PhD Thesis - Universität Augsburg

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-