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Chinese Physics B

Slater determinant and exact eigenstates of the two-dimensional Fermi–Hubbard model

Jun-Hang Ren, Ming-Yong Ye, Xiu‐Min Lin

Published Jul 1, 2018• 2 citations
Publisher

Abstract

We consider the construction of exact eigenstates of the two-dimensional Fermi–Hubbard model defined on an L × L lattice with a periodic condition. Based on the characteristics of Slater determinants, several methods are introduced to construct exact eigenstates of the model. The eigenstates constructed are independent of the on-site electron interaction and some of them can also represent exact eigenstates of the two-dimensional Bose–Hubbard model.

DOI
10.1088/1674-1056/27/7/073102

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Two-dimensional Hubbard model is very important in condensed matter physics. However it has not been resolved though it has been proposed for more than 50 years. We give several methods to construct eigenstates of the model that are independent of the on-site interaction strength UUU.
Exact solutions of the Hubbard model and the periodic Anderson model in the limit of infinite interaction strength are presented. Both models are studied on a D-dimensional decorated hypercubic lattice with periodic boundary conditions for any dimension D\ensuremath{\ge}2 and arbitrary size. The lattice is very similar to the perovskite lattice. In addition to the ground-state energy, a corresponding eigenstate is constructed. This ground state contains at least two particles per unit cell. For the Anderson model, the exact solution is restricted to a surface in the (Ef{\mathit{E}}_{\mathit{f}}Ef​,V) parameter space; however, the resulting relation V(Ef{\mathit{E}}_{\mathit{f}}Ef​) does not lead to unphysical parameters.
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.
We present exact explicit analytical results describing the exact ground state of four electrons in a two-dimensional square Hubbard cluster containing 16 sites taken with periodic boundary conditions. The presented procedure, which works for arbitrary even particle number and lattice sites, is based on explicitly given symmetry adapted base vectors constructed in r space. The Hamiltonian acting on these states generates a closed system of 85 linear equations, providing by its minimum eigenvalue, the exact ground state of the system. The presented results, described with the aim to generate further creative developments, not only show how the ground state can be exactly obtained and what kind of contributions enter in its construction, but also emphasize further characteristics of the spectrum. On this line (i) possible explications are found regarding why weak coupling expansions often provide a good approximation for the Hubbard model at intermediate couplings, or (ii) explicitly given low-lying energy states of the kinetic energy, avoiding double occupancy, suggest new roots for pairing mechanism attracting decrease in the kinetic energy, as emphasized by kinetic energy driven superconductivity theories.
Abstract We investigate Hubbard models with bond–charge interactions on general graphs. For a Hamiltonian H of such a model, we provide the condition on its parameters under which the η -pairing method can be employed to construct its exact eigenstates. We arrive at this condition by finding that the requirement for the η -pairing state ( η † ) N ∣ 0 〉 to be an eigenstate of H is identical to the requirement for it to be an eigenstate of a Hubbard-type Hamiltonian H m without bond–charge interactions. When the condition for ( η † ) N ∣ 0 〉 to be an eigenstate of the Hubbard-type Hamiltonian H m is satisfied, we demonstrate that there are additional states, distinct from ( η † ) N ∣ 0 〉 , which are also exact eigenstates of H m . Our results enhance the understanding of Hubbard models on general graphs, both with and without bond–charge interactions.