the Suzuki-Trotter t-DMRG is the fact that all errors can be
controlled directly. The error due to Suzuki-Trotter approximation of the time evolution
operator (algorithmic error) is defined by the order of the approximation and the time
step size (3.44). The same holds for the Arnoldi t-DMRG, where the upper bound of the
error of the Arnoldi approximation of the time evolution operator is well defined (3.25)
and can be reduced below the given value during the simulations. The DMRG truncation
error can be calculated and controlled easily at every DMRG step in both considered

3.5. Accuracy of adaptive time-dependent DMRG 119
algorithms. The key difference between Suzuki-Trotter t-DMRG and Arnoldi t-DMRG lies
in the effective state-space readaption process. In the Suzuki-Trotter t-DMRG at every
iteration of the time-step sweep, the current left and right blocks state-spaces are locally
enlarged considering the sites connecting these blocks explicitly (two free sites in blocksite-site-block
configuration). Then the local time-evolution operator acting on these two
sites is applied exactly and a new reduced effective state-space basis is constructed for
the left- or right-block depending on the sweep direction. Because of this procedure the
effective block state-space basis is perfectly readapted at every DMRG step in order to
accurately represent the state resulting from the application of the local time-evolution
operator. If the time-step size is small enough, then the effective bases obtained after a
full time step are also globally optimal, representing accurately the Suzuki-Trotter timeevolved
state. This space readaption procedure is fundamentally different in the Arnoldi
t-DMRG algorithm. In this algorithm the effective state-space basis is readapted in order to
accurately represent the entire time step interval and not just one time-evolving state. This
is typically achieved by targeting so called initial, final, and possibly some intermediate (in
the considered cases two of them) time-evolved states. The final and all the intermediate
states are reobtained from the initial one at each iteration of the time-step finite-system
DMRG sweeps. Since only one defective basis is reconstructed per iteration, several or at
least one complete finite-system DMRG sweep are required to readapt all bases to optimally
represent the current time-step interval. However, since there is now an extra procedure
which allows to reobtain the initial state at each iteration, this sweeping only deteriorates
the accuracy of this state and consequently the accuracy of all time-evolved states obtained
from it. This is why additional sweeps do not increase but decrease the accuracy of the
algorithm/simulation.
There might be some extra complications during the time-evolution simulations with
Arnoldi t-DMRG, namely the optimality of the representation of the Hamiltonian driving
the time evolution in the readapting effective state-space basis (see Section 3.5.2). This is
particularly hard to control.
The problem of the accuracy deterioration and the optimality of the representations can
be overcome with an extra computational cost using a closely related algorithm, namely
the time evolution of the Matrix Product State variational ansatz (vMPS) [76]. There, the
time-evolved state can be obtained by applying the approximated time-evolution operator
to a state in a fixed basis. The approximation to the time-evolution operator has a well
controlled error that can be chosen smaller than the truncation error. The truncation can
then be carried out maximizing the overlap of the truncated state with the time-evolved
state leading to a controlled truncation error.
Due to the above considered advantages of the adaptive t-DMRG based on the Suzuki-
Trotter approximation of the time-evolution operator, we will prefer to use this algorithm
when possible.

120 3. Real-time evolution using DMRG
3.6 Conclusions
In this chapter we have described two different DMRG algorithms which can be used
to study the time evolution of pure states. We have also performed an extensive accuracy
analysis for each of them using exactly solvable nontrivial time-evolution examples
(Section 3.5). The first algorithm, the adaptive t-DMRG based on the Suzuki-Trotter
approximation of the time-evolution operator (Section 3.3), is quite efficient and requires
moderate computational resources. It also allows explicit control of the errors produced
during the time-evolution simulations, but it is limited to systems with nearest-neighbor interactions.
In other cases the time-step targeting adaptive t-DMRG based on the Arnoldi
approximation of the time-evolution operator (one of the Krylov-subspace methods, see
Section 3.4) can be used, which is as efficient as the previous algorithm, but is free of the
above mentioned limitation. With the devised algorithms one can simulate in principle the
time evolution under arbitrary Hamiltonians for the systems which are mapped onto the
one-dimensional chain.
The considered algorithms have been successfully used to study different time-evolution
setups in spin as well as in fermionic and bosonic systems. The time evolution of Gaussian
wave packets or the dynamics of the density perturbations of the Gaussian form have
been used to study nonequilibrium transport through small interacting nanostructures
[4, 106, 208], Andreev-like reflections [39] and the phenomenon of spin-charge separation
in one-dimensional interacting systems [139, 140, 142, 143, 239]. Computing the real-time
Green’s functions, it has been possible to investigate the spectral properties of several
one-dimensional systems [65, 67, 195, 258]. Both algorithms have been employed far from
equilibrium dynamics, and local and global quantum quenches [34, 84, 161, 162, 200].
Although the algorithms used are powerful, they are limited to a certain class of systems.
As for the static cases investigated with the conventional DMRG, the efficiency of the
present algorithms is also controlled by the amount of entanglement of the time-evolving
state. The difference in the time-dependent cases is an extra entanglement produced with
time. This either requires a continuous increase of the effective state-space dimensions —
exponential in the entanglement entropy — with time or produces an increasing truncation
error. Therefore, there will be an additional limitation to the time scales accessible with the
numerical simulations. For systems with local Hamiltonians an upper theoretical bound for
the entanglement entropy can be obtained. All assessments are based on the Lieb-Robinson
theorem [101, 153, 177] which states that information can propagate in a system only with
a finite group velocity specific to the studied problem. In other words, correlations beyond
a “light cone” of width 2c g t, where c g is a maximal group velocity, decay exponentially
fast. p The physical light cone and the spatial decay of correlations is the basis of a very
interesting algorithmic extension [102, 103]. There are mathematical physics methods
p Recall that in relativistic physics the light cone imposes a hard cut.

3.6. Conclusions 121
2.5
2
S vN
(i,t)
1.5
1
0.5
SP, i=50
SP, i=35
DW, i=50
DW, i=35
0
0 10 20 30 40
time (t)
Figure 3.21: Von Neumann entropy of the bipartition entanglement S vN (i, t) as a
function of time t and cutting bond index i. SP: time evolution of a single particle
added at position i = 51 and time t = 0 to the ground state of the system of N e = 50
fermions on an open L = 100 chain. DW: time evolution of a fermionic domain wall
of size 50 residing in the left half of the open L = 100 chain. Results were obtained
by adaptive t-DMRG based on S-T(2) with the discarded-weight thresholds ε = 10 −16
(SP) and ε = 10 −14 (DW), and the time step δt = 0.005. The hopping amplitude is
J = 0.5.
giving the general upper bounds on the entanglement growth in time [25, 51]. Using
conformal field theory, Calabrese and Cardy found [28, 29] (see also Ref. [52]) that the
entanglement entropy grows linearly with time for the so-called global quench (i.e., when
the initial state differs globally from the ground state and the excess of energy is extensive),
while the growth is at most logarithmic for the local quench (i.e., when the initial state
has only a local difference from the ground state and therefore a small excess energy).
A linear growth of the Von Neumann entropy of a block of spins has been obtained in
the t-DMRG simulation of the Heisenberg chain after a sudden quench in the anisotropy
parameter (global quench) [34]. The examples considered in this thesis exhibit at worst
a logarithmic growth of the entanglement entropy in time (see Fig. 3.21; for the “light
cone” structure see Fig. 3.16 and Fig. 3.10 ). The second example, the time evolution of
a particle added to the ground state of a spinless fermion system falls into the class of
local quenches, while the first one, the time-evolution of the fermionic domain wall can

122 3. Real-time evolution using DMRG
be categorized to both of them. It can be interpreted as two chains in their own ground
states, which are connected at t = 0 (local quench) or the domain wall (t < 0) created by
a step like potential which is switched off at t = 0 (global quench). Therefore, it is more
suitable to adopt the definitions by Hastings [102], namely: “logarithmic entropy growth
tends to occur in cases where we can divide the chain into a small number of subchains
such that the initial state is an eigenstate of the Hamiltonian on each subchain ...whereas
the linear entropy growth tends to occur in cases where the initial state differs from an
eigenstate of the Hamiltonian on every subsystem of the full chain”. As a consequence, a
local quench can be easily simulated by means of t-DMRG, while a global one is harder
and numerics must be limited to relatively small system sizes and times. However, this is
not too relevant if phenomena of interest occur within accessible time intervals.
In addition, the above discussed time-evolution algorithms can be generalized to study
systems at finite temperatures. One can carry out the evolution in imaginary time on
purified mixed states [184, 238]. The purification can be obtained by introducing an ancillary
site to each site of the real system, doubling the system size. An initial state,
corresponding to infinite temperature, is constructed by preparing pairs of the physical
and the corresponding auxiliary sites in a maximally entangled state. The density matrix
of the physical system is then obtained by tracing out the auxiliary system. The obtained
algorithm [17, 66] is neither limited to large temperatures nor to homogeneous systems and
opens the possibility of simulating finite temperature, dissipation and decoherence effects.

123
4. NATURE OF THE BAND- TO MOTT-INSULATOR
TRANSITION IN ONE-DIMENSION
In this chapter we investigate the ground-state phase diagrams of the one-dimensional
“ionic” Hubbard and the adiabatic Holstein-Hubbard models at half-filling. For this we
use numerical diagonalization of finite systems with the Lanczos and the density matrix
renormalization group (DMRG) methods (the latter was discussed in Chapter 2). In the
ionic Hubbard model, which we consider in Section 4.1, an insulator-insulator phase transition
is identified from a band to a correlated insulator with simultaneous charge and
bond-charge order. The transition point is characterized by the vanishing of the optical
excitation gap while simultaneously the charge and the spin gaps remain finite and equal.
There is strong evidence for a possible second transition into a Mott-insulator phase. In the
following section (Section 4.2), the results obtained for the ionic Hubbard model are used
in order to clarify the physics of the crossover from a Peierls band insulator to a correlated
Mott-Hubbard insulator in the adiabatic limit of the half-filled one-dimensional Holstein-
Hubbard model. This transition is connected to the band to Mott insulator transition of
the ionic Hubbard model. Depending on the strength of the electron-phonon coupling and
the Hubbard interaction the transition is either first order or evolves continuously across
an intermediate phase with finite spin, charge, and optical excitation gaps. Both sections
contain separate introductions and the obtained results are summarized in the final section.
4.1 Ionic Hubbard model
4.1.1 Introduction
For more than two decades the correlation induced metal-insulator transition and its characteristics
has been one of the challenging problems in condensed matter physics [121].
This metal-insulator transition is often accompanied by a symmetry breaking and the development
of long range order [149]. In one dimension this ordering can only be related
to the breaking of a discrete symmetry. Examples include commensurate charge density
waves (CDWs) and Peierls dimerization (bond-order wave, BOW) phenomena. In contrast,
the transition into the Mott insulating (MI) phase in one dimension is not connected with
the breaking of a discrete symmetry [155]. In the MI phase the gapped charge degrees of

124 4. Nature of the Band- to Mott-insulator transition in one-dimension
freedom are uniformly distributed in the system, while the gapless spin degrees of freedom
are described by an effective S = 1/2 Heisenberg chain [165].
Due to the different symmetries of the CDW, BOW, and MI phases it is natural to expect
that these phases are mutually exclusive. The extended Hubbard model at half-filling
with an on-site (U) and a nearest-neighbor (V ) Coulomb repulsion provides a prominent
example with a transition from a MI to a CDW insulator in the vicinity of the U = 2V line
in the phase diagram [53]. Remarkably, the transition at weak coupling may involve an
intermediate BOW phase [180, 181, 216]. A similar phase-diagram structure was recently
also discovered for the Holstein-Hubbard model [63]. The tendency towards BOW order is
even more profound for the Hubbard model with an explicit bond-charge coupling where
the CDW and MI phases are often separated by a long range ordered Peierls dimerized
phase [126, 127].
In recent years particular attention has been given to another example for an extension
of the Hubbard model which includes a staggered potential term [8, 57, 58, 81, 157, 178, 179,
199, 224, 232, 259]. The corresponding Hamiltonian has been named the “ionic Hubbard
model” (IHM); in one dimension it is given by
H = − t ∑ )
(1 + (−1) i δ)
(c † iσ c i+1σ + h.c.
i,σ
+ U ∑ i
n i↑
n i↓
+ ∆ 2
∑
(−1) i n iσ , (4.1)
i,σ
where c † iσ creates an electron on site i with spin σ and n iσ = c† iσ c iσ . ∆ is the potential
energy difference between neighboring sites, and δ a Peierls modulation of the hopping
amplitude t. In the limit ∆ = δ = 0, Eq. (4.1) reduces to the ordinary Hubbard model,
the limit ∆ = 0 and δ > 0 is called the Peierls-Hubbard model, and the limit ∆ > 0 and
δ = 0 is usually referred to as the IHM. In the following, we will focus mainly on the effect
of the on-site modulation ∆, so we implicitly assume δ = 0 except where stated otherwise.
The IHM was first proposed and discussed almost thirty years ago in the context of
organic mixed-stack charge-transfer crystals with alternating donor (D) and acceptor (A)
molecules (. . .D +ρ A −ρ . . .) [178, 179, 233, 234]. These stacks form quasi-1D insulating
chains, and at room temperature and ambient pressure are either mostly ionic (ρ ≈ 1)
or mostly neutral (ρ ≈ 0) [233, 234]. However, several systems undergo a reversible neutral
to ionic phase transition, i.e., a discontinuous jump in the ionicity ρ upon changing
temperature or pressure [173, 174, 226, 228, 229, 230]. Later the IHM has been used in
a similar context to describe the ferroelectric transition in perovskite materials such as
BaTiO 3 [50, 122] or KNbO 3 [182].
The very presence of at least one transition in the ground state phase diagram of the
half-filled IHM model is easily traced by starting from the atomic limit [81, 189]. For t = 0,
it is obvious that at U < ∆ the ground state of the IHM has two electrons on the odd sites,

4.1. Ionic Hubbard model 125
and no electrons on the even sites corresponding to CDW order with maximum amplitude.
On the other hand, for U > ∆ each site is occupied by one electron and the ground state
has infinite spin degeneracy. Thus, for t = 0 a transition occurs at a critical value U c = ∆.
This transition is expected to persist for finite hopping amplitudes t > 0.
A renewal of interest in the IHM started with the bosonization analysis of Fabrizio et
al. (FGN) [57, 58], where a two-transition scenario for the ground-state phase diagram
of the 1D IHM was proposed. The key features of the FGN theory are the presence of
an Ising-type transition from a CDW band-insulator phase at U < Uch c into a BOW phase
at Uch c < U < Uc sp, and a continuous Kosterlitz-Thouless like transition into a MI phase
at U > Usp c . In this scenario the charge gap vanishes only at U = Uc ch and the system
might be “metallic” at this point. The second transition at U = Usp c is connected with the
closing of the spin gap, which is finite for all U < Usp c and vanishes for U > Uc sp . Thus, the
bosonization phase diagram essentially supports the "exclusion principle" of the ground
states.
Later on various attempts based on numerical tools have been performed to verify the
FGN phase diagram for the half-filled IHM. In particular, exact diagonalization [81, 232],
valence bond techniques [8], quantum Monte Carlo [259], and DMRG [157, 224] were used.
Unfortunately, conflicting results have so far been reported in these studies regarding the
nature of the transition and the insulating phases, the possibility of two rather than one
critical point, or the appearance of BOW order.
Given the numerous unresolved issues we reinvestigate the ground-state properties of
the IHM using the exact diagonalization Lanczos technique and the DMRG method. We
verify the presence of at least one transition at a critical coupling U c (∆) from a bandinsulator
(BI) to a correlated insulator (CI) phase. On finite systems the transition originates
from a ground-state level crossing with a change of the site-parity eigenvalue, which
implies the vanishing of the optical excitation gap at U c . Our DMRG results show that
the spin and charge gaps remain nevertheless finite and equal at the transition. Above U c
the charge and spin gaps split, the charge gap increases, while the spin gap decreases and
we identify long-range BOW order with a spontaneous site-inversion symmetry breaking.
The existence of a second transition is not unambiguously resolved within the accuracy of
our DMRG data. Yet, the scaling of the BOW order parameter changes qualitatively with
increasing U indicative for a possible second smooth transition point where the spin gap
closes.
We show that at U < U c (∆) the CDW-band insulator phase is realized. In this phase
BOW and spin density wave (SDW) correlations are strongly suppressed, and the spin and
charge gaps are equal and finite. The characteristic feature of the ground-state phases for
U > U c (∆) is the coexistence of long range CDW order with either a long range BOW (see
Fig. 4.1) or algebraically decaying BOW and SDW correlations.

126 4. Nature of the Band- to Mott-insulator transition in one-dimension
Figure 4.1: Illustration of two degenerate CDW+BOW patterns (a) and (b). Different
size spheres represent different on-site charge density, while different size ellipses
correspond to different bond charge densities. In the case of open boundary conditions
(a) is more preferable and the degeneracy is lifted.
4.1.2 Symmetry analysis
A good starting point for understanding the existence of a phase transition in the IHM is
to study the symmetry of the model manifestly seen in the limiting cases U ≪ ∆, t and
U ≫ ∆, t. The IHM is invariant with respect to inversion at a site and translation by two
lattice sites. If we denote the site inversion operator by ˆP, defined through
ˆPc † ˆP iσ † = c † L−i,σ
, for i = 0, · · · , L − 1 , (4.2)
and ˆT j for a translation by j sites, then any nondegenerate eigenstate |ψ n 〉 of H must obey
ˆP |ψ n 〉 = ±|ψ n 〉 and ˆT 2 |ψ n 〉 = |ψ n 〉. Because [H, ˆT 1 ] ≠ 0, any non-degenerate eigenstate
|ψ n 〉 of H is not an eigenstate of ˆT 1 .
For the half-filled Hubbard model (∆ = δ = 0) the ground state has P = +1 only for
U = 0, and P = −1 for any U > 0 [81]. However, in the IHM the phase transition from a
renormalized BI to a CI occurs at some finite U c > 0. This suggests that the parity of the
ground state remains even not only for U = 0, but for all U < U c . At U c , a ground-state
level crossing occurs on finite chains, as confirmed by exact diagonalization studies (see
below), connected with a site-parity change.
For U = 0 the ground state at half-filling is a CDW-BI. The alternating potential defines
two sublattices, doubling the unit cell and opening up a band gap ∆ for U = 0 at k = ±π/2.
The elementary spectrum consists of particle-hole excitations over the band gap. The
charge (∆ C ) and spin (∆ S ) excitation gaps are equal: ∆ C = ∆ S = ∆. We consider a
system to be a BI when the criterion ∆ S = ∆ C holds, where the spin and the charge gaps

4.1. Ionic Hubbard model 127
are given by
∆ S = E 0 (N = L, S z = 1)
− E 0 (N = L, S z = 0) ,
∆ C = E 0 (N = L + 1, S z = 1/2)
+ E 0 (N = L − 1, S z = 1/2)
− 2E 0 (N = L, S z = 0) ,
(4.3a)
(4.3b)
respectively. E 0 (N, S z ) is the ground-state energy, L the system length, N the number of
electrons, and S z the z-component of the total spin. As we show below the BI phase is
realized in the ground state of the IHM at U < U c .
In the strong-coupling limit U ≫ ∆, t, the low-energy physics of the IHM is described
by the following effective Heisenberg spin model [178, 179]
H eff = J ∑ i
S i · S i+1 + J ′ ∑ i
S i · S i+2 . (4.4)
In Eq. (4.4) the exchange couplings are given by
[
]
J = 4t2 1
U 1 − x − 4t2 1 + 4x 2 − x 4 )
,
2 U 2 (1 − x 2 ) 3
J ′ = 4t4
U 3 (1 + 4x 2 − x 4 )
(1 − x 2 ) 3 , (4.5)
where x = ∆/U. This result (4.4) implies that in the strong-coupling limit of the IHM the
low-energy physics is qualitatively similar to that of the Hubbard model, with modified
exchange coupling constants J and J ′ . For next-nearest neighbor couplings J ′ < 0.24J
the spin gap vanishes [31, 97, 190]. The coupling constants (4.5) satisfy the condition
J ′ < 0.24J at least for U > 3.6t for ∆ ≤ t and U > 3.6∆ for ∆ > t.
The effective spin model (4.4) is invariant with respect to translations by one lattice
spacing, whereas the original IHM is invariant only with respect to translations by two
lattice spacings. However, the doubling of the unit cell is ensured due to the charge
degrees of freedom. For the standard Hubbard model at arbitrary U ≠ 0 the number of
doubly occupied sites D in the ground state is finite. The exact Bethe-ansatz solution tells
that D scales as (t/U) 2 in the strong coupling limit and is given by [30]
D = ∑ i
〈n i↑
n i↓
〉 ≃ NA(t/U) 2 [1 + O ( (t/U) 2) ] , (4.6)
where A = 4 ln2. Contrary to the Hubbard model, where doublons are equally distributed
on all sites of the system, the non-equivalence of sites in the IHM leads to different probabilities
for finding a doublon on even or odd sites. Since doublons are spin singlets, their

128 4. Nature of the Band- to Mott-insulator transition in one-dimension
distribution is not influenced by spin fluctuations. Since the energies of doublons on even
and odd sites differ in ∆ and assuming the scaling for the density of doublons as in Eq. (4.6),
one easily obtains for the amplitude of the ionicity induced CDW in the strong coupling
limit
[
]
1
N (D t 2 1
odd − D even ) ≃ A 1
U 2 (1 − x) − 1
2 (1 + x) 2
= 4A 1
t 2
U 2
x
(1 − x 2 ) 2[1 + O ( (t/U) 2) ] (4.7)
where A 1 is a constant of order unity. Thus, although the effective spin Hamiltonian has
a higher symmetry than the original model from which it was derived, the translational
symmetry of the IHM is recovered due to the long range CDW pattern arising from the
staggered doublon and holon distribution.
4.1.3 Exact diagonalization results
In order to explore the nature of the spectrum and the phase transition, we have diagonalized
numerically small systems by the Davidson method [40, 41, 202, 204] similarly to
earlier exact diagonalization calculations [81, 199]. The energies of the few lowest eigenstates
were obtained for finite chains with L = 4n and periodic boundary conditions or
L = 4n + 2 with antiperiodic boundary conditions.
We first analyze short chains; for chain lengths L ≤ 16 finite-size effects do not change
the qualitative behavior discussed below. In Fig. 4.2, the lowest eigenenergies of the IHM
for ∆ = 0.5t, L = 8 and periodic boundary condition are shown as a function of U. At
U = 1.3t, a level crossing of the two lowest eigenstates occurs. A non-degenerate eigenstate
of the IHM has a well defined site-parity, so a ground-state level-crossing transition
necessarily corresponds to a change of the site-parity eigenvalue.
For U = 0, the IHM is easily diagonalized in momentum space by introducing
fermionic creation operators d γ,†
kσ
with a band index γ = A, B for the lower and upper
bands, respectively, with the dispersion E A/B (k) = ∓ √ 4t 2 cos 2 (k) + (∆/4) 2 for momenta
−π/2 < k ≤ π/2. For U = 0 the first two degenerate excited states at half-filling always
have negative site parity, because the ground state has P = +1, and the operator d B,†
qσ d A qσ
with q = π/2 obeys
ˆPd B,†
qσ dA qσ = −dB,† qσ dA qσ ˆP . (4.8)
The first two excited states shown in Fig. 4.2 are the spin singlet (S = 0, S z = 0) and triplet

4.1. Ionic Hubbard model 129
Figure 4.2: Lowest-energy eigenvalues of the IHM at half-filling for L = 8 sites,
periodic boundary conditions, and ∆ = 0.5t.
excitations (S = 1, S z = 0), created from the ground state by applying the operators
1
(
)
√ d B,†
q↑ 2 dA q↑ − dB,† q↓ dA q↓
,
1
(
)
√ d B,†
q↑ 2 dA q↑ + dB,† q↓ dA q↓
, (4.9)
respectively. Thus both excited states have total momentum k tot = 0 and negative site
parity. For U > 0, these degenerate excited states split in energy. Exact diagonalization
of finite IHM rings therefore identifies one critical U c > 0, separating a BI with P = +1 at
U < U c from a CI with P = −1 for U > U c .
4.1.4 DMRG results
4.1.4.1 Excitation gaps
In order to access the transition scenario in the long chain-length limit, we have studied
chains up to L=512 using the DMRG method. The fact that the transition at U c is
connected to a change in inversion symmetry requires some caution when open boundary
conditions (OBC) are used in DMRG studies. For OBC and L = 2n the IHM is not
reflection symmetric at any site. Thus, the ground state does not have a well defined site
parity, and the level-crossing transition is absent. To overcome this problem, one might

130 4. Nature of the Band- to Mott-insulator transition in one-dimension
try to use chains with OBC and an odd number of sites L = 2n + 1, since the Hamiltonian
in this case is reflection symmetric with respect to the site i c in the center of the chain
(i c = (L − 1)/2), and a site inversion operator is well defined by
ˆPc † ˆP † iσ = c † L−1−i,σ
, i = 0, ..., L − 1 . (4.10)
To test whether this is an improved choice we have calculated the site parity of the ground
state for U = 0 analytically for different chain lengths L = 2n + 1 and found
ˆP |ψ 0 〉 = (−1) n |ψ 0 〉 . (4.11)
On the other hand, if one extends the idea of Gidopoulos et al. [81] for the determination
of the site parity to chains with L = 2n + 1 for U ≫ t, one obtains
ˆP |ψ 0 〉 = (−1) [P L−1
m=1 m] |ψ 0 〉 = (−1) n |ψ 0 〉 . (4.12)
Thus, the parity eigenvalue of the ground state is the same at U = 0 and U ≫ t for a
given chain length, and no level crossing occurs. Due to the fact that the sharp transition
at a well defined U c does not exist in the finite-chain results for OBC, the extrapolation
is a rather subtle problem, since a sharp transition feature has to be identified from the
extrapolation of smooth curves. This requires the use of quite long chains in the critical
region.
In Fig. 4.3 extrapolated results are shown for the spin and charge gaps ∆ S
and ∆ C , respectively. Calculations were performed with OBC for chains of lengths
L = {30, 40, 50, 60}, and additionally up to L = 512 in the transition region around the
estimated U c . We assume a scaling behavior of ∆ C and ∆ S of the form
∆ γ (L) = ∆ ∞ γ + A γ
L + B γ
L 2 , (4.13)
where γ ∈ {S, C}. The extrapolation for L → ∞ is then performed by fitting this polynomial
in 1/L to the calculated finite-chain results. We note that different finite-size scaling
formulas were proposed in the literature mainly when periodic or antiperiodic boundary
conditions were used [81].
As can be seen from the main plot in Fig. 4.3, extrapolating the results for
L = {30, 40, 50, 60} does indeed not give a sharp transition behavior. As illustrated in
the inset, adding results for L up to 512 in the critical region changes the picture considerably.
Within numerical accuracy the charge and spin gaps remain equal and finite up to a
critical U c ≈ 2.1t, where a sharp kink for ∆ C is observed. Importantly, ∆ C does not close
at the critical point. We emphasize that the magnitude of ∆ C and ∆ S at the transition
point is sufficiently larger than our numerical uncertainty in the finite size scaling analysis
and therefore allows for a safe conclusion. ∆ C = ∆ S > 0 at the transition is in fact not in

4.1. Ionic Hubbard model 131
Figure 4.3: Results for the spin (∆ S ) and charge (∆ C ) gaps of the IHM at halffilling
with ∆ = 0.5t as a function of U. Energies were obtained by DMRG calculations
on open chains with L = {30, 40, 50, 60} (main plot) and up to L = 512 (inset), and
extrapolated to the limit of infinite chain length.
conflict with an underlying ground-state level crossing. If the ground states of the different
site-parity sectors become degenerate, the only rigorous consequence is the closing of the
optical excitation gap. The selection rules for optical excitations allow only for transitions
between states of different site-parity. Furthermore, optical transitions occur within the
same particle number sector. The optical gap is therefore by definition distinct from the
charge gap Eq. (4.3a) which involves the removal or the addition of a particle. The critical
point U c of the IHM has the remarkable peculiarity that the optical gap closes while ∆ C
remains finite. Above U c the charge and spin gaps split indicating that the corresponding
insulating phase is no longer a BI. ∆ S continuously decreases with increasing U and
becomes unresolvable small above U ∼ 2.5t within the achievable numerical accuracy.
The result, that ∆ C and ∆ S remain finite and equal at the transition is in agreement
with the data obtained by Qin et al. [157]. These authors performed DMRG calculations
for the IHM with ∆ = 0.6t for open chains up to L = 600 sites. They observed a surprising
non-monotonic scaling behavior of ∆ S with L for values of U close to the critical U c , i.e.,
for chain lengths L > 300 ∆ S started to increase again. It remains unclear whether this is
due to a loss of DMRG accuracy with increasing chain lengths and keeping a fixed number
of states in the DMRG algorithm. In contrast, our data always show a monotonous scaling
with 1/L. DMRG calculations for the IHM with ∆ = t have also been performed by Takada

132 4. Nature of the Band- to Mott-insulator transition in one-dimension
Figure 4.4: Results for the spin (∆ S ) and charge (∆ C ) gaps vs. U of the Peierls-
Hubbard model at half-filling with a modulation of the hopping amplitude δ = 0.5.
Energies were obtained by DMRG calculations on open chains with L = {30, 40, 50, 60}.
and Kido for chains up to L = 400 sites [224]. These authors interpret their results in the
region close to U c in favor of a two-transition scenario similar to that of FGN [57, 58].
However, as we will show below the accuracy of the currently available DMRG data is not
sufficient to provide a stringent argument in favor of this interpretation.
For comparison we show in Fig. 4.4 the spin and charge gaps versus U in the Peierls
Hubbard model. As we observe this model is distinctly different from the IHM. This is
also a band-insulator at U = 0, but in contrast to the IHM has ∆ C > ∆ S > 0 for any value
U > 0, i.e., the phase transition from the Peierls band-insulator to the correlated insulator
occurs at U c = 0. So although the Peierls and the ionic BI for U = 0 similarly possess an
excitation gap at the Brillouin zone boundary, applying a Coulomb U leads to distinctly
different behavior in both cases. The origin of the different behaviors must be traced to
the fact that the Hubbard interaction and the ionic potential compete locally on each site,
while the Peierls modulation of the hopping amplitude tends to move charge to the bonds
between sites, thereby avoiding conflict with the Hubbard term.
The spin-Peierls physics of the Peierls Hubbard model at large U evolves smoothly with
decreasing U into the physics of a spin-gapped CI-BOW state in the weak-coupling limit,

4.1. Ionic Hubbard model 133
0.4
0.2
L=32
〈(n(r)-1)〉
0
-0.2
U=0.8t, ∆=0.5t
U=4.0t, ∆=0.5t
-0.4
0 5 10 15 20 25 30
r
Figure 4.5: Electron-density distribution in the ground state of the IHM for ∆ = 0.5t
and U = 0.8t (diamonds) and U = 4t (circles). Results were obtained by DMRG calculations
on an open L = 32 chain.
which is characterized by long-range staggered bond-density correlations
g b (r) = 1 ∑
〈ψ 0 | b(i)b(i + r) |ψ 0 〉 , (4.14)
L
i
b(i) = ∑ )
(c † iσ c i+1σ + h.c. . (4.15)
σ
For U ≫ t the low-energy physics of the Peierls Hubbard model is described by the spin-
Peierls Heisenberg Hamiltonian with a staggered exchange interaction and a dimerization
induced spin gap [36].
4.1.4.2 Correlation functions
The important question remains about the nature of the insulating phase of the IHM for
U > U c . To further analyze the BI and CI phases below and above U c , we have evaluated
site- and bond-charge distribution functions as well as spin-spin correlation functions. In
Fig. 4.5 we show the charge distribution 〈0|(n(r) − 1)|0〉 (n(r) = n r,↑ + n r,↓ ) in the ground
state |0〉 for the IHM at U = 0.8t < U c and U = 4t > U c for a L = 32 chain. The alternating
pattern in the density distribution is well pronounced not only in the BI but also in the
CI phase far beyond the critical point at U ≫ U c . For the L = 32 chain the CDW is

134 4. Nature of the Band- to Mott-insulator transition in one-dimension
|〈n(L/2-1)-n(L/2)〉|/2
0.25
0.2
0.15
0.1
0.05
0
0.12
0.1
∆=0.5t
0.08
1.8 2 2.2 2.4
|〈n(L/2-1)-n(L/2)〉|/2
0.3
0.2
0.1
U=0
U=2.1t
U=4.0t
0
0 0.01 0.02 0.03
1/L
0 1 2 3 4
U/t
Figure 4.6: Staggered charge-density component vs. U, for ∆ = 0.5t and L = 512
(main plot). Its scaling behavior, for U = 0 (diamonds), U = 2.1t (circles) and
U = 4.0t (triangles) is shown in the inset.
well established at distances l ∼ L/2 even at U = 4t and its amplitude remains almost
unchanged in the finite-size scaling analysis (see the upper inset in Fig. 4.6). The main plot
in Fig. 4.6 shows that the staggered component of the charge density decreases smoothly
with increasing U. Close above the transition point near U = 2t one observes an anomaly,
i.e., a slight decrease of the CDW amplitude accompanied by a change in curvature; this
anomaly can be clearly identified in the enlargement shown in the lower inset. The anomaly
will find a natural explanation in the discussion below. Our numerical data show that the
alternating pattern in the electron density distribution in the IHM remains for arbitrary
finite U. Thus, the ionic potential enforces long range CDW order for all interaction
strengths.
Fig. 4.7 shows the DMRG results for the spin-spin correlation function
〈0|S z (L/2)S z (L/2 + r)|0〉 for the IHM at U = 0.8t < U c and U = 4t > U c in comparison
with the Hubbard chain. In the BI phase at U = 0.8t the SDW correlations are quickly
suppressed after a few lattice spacings. At U = 4t the amplitude of the SDW correlations
in the CI phase of the IHM is slightly reduced in comparison to the Hubbard model at the
same value of U. However, the large distance behaviors of the spin correlations in the CI
phase of the IHM and the MI phase of the Hubbard model are quite similar and become
almost indistinguishable (see the data in the inset for U = 4t). On the other hand, the

4.1. Ionic Hubbard model 135
〈S z (L/2)S z (L/2+r)〉
0.04
0.02
0
-0.02
-0.04
L=32
|〈S z L/2 Sz L/2+r 〉|
0.006
0.004
U=2.1t, ∆=0
U=2.1t, ∆=0.5t
U=4.0t, ∆=0
U=4.0t, ∆=0.5t
U=0.8t, ∆=0
U=0.8t, ∆=0.5t
U=4.0t, ∆=0
U=4.0t, ∆=0.5t
r = L/4+1
0.002
1/r
0
0 0.02 0.04 0.06 0.08 0.1 0.12
0 2 4 6 8 10 12 14 16
r
Figure 4.7: Spin-spin correlation function in the ground state of the IHM for ∆ = 0.5t
(open symbols) and the Hubbard model (∆ = 0) (full symbols) at U = 0.8t (diamonds)
and U = 4t (circles). Chain length L = 32 (main plot). Inset: Scaling of the spin-spin
correlation function at r = L/4 + 1 for U = 2.1t (up and down triangles) and U = 4t
(circles) for the IHM (open symbols) and the Hubbard model (full symbols).
long distance behavior of the spin-spin correlation function at U = 2.1t and ∆ = 0.5t, i.e.,
close above the transition, manifestly supports the finiteness of the spin gap (see inset).
This may be viewed as an indication for the existence of two different phases above U c .
But we cautiously point out that it is hard to judge on the persistence or vanishing of ∆ S
far above U c in the CI phase from the finite chain spin correlators alone.
To address the BOW ordering tendencies in the CI phase we have calculated the groundstate
distribution of the bond-charge density Eq. (4.14). Fig. 4.8 shows the results of the
DMRG calculations for the L = 32 IHM and the Hubbard chain at U = 0.8t and U = 2.6t.
The boundary effect for an open chain is strong and leads to a modulation of the bond
density already for the pure Hubbard model. Interestingly, the same behavior was also
observed previously for the bond expectation value 〈S i · S i+1 〉 in open antiferromagnetic
Heisenberg spin-1/2 chains [256]. The detailed comparison with the Hubbard chain in
Fig. 4.8 reveals, that the ionic potential leads to a reduction of the bond-density oscillations
at U < U c , while in the CI phase at U > U c their amplitude slightly increases. The
enhancement of BOW correlations above U c must simultaneously weaken the CDW amplitude.
This naturally explains the slight downward curvature near U c in the staggered

136 4. Nature of the Band- to Mott-insulator transition in one-dimension
1.8
1.6
UU c
U=0.8t, ∆=0
U=0.8t, ∆=0.5t
U=2.6t, ∆=0
U=2.6t, ∆=0.5t
1.2
1
0.8
0 5 10 15 20 25 30
r
Figure 4.8: Bond-charge density of the IHM for ∆ = 0.5t (open symbols) and the
Hubbard model (full symbols) at U = 0.8t (diamonds) and U = 2.6t (circles).
charge density component shown in Fig. 4.6.
In order to explore the possibility towards true long range BOW ordering in the IHM
above U c we plot in Fig. 4.9 the staggered bond density versus U in the center of long, open
chains with L = 256 and L = 512. In the BI phase this quantity is essentially zero. At
the transition point the staggered component of the bond density increases rapidly, and on
further increasing U it starts to decrease smoothly. However, the staggered bond density
remains finite for any U > U c on these long but finite chains and vanishes only in the limit
L → ∞.
Naturally it is necessary to perform a finite-size scaling analysis for this quantity.
Fig. 4.10 shows its chain length scaling behavior for U = 0, U = 2.1t, and U = 4t. Two
conclusions can be drawn from these results: In the absence of the interaction the staggered
bond density clearly extrapolates to zero - as expected for a conventional BI. In the
CI phase close above the transition the upward curvature of the staggered bond density
vs. 1/L points to a finite value in the infinite chain length limit, i.e., long range BOW
order. However, the scaling behavior changes in the CI phase far above the transition
point. Interestingly, the scaling behavior in the IHM far above U c starts to resemble the
results for the pure Hubbard model, for which the staggered bond density has to vanish in
the thermodynamic limit. The qualitative change in the scaling behavior of the staggered
bond density may again be viewed indicative for a possible second phase transition. Sum-

4.1. Ionic Hubbard model 137
0.12
0.1
L=256, ∆=0.5t
L=512, ∆=0.5t
|〈b(L/2-1)-b(L/2)〉|
0.08
0.06
0.04
0.02
0
0 1 2 3 4
U/t
Figure 4.9: Staggered bond-density component vs. U near the center of the L = 256
(circle, dotted line) and L = 512 (diamonds, solid line) chain.
0.2
|〈b(L/2-1)-b(L/2)〉|
0.15
0.1
0.05
U=0, ∆=0.5t
U=2.1t, ∆=0.5t
U=2.1t, ∆=0
U=4.0t, ∆=0.5t
U=4.0t, ∆=0
0
0 0.01 0.02 0.03
1/L
Figure 4.10: Scaling behavior of the staggered bond-density component in the IHM
near the center of the chain for U = 0 (BI, diamonds), U = 2.1t (CI close above the
transition, circles), U = 4.0t (CI far above the transition, up triangles) and in the
Hubbard model for U = 2.1t (squares) and U = 4.0t (down triangles).

138 4. Nature of the Band- to Mott-insulator transition in one-dimension
marizing the results for the SDW and BOW correlations we conclude that there is evidence
for two phases for U > U c . Close above U c long range BOW order develops in the ground
state of the IHM, while far above the transition point the correlation functions become
almost identical to those of the Hubbard model. Yet, long range CDW order exists for all
U. A precise location of a second transition point is however very difficult to resolve from
the currently available DMRG data.

4.2. Adiabatic Holstein-Hubbard model 139
4.2 Adiabatic Holstein-Hubbard model
4.2.1 Introduction
In quasi-one-dimensional materials like halogen-bridged transition-metal chain complexes,
conjugated polymers, organic charge transfer salts, or inorganic blue bronzes the itinerancy
of the electrons strongly competes with electron-electron and electron-phonon interactions,
which tend to localize the charge carriers by establishing commensurate spin- (SDW) or
charge-density wave (CDW) ground states (GSs). These compounds are particularly rewarding
to study for a number of reasons. They exhibit a remarkably wide range of
strengths of competing forces and, as a result, physical properties. These systems share
fundamental features with higher dimensional novel materials, such as high-temperature
superconductors, charge-ordered nickelates or colossal magneto resistance manganites, i.e.,
they are complex enough to investigate the interplay of charge, spin, and lattice degrees
of freedom which is important for strongly correlated electronic systems in two and three
dimensions as well. Nevertheless they are simple enough to allow for microscopic modeling.
Thus they are suitable modern systems to develop and test new theoretical methods by
bringing together techniques from quantum chemistry, electronic band structure investigations,
and many-body physics.
At half-filling, Peierls (PI) or Mott (MI) insulating phases are favored over the metallic
state. Quantum phase transitions between the insulating phases are possible and the
character of the electronic excitation spectra reflects the properties of the different insulating
GSs. Phonon dynamical effects, which are known to be particularly important in
low-dimensional materials [62, 131] may further modify the transition. Another interesting
feature of the materials mentioned is the existence of in-gap states, that may be caused
by donating electrons to or accepting electrons from the half-filled host material. Those
charged gap states are related to local lattice distortions (polarons or bipolarons). Another
possibility is “photo-doping”, i.e., the creation of neutral excitations (e.g. excitons).
In this section we study the PI-MI quantum phase transition in the adiabatic limit of
the Holstein-Hubbard model (HHM) at half-filling. Exact numerical methods [18, 252] are
used to diagonalize the HHM on finite chains, preserving the full dynamics of the phonons,
and the density matrix renormalization group (DMRG) technique (see Chapter 2) is applied
to the adiabatic HHM. Results for the HHM were obtained by Holger Fehske (now
at Ernst-Moritz-Arndt University Greifswald) and Gerhard Wellein (now at Erlangen Regional
Computing Center, RRZE). In the adiabatic limit two different scenarios emerge
with a discontinuous transition at strong coupling and two subsequent continuous transitions
in the weak coupling regime with the possibility for an intermediate insulating phase
with finite spin, charge, and optical excitation gaps.

140 4. Nature of the Band- to Mott-insulator transition in one-dimension
4.2.2 Theoretical models
The paradigm for correlated electron-phonon systems has usually been the one-dimensional
HHM defined by
∑
H = H t−U − gω 0 (b † i + b i )n ∑
iσ + ω 0 b † i b i , (4.16)
H t−U = −t ∑ i,σ
i,σ
(c † iσ c i+1σ + h.c.) + U ∑ i
i
n i↑ n i↓ . (4.17)
H t−U constitutes the conventional Hubbard Hamiltonian with hopping amplitude t and onsite
Coulomb repulsion strength U; c † iσ creates a spin-σ electron at site i and n iσ = c † iσ c iσ.
In (4.16), the second term couples the electrons locally to a phonon created by b † i . Here
g = √ ε p /ω 0 is a dimensionless electron-phonon coupling constant, where ε p and ω 0 denote
the polaron binding energy and the frequency of the optical phonon mode, respectively.
The GS of the Holstein model for U = 0 is a Peierls distorted state with staggered
charge order in the adiabatic limit ω 0 → 0 for any finite ε p . As in the Holstein model of
spinless fermions [27, 251], quantum phonon fluctuations destroy the Peierls state for small
electron-phonon interaction strength [131] — an issue which has remained unresolved in
early studies of the Holstein model using Monte Carlo techniques [110]. Above a critical
threshold g c (ω 0 ), the Holstein model describes a PI with equal spin and charge excitation
gaps — the characteristic feature of a band insulator (BI). a
The adiabatic limit of the HHM takes the form
H = H t−U − ∑ ∆ i n iσ + K ∑
∆ 2 i (4.18)
2
i,σ
i
(termed adiabatic HHM or AHHM); it includes the elastic energy of a harmonic lattice
with a “stiffness constant” K. In this frozen phonon approach, ∆ i = (−1) i ∆ is a measure
of the static, staggered density modulations of the BI phase. Eq. (4.18) with K = 0 and
fixed ∆ is the ionic Hubbard model (IHM) for which an insulator-insulator transition was
already established before (see Section 4.1) Interestingly, the IHM was motivated originally
in quite different contexts, i.e. for the description of the neutral to ionic transition in charge
transfer salts [178, 179] and ferro-electricity in transition metal oxides [50, 199].
4.2.3 Numerical results
4.2.3.1 Charge- and spin-structure factors
In order to establish the GS properties of the above models and the existence of the BI-MI
transition we start with the evaluation of the staggered charge- and spin-structure factors
a Although in the literature PI is more frequently used in connection with the Holstein-Hubbard models,
we will mostly use BI to stay with the definitions introduced in the previous section.

4.2. Adiabatic Holstein-Hubbard model 141
0.9
S c
(p)
0.6
0.3
ε p
=2t, ω 0
=1t (HHM)
ε p
=0.7t, ω 0
=0.1t (HHM)
K t=0.74 (AHHM L=8)
K t=0.74 (AHHM L=64)
0.0
S s
(p)
0.05
0.0
0.0 1.0 2.0
U/2ε p
Figure 4.11: Staggered charge- (upper) and spin-structure factors (middle panel) vs.
the rescaled Hubbard interaction U/2ε p . Lanczos results for the HHM on an 8-site ring
are given in the adiabatic (triangles) and non-adiabatic (squares) regimes. Lanczos
(L = 8 ring, down-triangle) and DMRG (open 64-site chain, stars) results are shown
for the AHHM with K = 0.74/t. Open (closed) symbols belong to GSs with site-parity
P = −1 (+1).
S c (π) and S s (π), respectively,
S c (π) = 1 ∑
(−1) j 〈(n iσ − 1 L
2 )(n i+j,σ ′ − 1 2 )〉 ,
j,σσ ′
S s (π) = 1 ∑
(−1) j 〈Si z L
Sz i+j 〉 , Sz i = 1 2 (n i↑ − n i↓ ) .
j
Results for the U-dependence of S c (π) and S s (π) on an 8-site HHM ring are shown in
Fig. 4.11 for two different phonon frequencies, corresponding to adiabatic and non-adiabatic
regimes. The BI regime is characterized by a large (small) charge- (spin-) structure factor.
Also shown in Fig. 4.11 are results for the AHHM [L = 8 (Lanczos), L = 64 (DMRG)]
which will be discussed in Section 4.2.4. Increasing U at fixed ε p and ω 0 , Peierls CDW
order is suppressed as becomes manifest from the rapid drop of S c (π) which decreases nearly
linearly in the adiabatic regime, but its initial decrease is significantly weaker for higher
phonon frequencies. The disappearance of the charge ordering signal is accompanied by a
steep rise in S s (π) indicating enhanced antiferromagnetic correlations in the MI phase. The

142 4. Nature of the Band- to Mott-insulator transition in one-dimension
σ reg (ω), S reg (ω)/S reg (∞)
2.5
2.0
1.5
1.0
0.5
0.0
2.0
1.5
1.0
0.5
0.0
2.0
1.5
1.0
0.5
0.0
PI
MI
U/2ε p
=0
U/2ε p
=0.93
U/2ε p
=2.14
0 2 4 6 8
ω/t
Figure 4.12: Optical conductivity in the 8-site HHM for ω 0 =0.1t and g 2 =7. Top
panel: BI phase for U = 0; middle panel: near criticality U ∼ U opt ; lower panel: MI
phase for U = 3t. Dashed lines give the normalized integrated spectral weights S reg (ω).
The lower two panels include σ reg for g = 0 (dotted black lines), i.e. for the pure
Hubbard chain.
data for S c (π) provide evidence for a critical point U c at which the CDW order disappears:
rather abruptly for adiabatic and smoothly for non-adiabatic phonon frequencies. Above
U c the low-energy physics of the system is qualitatively similar to the pure Hubbard chain;
it is governed by gapless spin and massive charge excitations.
4.2.3.2 Optical response
Valuable insight into the nature of the BI-MI transition is obtained from symmetry considerations
(see Section 4.1.2). The BI-MI transition of the IHM on finite lattices was shown
to be connected to a GS level crossing with a site-parity change, where the site inversion
symmetry operator ˆP is defined by ˆPc † ˆP iσ † = c † L−iσ
with L = 4n for i = 0, . . ., L − 1. This
feature will become evident in the regular part of the optical conductivity at T = 0,

4.2. Adiabatic Holstein-Hubbard model 143
σ reg (ω)= π L
∑
m≠0
|〈ψ 0 |ĵ|ψ m 〉| 2
E m − E 0
δ(ω−E m +E 0 ) . (4.19)
Here, |ψ 0 〉 and |ψ m 〉 denote the GS and excited states, respectively, and E m the corresponding
eigenenergies. Importantly, the current operator ĵ = −iet ∑ iσ (c† iσ c i+1 σ − c † i+1 σ c iσ) has
finite matrix elements between states of different site-parity only.
The evolution of the frequency dependence of σ reg (ω) from the BI to the MI phase
with increasing U is illustrated in Fig. 4.12. In the BI regime the electronic excitations are
gapped due to the pronounced CDW correlations. The broad optical absorption band for
U = 0 results from particle-hole excitations across the BI gap which are accompanied by
multi-phonon absorption and emission processes. The shape of the absorption band reflects
the phonon distribution function in the GS. Excitonic gap states may occur in the process
of structural relaxation. At U opt the optical gap ∆ opt closes, and due to the selection rules
for optical transitions this necessarily implies a GS level crossing with a site-parity change.
We have explicitly verified that the GS site parity in the BI phase is P = +1 and P = −1
in the MI phase (see also Fig. 4.11). For the HHM on finite rings U opt is identical to the
critical point where S c (π) sharply drops.
For the adiabatic phonon frequency used in Fig. 4.12 the phonon absorption threshold
is small and since the GS is a multi-phonon state, we find a gradual linear rise of the
integrated spectral weight S reg (ω) = ∫ ω
0 σreg (ω ′ ) dω ′ . S reg (ω)/S reg (∞) is a natural measure
for the relative weight of the different optical absorption processes. In contrast, in the
non-adiabatic regime (ω 0 ≥ t), the lowest optical excitations have mainly pure electronic
character in the vicinity of U opt . As a result the gap is closed by a state having large
electronic spectral weight.
In the MI phase the optical gap is by its nature a correlation gap. The lower panel in
Fig. 4.12 shows clearly that σ(ω) of the HHM in the MI phase is dominated by excitations
which can be related to those of the pure Hubbard model. In addition, phononic sidebands
with low spectral weight and phonon-induced gap states appear.
4.2.4 Phase diagram in the adiabatic limit
The above results for the HHM establish the BI-MI phase transition scenario on small rings
and trace it to the level crossing of the two site-parity sectors. In order to draw conclusions
about the phase diagram in the adiabatic regime we exploit the connection to the AHHM.
The magnitude of S c (π) in the HHM for U = 0 and ω 0 = 0.1t allows a straightforward
way to fix the stiffness constant K in Eq. (4.18). Using the result of the AHHM for S c (π)
at U = K = 0, we determine first the ionic potential strength ∆ 0 by the requirement that
Sc
IHM (π, ∆ 0 ) = Sc
HHM (π) for the same chain length and periodic boundary conditions. In a
second step, the GS energy of the AHHM, E 0 (K, ∆, U = 0), determines K by the criterion

144 4. Nature of the Band- to Mott-insulator transition in one-dimension
0.8
0.6
∆(U,K=0.74), L=8
∆ cr
(U), L=8
∆(U,K=0.74), L=64
∆ cr
(U), L=64
∆ cr
/t, ∆/t
0.4
0.2
0
0 0.5 1 1.5
U/2ε p
Figure 4.13: Level-crossing line ∆ cr (U) of the IHM for an 8-site ring (diamonds) and
from extrapolating Lanczos data for L ≤ 14 to a 64-sites chain (circles). In addition:
ionic potential strength ∆(U, K) of the AHHM for an 8-site ring (triangles) and on an
open 64-site chain (stars, DMRG results) for K = 0.74/t.
that E 0 is minimized for ∆ = ∆ 0 . We thereby obtain K = 0.74/t, which is henceforth kept
fixed when the interaction U is turned on. For each value of U, the ionic potential strength
of the AHHM is then obtained by minimizing E 0 (K, ∆, U) with respect to ∆, yielding
∆ = ∆(U, K) as shown in Fig. 4.13 (triangles). The resulting structure factors for the
AHHM are plotted in Fig. 4.11, too, and agree very accurately with the 8-site HHM ring
data for ω 0 /t = 0.1. This agreement reconfirms numerically that the AHHM is indeed the
appropriate effective model to describe the CDW phase of the HHM in the adiabatic limit.
The drop in S c (π) at the transition point results from a discontinuous vanishing of ∆(U, K)
(see Fig. 4.13). The large charge-structure factor S c (π) below U c and the enhancement of
the spin structure factor S s (π) above U c as well as the sharp changes at the transition point
find a natural explanation with the results for ∆(U, K) in Fig. 4.13. Below the transition
∆ is finite implying long range CDW order in the GS. At the transition point ∆ vanishes
discontinuously and thereby the AHHM reduces to the pure Hubbard model (∆ = 0).
Given the value for the stiffness constant K we also plot in Fig. 4.13 ∆(U, K) obtained
from DMRG on an open chain of length L = 64. For comparison, the corresponding results
for S c (π) and S s (π) in the AHHM are shown in Fig. 4.11, too (stars). S c (π) decreases
smoothly and almost linearly; although unresolved on the vertical scale in Fig. 4.11 the
transition remains discontinuous as a consequence of the results for ∆(U, K) in Fig. 4.13.

4.2. Adiabatic Holstein-Hubbard model 145
In contrast to the behavior of the 8-site chain, ∆(U, K) here decreases more smoothly with
increasing U and vanishes discontinuously near U/2ε p ≈ 0.75. The small discontinuous
increase in S s (π) at the transition is also hardly resolved for the 64-site chain in contrast
to the 8-site chain data. The discontinuous nature of the BI-MI transition in the AHHM
for ∆ i = (−1) i ∆ is obvious in the atomic limit t = 0 where ∆ = 1/K for U < U c = 1/K
and ∆ = 0 for U > U c . As verified above, the first order nature persists for finite small t,
i.e. in the strong coupling regime U, K −1 ≫ t.
Also shown in Fig. 4.13 is the level crossing line ∆ cr (U) of the IHM for L = 8 (diamonds)
and L = 64 (circles) chain. ∆ cr (U) for L = 64 was obtained from extrapolating Lanczos
results for rings of up to 14 sites to a 64-site chain. b Importantly, ∆(U) and ∆ cr (U) do
not intercept because ∆(U, K) jumps to zero before reaching the level crossing point of the
IHM.
The DMRG results presented in Fig. 4.13 for L = 64 raise the question whether the
discontinuous transition in the AHHM can turn into a continuous transition on approaching
the weak coupling regime by increasing the stiffness constant K. Indeed, as we have
explicitly verified by exact diagonalization of a periodic (and open too) AHHM ring of
length L = 14, the transition is second order in the regime U, K −1 ≪ t. The corresponding
Lanczos results for the variation of the GS energy vs. ∆ in the (K, U)-parameter plane are
summarized in Fig. 4.14. Detailed K-scans were performed for weak (U = 0.3t) and strong
(U = 5t) Hubbard interaction. The evolution of E(∆) in the AHHM in fact reveals that the
transition from the BI to the MI phase (sequence (2) − (3) − (4)) occurs discontinuously at
strong coupling K −1 , U ≫ t, while the transition follows a Ginzburg-Landau-type behavior
for a second order phase transition at weak coupling (sequence (1) − (5) dashed line in
Fig. 4.14).
Due to the continuous decrease of ∆(U) at weak coupling ∆(U) necessarily intercepts
the ∆ cr (U) line of the IHM. This intercept marks the point U opt when the site-parity sectors
become degenerate and the optical absorption gap ∆ opt closes. This situation therefore
implies the existence of an intermediate region U opt < U < U s with finite ∆, where U s marks
the point where ∆ continuously vanishes. Since ∆ = 0 for U > U s , i.e. when the AHHM
reduces to the Hubbard model, the spin gap vanishes at U s . The intermediate insulating
phase thus has finite spin, charge, and optical excitation gaps. For weak coupling the BI-MI
transition therefore evolves across two critical points U opt and U s . The U vs. K −1 phase
diagram contains a multicritical point at which a first order line splits into two continuous
transition lines. The additional transition line is also indicated in Fig. 4.14 (dotted line).
For weak U at fixed K the transition at U opt corresponds to the merging of the energies of
the two site-parity sectors; the CDW vanishes in a second order type transition at U = U s .
The E(∆) behavior, however, can only detect the boundary to the MI phase of the AHHM
b To obtain ∆ cr (U) for rings with L = 4m + 2 and L = 4m sites we used antiperiodic and periodic
boundary conditions, respectively.

146 4. Nature of the Band- to Mott-insulator transition in one-dimension
−1
K
1
PI
∆(U,K)>0
2
3
4
E
1
∆
5
∆(U,K)=0
MI
U
E
2
∆
E
5
E
4
E
3
∆
∆
∆
Figure 4.14: Side plots (1)-(5): Evolution of the ground-state energy vs. ∆ in the
AHHM in different regions of the (K −1 , U) parameter plane. From the variations in
E(∆) a crossover from a discontinuous BI (with ∆ > 0) to MI (∆ = 0) transition to
a second order transition is deduced. Main figure: Phase diagram of the AHHM; the
solid line represents a discontinuous, first order and the dashed line a continuous second
order transition. These results summarize Lanczos data for a 14-site AHHM chain with
periodic or open boundary conditions. Detailed runs where performed for U = 0.3t
and U = 5t. A possible additional continuous transition (dotted line) between two
insulating phases with finite ∆ is indicated as well.
where the GS energy is minimized for vanishing ∆.
We summarize these findings in the diagram for the excitation gaps ∆ C and ∆ S (4.3a)
shown in Fig. 4.15, In the Peierls BI phase for U < U opt the spin and charge gaps are
equal and finite and remarkably ∆ opt ≠ ∆ C (for a similar conclusion in the IHM see
[157]). At U = U opt the site-parity sectors become degenerate, ∆ opt = 0, but remarkably
∆ C = ∆ S > 0. For U ≥ U s the usual MI phase of the half-filled Hubbard chain with
∆ opt = ∆ C > ∆ S = 0 is realized. For strong coupling, when the BI to MI transition is first
order, U opt = U s , the spin gap discontinuously disappears at the transition and the optical
gap jumps from zero to the finite charge gap value of the Hubbard chain. In weak coupling
there exists an intermediate region U opt < U < U s in which all excitation gaps are finite.
The CDW persists for all U < U s . The site-parity eigenvalue is P = +1 in the BI and
P = −1 in the MI phase.
The insulating, intermediate phase at weak coupling as identified above remains yet
to be characterized. For the insulator-insulator phase transition(s) in the IHM (see Sec-

4.2. Adiabatic Holstein-Hubbard model 147
energy gaps
BI
P=+1
CDW
(LRO)
(+BOW?)
SDW
(Power Law)
MI
P=−1
∆ opt
∆
=C
∆ opt
∆
S
U U
opt
s
U
Figure 4.15: Qualitative behavior of the excitation gaps versus U in the weak coupling
regime of the AHHM U, K −1 ≪ t. Solid line: optical excitation gap ∆ opt , dotted line:
charge gap ∆ c , dashed line: spin gap ∆ s . BI phase: ∆ c = ∆ s and site parity P = +1;
MI phase: ∆ opt = ∆ c and P = −1.
tion 4.1) the strong indications for the intermediate phase with a long range bond-order
wave (BOW) have been found. We have positive numerical evidence for enhanced BOW
correlations above the level crossing transition in the IHM. Therefore, the natural candidate
for these intermediate phase will be the one with long range CDW+BOW order. We
have nevertheless attempted to search for BOW correlations in the weak coupling regime
of the AHHM, where the continuous nature of the transition into the MI phase was established
by the Lanczos results on the periodic 14-sites chain, i.e. these calculations naturally
focused on the weak-U regime (U < t). As discussed (see Section 4.1.4.2), in the attempt
to search for BOW order the DMRG calculations, which for numerical accuracy reasons
are predominantly performed on open chains, suffer from the fact that Friedel-like bondcharge
density oscillations are induced by the chain ends already for the pure Hubbard
chain. The identification of BOW order in the IHM or AHHM by DMRG on open chains
therefore requires a delicate subtraction procedure to discriminate a BOW signal from the
edge induced bond-charge oscillations of the Hubbard chain. This weak coupling regime
is notoriously hard for numerical evaluations and unfortunately the numerical accuracy
needed to allow a firm conclusion about the presence or absence of a BOW signal could
not be achieved within our DMRG runs.
While a confirmation is thus still lacking BOW order remains a vivid candidate order

148 4. Nature of the Band- to Mott-insulator transition in one-dimension
in coexistence with a CDW to characterize the intermediate phase in the AHHM at weak
coupling. We furthermore note that if the existence of a BOW is verified in the AHHM, its
phase diagram would be remarkably similar to the extended Hubbard model with nearest
neighbor Coulomb repulsion with an intervening BOW phase in the crossover between the
CDW and MI phases at weak coupling [180, 181, 216].
4.3 Conclusions
From the finite chain DMRG studies and finite size scaling analysis we draw the following
conclusions for the ground-state phase diagram of the IHM: the ionic potential leads to
long range CDW order for all interaction strengths. The data resolve one of the transition
points, namely from the BI to the CI phase. Remarkably, at the transition ∆ C = ∆ S and
both remain finite. Close above the transition, i.e., for 0 < (U/U c ) − 1 ≪ 1, we identify
a clear signal for long range BOW order. With increasing U above U c the finite size
scaling behavior of the staggered bond density and spin-spin correlation function changes
qualitatively and approaches the scaling behavior of the Hubbard model. With the current
chain length and DMRG accuracy limitations it was not possible to precisely identify and
locate the second transition point. The phase with BOW order necessarily has a finite spin
excitation gap. If BOW ceases to exist above a second critical value of U, the spin gap
has to vanish simultaneously leading to identical large distance decays of SDW and BOW
correlations.
The insulator-insulator transition at U c (∆) on finite periodic chains results from a
ground-state level crossing of the two site-parity sectors. The optical excitation gap therefore
has to vanish at U c ; remarkably, the DMRG data reveal that at the critical point ∆ C
and ∆ S remain both finite and equal. This itself clearly indicates the existence of a CI
phase with ∆ C > ∆ S > 0 which originates from the appearance of long range staggered
bond-density order. The distinction between the optical and the charge gap is therefore
of key importance for the structure of the insulating phases and the phase transitions of
the IHM. The investigation of the optical conductivity in the critical region is therefore a
demanding task for future work on the complex physics of the ionic Hubbard model.
In the adiabatic limit of the Holstein-Hubbard model two scenarios emerge with a
discontinuous BI-MI transition for U, K −1 ≫ t, and two continuous transitions for weak
coupling U, K −1 ≪ t with an intermediate phase where CDW order persists. Below the
transition point, U < U c , the ionic potential ∆ is finite and the system is in a BI phase,
similar to the BI phase of the IHM, implying long range CDW order in the GS. In the first
scenario the transition form the BI to the MI phase is characterized by a discontinuous
vanishing of ionic potential ∆(U, K) at the transition point U c , thereby the model reduces
to the pure Hubbard model with dominant SDW correlations. In the second scenario, the

4.3. Conclusions 149
BI-MI transition rather follows a Ginzburg-Landau-type behavior. Due to the continuous
decrease of ∆(U, K), ∆(U, K) naturally intercepts the BI-CI transition point of the IHM.
Therefore at this point U opt the site-parity sectors becomes degenerate and the optical
absorption gap disappears. This therefore implies the existence of an intermediate region
U opt < U < U s with a finite ∆. Due to strong similarities with IHM the intermediate phase
should be characterized with long range CDW+BOW order in the GS. The numerical
confirmation for this is out of reach at the moment, though. At U s , when the AHHM
reduces to the Hubbard model, a transition from the intermediate CI phase to the MI
phase occurs. In contrast to the IHM for U ≫ ∆ the transition point to the MI phase (U s )
can be identified and there is no long range CDW in the MI phase of AHHM.
The result presented in this chapter have already been published [63, 134].

150 4. Nature of the Band- to Mott-insulator transition in one-dimension

151
5. REAL-TIME DYNAMICS OF SPIN AND CHARGE
DENSITIES IN THE ONE-DIMENSIONAL
HUBBARD MODEL
In this chapter we study the real-time dynamics of spin- and charge-densities in the
one-dimensional Hubbard model using the time-dependent density matrix renormalization
group (t-DMRG) methods devised in Chapter 3. First we shortly recapitulate the
known facts about the one-dimensional Hubbard model (Section 5.1). In Section 5.2 we
describe the applied computational setup including the preparation of an initial state and
the simulation parameters used. We discuss the non-interacting limit of the Hubbard model
(the tight-binding model) in Section 5.3 for which the exact solution for the time evolution
can be obtained. Section 5.4 contains the results of the space-time evolution of the charge
and spin densities due to the addition of an electron to the ground state of the Hubbard
model in the Mott-insulating phase (at half-filling) and in the metallic phase close to it
(one electron less). In Section 5.5 we address the quality of the charge- and spin-excitation
dynamics for both considered cases. Conclusions including some short outlook close the
present chapter.
5.1 One-dimensional Hubbard model
Our starting point is the 1D Hubbard Hamiltonian which in the usual notation reads
H = −J ∑ i,σ
(c † i,σ c i+1,σ + h.c.) + U ∑ i
(n i↑ − 1/2)(n i↓ − 1/2) , (5.1)
where c † iσ (c iσ ) creates (annihilates) an electron on site i with spin σ and n iσ = c† iσ c iσ . Here
the nearest-neighbor hopping amplitude is denoted by J in order to avoid mixing with t
which will stand for the time in this chapter. In the case of vanishing on-site interaction
(U = 0) the Hubbard model reduces to the tight-binding model with a cosine dispersion
and a bandwidth W = 4J, which describes an ideal metal. Since the total number of
particles N e as well as the total numbers of particles with up- and down-spin (N ↑ and N ↓ ,
respectively) are conserved (see Section 1.2) only canonical ensembles are considered —
N ↑ and N ↓ are given — and hence the term with a chemical potential is already omitted

152 5. Real-time Dynamics of Spin and Charge Densities in the One-dimensional Hubbard Model
from the Hamiltonian (5.1). Furthermore, the lattice spacing constant a is used as the unit
length (a = 1) and as well as J are set to one; the time is thus measured in units of /J.
The exact solution of the 1D Hubbard model (later simply the Hubbard model) was
first obtained by Lieb and Wu in 1968 using the Bethe ansatz [155, 156]. Despite the fact
that the Hubbard model is integrable and the exact solution exists [54], the structure of
Bethe ansatz wave functions is so complex that it does not allow a direct calculation of
correlated functions. Here we only recapitulate the facts about the Hubbard model which
are relevant for the following discussion, further details on the model can be found in
Ref. [54] and references therein (see also Chapter 1).
At half-filling, i.e., n ≡ N e /L = (N ↑ + N ↓ )/L = 1, the ground state of the Hubbard
model is Mott-insulating for all values of U > 0, whereas for U = 0 or fillings different from
half-filling it is metallic. Strictly speaking the metal-insulator transition at U c = 0 + in the
half-filled model does not describe the Mott transition because the metallic phase does not
exist for finite interaction strengths. On the other hand, for all finite interaction strengths
(U > 0) if the electron density approaches the critical value n c of half-filling (n → n c = 1)
the system undergoes yet another metal-insulator transition which is an example of the
commensurate-incommensurate phase transition [44, 96, 197, 215].
The general classification of excitations of the Hubbard model was first proposed in
Refs. [55, 56]. It was shown that the excitation spectrum at half filling is given by scattering
states of four elementary excitations: holon and antiholon with spin 0 and charge
±e and charge neutral spinons with spin up or down respectively. The real “physical”
excitations can be then given only as SO(4) multiplets (see also Section 1.2) of these four
particles. An important consequence of this classification is that at half filling the spincharge
separation on the level of the quantum numbers of elementary excitations holds not
only at low energies, but extents to any finite energy in the thermodynamic limit. However
at finite energies the interaction between holons and spinons becomes nontrivial. It is
worthwhile to mention that these four quasiparticles are collective excitations involving an
extensive number of degrees of freedom and can be approximately related to simple realspace
pictures of a “hole” and an unpaired spin only in the strong coupling limit. Away
from half filling the number of elementary excitations is infinite [42]. The gapless charge
and spin excitations exist for all U > 0 in the case of n < 1 (see e.g., [6]) where the system
is an ideal metal. In contrast, in the Mott-insulating phase a gap opens for the charge mode
which causes the insulating behavior. The upper and lower Hubbard bands are separated
for all U > 0, but the gap is only of an appreciable size for U 2J. The spin mode remains
gapless in the Mott-insulating phase too. In the limit of infinitely strong interaction the
two modes decouple not only for low energies and the spin part of the Hamiltonian becomes
equivalent to the antiferromagnetic Heisenberg model with an exchange constant 4J 2 /U
(see also Section 1.3). In general, the bandwidth of the (anti)holon band does not depend
on on-site interaction U and equals 4J, whereas the bandwidth of the spinon band depends

5.1. One-dimensional Hubbard model 153
sensitively on U. This should be the case as in the large-U limit the effective Heisenberg
exchange is 4J 2 /U. a
In the thermodynamic limit and at low excitation energies the Hubbard model belongs
to the generic class of Luttinger liquids [93, 94, 95] and can be solved using field-theoretical
methods such as Bosonization [80, 85]. We will not present the detailed derivation here,
but for the sake of completeness we will stress the relevant results: The low-energy largedistance
behavior of a one-dimensional fermion system with spin independent interaction
away from the half-filling is described by the Hamiltonian
where
∫
H = H c + H s + ˜g
H ν = u ν
2
∫
dx cos( √ 8φ s )
dx
(K ν (∂ x θ ν ) 2 + 1 )
(∂ x φ ν ) 2
K ν
(5.2a)
(5.2b)
for ν = c, s. Here φ ν (ν = c, s) are canonical Bose fields describing the density fluctuations
of the spin ν = s and the charge ν = c, respectively, and θ ν are corresponding dual fields.
The parameters u ν are velocities, while K ν are related to the long-distance decay of the
correlation functions. These coefficients play a role similar to the Landau parameters of
(three-dimensional) Fermi liquid theory. For ˜q = 0 the Hamiltonian (5.2) is decomposed
into commuting parts H c and H s describing independent long-wave oscillations of the
collective charge and spin densities with linear dispersion relation ω c = u c |k| and ω s = u s |k|,
respectively, and the system is conducting. This model is usually called the Tomonaga-
Luttinger model [80, 159, 231]. The only nontrivial interaction effects come from the third
term of the Eq. 5.2a with ˜g ∝ U b which describes the backscattering processes. However,
for the repulsive interactions (U > 0) and the isotropic (SU(2) rotation invariant) Hubbard
model as discussed here (see also Section 1.2), K s = 1 and this term is marginally irrelevant.
The three remaining parameters completely determine the long-distance and low-energy
properties of the system. In the limit of an infinitesimal perturbation much broader than
the average interparticle spacing, both spin and charge velocities, as well as K c , can be
obtained from the Bethe ansatz; they depend on the interaction strengths and on the
charge density in the system. At half-filling, there is an additional contribution to the
charge sector due to Umklapp processes (UP) describing the scattering involving a finite
momentum transfer to the lattice equal to the reciprocal lattice vector 2π,
∫
H → H − ˜g
dx cos( √ 8φ c ). (5.3)
a Recall that J stands for the electron hopping amplitude (see Eq. (5.1)) instead of t which is used to
denote the time.
b the precise form does not matter for the presented discussions.

154 5. Real-time Dynamics of Spin and Charge Densities in the One-dimensional Hubbard Model
Below half filling UP involve high-energy degrees of freedom and as a result play no role in
the low-energy effective theory. On the other hand, at half filling UP involve only modes
in the vicinity of the Fermi points because 4k F = 2π (the so-called perfect nesting) and
lead to a Mott insulating phase already at U c = 0 + .
One of the most spectacular consequences of the Hamiltonian (5.2) is the complete
separation of the dynamics of the spin and charge degrees of freedom. In general one has
u c ≠ u s , i.e., the charge and spin oscillations propagate with different velocities. Only in the
non-interacting system one has u c = u s = v F . This implies that an extra particle created
in the ground state, at t = 0 and at the spatial coordinate x 0 , will split up into charge and
spin quasiparticles and after some time this charge and spin will be located at different
points in space (x c = x 0 + u c t and x s = x 0 + u s t, respectively), i.e., charge and spin will
separate completely. However, if the energy-momentum relation is only almost linear and
hence has some curvature (as is necessarily the case in lattice systems) the spin and charge
distributions will distort with time. One should also keep in mind that the local excitation,
like the one discussed above, will not fit in the low-energy/large-wavelength limit of the
Hubbard model.
Since practically every experiment deals with systems of finite extension, either due to
the geometry of the probe or due to the presence of imperfections, in what follows we will
focus on the Hubbard model on a finite chain with open boundary conditions. Although
many exact results have been derived for this case using the Bethe ansatz, properties like
e.g., the local densities cannot yet be calculated by exact methods. Therefore, numerical
tools such as t-DMRG are used to study the space-time evolution of the charge and spin
perturbations caused by adding a single electron to the ground state of the Hubbard model.
The first numerical simulations of spin-charge separation were performed by Jagla at
al. [124]. Using exact diagonalisation techniques they studied the space-time evolution of
the Gaussian wave packet with the predefined momentum k created in the ground state of
the system with up to L = 16 sites. Different velocities for the propagation of the charge
and spin wave packets have also been obtained by Kollath et al. [142, 143] in numerical t-
DMRG studies of the (inhomogeneous) Hubbard model on relatively large chains. In their
set-up the initial system was subjected to an external attractive field of Gaussian shape
and after releasing this field, the space-time evolution of the created charge- or spin-density
profiles were followed in time. Highly accurate t-DMRG simulations were performed by
Schmitteckert and Schneider [209] in order to show spin-charge separation in an L = 33 site
system with periodic boundary conditions which avoid Friedel oscillations. Quite recently
Ulbricht and Schmitteckert presented a two-terminal set-up where an excited state was
created by explicitly adding one electron Gaussian wave packet which was then transmited
through the interacting part of the chain which was modeled by the Hubbard Hamiltonian.
All these investigations where dealing with the Hubbard model far away from the Mott
insulating phase. In the studies presented here we will investigate the situations where

5.2. Computational setup 155
the electron is inserted in the Mott-insulating ground state of the Hubbard model or in
the ground state of the Hubbard model with one extra hole. Furthermore, an initially
spatially-localized electron will also probe the entire quasiparticle bands of the system.
5.2 Computational setup
5.2.1 Preparation of the initial state
In order to study the real-time dynamics of spin- and charge-densities in the 1D Hubbard
model, we first prepare the system in its ground state (GS)
H(N ↑ , N ↓ , U/J)|ψ GS 〉 = E GS |ψ GS 〉 . (5.4)
Here E GS and |ψ GS 〉 denote the ground state energy and the ground state wavefunction
of the Hubbard model (5.1) with N ↑ and N ↓ the total numbers of the up- and down-spin
electrons, and on-site interaction U. The initial state is then obtained by adding a single
particle with up-spin on site j at time t = 0 to the ground state of the system
|φ j (t = 0)〉 = α c † ↑ |ψ GS〉 . (5.5)
Here α denotes a normalization constant, chosen to ensure 〈φ j (0)|φ j (0)〉 = 1. The obtained
state is not an eigenstate of H and hence it will evolve in time
|φ j (t)〉 = e −iHt |φ j (0)〉. (5.6)
By adding an electron on a single site a localized wave packet consisting of all wave vectors
is constructed which then spreads out. Different components will move at different speeds,
given by the group velocity, determined as the slope of the dispersion curve at k. The
entire process is also schematically depicted in Fig. 5.1. Since the initial state (5.5) is
not an eigenstate of H, it will also not be the lowest energy state in this particle sector.
Moreover, due to the non-dissipative Hamiltonian dynamics the energy of the constructed
state will not change during the time evolution, hence it will never relax to the ground
state.
Since for the Hubbard model we already face the difficulty of an adequate analytical
description of the wavefunction at t = 0, numerical tools like DMRG and t-DMRG are
employed in order to determine the initial states of the system and to study their timeevolution.
At the beginning the ground state and the initial states are obtained using
“conventional” DMRG algorithms (infinite- plus finite-system algorithms, see Chapter 2)
by targeting both states with an equal weight. Later the adaptive t-DMRG algorithm (see
Chapter 3) is employed to carry out the time evolution of the initial state.

156 5. Real-time Dynamics of Spin and Charge Densities in the One-dimensional Hubbard Model
Figure 5.1: Schematic picture representing the creation of an initial state by adding
an up-spin electron to the ground state of the system and the subsequent space-time
evolution of the induced charge and spin densities.
5.2.2 Measurements
In order to probe the temporal effects of the added electron on the properties of the system,
we calculate the expectation values of local operators Ôi at site i
• in the ground state (equilibrium value):
〈Ôi〉 GS
:= 〈ψ GS |Ôi|ψ GS 〉 ; (5.7)
• in the time-evolved state:
〈Ôi〉 t (t) := 〈ψ(t)|Ôi|ψ(t)〉 ; (5.8)
• and the change from the corresponding equilibrium value:
at given times t.
δÔi(t) := 〈ψ(t)|Ôi|ψ(t)〉 − 〈ψ GS |Ôi|ψ GS 〉 = 〈Ôi〉 t (t) − 〈Ôi〉 GS (5.9)
Typically, the local operators considered here are the on-site charge n i = n i↑ + n i↓ and
spin S z i = 1 2 (n i↑ − n i↓ ). DMRG measurements are carried out at the end of the conventional
DMRG algorithm (〈Ôi〉 GS ) and at the end of the time-step sweeps for given times t
(〈Ôi〉 t (t)).

5.3. Real-time evolution in the tight-binding model 157
In the following we study the time evolution of the (up-spin) electron created in the
middle of the chain in the ground state of the half-filled and close to the half-filled system.
We consider the systems with open boundary conditions (OBC) which are preferable for
the used methods (DMRG and t-DMRG). The considered chain lengths range from L = 63
to L = 131 and the Hubbard interactions are taken to be U/J = 2, 4, 8, 20. For U = 2J the
bandwidth of the (anti)holon band is larger than the Hubbard gap, whereas for U 8J
the gap is considerably bigger.
5.2.3 Simulation parameters
In order to study the time evolution of the state obtained by creating a single electron
in the ground state of the given system we use the adaptive t-DMRG algorithm based
on the second order Suzuki-Trotter approximation (S-T(2)) of the time-evolution operator
(see Section 3.3). Most of the simulations were performed with the fixed discarded
weight threshold ε = 10 −10 and the Suzuki-Trotter time step δt = 0.02. In order to avoid
the state-space size blow-up during the simulations (with the possible crash of the executable
program), the number of DMRG kept states m was restricted to the interval
m ∈ [256, 1024]. To check the reliability of the data additional control runs with ε = 10 −12
and m ∈ [256, 2048], as well as a smaller time step δt = 0.01 were made. In a few cases the
control runs were also performed with the time-step targeting t-DMRG method based on
the Arnoldi approximation of the time-evolution operator (see Section 3.4). In this case the
number of DMRG kept states was limited to m max = 4048, the time step was chosen to be
δ = 0.5 and two intermediate states at times t + δt/3 and t + 2δt/3 with weights 1/6 each
were additionally targeted with the initial and final states of the time step with weights
1/3 each. The initial and the ground states were obtained using the conventional DMRG
method where both of them were targeted with the same weight 1/2. The evaluation of the
desired operators (DMRG measurements) was carried out at the end of the conventional
DMRG sweeps (〈Ôi〉 GS ) and at the final complete half sweeps of the time-step for a given
time t. The largest time scale considered is t = 20. The largest errors in the obtained data
are of the order O(10 −4 ) for the considered simulation parameters.
5.3 Real-time evolution in the tight-binding model
Before we turn to the consideration of the results obtained for the interacting U ≠ 0 (correlated)
system, let us first, for the sake of completeness consider the uncorrelated case,
U = 0. In this case the Hubbard Hamiltonian (5.1) reduces to the tight-binding model
which has a diagonal form in momentum space
ˆT = −2J ∑ k,σ
cos(k) c † kσ c kσ . (5.10)

158 5. Real-time Dynamics of Spin and Charge Densities in the One-dimensional Hubbard Model
Here c † kσ (c kσ
) creates (annihilates) an electron with wavenumber k and spin σ, and is an
inverse Fourier transform of the corresponding real-space creation (annihilation) operators
c kσ
= √ 1 ∑
e −ikj c jσ ,
L
j
k = 2πn
L
, n = 1, 2, . . ., L , (5.11)
in the case of periodic boundary conditions (PBC), or inverse Fourier sine transform
c kσ
=
√
2 ∑
√ sin(kj) c jσ ,
L + 1
j
k = πn , n = 1, 2, . . ., L , (5.12)
L + 1
in the case of open boundary conditions (OBC). As mentioned previously we focus only
on OBC in this chapter. Since
[n kσ , ˆT] = 0 , (5.13)
where n kσ = c † kσ c kσ
is the number operator of electron with spin σ and “wavenumber” k, the
number of electrons with given spin and momentum is a conserved quantity. This in turn
allows us to construct the ground and excited eigenstates of the Hamiltonian by adding
electrons with given momentum and spin to the vacuum state |0〉. Furthermore, since upand
down-spin sectors are completely independent from each other, all these eigenstates
can be written as a product of the corresponding eigenstates of the free spinless fermion
model with L sites and N e = N ↑ or N e = N ↓ , respectively. The model of free spinless
lattice fermions together with the time evolution of the different initial states is studied in
Appendix A. c Here we only use the results obtained there. The ground state of the system
of N ↑ and N ↓ electrons can be then written as a Fock state in momentum-space
|ψ GS 〉 =
( π
L+1 N ↑
∏
k= π
L+1
c † k↑
)( π
L+1 N ↓
∏
k= π
L+1
c † k↓
)
|0〉 . (5.14)
The simplest excitations above the ground state are obtained by adding (“particle” excitation
with charge −e and spin σ) or removing (“hole” excitation with charge e and spin
−σ) one electron. Multiparticle excited states are scattering states of particle and hole
excitations. The state obtained after adding an up-spin electron on site j at time t = 0 to
the ground state |ψ GS 〉 is given by
√
2
|φ j (0)〉 = αc † j↑ |ψ GS〉 = α√ L + 1
π
L+1 L
∑
k= π(N ↑ +1)
L+1
sin(kj)c † k↑
( π
L+1 N ↑
∏
k= π
L+1
c † k↑
)( π
L+1 N ↓
∏
k= π
L+1
c † k↓
)
|0〉 , (5.15)
c Note, that the hopping amplitude considered in Appendix A is J = 0.5 instead of J = 1 in this chapter.
Therefore, t there is equivalent to t/2 in this chapter.

5.4. Real-time evolution in the Hubbard model 159
√
where α = 1/〈ψ GS |c j↑
c † j↑ |ψ GS〉. The expectation values of local operators like charge and
spin densities are given as the sum and the difference of the corresponding on-site charge
densities of the spinless fermion model, e.g., the equilibrium values of the on-site charge
and spin densities are given by
and
〈n i 〉 GS = 〈n i↑ 〉 GS + 〈n i↓ 〉 GS = 〈n i 〉 Ne=N ↑
GS
〈S z i 〉 GS = 1 2
( ) 1
(
〈ni↑ 〉 GS − 〈n i↓ 〉 GS = 〈n i 〉 Ne=N ↑
GS
2
+ 〈n i 〉 Ne=N ↓
GS
(5.16)
)
− 〈n i 〉 Ne=N ↓
GS
, (5.17)
respectively.
One of the most important observations made in Appendix A, which can be generalized
for the present case of both up- and down-spin species, is that the time evolution of the
charge and spin densities, due to the creation of a single up-spin electron on site j at time
t = 0 in the ground state of the system with N ↑ and N ↓ electrons, is completely determined
by the momentum-space states of the up-spin electrons which are unoccupied in the ground
state of the host system. Hence the time evolution of the charge and spin densities are
identical in the tight-binding model.
5.4 Real-time evolution in the Hubbard model
Now we turn to the case when U ≠ 0. In this section we consider the time evolution of
the spin and charge densities due to the creation of an (up-spin) electron in the ground
state of the Hubbard model. In contrast to the case of non-interacting electrons U = 0,
excitations for U ≠ 0 cannot be described in terms of electronic degrees of freedom, i.e.,
electrons and holes, and an analytical treatment of the problem proves to be difficult if
not impossible. Therefore, we directly move to the analysis of the results obtained using
t-DMRG, and in the following subsections we consider the cases where the host system is
i) at half-filling (Mott insulator) (Section 5.4.1) or ii) close to it (“metal”) (Section 5.4.2).
Later, in Section 5.5 we address the quality of the charge- and spin-excitation dynamics
for both considered cases.
5.4.1 System at half-filling
In this section we consider the space-time evolution of the charge and spin densities induced
by creating an up-spin electron in the ground — Mott insulating — state of the half-filled
Hubbard model. Because of the particle-hole symmetry of the model Hamiltonian (5.1)
(see Chapter 1) this set-up is equivalent to the removal of the up-spin electron (creation of
the down-spin hole) from the ground state of the Hubbard model at half-filling. We prefer

160 5. Real-time Dynamics of Spin and Charge Densities in the One-dimensional Hubbard Model
the former formulation since in the following section we consider the one-hole ground state
to which an up-spin electron is added.
We start with the system on an open L = 64 chain with N ↑ = N ↓ = 32 up- and downspin
electrons, respectively, and the Hubbard interaction U = 8J. The total number of
particles N e = N ↑ + N ↓ = L (half-filling) and the z-projection of the total spin S z = 1 2 (N ↑−
N ↓ ) = 0. The initial state is then obtained by adding the up-spin electron on site j = 32 at
time t = 0 to the ground state of the system. Different results for this case are collected in
the six plots of Fig. 5.2 where the expectation values of the on-site charge and spin densities
in the ground state (blue) the initial state (cyan), and the difference between them (red)
(left column) are shown, as well as the space-time evolution of the change of the on-site
charge and spin densities from their corresponding equilibrium values (right column).
The perturbation in the charge density induced by the creation of the (up-spin) electron
at site 32 is well localized and a strong peak with a height almost 1 is observed in δn i (t = 0)
at i = 32 (see the red curve in Fig. 5.2a). On the other hand, the perturbation induced in
the spin density is extended over the entire system (see the red curve in Fig. 5.2c). These
different behaviors are explained by a (large) gap in the charge excitation spectrum causing
the exponential decay of the charge-charge correlations in the system and zero spin gap
(gapless spin excitation) leading to a power-law decay of the spin-spin correlation functions.
The instant response encompassing the entire system is due to the spin-spin correlations
already “encoded” in the ground state of the system. This becomes more obvious when the
ground state is written as a superposition of two unnormalized states
|ψ GS 〉 = n j↑ |ψ GS 〉 + (1 − n j↑ )|ψ GS 〉 = |ψ GS 〉 1 + |ψ GS 〉 0 , (5.18)
where |ψ GS 〉 1 := n j↑ |ψ GS 〉 and |ψ GS 〉 0 := (1 − n j↑ )|ψ GS 〉 are complementary projections of
the ground state with and without up-spin electron on the site j, respectively. Recall, that
n j↑ and hence (1 − n j↑ ) are projection operators, n 2 j↑ = n j↑ and (1 − n j↑) 2 = (1 − n j↑ ).
Now, creation of the up-spin electron on the site j destroys the projection |ψ GS 〉 1 , i.e.,
|φ j (0)〉 = αc † j↑ |ψ GS〉 = α(c † j↑ |ψ GS〉 1 + c † j↑ |ψ GS〉 0 ) = αc † j↑ |ψ GS〉 0 , (5.19)
and hence exposes the correlations in the ground state when the on-site densities are
measured in the obtained (initial) state.
Nevertheless, as the space-time evolution of the change of the on-site charge (δn i (t))
and spin (δSi z (t)) densities from the corresponding equilibrium values show (see Fig. 5.2b
and Fig. 5.2d) the charge- as well as the spin-perturbations are propagating from the small
region centered about site j = 32 and the value of the on-site spin density obtained at
t = 0 + at any given site stays unchanged until the spin or charge wave fronts reach this
site. False-color plots show in Fig. 5.2b and Fig. 5.2d, respectively, the initial charge and
spin perturbations “melting” and later splitting up into counter-propagating excitations.

5.4. Real-time evolution in the Hubbard model 161
〈n i
〉 GS
, 〈n i
〉 t
(t=0), δn i
(t=0)
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
L=64
N ↑
=32(+1), N ↓
=32
U = 8J
〈n i
〉 GS
〈n i
〉 t
(t=0)
δn i
(t=0)
-30 -20 -10 0 10 20 30
i-32
(a)
(b)
0.5
〈S i
z
〉GS , 〈S i
z
〉t (t=0), δS i
z
(t=0)
0.4
0.3
0.2
0.1
0
-0.1
L=64
N ↑
=32(+1), N ↓
=32
U = 8J
〈S i
z
〉GS
〈S i
z
〉t (t=0)
δS i
z
(t=0)
-0.2
-30 -20 -10 0 10 20 30
i-32
(c)
(d)
0.5
〈S i
z
〉GS , 〈S i
z
〉t (t=0), δS i
z
(t=0)
0.4
0.3
0.2
0.1
0
-0.1
L=64
N ↑
=32(+1), N ↓
=32
U = 8J
〈S i
z
〉GS
〈S i
z
〉t (t=0)
δS i
z
(t=0)
-0.2
-30 -20 -10 0 10 20 30
i-31.5
(e)
(f)
Figure 5.2: Plots (b,d,f): space-time evolution of the additional charge (b), spin (d)
and bond-averaged spin (f) densities due to the creation of up-spin electron at t = 0 on
site j = 32 of open L = 64 chain in the ground state of the half-filled Hubbard model
with U = 8J, and N ↑ = N ↓ = 32 up- and down-spin electrons, respectively. Plots
(a,c,e): charge- (a), spin- (c) and bond-averaged spin-density (e) distributions in the
ground state (blue), the initial state (cyan), and the difference between them (red).

162 5. Real-time Dynamics of Spin and Charge Densities in the One-dimensional Hubbard Model
The charge- and spin-density wave fronts can be clearly identified on these plots as the
sides of isosceles triangles. An additional structure left behind these fronts is also seen, as
well as in the course of time reducing value of the maximum on-site density within the wave
front. One of the main reasons for these effects, also present in the non-interacting case
(see Appendix A, or Fig. 5.7), is the non-linear energy dispersions of the corresponding
quasiparticles and the finite-energy bandwidths.
Due to unequal lengths of chain parts in which the system is divided by the site j = 32
an appreciable asymmetry with respect to this position is present in the spin distribution
in the initial as well as in the time evolved states, whereas the charge distribution is less
affected. This asymmetry can be avoided by considering the systems with an odd number
of sites. In this case, however, the z-projection of the total spin is always nonzero (S z ≠ 0)
in the half-filled system (N e = L) and extra effects are already expected in the spin-density
distribution in the ground state. Nonetheless, to measure the velocities of the spin and
charge wave fronts, the symmetric propagation is quite helpful. This will also be important
in the next subsection where the host systems close to half-filling are studied. Therefore,
in the following we switch to the systems with an odd number of lattice sites.
Since the charge-density distribution is not affected by N ↑ ≠ N ↓ , in Fig. 5.3 we only
show the on-site spin density as a function of position i in the ground state 〈Si z〉 GS (blue),
the initial state 〈Si z〉 t (0) (cyan) and the difference between them δSz i (0) (red) for three
different chain sizes L = 63, 64, 65 and S z = − 1, 0, 2 −1 , respectively. A well defined antiferromagnetic
spin-density wave (SDW) pattern can be identified in the ground states of the
2
systems with an odd number of sites (see the blue curves in Fig. 5.3a and Fig. 5.3e). This
pattern might be associated with the spin quasiparticle (quasiparticles) which is (are) well
delocalized in the system and hence repelled form the open (sometimes also called reflecting)
chain ends. At least for the spin-1/2 Heisenberg model (see Section 1.3), which gives
an effective description of the studied system in the limit of large interactions U/J → ∞,
it is well known [59] that the ground state of the system is a single spinon state in the
case of odd chain length and periodic boundary conditions (PBC). Typically, SDW is not
seen in the system with PBC, because of translational invariance, but in the case of OBC
spinons might be repelled or pinned by the boundaries (repelled for the considered system)
and the SDW becomes visible in the ground state. The ground-state distribution of
the spin-density also shows that one has to distinguish between the systems with L − 1
divisible by 4 or giving a residual equal to 2. In the following we refer to them as “4-even”
and “4-odd” systems, respectively. In the 4-even systems, like L − 1 = 64, the on-site spin
density in the middle of the chain (the site 33) is negative, whereas for the 4-odd systems,
like L − 1 = 62 it is positive (the site 32). This implies that in 4-odd (4-even) systems
the antiferromagnetic oscillation in the initial state is enhanced (reduced) with respect to
the SDW in the ground state (compare the cyan and blue curves in Figs. 5.3a and 5.3e).
Moreover, in the 4-odd systems two antiphase domain walls (“π” shifts) appear at 28 − 29

5.4. Real-time evolution in the Hubbard model 163
0.5
0.5
〈S i
z
〉GS , 〈S i
z
〉t (t=0), δS i
z
(t=0)
0.4
0.3
0.2
0.1
0
-0.1
L=63
N ↑
=31(+1), N ↓
=32
U = 8J
〈S i
z
〉GS
〈S i
z
〉t (t=0)
δS i
z
(t=0)
〈S i
z
〉GS , 〈S i
z
〉t (t=0), δS i
z
(t=0)
0.4
0.3
0.2
0.1
0
-0.1
L=63
N ↑
=31(+1), N ↓
=32
U = 8J
〈S i
z
〉GS
〈S i
z
〉t (t=0)
δS i
z
(t=0)
-0.2
-30 -20 -10 0 10 20 30
i-32
(a)
-0.2
-30 -20 -10 0 10 20 30
i-31.5
(b)
0.5
0.5
〈S i
z
〉GS , 〈S i
z
〉t (t=0), δS i
z
(t=0)
0.4
0.3
0.2
0.1
0
-0.1
L=64
N ↑
=32(+1), N ↓
=32
U = 8J
〈S i
z
〉GS
〈S i
z
〉t (t=0)
δS i
z
(t=0)
〈S i
z
〉GS , 〈S i
z
〉t (t=0), δS i
z
(t=0)
0.4
0.3
0.2
0.1
0
-0.1
L=64
N ↑
=32(+1), N ↓
=32
U = 8J
〈S i
z
〉GS
〈S i
z
〉t (t=0)
δS i
z
(t=0)
-0.2
-30 -20 -10 0 10 20 30
i-32
(c)
-0.2
-30 -20 -10 0 10 20 30
i-31.5
(d)
0.5
0.5
〈S i
z
〉GS , 〈S i
z
〉t (t=0), δS i
z
(t=0)
0.4
0.3
0.2
0.1
0
-0.1
L=65
N ↑
=32(+1), N ↓
=33
U = 8J
〈S i
z
〉GS
〈S i
z
〉t (t=0)
δS i
z
(t=0)
〈S i
z
〉GS , 〈S i
z
〉t (t=0), δS i
z
(t=0)
0.4
0.3
0.2
0.1
0
-0.1
L=65
N ↑
=32(+1), N ↓
=33
U = 8J
〈S i
z
〉GS
〈S i
z
〉t (t=0)
δS i
z
(t=0)
-0.2
-30 -20 -10 0 10 20 30
i-33
(e)
-0.2
-30 -20 -10 0 10 20 30
i-32.5
(f)
Figure 5.3: On-site (a,c,e) and bond-averaged (b,d,e) spin-density distributions in
the ground state (blue), initial state (cyan), and difference between them (red). An
up-spin electron is created on site j at t = 0 in the ground state of the half-filled Hubbard
model with U = 8J. Further parameters of the system are: j = 32, L = 63,
N ↑ = 31, N ↓ = 32 (a,b); j = 32, L = 64, N ↑ = N ↓ = 32 (c,d); j = 33, L = 65,
N ↑ = 32, N ↓ = 33 (e,f).

164 5. Real-time Dynamics of Spin and Charge Densities in the One-dimensional Hubbard Model
(a)
(b)
(c)
(d)
(e)
(f)
Figure 5.4: Space-time evolution of the spin density (a,b) and of the change of onsite
(c,d) and bond-averaged (e,f) spin-densities from their corresponding equilibrium
values due to the creation of an up-spin electron on site j = 33 at t = 0 in the ground
state of the half-filled Hubbard model with U = 8J. Further parameters of the system
are: j = 32, L = 63, N ↑ = 31, and N ↓ = 32 (a,c,e); j = 33, L = 65, N ↑ = 32, and
N ↓ = 33 (b,d,f).

5.4. Real-time evolution in the Hubbard model 165
and 35 − 36 sites, close to the position j = 32 where the up-spin electron was added to
the system (see the cyan curve in Fig. 5.3a). Note, that these features of 4-odd and 4-even
systems depend on whether the sign of the local spin density on the site where the up-spin
electron is added is positive or negative.
In order to even out the above discussed antiferromagnetic oscillations, in the following
we also consider the bond-averaged spin density, given by
¯S z i+1/2 = 1 2 (Sz i + Sz i+1 ) . (5.20)
However, to avoid bad surprises due to this procedure, e.g., a complete disappearance of the
spin perturbation signal, we first perform a comprehensive analysis of the possible effects.
In Fig. 5.4 and Fig. 5.2 we present the space-time evolutions of 〈Si z 〉 t (t), δSi z (t) and δ ¯S i z (t)
for L = 63 (4-odd), L = 65 (4-even), as well as L = 64, system sizes. d On the false-color
plots displaying the space-time evolution of 〈Si z 〉 t (t) a reduction of the antiferromagnetic
oscillations can be identified behind the charge wave fronts for all considered chain sizes
(see Figs. 5.4a, 5.2d, and 5.4b for L = 63, 64, and 65, respectively). This can be explained
by a remaining interaction between spin and charge quasiparticles. The main “signal”
corresponding to the maxima in the on-site spin density is also well distinguishable in
all three cases. If the space-time evolution of the spin density is followed by studying
its change from the corresponding equilibrium value (δSi z (t)) (see Figs. 5.4c, 5.2b, and
5.4d), one encounters an enhancement instead of the above mentioned reduction of the
antiferromagnetic oscillations behind the charge wave fronts. Moreover, for 4-even system
this enhancement is supplemented by an additional complete spin flip (π-shift). One should
admit, however, that the main signal still remains distinguishable for all three considered
chain sizes. All these complications are absent in the space-time evolution of the change
of bond-averaged spin-densities δ ¯S i z (t), as defined by Eqs. (5.20) and (5.9), and it allows
us to easily follow the space-time evolution of the maxima in the on-site spin densities (see
Figs. 5.4e, 5.2f and 5.4f).
Now we can proceed and investigate the evolution of the spin- and charge-density
perturbations induced by adding an (up-spin) electron to the ground state of the half-filled
Hubbard model in the middle (site j = (L + 1)/2) of an open chain with an odd number
of sites L. We show in Fig. 5.5 the space-time evolution of the change of the on-site
charge δn i (t) (left column) and bond-averaged spin δ ¯S i z (t) (right column) densities from the
corresponding equilibrium values for three different interaction strengths U/J = 4, 8, 20.
The size of the considered system is L = 65 and S z = − 1 in the host system. As one sees,
2
both perturbations — charge and spin — induced at t = 0 + melt and later split up into
the counter-propagating excitations. Well defined density fronts can be identified, as the
sides of isosceles triangles on the corresponding plots in Fig. 5.5, for both densities and for
d For L = 64 〈S z i 〉 t (t) = δSz i (t) since 〈Sz i 〉 GS = 0.

166 5. Real-time Dynamics of Spin and Charge Densities in the One-dimensional Hubbard Model
(a)
(b)
(c)
(d)
(e)
(f)
Figure 5.5: Space-time evolution of the charge- (a,c,e) and bond-averaged spindensity
(b,d,e) deviations from their corresponding equilibrium values due to the creation
of an up-spin electron on site 33 at t = 0 in the ground state of the half-filled
Hubbard model on an open L = 65 chain, with N ↑ = 32 and N ↓ = 33 up- and downspin
electrons, respectively, and for U = 4J (a,b), U = 8J (c,d), and U = 20J (e,f)
interaction strengths.

5.4. Real-time evolution in the Hubbard model 167
0.12
U = 8J
0.08
δn i
(t)
0.04
0
δS _ i (t)
0.12
0.08
0.04
L=131, N ↑
=65(+1), N ↓
=66, j = 66
L=65, N ↑
=32(+1), N ↓
=33, j = 33
t = 5, L = 131
t = 5, L = 65
t = 10, L = 131
t = 10, L = 65
0
-30 -20 -10 0 10 20 30
i - j
(a)
δn i
(t)
0.12
0.1
0.08
0.06
0.04
0.02
0
U = 8J
|i - j| = 10, L = 131
|i - j| = 20, "
|i - j| = 10, L = 65
|i - j| = 20, "
δS _ i (t)
0.05
0.04
0.03
0.02
0.01
0
L=131, N ↑
=65(+1), N ↓
=66, j = 66
L=65, N ↑
=32(+1), N ↓
=33, j = 33
0 2 4 6 8 10 12 14 16 18 20
time (t)
(b)
Figure 5.6: Plot (a): Snapshots of the time evolution of the change of the charge and
bond-averaged spin densities from their corresponding equilibrium values at times t = 5
and t = 10. Plot (b): time-evolution of the on-site charge- and bond-averaged spindensity
deviations from their corresponding equilibrium values at sites i with |i − j| = 10
and |i − j| = 20. Arrows of the corresponding colors indicate the main (typically the
first) maxima in the on-site charge and bond-averaged spin densities. An up-spin
electron was added at t = 0 to the ground state of the half-filled Hubbard model in the
middle of open L = 65 and L = 131 chains. Interaction U = 8J and S z = − 1 in the
2
ground state.

168 5. Real-time Dynamics of Spin and Charge Densities in the One-dimensional Hubbard Model
all considered values U. Their respective velocities are different and the spin and charge
densities evolve more or less as non-interacting perturbations, i.e., the spin and the charge
have separated. This can be also identified in Fig. 5.6 where snapshots of δn i (t) and δ ¯S i z(t)
at times t = 5 and t = 10 (upper plot), as well as time-profiles at sites |i − j| = 10 and
|i − j| = 20 (bottom plot) are shown for U = 8J. Equivalently, these plots represent the
vertical (Fig. 5.6a) and the horizontal (Fig. 5.6b) cross-sections of the false-color plots in
Fig. 5.5c and Fig. 5.5d at times t = 5 and t = 10, and positions |i − j| = 10 and |i − j| = 20,
respectively. Plots in Fig. 5.6 also contain the data obtained for a two times larger system,
namely the half-filled Hubbard model with L = 131 sites and N ↑ = 65 and N ↓ = 66 upand
down-spin electrons in the ground state. Surprisingly, the charge-density profiles are
almost independent of the chain size and the propagation of the charge perturbation is
indistinguishable in both systems until the wave fronts reach the chain boundaries (see the
upper plot in Fig. 5.6a). The same holds for other chain sizes, too. The bond-averaged
spin-density profiles depend on the system size, although only weakly and mainly due
to 4-even 4-odd chains — L = 65 is 4-even and L = 131 is 4-odd, respectively. Later in
Section 5.5 it is shown that the induced charge and spin perturbations spread ballistically
before the boundaries are reached. In time evolutions of the change of the charge and the
bond-averaged spin densities, see Fig. 5.6b, small dips (the green and cyan solid curves,
L = 131) and rises (the blue dashed and black dash-dot curves, L = 65) are encountered
in δ ¯S i z(t) just after the maxima in the on-site charge density δn i (t) pass the corresponding
site/bond. These “extra” deviations, already discussed above, should be the signatures of
the remaining interactions between the charge and spin degrees of freedom. Irrespective of
this, even for extremely localized spin and charge perturbations such as a single electron
inserted in the ground state of the system on a well defined site, robust effects of spin-charge
separation are observed in the obtained numerical data. From that it can be deduced that
the added electron shortly after the insertion decays into spin and charge excitations.
The speed of the charge- (vc wf ) and spin-density (vwf s ) wave fronts can be determined
by measuring the time at which the main (typically the first) maximum in the deviation
of the on-site charge δn i (t) and the bond-averaged spin δ ¯S i z (t) densities arrive at different,
but well defined positions; the procedure similar to the one shown in Fig. 5.6b where
the arrows of the corresponding color indicate the appearance of the main maxima. An
alternative set-up would be to measure the time at which the change in the local density
becomes larger than an initially defined small threshold, that must however be significantly
larger than the error in the obtained data. In this thesis the former approach is used.
The velocities of the charge vc
wf and spin vs
wf wave-fronts for four different values of the
interaction U/J = 2, 4, 8, 20, as well as for the non-interacting system U = 0, and for the
limit of large interaction U/J → ∞, are listed in Tab. 5.1. The space-time evolution of
the on-site charge density for the latter two cases are shown in Fig. 5.7. As discussed
previously (see Section 5.3), in the case of the non-interacting system, the time evolutions

5.4. Real-time evolution in the Hubbard model 169
U/J
v wf
c
v wf
s D c 2 D s 2
0 1.90 1.90 1.98 1.98
2 2.11 1.53 3.12 1.50
4 2.07 1.12 2.64 0.90(6)
8 1.98 0.67(2) 2.22 0.33(7)
20 1.91 0.24(6) 2.02 0.07(0)
∞ 1.91 − 2.00 −
Tab. 5.1: Velocities of the charge- vc
wf and the spin-density vs
wf wave-fronts due to
the creation of an up-spin electron on site 33 at t = 0 in the ground state of the halffilled
Hubbard model on an open L = 65 chain with N ↑ = 32 and N ↓ = 33 up- and
down-spin electrons, respectively. Also listed are the rates of the ballistic spreading of
the induced charge and spin densities D2 c and D2, s respectively.
of the charge and spin perturbations coincide. The maximum group velocity available in
this system is vc
max = vs
max = 2J (see Appendix A). The obtained wave-front speed is yet
smaller, vc
wf = vs
wf ≈ 1.90, because of the definition/set-up used to determine it — position
of the density maximum in the wave front. In the U/J → ∞ limit, as far as only the
charge-density is considered, the system behaves as if the up-spin electron was created
in the middle of a fully polarized ferromagnetic system with N e = L down-spin electrons.
Therefore, a created up-spin electron in the U/J → ∞ limit behaves similar to the single
free spinless fermion spreading in the empty system, or equivalently as a single free spinless
hole spreading in the completely filled system. Recall that the Hubbard Hamiltonian
(5.1) is particle-hole symmetric. The speed of the charge wave-fronts is vc
wf ≈ 1.91. For
intermediate values of U the enhancement of the charge-spreading velocity up to 10%
is observed and the expansion depends on U only moderately, whereas the speed of the
spin wave-front spreading is strongly influenced by U, as it was already anticipated in
Section 5.1. This is in strong contrast to the charge and spin dynamics in the ionic chain
(see Appendix B), which represents the simplest model describing the band insulator at
half-filling. As in the case of the tight-binding model, the eigenstates of this model are
constructed by well defined quasiparticles with charge and spin, and the time evolution of
the charge- and spin- density perturbations coincide. The gap between the filled valence
and the empty conduction bands is equal to the strength of the alternating on-site energy
modulation ∆ — the so-called ionic potential. An increase of the ionic potential reduces
the bandwidth of the lower (valence) and upper (conducting) energy subbands, strongly
renormalizes the maximum group velocity, and consequently the speed of the charge/spin
wave-fronts in the system.
The error in vc wf , also taking into account the systems with different chain sizes, is up

170 5. Real-time Dynamics of Spin and Charge Densities in the One-dimensional Hubbard Model
(a)
(b)
Figure 5.7: Space-time evolution of the charge-density deviation from its corresponding
equilibrium value due to the creation of an up-spin electron on site 33 at t = 0
in the ground state of the half-filled (N ↑ = 32 N ↓ = 33) and empty (N ↑ = N ↓ = 0)
non-interacting (U = 0) Hubbard model on an open L = 65 chain in plots (a) and (b),
respectively.
to 3% (largest for U = 0), whereas the error in v wf
s reaches ≈ 10% in the case of U = 20J.
The latter is due to large time scales required for strong interactions in order to obtain
more accurate results for the speed of the spin wave fronts — in the presented simulations
the time evolution was only carried out until t max = 20.
5.4.2 System close to half-filling
In this section we consider a set-up where the system to which the (up-spin) electron is
added is not half-filled but contains an extra hole in the ground state. We only consider
systems with an odd number of lattice sites, for which only the modulations in the on-site
charge-density are present in the ground state, whereas for the chains with an even number
of sites extra spin-density modulations are also expected since S z ≠ 0. These charge and
spin-density modulations, which are typically not seen in the case of PBC because of the
translational invariance of the system, are pinned or repelled by boundaries in the case of
OBC and hence become visible in the ground-state spin- and charge-density distributions.
The low-energy properties of the one-hole states have been extensively investigated in
the context of the isotropic and the anisotropic t − J models, see e.g., Refs. [217, 220,
248, 262] and references therein. The former is particularly interesting since it gives an
effective description of the Hubbard model in the strong coupling limit, U/J → ∞ (see
Chapter 1). It was shown in Ref. [262] that the ground state of the t − J model with PBC

5.4. Real-time evolution in the Hubbard model 171
〈n i
〉 GS
1.01
1
0.99
0.98
U = 0
U = 2J
U = 4J
U = 8J
U = 20J
U → ∞
0.97
0.96
L=65, N ↑
=N ↓
=32
-30 -20 -10 0 10 20 30
i - 33
Figure 5.8: Charge density distribution in the ground state of the Hubbard model on
an open L = 65 chain with N ↑ = N ↓ = 32 up- and down-spin electrons, respectively,
and different interaction strengths U/J = 0, 2, 4, 8, 20, as well as the U/J → ∞ limit.
is the one-holon state when the system has an odd number of sites, whereas for even chain
sizes it is given as a coupled spinon-holon ground state, not a bound state though.
As in the previous subsection, here we mainly consider the Hubbard model on an
open L = 65 chain. The total numbers of up- and down-spin electrons are N ↑ = N ↓ = 32,
respectively (S z = 0). At first, we study the charge-density distribution in the ground state
of the system and in Fig. 5.8 show 〈n i 〉 GS
as a function of position i for four different values
of the interaction U/J = 2, 4, 8, 20, as well as for the non-interacting system U = 0, and
for the limit of large interaction U/J → ∞. For the latter two cases an analytical form
of the on-site charge density can be obtained. In the non-interacting case (U = 0) from
Eqs. (5.16) and (A.15) follows
〈n i 〉 GS
= 〈n i 〉 Ne=N ↑
GS
+ 〈n i 〉 Ne=N ↓
GS
= L
L + 1 − 1 sin(2k F,↑ i) − sin(2k F,↓ i)
2(L + 1) sin ( ) (5.21)
πi
L+1
and for the considered system parameters: odd L and N ↑ = N ↓ = (L + 1)/2,
〈n i 〉 GS
= 〈n i 〉 Ne=N ↑
GS
+ 1 − 〈n i 〉 Ne=L−N ↓
GS
= L
L + 1 + cos(πi)
L + 1 . (5.22)
Here, the particle-hole transformation for the down-spin electrons and the symmetry of the
tight-binding model with respect to this transformation are used. In general, there are two
Friedel oscillations with wavenumbers 2k F,↑ = 2k F,↓ = 2π
L+1 (N ↑ + 1/2) in the charge-density

172 5. Real-time Dynamics of Spin and Charge Densities in the One-dimensional Hubbard Model
distribution resulting for the considered parameters in one standing wave with wavenumber
π, Eq. (5.22) (see also the grey asterisks in Fig. 5.8). In the case of U/J → ∞, as far as only
the charge-density is considered, the system behaves as a fully polarized ferromagnet with
N e = N ↑ + N ↓ down-spin electrons e for which one has a gas of N h = L − N e free holes.
Therefore, the ground-state charge-density distribution of the system with N e = L − 1
electrons in the U/J → ∞ limit is given by the fermion distribution of the free spinless
fermion model with the same filling (A.15)
〈n i 〉 GS = 〈n i 〉 Ne=L−1
GS
= 1 − 〈n i 〉 Ne=1
GS
= L
L + 1 + 1 ( ) 2πi
L + 1 cos , (5.23)
L + 1
and there is only a 2k F,c = 2(k F,↑ + k F,↓ ) oscillation in the system (the bold black curve in
Fig. 5.8). The hole is delocalized in the system, reflected from boundaries, and produces in
the charge-density distribution a shallow cosine-shaped trough of depth 2/(L + 1) centered
about the middle of the open chain (at site 33 for L = 65).
For intermediate values of U all three oscillations with wavenumbers 2k F,↑ , 2k F,↓ , and
2k F,c , respectively, are present in the on-site charge density (see e.g. Ref. [19, 214]), as is also
seen in Fig. 5.8, where effective oscillations with the wavenumber π are identified on top of
the background of a shallow trough of depth ≈ 2/(L + 1) centered about the middle of the
chain. The amplitude of “π”-oscillations reduces with increasing U and the ground-state
charge-density distribution for U = 20J becomes quite close to the results for U/J → ∞
limit (the blue circles in Fig. 5.8). This gradient in on-site charge density (directed from
the chain center to the chain ends) plays an important role in the propagation of the
charge-density perturbation as will be seen in the following.
In contrast to the on-site charge density, the on-site spin density is homogeneous and
equals zero for all values of the interaction U.
Now we proceed and consider the time evolution of the additional charge and spin
densities due to the creation of an up-spin electron on site J = 33 at t = 0 in the ground
state of the above considered system. We show in Fig. 5.10 the space-time evolution of
the change of the on-site charge δn i (t) (left column) and the bond-averaged spin δ ¯S z i (t)
(right column) densities for three different values of the interaction, U/J = 4, 8, 20. We
also display the charge- and spin-density distributions in the ground state (blue) initial
state (cyan), and the difference between them (red) for U = 8J in Fig. 5.9. Note that
δ ¯S z i (t) = 〈 ¯S z i 〉 t (t) since 〈 ¯S z i 〉 GS
= 0. At t = 0 + , the induced spin-density perturbation is
less extended in space, as is seen in Fig. 5.9b (the cyan curve), compared to the case with
the host system at half-filling considered in the previous section (Section 5.4.1). It melts
e In general, one does not distinguish between up- and down-spin electrons, since in the absence of
magnetic fields the model is — at least — invariant under the reversal of all spins, however this symmetry
is broken by predefining the spin of the electron inserted in the system at t = 0 — the up-spin in the
considered cases.

5.4. Real-time evolution in the Hubbard model 173
2
0.5
〈n i
〉 GS
, 〈n i
〉 t
(t=0), δn i
(t=0)
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
L=65
N ↑
=32(+1), N ↓
=32
U = 8J
〈n i
〉 GS
〈n i
〉 t
(t=0)
δn i
(t=0)
〈S i
z
〉GS , 〈S i
z
〉t (t=0), δS i
z
(t=0)
0.4
0.3
0.2
0.1
0
-0.1
L=65
N ↑
=32(+1), N ↓
=32
U = 8J
〈S i
z
〉GS
〈S i
z
〉t (t=0)
δS i
z
(t=0)
0
-30 -20 -10 0 10 20 30
i-33
(a)
-0.2
-30 -20 -10 0 10 20 30
i-33
(b)
Figure 5.9: On-site charge- (a) and spin-density (b) distributions in the ground state
(blue), the initial state (cyan), and the difference between them (red). An up-spin
electron is added on site 33 at t = 0 to the ground state of the Hubbard model on an
open L = 65 chain with N ↑ = N ↓ = 32 up- and down-spin electrons, respectively. The
Hubbard interaction is U = 8J.
and later splits up into counter-propagating excitations (see the right column in Fig. 5.10).
Well defined spin-density fronts can be identified, as the sides of isosceles triangles on
the corresponding false-color plots in Fig. 5.10, and the observed space-time evolution is
quite similar to the one studied in the previous section (compare to the right column of
Fig. 5.5). In contrast, the space-time evolution of the induced charge density is rather
different. At t = 0 + , it is well localized at site 33, as is seen in Fig. 5.9a, similar to the case
of a Mott-insulating host system (see Section 5.4.1), although in this case not all charge
excitations are gapped as this was the case in the preceding example. For short time scales,
t < 2, the space-time evolution is almost identical to the corresponding examples from the
previous section. The major differences set in later in time when the induced charge density
stays predominantly concentrated about the initial position rather than within the wave
fronts, which are also less pronounced in this case. The confinement around the initial
position becomes stronger with growing on-site interaction U. As will be shown in the
following section the spreading of the charge perturbation is comparatively subdiffusive in
these cases. The charge-density oscillations with the time period inverse proportional to
the Hubbard gap are also distinguishable in the charge-density wave fronts. Moreover, for
U = 4J the rise in the charge-density within the wave front starts to transform into a dip,
as is seen at t ≈ 3 and later at t ≈ 9 in Fig. 5.10a as the light blue spots. All observed
differences are related to the scattering of the propagating charge quasiparticles on the
shallow gradient in the on-site charge-density in the ground state of the host system. This

174 5. Real-time Dynamics of Spin and Charge Densities in the One-dimensional Hubbard Model
(a)
(b)
(c)
(d)
(e)
(f)
Figure 5.10: Space-time evolution of the charge- (a,c,e) and bond-averaged spindensity
(b,d,e) deviations from their corresponding equilibrium values due to the creation
of an up-spin electron on site 33 at t = 0 in the ground state of the half-filled
Hubbard model on an open L = 65 chain with N ↑ = N ↓ = 33 up- and down-spin electrons,
respectively, and for U = 4J (a,b), U = 8J (c,d), and U = 20J (e,f) interaction
strengths.

5.4. Real-time evolution in the Hubbard model 175
0.12
U = 8J
0.08
δn i
(t)
0.04
0
δS _ i (t)
0.12
0.08
0.04
L=131, N ↑
=65(+1), N ↓
=65, j = 66
L=65, N ↑
=32(+1), N ↓
=32, j = 33
t = 5, L = 131
t = 5, L = 65
t = 10, L = 131
t = 10, L = 65
0
-30 -20 -10 0 10 20 30
i - j
Figure 5.11: Plot (a): snapshots of the time evolution of the change of the charge
and bond-averaged spin densities from their corresponding equilibrium values at times
t = 5 and t = 10. An up-spin electron was added at t = 0 to the ground state of
the Hubbard model in the middle (j = 33 and j = 66, respectively) of open L = 65
and L = 131 chains. Interaction U = 8J and N ↑ = N ↓ = 32 and N ↑ = N ↓ = 65 for
L = 65 and L = 131, respectively.
explains why the confinement becomes stronger with increasing on-site interaction and why
it becomes weaker in the larger systems, as is seen in Fig. 5.11, where snapshots of δn i (t) at
times t = 5 and t = 10 are shown for L = 65 and L = 131 system sizes and U = 8J (upper
plot). The depth of the gradient in the charge-density distribution in the ground state of
the Hubbard model on an open L = 131 chain is twice smaller than in the system with
L = 65 (≈ 2/(L + 1)). The bond-averaged spin-density profiles are almost independent of
the chain size and the propagation of the spin perturbation is indistinguishable in both
systems until the wave fronts reach the chain boundaries (see also the bottom plot in
Fig. 5.11). The same holds for other odd chain sizes too. A similar behavior, namely
the reflection of the propagating charge perturbation and passing of the spin one, were
earlier observed for the Hubbard chain embedded in a harmonic trap [142, 143]; there a
Mott-insulating plateau is created in the middle of the system (for appropriately chosen
parameters) and the charge-density gradients are adjacent to it, which reflect to this plateau
approaching charge wave fronts.
In the case of the ionic-chain close to half-filling the situation is different. Although
the charge modulation with wavenumber π is always present in this system, the reason for

176 5. Real-time Dynamics of Spin and Charge Densities in the One-dimensional Hubbard Model
(a)
(b)
Figure 5.12: Space-time evolution of the change of charge (a) and bond-averaged
spin densities from corresponding equilibrium values due to the creation of up-spin
electron on site 17 at t = 0 in the ground state of fully polarized Hubbard model on an
open L = 65 chain with N ↑ = N ↓ = 32 up- and down-spin electrons, respectively, and
U = 8J.
its existence is the alternating on-site potential and not the Friedel oscillations. Moreover,
since there is no interaction among the constituent parts of the system, an extra particle
added to the system at t = 0 only sees the Fermi sea and not the spatial distribution of
the electrons. The induced charge (equivalently spin) perturbation propagates freely in the
ionic-chain. However, there are some similarities, too: the oscillations with the time period
inverse proportional to the gap are present in the wave fronts (see Appendix B), which are
also less pronounced in this case as compared to the free spinless lattice fermion model
(tight-bindig model). Nevertheless, the charge-spin wave front spreads with the constant
speed determined by the strength of the ionic potential and not the filling — at least for
N e L (see Appendix B).
We close this section by considering the set-up where an additional (up-spin) electron
is created at site j = 17 and not in the middle j = 33 of the open L = 65 chain at t = 0.
The space-time evolution of δn i (t) and δ ¯S i z (t) for U = 8J are shown in 5.12. The two
important modifications of the previously observed charge and spin-density dynamics are:
a) the center of mass of the change of the charge-density distribution from its corresponding
equilibrium value slowly moves to the center of the system (minimal on-site charge-density
in the ground state of the host system) and b) when the propagating charge perturbation
starts to “feel” the increasing gradient in the on-site charge density it starts to scatter
and later at t ≈ 10 around the position i − j ≈ 10 the rise within the charge wave fronts
transforms into a dip, but the wave front keeps propagating with almost the same speed.

5.5. Ballistic vs. subdiffusive dynamics 177
The spin part remains more or less unchanged, although the asymmetry is noticeable in the
time-evolving states due to the different lengths of the chain parts into which the system
is divided by the site where the extra electron is added.
5.5 Ballistic vs. subdiffusive dynamics
In this section we try to qualitatively characterize the type of the spreading of charge
and spin perturbations induced by adding an (up-spin) electron to the ground state of
a given system. Following the detection scheme widely used in laboratory experiments,
e.g., on cold atomic gases in optical lattices [23, 152], we are interested in the form of the
spatial spreading of an initially localized perturbation, which we characterize by the second
moment (variance) given by
σ 2 (t) =
L∑
(i − µ(t)) 2 ρ i (t) , µ(t) =
i=1
L∑
i ρ i (t) . (5.24)
i=1
Here ρ i (t) are the occupation probabilities of lattice sites i, ∑ i ρ i (t) = 1 and µ(t) is the first
moment of the distribution. The time dependence of the variance allows us to distinguish
e.g., ballistic from diffusive regimes. The dynamics is considered to be ballistic, if the
variance (5.24) grows quadratically in time, which is the well known behavior of noninteracting
particles, while diffusive behavior manifests itself in the linear increase in time.
In the case of free spinless lattice fermions with a cosine energy dispersion (2J cos(k)),
if a single particle is spreading in an empty lattice, ρ i (t) ≡ 〈n i 〉 t (t) and
σ 2 FG(t) = 1 π
∫ π
0
dk (2Jt sin(k)) 2 = (2J)2
2
t 2 ,
(5.25a)
whereas in the case of partially filled host system, ρ i (t) ≡ δn i (t) (5.9), and
∫π
σFG 2 (t) ≈ 1 dk (2Jt sin(k)) 2 = σFG 2 π − k F
k F
(0) + DFG 2 t 2 , (5.25b)
where D FG
2 is a positive real number and k F is the Fermi momentum (see Section 5.3 or/and
Appendix A). Note that, the maximal group velocity for the system is v max = 2J, whereas
the average group velocity in the case of an empty host is ¯v = 2J/ √ 2. For massless Dirac
fermions with linear energy dispersion ǫ k
= v D F k and
σ 2 D (t) = σ2 D (0) + (vD F )2 t 2 . (5.26)

178 5. Real-time Dynamics of Spin and Charge Densities in the One-dimensional Hubbard Model
In order to assess the quality of the expansion of the induced charge and spin perturbation
in the set-ups considered in the previous sections of the present chapter, we follow
the time evolution of the deviation of the spatial variance from its initial value
L
∆σc 2 (t) := σ2 c (t) − σ2 c (0) = ∑
L∑
(i − µ c (t)) 2 δn i (t) , µ c (t) = i δn i (t) (5.27)
i=1
in the case of the charge perturbation and
L
∆σs 2 (t) := σ2 s (t) − σ2 s (0) = ∑
L
(i − µ s (t)) 2 (2 δ ¯S i z(t)) , µ s (t) = ∑
i (2 δ ¯S i z (t)) (5.28)
i=1
for the spin perturbation. Here the additional factor 2 is used in order to assure the
normalization of the distribution, ∑ i 2 ¯S i z (t) = 1.f
A similar scheme was earlier used by Langer et al. [147] in the context of spin- 1 2
chains. Using the adaptive t-DMRG methods they studied the magnetization dynamics
after preparing the system in an inhomogeneous initial state. For instance the system
was subjected to an external magnetic field of Gaussian shape and, after releasing the
confining field, the evolution of the magnetization was followed in time. Analyzing the
time dependence of the spatial variance of the magnetization during the time evolution,
they found that in the critical regime of the XXZ-chain, the magnetization dynamics is
ballistic, whereas in the massive regime, their results indicated diffusive transport at half
filling and ballistic transport away from half filling. The advantage of this scheme with
respect to analyzing currents and their correlation functions directly, lies in the possibility
to be used beyond the linear response regimes, e.g., in systems substantially driven out of
equilibrium.
The time evolution of the deviation of the charge (c) and spin (s) spatial variances from
their corresponding initial values, ∆σc/s 2 (t) as defined by Eqs. (5.27) and (5.28), respectively,
are shown in Fig. 5.13 for an open L = 65 size system and different interaction strengths
U/J = 0, 2, 4, 8, 20 (left column), and for U = 8J and different chain sizes, from L = 63
to L = 131 (right column). The spatial variances of the charge perturbation for both the
above considered set-ups are displayed in Fig. 5.13 (first and last rows), whereas for the spin
perturbation only the results for the set-up with the host systems close to half-filling are
presented (the second row). As mentioned previously, the deviation of the bond-averaged
spin density from its corresponding equilibrium value spreads more or less identically in
both considered set-ups for systems with odd chain sizes.
For the of non-interacting system (U = 0) the time-evolutions of the spin and charge
perturbations, induced by adding an (up-spin) electron to the ground state, coincide. Moreover,
since electrons are not interacting the induced perturbation always spreads ballistically
in the system. The best fit of a power law to ∆σc/s 2 (t) yields ∆σ2 c/s (t) ≈ 2.03 t1.99 ,
f Recall that S z i = 1 2 (n i↑ − n i↓ ).
i=1
i=1

5.5. Ballistic vs. subdiffusive dynamics 179
∆σ c
2
(t)
∆σ s
2
(t)
∆σ c
2
(t)
900
800
700
600
500
400
300
200
100
900
800
700
600
500
400
300
200
100
900
800
700
600
500
400
300
200
100
L = 65, N ↑
= 32(+1), N ↓
= 33
U = 0
U = 2J
U = 4J
U = 8J
U = 20J
U = ∞
D c 2 t2
0
0 2 4 6 8 10 12 14 16 18 20
time (t)
(a)
L = 65, N ↑
= 32(+1), N ↓
= 32
U = 0
U = 2J
U = 4J
U = 8J
U = 20J
D s 2 t2
0
0 2 4 6 8 10 12 14 16 18 20
time (t)
(c)
L = 65, N ↑
= 32(+1), N ↓
= 32
U = 0
U = 2J
U = 4J
U = 8J
U = 20J
D c 2 t2
0
0 2 4 6 8 10 12 14 16 18 20
time (t)
(e)
∆σ c
2
(t)
∆σ s
2
(t)
∆σ c
2
(t)
900
800
700
600
500
400
300
200
100
160
140
120
100
U = 8J
L = 63, N ↑
= 31(+1), N ↓
= 32
L = 64, N ↑
= 32(+1), N ↓
= 32
L = 65, N ↑
= 32(+1), N ↓
= 33
L = 66, N ↑
= 33(+1), N ↓
= 33
L = 131, N ↑
= 65(+1), N ↓
= 66
2.22t 2
0
0 2 4 6 8 10 12 14 16 18 20
time (t)
80
60
40
20
500
400
300
200
100
(b)
U = 8J
L = 63, N ↑
= 31(+1), N ↓
= 31
L = 65, N ↑
= 32(+1), N ↓
= 32
L = 101, N ↑
= 50(+1), N ↓
= 50
L = 131, N ↑
= 65(+1), N ↓
= 65
0.33t 2
0
0 2 4 6 8 10 12 14 16 18 20
time (t)
(d)
U = 8J
L = 63, N ↑
= 31(+1), N ↓
= 31
L = 65, N ↑
= 32(+1), N ↓
= 32
L = 101, N ↑
= 50(+1), N ↓
= 50
L = 131, N ↑
= 65(+1), N ↓
= 65
0
0 2 4 6 8 10 12 14 16 18 20
time (t)
(f)
Figure 5.13: Time dependence of the deviation of the charge (c) and spin (s) spatial
variances from their initial values, ∆σν 2 (t) ν = c, s, as a function of time t (solid lines)
for an open L = 65 size system and different U (left column), and for U = 8J and
different L (right column). Plots (a) and (b) correspond to the half-filled host systems,
whereas Plots (c)-(f) show results for the host systems close to half filling (see text).
Also shown are results for U = 0 and L = 65 (dashed lines). The dashed-dot lines of
the corresponding colors are the quadratic fits to the data.

180 5. Real-time Dynamics of Spin and Charge Densities in the One-dimensional Hubbard Model
which is thus slightly different from the expected ballistic expansion ≈ 2 t 2 , but a deviation
smaller than 1% from the expected exponent of 2 is very much within the accuracy of
the numerical calculations (see the dashed and dash-dot gray curves in the left column of
Fig. 5.13). For the fitting only the data for t < 12 are used, which does not contain visible
effects due to the wave-front reflection from chain ends (see e.g., Fig. 5.7a). Typically,
these effects are appearing within a few sites from the boundaries.
The perfect square fit ∆σc(t) 2 = 2 t 2 is only obtained for a free spinless fermion spreading
in an empty system (see the dashed and dash-doted black curves in Fig. 5.13a). As
discussed previously (see Section 5.4.1), this case would correspond to the U/J → ∞ limit,
where the up-spin electron (or equivalently down-spin hole) is added to the ground state
of the fully polarized ferromagnet with N e = L down-spin electrons.
For intermediate values of the Hubbard interaction (U), when an electron is added
to the ground state of the half-filled system (Mott-insulating state) on a given site (see
Section 5.4.1), the induced charge perturbations spread ballistically in the system, as is
seen in Figs. 5.13a and 5.13b (compare the solid and dash-doted curves of the same color).
Although the best fits of a power law yield exponents always larger than 2 — up to 2.05 for
U = 2J — they are perfectly within the accuracy of the numerical data. Similarly to the
speeds of the wave-front, the rates of the ballistic expansion D2 c , obtained by a quadratic
fit to the data, are enhanced for U > 0 as compared to the U = 0 and U/J → ∞ limit.
The largest D2 c (U/J), among the U values considered, corresponds to U = 2J. Further
increase of U reduces the ballistic expansion rate to the one corresponding to the U/J → ∞
limit. D2 c (U/J) are almost independent of chain size, as is seen in Fig. 5.13b, where all
curves for U = 8J and L = 63, 64, 65, 66, 131 lie on top of each other for t < 13, indicating
that the propagation of the charge perturbation is almost indistinguishable in systems of
different lengths until the wave fronts reach the chain boundaries. The rates of the ballistic
expansion of the charge perturbation D2(U) c are listed in Tab. 5.1 (see page 169).
When an electron is created in the ground state of the Hubbard model with an extra hole
(the second set-up, see Section 5.4.2), the spreading of the induced charge perturbation is
rather unusual. As discussed in the previous subsection, the perturbation remains mainly
confined close to the creation point. Increasing U strengthens this confinement, as is
identified in Fig. 5.13e, where the ∆σc 2 (t) for open L = 65 chain and different values of U
are shown. On the other hand, in the system with a larger chain length and consequently
a smaller gradient in the ground-state charge-density distribution (see the discussion in
Section 5.4.2), the intensity and hence the spatial variance of the charge-perturbation
expansion is increased, as is confirmed in Fig 5.13f, where the results for U = 8J and chain
lengths up to L = 131 are presented. A careful analysis of the data reveals that at short
time scales the dynamics of the charge-perturbation spreading is always ballistic (see e.g.,
the solid and dash-dot red curves in Fig. 5.13e). Later, it first becomes diffusive and then
subdiffusive. For instance, ∆σc 2 (t) for U = 8J and L = 131 (see the gray curve in Fig. 5.13f)

5.6. Conclusions 181
shows almost quadratic growth in t (≈ 1.93 t 1.94 ) for t < 3, at t ≈ 7 the growth becomes
linear in t, and later at t ≈ 15 it changes to sublinear in t. Thus the dynamics of the
charge-perturbation expansion exhibits a crossover from ballistic to diffusive and later to
subdiffusive regimes. More realistically, it is rather a mixture of ballistic and subdiffusive
expansions, where the latter becomes more significant as time increases. The duration of
each regime depends on the interaction strength and the system size.
As discussed in Section 5.4.2, the space-time evolution of the change of the bondaveraged
spin density from its corresponding equilibrium value is more or less similar for
both considered set-ups. Moreover, the induced spin perturbation is spreading ballistically,
as is also seen in Fig. 5.13c, where ∆σs 2 (t) for the L = 65 site system and U/J = 2, 4, 8, 20
are shown for the second considered set-up (host system close to half-filling). Also for this
case the best fits of a power law to the data yield exponents that are tolerably larger than
2, up to 3% for U 8J. The largest deviation, up to 8%, is found for U = 20J, which
at the same time is the least accurate, since t = 20 is too small for the perturbation to
spread far enough in space. Ballistic expansion rates D2 s (U), obtained by a quadratic fit
to the data, differ only slightly from the corresponding ones for the first considered set-up
(the half-filled host system), which are listed in Tab. 5.1 (see page 169). As expected, an
increase of U reduces the expansion rate of the induced spin perturbation; D2(U) s decreases
systematically with U, becoming zero in the U/J → ∞ limit. The reason is that, in this
limit no spin-dynamics is possible at all. The weak dependence of the bond-averaged spin
density on the system size is also confirmed by the data shown in Fig. 5.13d, where the
spatial variance (∆σs 2 (t)) for U = 8J and different chain sizes — up to L = 131 — are
shown.
5.6 Conclusions
Analyzing the real-time dynamics of the spin and charge densities induced by adding an
electron to the ground state of a given system we have found that: When the host system
is the Hubbard model in the Mott-insulating phase (half-filling), induced perturbations
— charge and spin — propagate ballistically in the system. The respective wave-front
velocities and ballistic expansion rates of the charge- and spin-densities are different and
they evolve more or less as non-interacting perturbations. A slight reduction of the antiferromagnetic
oscillations in the induced on-site spin-density just behind the charge-density
wave fronts has nonetheless been identified. This should be the signature of the remaining
interactions between spin and charge quasiparticles. Irrespective of that, even in the case
of extremely localized spin and charge perturbations such as a single electron inserted in
the ground state of the system on a well defined site, robust effects of spin-charge separation
have been observed in the obtained numerical data. From that it can be deduced that

182 5. Real-time Dynamics of Spin and Charge Densities in the One-dimensional Hubbard Model
the added electron shortly after the insertion decays into spin and charge excitations. The
speed of the charge wave-fronts is enhanced by the Hubbard interaction as compared to the
non-interacting case (the tight-binding model). Larger enhancements are obtained in the
case of small interaction strength; for U = 2J the enhancement is about 10%. For larger
interactions it approaches from above the speed and the ballistic expansion rate of the free
spinless particle that is spreading in an empty system, which is the effective picture in the
U/J → ∞ limit. The speed and the ballistic expansion rates of the induced spin perturbation
decrease systematically with U, and become zero in the U/J → ∞ limit, which is
in perfect agreement with expectations since no spin dynamics is expected at U/J = ∞.
When the electron is added to the ground state of the Hubbard model with an extra hole
(metallic phase), the induced spin perturbation still propagates ballistically in the system.
Moreover, its space-time evolution followed using the bond-averaged spin-densities is quite
similar to the one observed in the previous case. The dynamics of the charge-perturbation
spreading is rather unusual. Open boundary conditions in combination with the Hubbard
interaction U create a cosine-shaped trough in the ground-state charge-density distribution
of the host system. The induced charge perturbation scatters on this gradient during
the spreading and thus predominantly stays closer to the bottom of this trough. This
conjecture is supported by the observed enhancement of the spatial variance with reducing
on-site interaction strengths or increasing chain size. The latter reduces the depth of
the gradient, which is inverse proportional to the chain length. At short time-scales the
dynamics of the charge-perturbation spreading is always ballistic. Later in time it exhibits
a crossover from ballistic to diffusive and later to subdiffusive regimes. More realistically,
it is rather a mixture of ballistic and subdiffusive expansions, where the latter becomes
more significant as time increases. The duration of each regime depends on the interaction
strength and the system size. A similar behavior is also observed when not only one but
a few electrons are missing in the host system. Whether the long-time limit character of
the charge-perturbation expansion is subdiffusive in the thermodynamic limit and whether
this type of crossover — from ballistic to diffusive and then to subdiffusive — is observed
in the case of periodic boundary conditions, remains an open question and requires further
investigations. These analyses are beyond the scope of this thesis. In practical situations,
however, experiments deal with finite size systems, either due to the geometry of samples
(e.g., nanodevices, heterostructures) or due to the presence of imperfections. Other studies
in the spin chain systems (see e.g., Ref. [147]) have also identified the crossover from ballistic
to diffusive dynamics in the vicinity of phases with massive spin excitations.
Yet another issue that has to be taken into account in the case of finite systems is
the parity of the chain length. As it was discussed in this chapter, the parity of the
chain length already determines the features seen in the ground state of the host system
(e.g., antiferromagnetic SDW pattern). Moreover, in the case of odd chain sizes L, further
distinction between the chains according to the divisibility of L − 1 by 4 is required for

5.6. Conclusions 183
the spin perturbation. This divisibility by 4 determines whether the antiferromagnetic
oscillations that were already present in the ground state of the host systems are enhanced
or reduced by adding an electron to the ground state and whether domain walls appear
that spread either with charge or spin wave fronts.
The final thing that should be noted about finite size effects is that as long as one stays
in the ballistic regime, the length of the chain does not matter at least for times smaller
than the time needed to reach the boundary.
It is worthwhile to mention, that the observed behavior is in contrast to the free and
band insulator cases also studied in Appendix A and Appendix B, respectively, where the
spreading is always ballistic with no spin-charge separation.

184 5. Real-time Dynamics of Spin and Charge Densities in the One-dimensional Hubbard Model

185
6. SUMMARY AND OUTLOOK
In this thesis we have discussed the physics of insulator-insulator transitions in the onedimensional
strongly correlated systems. In addition we have also studied real-time dynamics
of the charge and spin degrees of freedom when a single electron is inserted in the
insulating ground state of a system. At the same time our goal has been the development
and improvement of numerical techniques that can be used to extract information from
the many-particle Hamiltonians used to model quantum strongly correlated systems. In
the last years solid-state physics has increasingly benefited from scientific computing and
the importance of numerical techniques is growing quickly in this field. Because of the
high complexity of the realistic systems, full understanding of their properties cannot be
developed using analytical methods only. Numerical simulations do not only provide quantitative
results for the properties of specific materials but are also widely used to test the
validity of theories and analytical approaches.
Density-Matrix Renormalization Group
Since its invention the DMRG has been under constant development. The algorithm was
rapidly extended and adapted to different situations, becoming the most reliable and versatile
method for 1D systems. Its field of applicability has now extended beyond condensed
matter physics.
In this thesis we have provided the theoretical foundation that is necessary to understand
how DMRG computations are performed. We have presented the standard DMRG
algorithms for calculating ground state properties of quantum lattice many-body system
such as infinite- and finite-system DMRG methods and most of the possible improvements
that are required for the development of the efficient computer code and provided
the possibilities for further extensions of the method (e.g., to investigate time-dependent
problems).
Based on the presented descriptions, an efficient and flexible DMRG code was designed
and developed which allows to study the ground state and low-energy properties of systems
that can be mapped on the one-dimensional chain with (possibly) arbitrary range
interactions.
The principal limitation of the DMRG method is the rapid increase of the computational
effort with the system size in dimension larger than one and with the spatial extent of the

186 6. Summary and Outlook
interactions. Therefore, the majority of systems investigated with DMRG until now have
been (quasi-) one-dimensional systems with short-range interactions [99, 100, 210].
Real-time evolution using DMRG
In order to study the time dependent problems we have devised two different timedependent
DMRG algorithms which can be used to study the time evolution of pure
states. Other different variants have been also discussed. The first algorithm, the adaptive
t-DMRG based on Suzuki-Trotter approximation of the time-evolution operator, is quite
efficient and requires moderate computational resources. It also allows explicit control of
errors produced during the time-evolution simulations, but unfortunately is limited to the
systems with nearest-neighbor interactions. The second algorithm, the time-step targeting
adaptive t-DMRG based on the Arnoldi approximation for the time-evolution operator (one
of the Krylov-subspace methods) is as efficient as the previous algorithm, but is free from
the above mentioned limitation. We have also performed extensive analysis of accuracy
of both considered time dependent DMRG methods using two nontrivial time-evolution
examples which have been separately considered in Appendix A. In contrast to the previous
studies it has been shown that the weight of states discarded at each iteration can be
used in order to assess the errors due to the DMRG state-space truncations. Analytical
expressions for the estimated error have been provided and later verified on the studied
exactly solvable examples. Moreover, it has been shown that the idea originally proposed
in Ref. [84], of two different regimes in the error development is strongly dependent on the
studied problem and in general is error prone. The efficiency of the presented algorithms
is controlled by the amount of entanglement of the time-evolving state that usually grows
in time. As a consequence, a local quench (like in the examples studied in this thesis)
can be easily simulated by means of t-DMRG, while a global quench is harder and the
numerics must be limited to relatively small system sizes and times. However, this is not
too relevant if phenomena of interest occur within accessible time intervals. The flexible
computer codes for both variants of the time-dependent DMRG algorithms have been also
developed.
The described time-evolution methods are flexible enough and allow to investigate
— with high accuracy — diverse time-evolution out-of-equilibrium problems that can be
mapped to a one-dimensional chain. They also provide advanced tools to elaborate numerical
experiments. Since new algorithms do not suffer from the sign problem, they are
suitable to study problems beyond the reach of Monte Carlo techniques. The considered
algorithms have recently been successfully used to study different time-evolution setups
in spin systems as well as in fermionic and bosonic ones. Time evolution of Gaussian
wave packets or dynamics of density perturbations with Gaussian form have been used
to study nonequilibrium transport through small interacting nanostructures as well as the

187
phenomenon of spin-charge separation [208, 4, 106, 143, 142, 139, 140, 239]. Computing
the real-time Green’s functions, it has been possible to investigate the spectral properties
of several one-dimensional systems [258, 67, 195, 65]. Both algorithms have been employed
to study far from equilibrium dynamics, and local and global quantum quenches
[84, 161, 34, 200, 162]. In addition, algorithms can be generalized to study systems at finite
temperatures [66, 17].
Band- to Mott-insulator transition in 1D
Motivated by controversial results on the ground-state phase diagram we have reinvestigated
the ionic Hubbard model (IHM) using exact diagonalization and DMRG methods.
From the finite-chain DMRG studies and finite-size scaling analysis we have found that the
ionic potential leads to long range CDW order for all interaction strengths. The obtained
results clearly resolve one of the expected two transition points, namely from the bandinsulator
(BI) to the correlated-insulator (CI). Remarkably, at the transition the charge
∆ C and the spin ∆ S gaps are equal and both remain finite. This itself clearly indicates
the existence of a CI phase with ∆ C > ∆ S > 0. Analyzing finite periodic chains we have
found that this insulator-insulator transition results from a ground-state level crossing of
the two site-parity sectors. The optical excitation gap therefore has to vanish at the transition
point. The distinction between the optical and the charge gap is therefore of key
importance for the structure of the insulating phases and the phase transitions of the IHM.
Analyzing the correlation function obtained using the DMRG method, close above the transition
point, we have identified a clear signal for long range bond order wave (BOW). With
further increase of the on-site interaction the finite size scaling behavior of the staggered
bond density and spin-spin correlation function changes qualitatively and approaches the
scaling behavior of the Hubbard model. This strongly suggests the existence of the second
transition point, namely from CI to the Mott-insulating (MI) phase. The existence of the
BOW order would necessarily imply a finite spin excitation gap. If BOW ceases to exist
above a second critical value of U, the spin gap has to vanish simultaneously leading to
identical large distance decays of SDW and BOW correlations. Unfortunately, with the
current chain length and due to the DMRG accuracy limitations it was not possible to
precisely identify and locate the second transition point.
The second topic considered in this chapter was the adiabatic limit of the Holstein-
Hubbard model (AHHM). In this model the alternating modulation of the on-site energy is
caused by the static lattice distortion which effects in extra lattice elastic energy. Using exact
diagonalization and DMRG methods we have found that two different phase transition
scenarios emerge with a discontinuous BI-MI transition for strong coupling U, K −1 ≫ t
and two continuous transitions for weak coupling U, K −1 ≪ t with an intermediate phase
where CDW order persists. Below the transition point, U < U c , the ionic potential ∆ is

188 6. Summary and Outlook
finite and hence the system is in a BI phase, similar to the BI phase of the IHM. Analyzing
the dependence of the ground state energy of the IHM on the ionic potential strengths we
have established that in the first scenario the transition form the BI to the MI phase is
characterized by a discontinuous vanishing of the ionic potential ∆(U, K) at the transition
point U c . Thereby the model reduces to the pure Hubbard model. In the second scenario,
the BI-MI transition rather follows a Ginzburg-Landau-type behavior. Due to its continuous
decrease ∆(U, K) naturally intercepts the BI-CI transition point of the IHM where the
optical absorption gap disappears. This therefore implies the existence of an intermediate
region with a finite ∆. Relying on the strong similarities with the IHM we have concluded
that the intermediate phase should be characterized by long range CDW+BOW order in
the ground state. The numerical confirmation for this has been out of reach, though. At
the second transition point, where the AHHM reduces to the Hubbard model, a transition
from an intermediate CI phase to the MI phase occurs, which in contrast to the IHM model
can be identified and there is no long range CDW in the MI phase of the AHHM.
Real-time dynamics of spin- and charge-densities in the
one-dimensional Hubbard model
Using t-DMRG we have studied the time evolution of the local additional charge and spin
densities due to the creation of a single electron on a given site in the ground state of the
Hubbard model: i) in the Mott-insulating phase (half filling) and ii) in the metallic phase
close to it (one electron less). Even in the case of extremely localized and therefore highenergy
initial perturbations such as a single electron inserted on a well defined site into the
ground state of the system, we have clearly observed robust effects of the separation of the
spin and charge degrees of freedom in the obtained numerical data. Analyzing the spacetime
evolution of the induced spin and charge densities, we have found: that in case i) the
speed of the charge-density wave fronts is enhanced by the interaction as compared to the
non-interacting case and the speed of the spin-density wave fronts reduces systematically to
zero with growing interaction strength. In case ii) the space-time evolution of the induced
spin perturbation, if it is probed using the bond-averaged spin-densities, is quite similar
to the one observed in case i). The induced charge perturbation, however, scatters on the
shallow gradient in the on-site charge-density distribution and thus predominantly stays
closer to the creation point.
We have also outlined the scheme which allows to characterize spreading dynamics of
the charge and spin perturbations. Analyzing the time dependence of the spatial-variance
of each of them it has been possible to distinguish between the ballistic, diffusive, and
subdiffusive regimes. This is a rather new scheme that can also be employed in the situations
where a system is far from equilibrium and the usual tools of the linear response
theory are thus not applicable. Using it we have found that the expansion of the induced

spin-perturbation is ballistic in both considered cases. The induced charge perturbation
spreads also ballistically in the case with Mott-insulating host, whereas in the case with
metallic host its expansion exhibits crossovers from ballistic into diffusive and then into
subdiffusive regimes with time. A careful analysis of the obtained results has revealed that
at short time scales the dynamics of the charge-perturbation spreading is always ballistic.
The duration of each regime depends on the interaction strength and system size. We
have also found that as along as one stays in the ballistic regime the chain length does not
matter for both considered cases, at least for times smaller than the time needed to reach
the boundary.
Whether the long-time-limit character of the charge-perturbation expansion is subdiffusive
in the thermodynamic limit and whether this type of crossover (from ballistic into
diffusive and then into subdiffusive) is observed in the case of periodic boundary conditions,
are left for future investigations. Nevertheless, since in practical situations experiments
deal with finite size systems, either due to the geometry of probes (e.g., nanodevices, heterostructures)
or due to the presence of imperfections, the finite-size effects caused by the
parity of the chain length L or divisibility of L − 1 by 4 have been extensively investigated
in the corresponding sections.
In addition, in Appendix B we have presented results for the space-time evolution of
the charge density in the so-called ionic chain, which is the simplest model describing the
band insulator at half filling. In contrast to the observations made for the Hubbard model,
in the case of the ionic-chain the induced charge perturbation, which would also coincide
with the induced spin perturbation, spreads always ballistically. We have found that an
increase of the ionic potential ∆, which also defines the gap of the charge/spin excitations,
reduces the bandwidths of the valence and conduction bands, strongly renormalizes the
maximum group velocity, and consequently the speed of the charge/spin wave front in the
system. How the interaction between the electrons influences this dynamics and whether
induced spin and charge perturbations spread equally in the case of band- or correlatedinsulator
phases is an interesting topic for the future studies. The ionic Hubbard model
investigated in this thesis provides a nice playground for these investigations.
189

190 6. Summary and Outlook

Appendices
191

193
Appendix A
FREE SPINLESS LATTICE FERMIONS IN
ONE-DIMENSION
In this appendix we consider the system of free spinless lattice fermions in one-dimension
and investigate the real-time evolutions in different setups. We consider the time evolutions
of wave packets added to the ground state of the system at time t = 0 and the time
evolutions of the position-space Fock states. We mainly derive those quantities which are
used to analyze the accuracy of the time-evolution methods based on DMRG (Section 3.5).
Most of the formulas obtained in this appendix are later adapted for the ionic-chain in
Appendix B.
as
The Hamiltonian of the one-dimensional system of free spinless lattice fermions is given
H = −J
∑L−1
(c † i c i+1 + h.c.) , (A.1)
i
where L is the chain length, J is the nearest-neighbor hopping amplitude, and c † i (c i )
creates (annihilates) a spinless fermion on the lattice site i. We consider the system with
open boundary conditions (OBC), which are preferable in case of DMRG simulations. OBC
imply that the fermion densities vanish identically on “phantom” sites 0 and L + 1. In this
appendix as in the whole thesis we assume a lattice spacing constant a = 1, we set = 1,
and measure the time in the units of inverse energy. The model considered is equivalent
to the XX spin- 1 chain, which was exactly solved almost half a century ago by Lieb et
2
al. [154] (see also Refs. [14, 15, 16]). These two problems map onto each other via the
Jordan-Wigner transformation. Using the following discrete Fourier sine transformation,
which is more suitable in the case of OBC a
c j =
√
2 ∑
√
L + 1
k
sin(kj) c k
with k = πn , n = 1, 2, . . ., L , (A.2)
L + 1
a In the case of periodic boundary conditions (PBC) j + L ≡ j and the “normal” discrete Fourier transformation
c j = √ 1
L
∑k eikj c k
with k = 2πn , n = 0, . . .,L − 1 is used.
L

194 Appendix A. Free Spinless Lattice Fermions in One-Dimension
Hamiltonian (A.1) can be brought to a diagonal form
H = ∑ k
ǫ(k) c † k c k = ∑ k
ǫ(k) n k
,
(A.3)
where
ǫ(k) = −2J cos(k) ,
(A.4)
is the energy dispersion, c † k (c k
) creates (annihilates) a spinless fermion with the “wavenumber”
k, and n k
= c † k c k
is the fermion number operator. In this form, it becomes clear that
any non-vanishing product of c † k
generates an eigenstate of the Hamiltonian (A.3) and that
the operators c k
have the trivial time-evolution form
c k
(t) = e −iǫ(k)t c k
(0) .
(A.5)
The ground state of the system of N e fermions can be written as
|ψ GS 〉 =
( πNe
L+1
∏
k= π
L+1
c † k
)
|0〉 =
( ∏
k
f(ǫ k )c † k
)
|0〉 , (A.6)
and the ground state energy is
E GS =
πNe
L+1
∑
k= π
L+1
ǫ(k) = ∑ k
f(ǫ k )ǫ(k) ,
(A.7)
where |0〉 is the vacuum state and f(ǫ) is a “Fermi” function
f(ǫ) =
{ 1 ǫ < EF
, E F = −2J cos(k F ), k F = (N e + 1) π 2
. (A.8)
0 ǫ > E F L + 1
In the following we study the time evolution in two different setups:
1. time evolution of a fermionic wave packet added to the ground state of the system of
N e fermions at time t = 0, e.g., a single fermion can be created on some lattice site
at time t = 0, which would correspond to the localized version of the wave packet
(see Fig. A.1),
2. time dynamics of the state constructed with a non-vanishing product of fermion
creation operators in the real space (Fock state), for example the time evolution of
the fermionic domain wall residing in the left half of the chain at time t = 0 (see
Fig. A.7).

A.1. Time evolution of a wave packet 195
Figure A.1: Schematic pictures of the initial states. Gaussian wave packet (top) or a
single particle (bottom) added to the ground state of the system.
A.1 Time evolution of a wave packet
Initial states
We start with the time evolution of some wave packet added at time t = 0 to the ground
state of the system of N e fermions (see schematic picture in Fig. A.1). The initial state
|Φ(0)〉 is given by
|Φ(0)〉 = ∑ √
2
A j c † j |ψ ∑∑
GS〉 = √ A j sin(kj)c †
j
L + 1
k |ψ GS〉 (A.9)
j k
∑
with A ∗ jA j = 1.
j
In the case of a Gaussian wave packet
√
( )
1 (j −
A j = exp (ip 0 (j − j 0 ))
σ √ 2π exp j0 ) 2
2σ 2
(A.10)
where p 0 is the group velocity of the wave packet. The state obtained after the creation of
a single particle with zero initial velocity on some given site j (A i = δ ij ) is
√
2
|φ j (0)〉 = c † j |ψ ∑
GS〉 = √ sin(kj)c † k |ψ GS〉 .
(A.11)
L + 1
In this case a localized wave packet consisting of all wave vectors is constructed, which
then spreads out.
It is more convenient to work in momentum space, since the Hamiltonian (A.3) has the
diagonal form in this space and the ground state of the model can be written as a single
k

196 Appendix A. Free Spinless Lattice Fermions in One-Dimension
1
0.5
e(k), v g
(k)
0
-0.5
e(k)
v g
(k)
-1
0 π/2 π
k
Figure A.2: Energy dispersion ǫ(k) and group velocity v g (k) = ∂ǫ(k)/∂k for a system
of free spinless fermions on a 1D open chain. The hopping amplitude is J = 0.5.
product of fermion creation operators (A.6). The time-evolved states can then be written
in momentum space as
|Φ(t)〉 = e −iHt |Φ(0)〉 =
√
2 ∑ ∑
√ A j e −i(ǫ(k)+EGS)t sin(kj)c † k |ψ GS〉
L + 1
j k
(A.12)
and
|φ j (t)〉 = e −iHt |φ j (0)〉 =
√
2 ∑
√ e −i(ǫ(k)+EGS)t sin(kj) c † k |ψ GS〉 .
L + 1
k
(A.13)
Since c † k |ψ GS〉 = 0 for k < k F , only the states with energies above the Fermi energy (E F )
form and participate in the time evolution of the packet. The different components propagate
at different speeds, given by the group velocity, determined as the slope of the
dispersion curve at k
v g (k) = ∂ǫ(k)
∂k
= 2J sin(k) .
(A.14)
In Fig. A.2 we show the energy dispersion ǫ(k) and the group velocity v g (k) for a hopping
amplitude J = 0.5. The maximal group velocity equals 2J = 1 and it is acquired at
k = π/2.

A.1. Time evolution of a wave packet 197
Expectation values
Before presenting the results, let us first calculate the expectation values of some operators.
The distribution of fermions in the ground state is given by
〈n i 〉 GS := 〈ψ GS |n i |ψ GS 〉 = 2 ∑
sin(ki) sin(k ′ i)〈ψ GS |c † k
L + 1
c k
|ψ ′ GS 〉
k,k ′
= 2
L + 1
∑
sin 2 (ki)f(ǫ(k))
k
= N e + 1 2
L + 1 − 1
2(L + 1)
sin(2k F i)
sin ( ) .
πi
L+1
(A.15)
In order to follow the time evolution of the wave packet we study how the local (on-site)
fermion density develops in time. The on-site fermion density at position i and time t is
given by
where
〈Φ(t)|c † i c i |Φ(t)〉 = 4
(L + 1) 2 ∑
j,j ′
〈n i 〉 t := 〈Φ(t)|n i|Φ(t)〉
〈Φ(t)|Φ(t)〉
∑
, (A.16a)
k,k ′ ,q,q ′ e −i(ǫ(k)−ǫ(k′ ))t sin(k ′ j ′ ) sin(qi) sin(q ′ i) sin(kj)A ∗ j ′A j
× 〈ψ GS |c k ′c † qc q ′c † k |ψ GS〉 ,
which using Wick’s theorem for 〈ψ GS |c k ′c † qc q ′c † k |ψ GS〉 can be rewritten as
〈Φ(t)|c † i c i |Φ(t)〉 =
4 ∑ ∑
=
e −i(ǫ(k)−ǫ(q))t sin(qj ′ ) sin(qi) sin(ki) sin(kj)A ∗
(L + 1) 2 j ′A j
j,j ′
q
k,q
× [1 − f(ǫ(q))][1 − f(ǫ(k))]
4 ∑
+ sin 2 (qi)f(ǫ(q)) ∑ ∑
sin(kj ′ ) sin(kj)A ∗
(L + 1) 2 j ′A j [1 − f(ǫ(k))] . (A.16b)
j,j ′
k
One should not forget that the states |Φ(t)〉 and |φ j (t)〉 are not normalized
〈Φ(t)|Φ(t)〉 = 〈Φ(0)|Φ(0)〉 = ∑ j,j ′
A ∗ j ′A j 〈ψ GS|c j ′c † j |ψ GS〉
= 2
L + 1
∑ ∑
sin(kj ′ ) sin(kj)A ∗ j ′A j [1 − f(ǫ(k))] . (A.16c)
j,j ′
k

198 Appendix A. Free Spinless Lattice Fermions in One-Dimension
The last time-independent term in (A.16b) is 〈Φ(0)|Φ(0)〉〈ψ GS |n i |ψ GS 〉 and hence it reassembles
the ground-state on-site density of fermions in Eq. (A.16a).
In the following we focus only on the case of a single fermion created on a given site j
(A.11). This example is used for the error analysis in the time-evolution simulations using
adaptive DMRG algorithms (Section 3.5). The on-site density at position i and time t is
〈n i 〉 t = 〈φ j(t)|n i |φ j (t)〉
〈φ j (t)|φ j (t)〉
, (A.17a)
where
4 ∑
〈φ j (t)|n i |φ j (t)〉 = e −i(ǫ(k)−ǫ(q))t sin(qj) sin(qi) sin(ki) sin(kj)
(L + 1) 2
k,q
× [1 − f(ǫ(q))][1 − f(ǫ(k))]
4 ∑
+ sin 2 (qi)f(ǫ(q)) ∑ sin 2 (kj)[1 − f(ǫ(k))]
(L + 1) 2 q
k
(A.17b)
and
〈φ j (t)|φ j (t)〉 = 〈φ j (0)|φ j (0)〉 = 〈ψ GS |(1 − n j )|ψ GS 〉
= 2 ∑
sin 2 (kj) [1 − f(ǫ(k))] .
L + 1
k
(A.17c)
In the simple case of an empty initial system (N e = 0), the expectation value Eq. (A.17b)
at time t = 0 is one if i = j and zero otherwise. In general, the first — time-dependent —
term in (A.17b) at time t = 0 describes the induced local densities caused by the creation of
the fermion on site j. It also reassembles the density-density correlations in the system at
t = 0. The last time-independent term in (A.17b) is 〈φ j (0)|φ j (0)〉〈ψ GS |n i |ψ GS 〉 and hence
it reproduces in Eq. (A.17a) the on-site density of fermions in the ground state.
Results
Let us look into this matter more precisely. Consider a lattice with L = 100 sites and
hopping amplitude J = 0.5. We start with an empty system, N e = 0, and study the time
evolution of a single fermion created at time t = 0 on site j = 51. The results are shown in
Fig. A.3, where the false color codes the expectation value of the particle number operator
(A.17) on site i and time t. There is a strong peak at position i = 51 at t = 0 corresponding
to a created fermion, which then splits in two pronounced density fronts (wave fronts)
propagating in both directions with the constant velocity v fr ≈ 1 (isosceles triangle, with
corners at (t = 0 , i − 51 = 0) and (t = 40 , i − 51 = ±40), see Fig. A.3). There are several
subfronts of similar structure behind the main ones which propagate with smaller velocities.

A.1. Time evolution of a wave packet 199
Figure A.3: Space-time evolution of a single spinless fermion on L = 100 open chain.
One might naively expect that the initial packet expands like a Gaussian wave packet
centered around the creation site or splits into two Gaussians propagating to the left and
to the right from the creation point until the lattice boundaries are reached (when the initial
system is not empty), but as mentioned above extra structures are left behind the wave
fronts. The observed behavior is due to the nonlinear energy dispersion ǫ(k) = −2J cosk
(A.4) and thus k-dependent group velocity (A.14) (see also Fig. A.2). Note, that the front
velocity v fr corresponds to the maximal group velocity ≈ 1 which is acquired at k = π/2.
Next we consider the cases where the initial system is not empty, but contains N e = 25,
N e = 50, and N e = 75 fermions. In the left column in Fig. A.4, we show the on-site density
in the ground state of the system 〈n i 〉 GS (A.15) (blue), after the creation of a fermion at
site j = 51 〈n i 〉 t=0 (A.17) (cyan), and the difference between them (red) for different fillings
N e = 25, 50, 75. The right column of the same figure shows the space-time evolution of the
difference between the time evolved fermion density at site i and time t (A.17) and the
fermion density in the ground state of the system (A.15). In Fig. A.4a and Fig. A.4e, which
correspond to N e = 25 and N e = 75, respectively, Friedel oscillations with wavenumber 2k F
can be identified in the on-site fermion density in the ground state (blue curves). These
oscillations are absent in the case of N e = 50 (see Fig. A.4c). In all considered cases, the
addition of a fermion creates a wave packet that has a finite extent. One can also identify a
slight asymmetry in the difference in the on-site fermion density between the initial and the
ground states with respect to position 51 (the site at which the fermion was added), due
to the unequal lengths of the chain parts to the left and to the right of this position. For

200 Appendix A. Free Spinless Lattice Fermions in One-Dimension
〈n i 〉 GS , 〈n i 〉 t=0 , 〈n i 〉 t=0 - 〈n i 〉 GS
1
0.8
0.6
0.4
0.2
N e =25(+1)
〈n i 〉 GS
〈n i 〉 t=0
〈n i 〉 t=0 -〈n i 〉 GS
0
-50 -40 -30 -20 -10 0 10 20 30 40
i-51
(a)
(b)
〈n i 〉 GS , 〈n i 〉 t=0 , 〈n i 〉 t=0 - 〈n i 〉 GS
1
0.8
0.6
0.4
0.2
N e =50(+1)
〈n i 〉 GS
〈n i 〉 t=0
〈n i 〉 t=0 -〈n i 〉 GS
0
-50 -40 -30 -20 -10 0 10 20 30 40
i-51
(c)
(d)
1
N e =75(+1)
〈n i 〉 GS , 〈n i 〉 t=0 , 〈n i 〉 t=0 - 〈n i 〉 GS
0.8
0.6
0.4
0.2
〈n i 〉 GS
〈n i 〉 t=0
〈n i 〉 t=0 -〈n i 〉 GS
0
-50 -40 -30 -20 -10 0 10 20 30 40
i-51
(e)
(f)
Figure A.4: Plots (a,c,e): on-site density as a function of position i in the ground
state (blue), initial state (cyan), and the difference between them (red). Plots (b,d,f):
space-time evolution of the difference in the on-site fermion density between the timeevolved
and the ground states. A single fermion is created on the site j = 51 at t = 0
in the ground state of the system of N e = 25, 50, 75 fermions on an open L = 100
chain.

A.1. Time evolution of a wave packet 201
Figure A.5: Space-time evolution of a massless Dirac fermion on L = 100 open chain.
The energy dispersion is ǫ(k) = −v F k with v F = 1.
〈n i 〉 GS , 〈n i 〉 t=0 , 〈n i 〉 t=0 - 〈n i 〉 GS
1
0.8
0.6
0.4
0.2
N e =25(+1)
〈n i 〉 GS
〈n i 〉 t=0
〈n i 〉 t=0 -〈n i 〉 GS
0
-50 -40 -30 -20 -10 0 10 20 30 40
i-51
(a)
(b)
Figure A.6: (a) On-site density as a function of position i in the ground state (blue),
initial state (cyan), and the difference between them (red). (b) Space-time evolution of
the difference between the on-site fermion densities in the time-evolved and the ground
states. A Gaussian wave packet with an initial velocity p 0 = π/2 and σ = 4 is created
in the ground state of the system of N e = 25 fermions on an open L = 100 chain.

202 Appendix A. Free Spinless Lattice Fermions in One-Dimension
Figure A.7: Schematic pictures of the initial states: arbitrary real-space Fock state
(top) and domain wall of fermions residing in the left half of the chain (bottom).
cases where the ground state contains N e L/2 = 50 fermions, the wave fronts propagate
with the same velocity as in the case of the empty initial system. The wave-front velocity
starts to decay as soon as N e > 50 in the ground state (see e.g., Fig. A.4f). This is due to
the fact that only the states with energies above the Fermi energy are participating in the
time evolution and thus the actual velocity of the wave-fronts correspond to the maximum
available, v wf = ± max k [v g (k)(1 − f(ǫ(k))].
A simple picture of split Gaussians propagating in both directions will occur for particles
with linear energy dispersion, as in the Luttinger model [159] where massless Dirac fermions
with linear dispersion are considered (see Fig. A.5). In this case the group velocity is k-
independent, ǫ(k) = −v F k. b One can also create a well formed Gaussian wave packet with
a finite width and an initial group velocity p 0 = π/2 in order to pick up an almost linear
branch of the energy dispersion of free fermions (see Fig. A.6).
A.2 Time evolution of Fock states
In this section we consider the time evolution of the Fock states, that are created with
a non vanishing product of the fermion creation operators in real space (see schematic
picture in Fig. A.7)
)
|ψ〉 = |0〉 . (A.18)
( ∏
{j}
Here {j} denotes an arbitrary set of real-space indices. An example of the state of this
kind is a domain wall of N e fermions residing in the left part of the chain (see also Fig. A.7)
( ∏
)
|ψ〉 = |0〉 . (A.19)
c † j
jN e
c † j
More detailed studies of the dynamics of a fermion domain wall, which also correspond
to magnetization domain wall dynamics in the XX-Heisenberg chain can be found in
b For Dirac fermions one can use the already obtained formulas (A.2)–(A.17) with ǫ(k) = −2J(k − π/2)
instead of ǫ(k) = −2J cosk.

A.2. Time evolution of Fock states 203
Figure A.8: Space-time evolution of the on-site fermion density for a domain wall of
N e = 50 fermions residing in the left half of the L = 100 open chain at t = 0.
[7, 84, 120]. This setup was already used by Gobert et al. [84] in the accuracy analysis
of the adaptive time-dependent DMRG based on the 2nd order Suzuki-Trotter decomposition
of the time-evolution operator. Fock states are particularly interesting, because
they correspond to the matrix-product states (2.21) with dimensions 1. They provide a
unique chance to study the errors produced with the time evolution method alone, since
the initial states are constructed exactly in this case. In this section we only evaluate the
expectation value of the fermion number operator at the position i at time t that is used in
the error analysis in the time-evolution simulations using the adaptive DMRG algorithms
(Section 3.5).
Using c k
(t) (see Eq. (A.5)) the time-evolved form of the annihilation operator in the
real space can be written as
√ √
2 ∑
2 ∑
c j (t) = √ sin(kj) c k
(t) = √ sin(kj) e iǫ(k)t c k
(0)
L + 1 L + 1
= 2
L + 1
The expectation value of n i (t) is given by
k
∑ ∑
sin(kj) sin(kl) e iǫ(k)t c l
(0) .
k
l
k
(A.20)
〈ψ|n i (t)|ψ〉 = 〈ψ|c † i (t)c i (t)|ψ〉
4 ∑
=
e −it(ǫ(k)−ǫ(k′ )) sin(ki) sin(kl) sin(k ′ i) sin(k ′ l ′ )〈ψ|c †
(L + 1) 2 l (0)c l
(0)|ψ〉 . (A.21)
′
∑
k,k ′ l,l ′

204 Appendix A. Free Spinless Lattice Fermions in One-Dimension
Taking into account the structure of |ψ〉 (A.18),
{
〈ψ|c † l (0)c δl,l ′ for l ∈ {j}
l
(0)|ψ〉 =
′ 0 else
, (A.22)
(A.21) can be written as
〈ψ|n i (t)|ψ〉 =
4 ∑
(L + 1) 2
k,k ′ ∑
{j}
e −it(ǫ(k)−ǫ(k′ )) sin(ki) sin(kj) sin(k ′ i) sin(k ′ j) .
(A.23)
In the case of the domain wall (A.19) {j} = {1, . . ., N e }. In Fig. A.8 we show the spacetime
evolution of the on-site fermion density for the case where the initial state is the
domain wall of N e = L/2 fermions residing in the left half of the L = 100 open chain. As
in the previous example the density fronts propagate with the maximal group velocity
v fr ≈ 1 for the considered system (isosceles triangle, with corners at (t = 0 , i − 51 = 0)
and (t = 40 , i − 51 = ±40), see Fig. A.8). There are several subfronts of similar structure
behind the main ones that are propagating with smaller velocities.

205
Appendix B
IONIC-CHAIN
In this appendix we consider another system which can be solved exactly, namely the
so-called ionic-chain. The interest in this system is further motivated by the fact that
at half-filling its ground state is a band insulator. For the ionic chain we adapt most of
the formulas derived for the free lattice fermion model from the previous appendix (see
Appendix A). We only consider the time evolution of a single particle added to the ground
state of the system.
The model Hamiltonian is given as
H = −J
∑L−1
(c † i c i+1 + h.c.) + ∆ 2
i=1
L∑
(−1) i n i ,
i=1
(B.1)
where L is the chain length, J is the nearest-neighbor hopping amplitude, and ∆ is the difference
in ionic potentials between neighboring sites; c † i (c i ) creates (annihilates) a spinless
fermion on site i and n i = c † i c i is the fermion number operator. The alternating potential
defines two sublattices, doubling the unit cell and opening up a band gap |∆| at k = ±π/2.
The first sublattice, denoted by A, consists of sites with odd indices, while the other,
called B, contains all sites with even indices (see Fig. B.1). In the following we assume a
lattice spacing constant a = 1 (the size of the unit cell is 2) and set = 1. Time is measured
in units of inverse energy. We consider the system with open boundary conditions
(OBC), which imply that the particle densities vanish identically on the “phantom” sites 0
and L + 1. Because of the OBC it is preferable to use the following discrete Fourier sine
transformation a instead of a normal one
2 ∑
c j = √ sin(κj) c γ(j)
κ
(B.2a)
L + 1
κ
with
κ = πn
[ ] L + 1
L + 1 , n = 1, 2, . . ., 2
(B.2b)
a The inverse Fourier transformation has the form: c A ∑
2
κ = [(L+1)/2]
√ L+1 i=1
sin(κ(2i − 1))c 2i−1
and
c B κ = √ 2 ∑ [(L+1)/2]
L+1 i=1
sin(κ 2i)c 2i .

206 Appendix B. Ionic-Chain
Figure B.1: Sketch of the ionic chain subdivided into the A and B sublattices with
different on-site energies ±∆/2 and ∆ > 0.
and
γ(j) =
{ A , for j odd
B , for j even .
(B.2c)
Here [·] denotes the integer part, γ(j) determines in which sublattice the fermion is created
by c † j , and cA/B,† κ (c A/B
κ ) creates (annihilates) a spinless fermion with “wavenumber” κ on
the A/B sublattice. Note that for an odd total site number L the state with κ = π/2
exists and to ensure the orthonormality of the transformation the corresponding Fourier
component should be multiplied with an extra √ 1
2
.
Using (B.2) one can rewrite Hamiltonian (B.1) in the following form
H = ∑ κ
( −
(c A†
∆
κ , cB† κ ) ǫ(κ)
2
ǫ(κ)
∆
2
) ( c
A
κ
c B κ
)
− (L mod 2) ∆ 2 cA† π/2 cA π/2 (B.3)
where ǫ(κ) = −2J cos(κ). The final term in (B.3) is nonzero only when the total number
of sites L is odd. Using the Bogolyubov transformation
c A,†
c B,†
κ = v κ a † κ − u κb † κ ,
κ = u κ a † κ + v κ b † κ ,
(B.4)
with
v κ =
√ (
1
1 − ∆/2 ) √ (
1
, u κ = 1 + ∆/2 )
, vκ 2 2 E κ 2 E + u2 κ = 1, (B.5)
κ
one can bring the Hamiltonian (B.3) to a diagonal form
H = ∑ (
)
E(κ)(a † κa κ − b † κb κ ) + (L mod 2)E(π/2) θ(∆)a † π/2 a π/2 − θ(−∆)b† π/2 b π/2
where
√ (∆
2
(B.6)
E(κ) = −
) 2
+ ǫ 2 (k), (B.7)
and θ(x) is the Heaviside step function. From (B.6) it is clear that there are two well
defined energy subbands with dispersions E a/b (κ) = ∓√ (∆
2
) 2
+ ǫ 2 (κ) and a band gap,

207
1.5
1
E a/b (κ)
0.5
0
-0.5
∆=0.02
∆=0.02, L=100
∆=0.5
∆=0.5, L=100
∆=2.0
∆=2.0, L=100
-1
-1.5
0 π/4 π/2
κ
Figure B.2: Energy dispersion E a/b (κ) in subbands a and b of the 1D ionic chain with
open boundary conditions. Lines correspond to the infinite chain length. The hopping
amplitude is J = 0.5.
1
∆=0.02
∆=0.02, L=100
0.5
v g
a/b
(κ)
0
-0.5
-1
∆=0.5
∆=0.5, L=100
∆=2.0
∆=2.0, L=100
0 π/4 π/2
κ
Figure B.3: Group velocity v a/b
g (κ) for a and b subbands of the 1D ionic chain with
open boundary conditions. Lines correspond to the infinite chain length. The hopping
amplitude is J = 0.5.

208 Appendix B. Ionic-Chain
equal to |∆|, at κ = π/2 (see Fig. B.2). a † κ and b† κ create quasiparticles with wavenumber
κ in the lower (a) and upper (b) energy subbands, respectively. For chains with odd
total number of sites one has an extra fermion creation operator on the A sublattice with
the wavenumber κ = π/2 and the corresponding Bogolyubov coefficients are v π/2 = 1 and
u π/2 = 0 for ∆ > 0 and vice versa for ∆ < 0.
In order to write everything in a more compact form, we will work in the extended
Brillouin zone and introduce the following notation:
w A k := ⎧
⎨
⎩
⎧
⎪⎨ a †
d † k
, for 0 < k < π/2
k := b † π−k
⎪ , for π/2 < k < π ; (B.8a)
⎩
θ(∆)a † k + θ(−∆)b† k , for k = π/2
v k , 0 < k < π/2
−u π−k , π/2 < k < π
1 , k = π/2
⎧
⎪⎨
E k :=
⎪⎩
S(k, j) := w γ(j)
k
×
; w B k := ⎧
⎨
⎩
u k , 0 < k < π/2
v π−k , π/2 < k < π
0 , k = π/2
; (B.8b)
(∆ ) 2
−√
+ ǫ
2 2 (k) , for 0 < k < π/2
√ (∆ ) 2
+ ǫ
2
2 (π − k), for π/2 < k < π
. (B.8c)
− ∆ , for k = π/2 2
Going one step further, combining the Fourier coefficients sin(kj) with the Bogolyubov
coefficients w A/B
k
, a new orthogonal transformation from real space to extended momentum
space can be formed:
⎧
⎪⎨
and
k
⎪⎩
sin(kj) , for 0 < k < π/2
sin((π
√
− k)j) , for π/2 < k < π (B.9)
1
sin(πj) , for k = π/2 2
2 ∑
c j = √ S(k, j) d k
with k = πn , n = 1, 2, . . ., L . (B.10)
L + 1 L + 1
These steps create a complete formal analogy with the free lattice fermion model (see
Appendix A). The model Hamiltonian (B.3) can be rewritten as
H = ∑ k
E k
d † k d k ,
(B.11)
and with the substitution of sin(kj) with S(k, j) and c k
with d k
, the expressions derived
for the free lattice fermion model can be used for the ionic chain. The ground state of the
system with N e particles is
( Neπ )
)
L+1
∏
|ψ GS 〉 =
|0〉 = f(E k )d † k
|0〉 , (B.12)
k= π
L+1
d † k
( ∏
k

209
1
|w A/B (k)|
0.75
0.5
|w A (k)|, ∆=0.5
|w B (k)|, ∆=0.5
|w A (k)|, ∆=2.0
|w B (k)|, ∆=2.0
0.25
L=100
0
0 π/4 π/2 3π/4 π
k
Figure B.4: Bogolyubov coefficients w A/B (k) for A and B sublattices of the 1D ionic
chain with open boundary conditions. The chain length is L = 100 and the hopping
amplitude J = 0.5.
where |0〉 is the vacuum state and
f(ǫ) =
{ 1 ǫ < EF
, E F := E kF , k F = (N e + 1 ) π 2
. (B.13)
0 ǫ > E F L + 1
The weight of each quasiparticle is asymmetrically distributed among the sublattices
a † κ = v κ c A,†
κ
b † κ = −u κc A,†
κ
+ u κ c B,†
κ
+ v κ c B,†
κ
or equivalently
d † k = wA k cA,† k
+ w B k cB,† k
(B.14)
(see Eqs. (B.8) and (B.5)). In Fig. B.4 we show the Bogolyubov coefficients w A/B (k) (B.8b)
vs. k for two different ionic potentials ∆ = 0.5 and ∆ = 2.0, in the case of the open L = 100
chain and the hopping amplitude J = 0.5. The weight of the quasiparticle from the lower
(upper) energy subband is partially shifted from sublattice B to sublattice A (from A to
B) and the amount of the shifted weight increases with growing |∆|. Note also that the
weights of the quasiparticles close to the upper edge of the lower energy subband and the
lower edge of the upper energy subband shifts almost completely. For ∆ > 0, one has
a negative ionic potential on sublattice A and positive on B and hence fermions in the
ground state predominantly populate the sublattice with the negative ionic potential. For
∆ < 0 the roles of w A (k) and w B (k) as well as A and B sublattices are exchanged.

210 Appendix B. Ionic-Chain
The on-site density of fermions in the ground state is given by
〈n i 〉 GS := 〈ψ GS |n i |ψ GS 〉 = 4 ∑
sin(κi) sin(κ ′ i) 〈ψ GS |cκ
γ(i),† c γ(i)
κ
|ψ
L + 1
′ GS 〉
κ,κ ′
= 4
L + 1
∑
S(k, i) 2 f(ǫ(k)) .
k
(B.15)
The state obtained after the creation of a particle on a given site j in the ground state of
the system and its time-evolution are given by
|φ j (0)〉 := c † j |ψ 2 ∑
GS〉 = √ sin(κj) c γ(j),†
κ |ψ GS 〉
L + 1
=
|φ j (t)〉 =
=
2 ∑
√ S(k, j) d † k |ψ GS〉 ,
L + 1
k
2
√ e ∑ −iHt S(k, j) d † k |ψ GS〉
L + 1
k
2 ∑
√ S(k, j) e −i(E k+E GS )t d † k |ψ GS〉 .
L + 1
k
κ
(B.16a)
(B.16b)
In this way a localized wave packet consisting of all wave vectors is formed, which then
spreads out. Since d † k |ψ GS〉 = 0 for k < k F , only the states with energies above the Fermi
energy (E F ) form and participate in the time evolution of the created particle. The different
components (quasiparticles) propagate at different speeds, given by the group velocity,
determined as the slope of the dispersion curve at κ
vg
a/b (κ) = ∂Ea/b (κ)
∂κ
= ±
√ (∆
2
2J 2 sin(κ)
) 2
+ (2J cos(κ))
2
(B.17)
(see also Fig. B.3).
The time evolution of the on-site fermion density is
with
〈n i 〉 t = 〈φ j(t)|n i |φ j (t)〉
〈φ j (t)|φ j (t)〉
(B.18a)
〈φ(t)|n i |φ(t)〉 = 〈ψ GS |c j e iHt c † i c i e−iHt c † j |ψ GS〉
16 ∑
=
S(k ′ , j)S(q, i)S(q ′ , i)S(k, j)e i(E
(L + 1) 2 k ′−Ek)t 〈ψ GS |d k ′d † q d q
d † ′ k |ψ GS〉
k,k ′ ,q,q ′
16 ∑
= S(q, j)S(q, i)S(k, i)S(k, j)e i(Eq−Ek)t [1 − f(E
(L + 1) 2 q )] [1 − f(E k )]
k,q
16 ∑
+ S(q, i) 2 f(E
(L + 1) 2 q ) ∑
q
k
S(k, j) 2 [1 − f(E k )]
(B.18b)

211
and
〈φ j (t)|φ j (t)〉 = 〈φ j (0)|φ j (0)〉 = 4
L + 1
∑
S(k, j) 2 [1 − f(E k )].
k
(B.18c)
In the simple case of an empty initial system (N e = 0), the expectation value Eq. (B.18b)
at time t = 0 is one if i = j and zero otherwise. In general, as in the case of the free
spinless lattice fermion model, the first — time-dependent — term in (B.18b) at time
t = 0 describes the induced local densities caused by the creation of the fermion on site
j. It also reassembles the density-density correlations in the system at t = 0. The last
— time-independent — term in (B.18b) reassembles the ground-state on-site density of
fermions in Eq. (B.18a).
In the following, we consider the difference in the on-site fermion density between the
time-evolved and the ground states
δn i (t) = 〈n i 〉 t − 〈n i 〉 GS
(B.19)
in order to study the space-time evolution of the fermion added to the ground state of a
given system.
Time evolution of a single fermion
We consider an open chain with L = 100 sites and J = 0.5, and investigate the spacetime
evolution of the spinless fermion added at site j = 51 at time t = 0 to the ground
state of the system of N e fermions. Several values of N e are particularly interesting:
N e = 0 which corresponds to the empty initial system, N e = L/2 for which the ground
state of the system is insulating, and two additional minimal numbers of fermions, Ne a(|∆|)
and Ne b (|∆|) for which the states that have the maximal group velocity are occupied in
both (lower and upper) energy subbands, respectively. We use four different values of
the ionic potential ∆ = 0.5, ∆ = −0.5, ∆ = 2.0, and ∆ = −2.0. In the case of |∆| = 0.5,
Ne a (0.5) = 35 and Ne(0.5) b = 66, while for |∆| = 2.0 Ne a (2.0) = 27 and Ne b (2.0) = 74 (see
also Fig. B.3). Furthermore, for |∆| = 0.5 the width of each energy subband is larger than
the gap between them (|∆|), whereas for |∆| = 2.0 the relation is opposite (see Fig. B.2).
Note also that in the case of an even chain length L, changing the sign of ∆ is equivalent
to a reflection with respect to the central bond (i → L − i + 1, i = 1, . . ., L).
First we study the space-time evolution of a single spinless fermion created at time t = 0
on site j = 51 in the empty system (N e = 0). In Fig. B.5 we show the space-time evolution
of the on-site fermion density 〈n i 〉 t (B.18) for ∆ = 0.5, ∆ = −0.5, ∆ = 2.0, and ∆ = −2.0
(plots (a), (b), (c), and (d), respectively). The wave fronts for the present model are less
pronounced as compared to the case of free lattice fermions (see Fig. A.3). Nonetheless, on
average the wave fronts can still be determined and the occurrence of additional oscillations
with the time period proportional to 1/|∆| can be identified in the space-time evolution of

212 Appendix B. Ionic-Chain
(a) ∆ = 0.5 (b) ∆ = −0.5
(c) ∆ = 2.0 (d) ∆ = −2.0
Figure B.5: Space-time evolution of a single spinless fermion on L = 100 open ionicchain
with ∆ = 0.5 (a), ∆ = −0.5 (b), ∆ = 2.0 (c), and ∆ = −2.0 (d).
the on-site fermion density. The averaged fronts propagate with maximal group velocity
which reduces with growing |∆|. Note that in general, growing |∆| reduces the width of
each energy subband as well as the value of the maximal group velocity; furthermore κ at
which the maximum group velocity is acquired shifts from κ ≈ π/2 (|∆| ≪ 1) to κ = π/4
(|∆| → ∞) (see Eqs. (B.7) and (B.17), and Figs. B.2 and B.3).
The first term of Eq. (B.18b) is invariant with respect to time reversal or/and exchange
of k and q. Moreover, for an empty initial system, the first term of Eq. (B.18b) is also
invariant with respect to the substitution of k with π − k or/and q with π − q, while the
second term is zero. Therefore Eq. (B.18b) and consequently 〈n i 〉 t (B.18) are symmetric
with respect to the reflection of ∆ (∆ → −∆) for an empty initial system (N e = 0). As
a result, the plots corresponding to ±∆ are identical (see Fig. B.5). This symmetry does
not exist when N e ≠ 0 and differences between the cases with ±∆ start to appear (see e.g.,

213
Figs. B.6–B.8).
A careful analysis of the state |ψ j (0)〉 (B.16), obtained after the addition of a fermion on
a given site j to the ground state of the system of N e fermions, reveals that the weights of
induced quasiparticles are asymmetrically distributed between the upper and lower energy
subbands (see also Eqs. (B.9), (B.8b), and (B.5), as well as Fig. B.4). This asymmetry
becomes more pronounced with growing |∆| (e.g., compare plots (a) and (b) with (c) and
(d) in Fig. B.5). To which subband weights are shifted depends on the sign of the ionic
potential on the site where the fermion was added; in the case of the negative (positive)
ionic potential on site j the quasiparticles in the lower (upper) energy subband have larger
weights. In addition, since quasiparticles from the lower (upper) energy subband have
larger weights in the sublattice with the negative (positive) ionic potential (see Eq. (B.14)
and Fig. B.4), the fermion predominantly spreads in the sublattice to which it was added.
Let us consider the case with N e = 32 < L/2 fermions in the ground state. For
|∆ = 2.0|, N e > Ne a(2.0) = 27 and the state with k = N e a (2.0)
π = 27 π that has the maximal
group velocity (see the green curve in Fig. B.3) is already occupied in the ground state,
L+1 101
whereas for |∆ = 0.5|, N e < Ne a 35
(0.5) = 35 and the state k = π with the maximal group
101
velocity (see the blue curve in Fig. B.3) is unoccupied. In the left columns of Fig. B.6
and Fig. B.7 we show the on-site density vs. position i in the ground state 〈n i 〉 GS (B.15)
(blue), the initial state 〈n i 〉 t=0 (B.18) (cyan), and the difference between them δn i (t = 0)
(B.19) (red). The right columns of the same figures show the space-time evolution of the
difference in the on-site fermion density between the time-evolved and the ground states
δn i (t). A single spinless fermion is added at the site j = 51 at t = 0. In Fig. B.6 we show
results for ∆ = 2.0 and ∆ = −2.0, while in Fig. B.7 results for ∆ = 0.5 and ∆ = −0.5 are
presented. The fermion distribution in the ground state shows a well established charge
density wave (CDW) produced by the alternating potential. Moreover, the CDW is modulated
by Friedel oscillations with wavenumber 2k F caused by the OBC. As expected, the
amplitude of the CDW increases with growing |∆|. The difference between the ±∆ cases
can already be identified in the fermion distribution in the initial state 〈n i 〉 t=0 (compare
the cyan curves on plots (a) and (c) in Fig. B.6 or Fig. B.7). At t = 0 the wave packet,
resulting from the creation of the fermion, has a finite extent. δn i (t = 0) exhibits a slight
asymmetry with respect to position 51 (site at which the fermion was added), because the
lengths of the chain parts to the left and to the right of this position are not equal (e.g.,
see the red curves in Fig. B.7a and Fig. B.7c).
Since for N e < L/2 the upper energy subband is empty in the ground state, the averaged
wave fronts propagate with a velocity that is maximal for the studied model. However,
for ∆ = 2.0 (the negative ionic potential on site j = 51) the maxima in δn i (t) are shifted
from the wave fronts to the internal parts (see Fig. B.6b). This is connected to the fact
that the quasiparticles with larger weights are generated in the lower energy subband
which is already partially filled (up to k F = Ne+1 2
π = 32.5 π); the quasiparticle state with
L+1 101

214 Appendix B. Ionic-Chain
〈n i 〉 GS , 〈n i 〉 t=0 , δn i (t=0)
1
0.8
0.6
0.4
0.2
N e =32(+1)
∆=2.0
〈n i 〉 GS
〈n i 〉 t=0
δn i (t=0)
0
-50 -40 -30 -20 -10 0 10 20 30 40
i-51
(a)
(b)
〈n i 〉 GS , 〈n i 〉 t=0 , δn i (t=0)
1
0.8
0.6
0.4
0.2
N e =32(+1)
∆=-2.0
〈n i 〉 GS
〈n i 〉 t=0
δn i (t=0)
0
-50 -40 -30 -20 -10 0 10 20 30 40
i-51
(c)
(d)
Figure B.6: Plots (a) and (c): on-site density as a function of position i in the ground
state (blue), the initial state (cyan), and the difference between them (red). Plots (b)
and (d): space-time evolution of the difference in the on-site fermion density between
the time-evolved and the ground states. A single fermion is created on the site j = 51
at t = 0 in the ground state of the system of N e = 32 fermions on an open L = 100
chain. Ionic potential ∆ = 2.0 (plots (a) and (b)) and ∆ = −2.0 (plots (c) and (d)).
the maximum group velocity — the state with κ = 27
101
π — is already occupied in the
ground state and hence does not participate in the time evolution of |ψ j (t)〉. Recall, that
only quasiparticles with energies above the Fermi energy are participating in the time
evolution of the state |ψ j (t)〉 (B.16). Therefore, the maxima in δn i (t) propagate with the
speed corresponding to the largest accessible empty state in the lower energy subband,
max kF

215
〈n i 〉 GS , 〈n i 〉 t=0 , δn i (t=0)
1
0.8
0.6
0.4
0.2
N e =32(+1)
∆=0.5
〈n i 〉 GS
〈n i 〉 t=0
δn i (t=0)
0
-50 -40 -30 -20 -10 0 10 20 30 40
i-51
(a)
(b)
〈n i 〉 GS , 〈n i 〉 t=0 , δn i (t=0)
1
0.8
0.6
0.4
0.2
N e =32(+1)
∆=-0.5
〈n i 〉 GS
〈n i 〉 t=0
δn i (t=0)
0
-50 -40 -30 -20 -10 0 10 20 30 40
i-51
(c)
(d)
Figure B.7: Plots (a) and (c): on-site density as a function of position i in the ground
state (blue), the initial state (cyan), and the difference between them (red). Plots (b)
and (d): space-time evolution of the difference in the on-site fermion density between
the time-evolved and the ground states. A single fermion is created on the site j = 51
at t = 0 in the ground state of the system of N e = 32 fermions on an open L = 100
chain. Ionic potential ∆ = 0.5 (plots (a) and (b)) and ∆ = −0.5 (plots (c) and (d)).
|∆| = 0.5 the difference between ±∆ is also noticeable (see Fig. B.7), however in this case
the quasiparticle state with k = 35 π, which has the maximum group velocity (see the blue
101
curve in Fig. B.3), is unoccupied in the ground state and the maxima in δn i (t) remain in
the wave fronts for both ∆ = ±0.5.
The additional oscillations in the on-site densities that have the time period ∼ 1/|∆| are
well traceable for ∆ = ±0.5 (see Fig. B.7), and for ∆ = 2.0 (see plot Fig. B.6b), while for
∆ = −2.0 these oscillations start to disappear (see Fig. B.6d). The nature of the observed

216 Appendix B. Ionic-Chain
〈n i 〉 GS , 〈n i 〉 t=0 , δn i (t=0)
1.4
1.2
1
0.8
0.6
0.4
N e =50(+1)
∆=2.0
〈n i 〉 GS
〈n i 〉 t=0
δn i (t=0)
0.2
0
-50 -40 -30 -20 -10 0 10 20 30 40
i-51
(a)
(b)
〈n i 〉 GS , 〈n i 〉 t=0 , δn i (t=0)
1.4
1.2
1
0.8
0.6
0.4
N e =50(+1)
∆=-2.0
〈n i 〉 GS
〈n i 〉 t=0
δn i (t=0)
0.2
0
-50 -40 -30 -20 -10 0 10 20 30 40
i-51
(c)
(d)
Figure B.8: Plots (a) and (c): on-site density as a function of position i in the ground
state (blue), the initial state (cyan), and the difference between them (red). Plots (b)
and (d): space-time evolution of the difference in the on-site fermion density between
the time-evolved and the ground states. A single fermion is created on the site j = 51
at t = 0 in the ground state of the system of N e = 50 fermions on an open L = 100
chain. Ionic potential ∆ = 2.0 (plots (a) and (b)) and ∆ = −2.0 (plots (c) and (d)).
behavior is connected to the specific structure of w A/B (k) (see Fig. B.4): the weights
of the quasiparticles generated close to the upper edge of the lower energy subband are
monotonically reaching the maximum ≈ 1 or vanishing for ∆ > 0 and ∆ < 0, respectively.
As N e → L/2, this causes the growing width of the wave packet obtained at t = 0 and
pronounced oscillations with the time period ∼ 1/|∆| in the case of ∆ > 0. For ∆ < 0, the
wave packet at t = 0 is more localized and the additional oscillations disappear gradually.
Finally we consider the case where the initial system contains N e = L/2 = 50 fermions.

In this case the ground state of the system has completely filled lower and completely
empty upper energy subbands and the system is insulating. In Fig. B.8 we show results
for two different ionic potentials ∆ = 2.0 (plots (a) and (b)) and ∆ = −2.0 (plots (c) and
(d)). As in the previous case a well established CDW can be identified in the ground
state, however for this particular filling the Friedel oscillations due to the OBC are absent
(see the blue curves in Fig. B.8a and Fig. B.8c). Moreover, due to the finite gap in the
excitation spectra the effects of the OBC as well as the effective width of the wave packet
obtained at t = 0 extend only over a few sites. In the case of ∆ = 2.0 (see Fig. B.8a and
Fig. B.8b), the sublattice to which the fermion is added is already quite well populated.
Consequently, most of the weight of the added particle is “redistributed” to the nearestneighbor
sites of the site 51 (see the red curve in Fig. B.8a) and the fermion predominantly
spreads in the complimentary sublattice. Since for the considered filling, only quasiparticles
in the upper energy subband are participating in the time evolution, irrespective of the
sign of ∆, oscillations with period ∼ 1/|∆| are completely absent and the averaged wave
fronts propagate with the maximal group velocity for the studied model (see Fig. B.8b and
Fig. B.8d).
In the case of N e > L/2 fermions in the initial system (results not shown here), the speed
by which the wave fronts propagate in the chain corresponds to the largest accessible for the
states left empty in the ground state of the system, max kF

218 Appendix B. Ionic-Chain

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List of publications 239
LIST OF PUBLICATIONS
Parts of this thesis have already been published in several articles:
• K. Byczuk, M. Sekania, W. Hofstetter, and A. P. Kampf
Insulating behavior with spin and charge order in the ionic Hubbard model
Phys. Rev. B 79, 121103 (2009)
• H. Fehske, G. Wellein, A. P. Kampf, M. Sekania, G. Hager, A. Weisse, H. Büttner,
and A. R. Bishop
One-Dimensional Electron-Phonon Systems: Mott- Versus Peierls-Insulators
High Performance Computing in Science and Engineering 2002, 339 (2003)
• A. P. Kampf, M. Sekania, G. I. Japaridze, and Ph. Brune
Nature of the insulating phases in the half-filled ionic Hubbard model
J. Phys.: Condens. Mat. 15, 5895-5907 (2003)
• H. Fehske, A. P. Kampf, M. Sekania, and G. Wellein
Nature of the Peierls- to Mott-insulator transition in 1D
Eur. Phys. J. B 31, 11 (2003)
• G. I. Japaridze, A. P. Kampf, M. Sekania, P. Kakashvili and Ph. Brune
Eta-pairing superconductivity in the Hubbard chain with pair hopping
Phys. Rev. B 65, 014518 (2002)

240 List of publications

Acknowledgments 241
ACKNOWLEDGMENTS
At this point I would like to express my gratitude to all those people who have supported
me during this time.
In the first place, I would like to thank my supervisor Prof. Dr. Arno P. Kampf who
gave me the opportunity to work in his group, for numerous helpful discussions, valuable
suggestions, and his guidance during my stay in Augsburg. Due to his support, I was able
to attend several international workshops and conferences.
I would also like to express my great gratitude to Prof. Dr. Dieter Vollhardt for his
generous and kind support during my stay in Augsburg and especially at the last stage of
the writing of my PhD thesis, which was particularly important. I appreciate very friendly
atmosphere and pleasant relations that he always manages to maintain in the group.
I want to thank Prof. Dr. Thilo Kopp who kindly agreed to co-report on this thesis
despite the tight schedule.
I am very grateful to my former supervisor Prof. Dr. Gia Japaridze for introducing
me to the field of computational solid-state physics, and for being not only an excellent
teacher, but first of all a good friend.
I also want to thank my collaborators Prof. Dr. Holger Fehske, Prof. Dr. Karen Hallberg,
and Prof. Dr. Krzysztof Byczuk for inspiring, motivating, and open discussions.
My sincere thanks are due to all, present and former, members of the group and for the
very friendly and productive atmosphere at the Center for Electronic Correlations and Magnetism
in Augsburg, among them Markus Schmid, Dr. Michael Sentef, Dr. Marcus Kollar,
Dr. Anna Kauch, Prof. Dr. Krzysztof Byczuk, Dr. Prabuddha Chakraborty, Dr. Jan Kunes,
Markus Greger, and Christian Gramsch, as well as Dr. Georg Keller, Prof. Dr. Stefan
Kehrein, Priv.-Doz. Dr. Ralf Bulla, Prof. Dr. Thomas Pruschke, Prof. Dr. Vladimir Anisimov,
and Dr. Hyung-Jung Lee.
I wish to thank Mrs. Barbara Besslich and Mrs. Anita Seidl, who helped me in
organizational matters with great tenderness.
I also want to thank my office mates Markus Schmid and Dr. Robert Zitzler for the
great fun, interesting and endless discussions on various topics, and a pleasant working
atmosphere in the office.
I owe my deepest gratitude to Dr. Anna Kauch and Prof. Dr. Liviu Chioncel for the
careful reading of the drafts of my thesis, for valuable comments and suggestions, and

242 Acknowledgments
invariable motivation during the writing of the thesis.
I would also like to thank Ralf Utermann for keeping the computer system and compute
cluster running, being open to questions and suggestions, and ready to help.
Most especially, I want to thank my great friends Dr. Ivan Leonov and Dr. Andrey
Katanin.
My cordial gratitude to my best and oldest friends — starting from the school time
— Dr. Irakli Titvinidze Dr. Paata Kakashvili, Nikoloz Kumsiashvili, Tengiz Tetunashvili,
Archil Razmadze, and Dr. George Titvinidze for constantly keeping in touch with me and
for all the fun we had and will still have together.
Especial part in these acknowledgments belongs to my grandparents, who took a great
part in forming me as a person whom I am now. Particularly to my grandfather Valerian
Kereselidze who was the greatest person in my life. I want to wish my beloved grandmother
Neli Kalandadze health and a long happy life with her lovely great grandchildren.
Finally, I would like to thank my parents and my brother with his family for endless
support and unconditional love.