PhD Thesis - Universität Augsburg

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