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Communications in Theoretical Physics

The Structure of Energy Matrix and the Ground State for Two-Dimensional Hubbard Model

Ke-Cai Zeng, Zhenqing Yang, Kong Fan–mei

Published Sep 15, 1996• 0 citations
Publisher

Abstract

By constructing a series of ideal matrices which possess the same properties as a real energy matrix of two-dimensional Hubbard model does, the dependence of the lowest energy eigenvdue on the strudture of matrix is studied. It is found that the disorder of the distribution of a large number of off-diagonal blocks trends to even the lowest energy state and the so-called d -state. Thus degenerate ground states of various will be obtained in the thermodynamic limit.

DOI
10.1088/0253-6102/26/2/135

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By constructing a series of ideal matrices which possess the same properties as a real energy matrix of two-dimensional Hubbard model does, the dependence of the lowest energy eigenvdue on the strudture of matrix is studied. It is found that the disorder of the distribution of a large number of off-diagonal blocks trends to even the lowest energy state and the so-called d -state. Thus degenerate ground states of various will be obtained in the thermodynamic limit.
It is shown that for a definite cluster in the two-dimensional U=\ensuremath{\infty} Hubbard model, there are states in any Sz{\mathit{S}}^{\mathit{z}}Sz sector, which are degenerate with the lowest state in the Smaxz{\mathit{S}}_{\mathrm{max}}^{\mathit{z}}Smaxz​ sector. It is shown further that these states become the lowest ones in their sectors as the cluster becomes large enough, keeping the hole concentration constant. Then degenerate ground states of various Sz{\mathit{S}}^{\mathit{z}}Sz are obtained. The properties of the ground state of any Sz{\mathit{S}}^{\mathit{z}}SzSmaxz{\mathit{S}}_{\mathrm{max}}^{\mathit{z}}Smaxz​ can be found through those of the ground state of Smaxz{\mathit{S}}_{\mathrm{max}}^{\mathit{z}}Smaxz​.
The energy level statistics of the Hubbard model for L\ifmmode\times\else\texttimes\fi{}L square lattices (L=3,4,5,6) at low filling (four electrons) is studied numerically for a wide range of coupling strength. All known symmetries of the model (space, spin, and pseudospin symmetry) have been taken into account explicitly from the beginning of the calculation by projecting into symmetry-invariant subspaces. The details of this group theoretical treatment are presented with special attention to the nongeneric case of L=4, where a particular complicated space group appears. For all the lattices studied, a significant amount of levels within each symmetry invariant subspaces remains degenerated, but except for L=4 the ground state is nondegenerate. We explain the remaining degeneracies, which occur only for very specific interaction-independent states, and we disregard these states in the statistical spectral analysis. The intricate structure of the Hubbard spectra necessitates a careful unfolding procedure, which is thoroughly discussed. Finally, we present our results for the level spacing distribution, the number variance ${\mathrm{\ensuremath{\Sigma}}}^{2}$, and the spectral rigidity ${\mathrm{\ensuremath{\Delta}}}_{3}$, which essentially all are close to the corresponding statistics for random matrices of the Gaussian ensemble independent of the lattice size and the coupling strength. Even very small coupling strengths approaching the integrable zero coupling limit lead to Gaussian ensemble statistics, stressing the nonperturbative nature of the Hubbard model.
Abstract. We study the influence of on-site disorder on the magnetic properties of the ground state of the infinite-U one-dimensional (1D) Hubbard model. We find that the ground state is not ferromagnetic. This is analysed in terms of the algebraic structure of the spin dependence of the Hamiltonian. A simple explanation is derived for the 1=N periodicity in the persistent current for this model. The Hubbard model is the simplest model one can study to examine the effects of correlations between electrons in narrow energy bands. The Hamiltonian consists of a nearest neighbour hopping term and an electron–electron repulsion, U, which acts only when two electrons are on the same site. The Hubbard model is also the canonical model for the study of itinerant ferromagnetism. The strong coupling regime is of special importance for the study of ferromagnetism, since a theorem by Nagaoka [1] states that in theU D 1 limit, the ground state (GS) is ferromagnetic given some connectivity property of the lattice (which holds in most cases for d> 1). The model is solvable in one dimension and it was shown [2] that for open boundary conditions (BCs) the GS for finiteU is a singlet, i.e. there can be no ferromagnetism unless one postulates explicitly spin- or velocity-dependent forces. For infinite U, the GS of
The inhomogeneous Hubbard model is investigated in one and two dimensions by using lattice density-functional theory (LDFT). The ground-state energy E=T+WE=T+WE=T+W is regarded as a functional of the single-particle density matrix $\ensuremath{\gamma}$, where $T[\ensuremath{\gamma}]$ represents the kinetic and crystal-field energy, and $W[\ensuremath{\gamma}]$ the interaction energy. Besides the known functional $T[\ensuremath{\gamma}]$ we propose a simple scaling approximation to the interaction energy $W[\ensuremath{\gamma}]$, which is based on exact results for the Hubbard dimer and on a scaling hypothesis within the domain of representability of $\ensuremath{\gamma}$. As applications we consider the Hubbard model on one- and two-dimensional bipartite lattices. Several ground-state properties are determined including the kinetic and Coulomb energy, density distribution, nearest-neighbor bond order, and charge gap. Comparison with exact Lanczos diagonalizations shows that LDFT with the scaled dimer approximation to $W[\ensuremath{\gamma}]$ yields a quite accurate description of the interplay between correlations and charge redistributions, from weak to strong coupling regimes, and for all band fillings. Goals and limitations of the present approach are discussed.
We investigate the ground state of the Hubbard model in one and two dimensions. Long perturbation expansions in t/U, i.e., about the atomic limit, are computed for the ground-state energy of finite lattices with N\ensuremath{\le}10 for arbitrary electron density. We also extend the series for the one-dimensional infinite lattice by expanding the Bethe-ansatz equations. Pad\'e-approximant analysis of the series shows a distribution of singularities that varies dramatically with the electron density n. We also investigate the effect of inclusion of next-nearest-neighbor hopping. This appears to have little effect at half-filling but is significant away from half-filling.