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International Journal of Modern Physics B

PRELIMINARY GROUP ANALYSIS OF A 4 X 4 TWO DIMENSIONAL HUBBARD MODEL

G. Fano, F. Ortolani, F. Semeria

Published Dec 1, 1989• 16 citations
Publisher

Abstract

A finite size two-dimensional Hubbard model is considered; it is shown that the consideration of the spatial symmetry group of the Hamiltonian reduces the dimension of the problem of two orders of magnitude in the case of a 4 X 4 lattice. Exact diagonalization of the Hamiltonian can then be performed with the aid of modern high-speed computers.

DOI
10.1142/s0217979289001184

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A finite size two-dimensional Hubbard model is considered; it is shown that the consideration of the spatial symmetry group of the Hamiltonian reduces the dimension of the problem of two orders of magnitude in the case of a 4 X 4 lattice. Exact diagonalization of the Hamiltonian can then be performed with the aid of modern high-speed computers.
The finite-size corrections in exact diagonalization studies of the one- and two-dimensional Hubbard model can be reduced systematically by a grand-canonical integration over boundary conditions. We present results for ground-state properties of the two-dimensional (2D) Hubbard model and an evaluation of the specific heat for the 1D and 2D Hubbard model. We find the reduction of the finite-size corrections to be substantial, especially in two dimensions. \textcopyright{} 1996 The American Physical Society.
We have used both an exact diagonalization method for finite-size systems and a perturbation technique to study the lattice instability due to dimerization in a two-dimensional Hubbard model in the large-U limit. We show that a critical electron-phonon coupling strength is required to form a dimerized state, in contrast to the 1-dimensional system. Our calculations suggest the phase transition to the dimerized state is of first order. The relevance to high-Tc{T}_{c}Tc​ superconductor materials is discussed.
The Lanczos algorithm has been applied to the Hubbard model on a 4\ifmmode\times\else\texttimes\fi{}4 square lattice and several equal-time correlation functions have been evaluated. This computation has been made possible by a classification of all the irreducible representations of the space symmetry group, which allowed a considerable reduction of the dimension of the Hilbert space of the system. Such an analysis is quite general and might be useful in other physical applications. Our results shed light on the effects of doping on the magnetic ordering, suggest a spin-charge decoupling at large U, and show a remarkable reduction of the quasiparticle weight at strong coupling.
The Hubbard model is exactly solved for two particles on $d\ensuremath{-}$dimensional hypercubes. It is shown that the spectrum can be separated into two parts: a trivial (UUU-independent) part resulting from symmetries of hypercubes and a nontrivial part expressed as a single-impurity problem on a set of finite chains of size ${d}^{\ensuremath{'}}+1$ ${(d}^{\ensuremath{'}}<~d).$ The exact expression for the one-particle Green's function is given. Finally, we discuss the extension of these results to standard hypercubic lattices with periodic boundary conditions.
An exact analytical diagonalization is used to solve the two-dimensional extended Hubbard model (EHM) for a system with finite size. We have considered an EHM including on-site and off-site interactions with interaction energies U and V , respectively, for a square lattice containing 4 × 4 sites at one-eighth filling with periodic boundary conditions, recently treated by Kovacs and Gulacsi (2006 Phil. Mag. 86 2073). Taking into account the symmetric properties of this square lattice and using a translation linear operator, we have constructed a r-space basis only with 85 state-vectors which describe all possible distributions for four electrons in the 4 × 4 square lattice. The diagonalization of the 85 × 85 matrix energy allows us to study the local properties of the above system as a function of the on-site and off-site interactions energies, where we have shown that the off-site interaction encourages the existence of the double occupancies at the first excited state and induces a supplementary conductivity of the system.