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Physical Review Letters

Hubbard and Anderson models on perovskitelike lattices: Exactly solvable cases

U. Brandt, Andreas Giesekus

Published Apr 27, 1992• 65 citations
Publisher

Abstract

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 (${\mathit{E}}_{\mathit{f}}$,V) parameter space; however, the resulting relation V(${\mathit{E}}_{\mathit{f}}$) does not lead to unphysical parameters.

DOI
10.1103/physrevlett.68.2648

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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.
We generalize an approach, which was recently introduced by Brandt and Giesekus to calculate the exact ground-state energy for strongly interacting particles on special perovskitelike lattices, to the periodic Anderson model in the dimension d=1 and to extended Emery models in d=1,2 on regular lattices for arbitrary spin degeneracy. For these models we calculate the exact ground-state energy for a restricted parameter regime in the strong-coupling limit. The ground-state energy shows a simple algebraic structure. We also present an eigenfunction of the Hamiltonian with the ground-state energy as its corresponding eigenvalue.
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.
We present results on the thermodynamic quantities, resistivity, and optical conductivity for the Hubbard model on a simple hypercubic lattice in infinite dimensions. Our results for the paramagnetic phase display the features expected from an intuitive analysis of the one-particle spectra and substantiate the similarity of the physics of the Hubbard model to those of heavy-fermion systems. The calculations were performed using an approximate solution to the single-impurity Anderson model, which is the key quantity entering the solution of the Hubbard model in this limit. To establish the quality of this approximation we compare its results, together with those obtained from two other widely used methods, to essentially exact quantum Monte Carlo results.