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Journal of Physics A Mathematical and General

Variational bounds for lattice fermion models. I. Spinless fermions

R J Busill, Colin J. Thompson

Published Jun 21, 1993• 1 citation
Publisher

Abstract

The authors assess the validity of mean-field theories for lattice fermion systems with attractive interactions by comparing variational upper bounds from two plausible mean-field approximations-a Van der Waals (independent particle) approximation and a pairing approximation of the BCS type-for the Helmholtz free energy of a spinless fermion model with attractive nearest-neighbour interactions. It is found that there is a crossover from the Van der Waals approximation giving the best bound to the pairing approximation giving the best bound as t, the hopping integral passes through a critical value t c . This crossover phenomenon exists in all dimensions as well as for the limiting d to infinity model. The authors conclude that there can be no simple mean-field theory which is valid for fermion systems with attractive interactions.

DOI
10.1088/0305-4470/26/12/014

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The authors assess the validity of mean-field theories for lattice fermion systems with attractive interactions by comparing variational upper bounds from two plausible mean-field approximations-a Van der Waals (independent particle) approximation and a pairing approximation of the BCS type-for the Helmholtz free energy of a spinless fermion model with attractive nearest-neighbour interactions. It is found that there is a crossover from the Van der Waals approximation giving the best bound to the pairing approximation giving the best bound as t, the hopping integral passes through a critical value t c . This crossover phenomenon exists in all dimensions as well as for the limiting d to infinity model. The authors conclude that there can be no simple mean-field theory which is valid for fermion systems with attractive interactions.
We study a microscopic model for four spinless fermions on the square lattice which exhibits a quartet bound state in the strong coupling regime. The four-particle quantum states are analyzed using symmetry arguments and by introducing a zoo of relevant lattice animals. These considerations, as well as variational and exact diagonalization calculations, demonstrate the existence of a narrow quartet band at small hopping and a first-order transition to delocalized fermions at a critical hopping parameter, in qualitative contrast to, e.g., the BCS-BEC crossover in the attractive Hubbard model. In the case of pure attraction, an intermediate phase is found, in which a more extended and presumably more mobile hybrid quartet dominates the ground state. We comment on the relevance of the spin degree of freedom and on the reasons why electron quartetting is rarely observed in real materials.
Conventional Hartree-Fock mean field theory is used, for the first time, to investigate the full two-dimensional t-J model. To date, all other nontrivial mean field approaches modify the Hamiltonian or violate the double occupancy constraint. Unlike conventional mean field theory, these methods do not give variational results for the t-J model. The mean field phase diagram of the t-J model predicts enhanced superconductivity at the crossover between an insulating phase at low doping and a metallic phase at moderate doping. This behavior is remarkably similar to that of high temperature superconductors, for which the model is expected to provide a good description. In the mean field approximation, the t-J model is described by self-interacting spinless fermions hopping on a background lattice with spiral antiferromagnetic order. The interaction is attractive and high temperature superconductivity arises from the narrow bandwidth at the insulator to metal crossover.
In order to develop our theoretical understanding of the superconducting state of strongly correlated electron systems, we investigate the many-body ground state of the attractive Hubbard model in infinite dimensions using the Gutzwiller-type projected BCS variational (GBCS) wave function. We first calculate various physical quantities by numerically evaluating analytical equations for systems in the thermodynamic limit. We then show that the GBCS wave function has the lowest energy when the gap parameter has a mixture of s wave and extended s wave pairing symmetry and that the ground state energy of the GBCS is lower than that of either the BCS or Fermi liquid at any electron density and any strength of the attractive interactions in our model. We also show that with infinite strength interactions, each pair of electrons with opposite spins forms a tightly bound pair on a lattice site (BEC-like state). We find a smooth crossover from a BCS-like state with large, overlapping Cooper pairs (for weakly attractive interactions) to a BEC-like state of composite bosons formed out of the tightly bound pairs of fermions (for strongly attractive interactions).
Abstract Very few statistical mechanical models have been solved exactly. In three dimensions not even the nearest neighbour spin-1/2 Ising model is tractable. Therefore it is necessary to resort to approximation methods. One of the most widely used of these is mean-field theory. A systematic way of deriving the mean-field theory for a given microscopic Hamiltonian is to start from the Bogoliubov inequality This gives the best possible approximation to the true free energy for a given choice of Ho because the inequality (4.1) insists that the meanfield free energy cannot fall below the true free energy. (This is analogous to the variational principle in quantum mechanics.) The usual choice for Ho is a free Hamiltonian -one with no interactions-as this enables explicit calculation of the right-hand side of (4.1).
In a recent contribution [P. Corboz, R. Or\'us, B. Bauer, and G. Vidal, Phys. Rev. B 81, 165104 (2010)] fermionic projected entangled-pair states (PEPSs) were used to approximate the ground state of free and interacting spinless fermion models, as well as the $t\text{\ensuremath{-}}J$ model. This paper revisits these three models in the presence of an additional next-nearest hopping amplitude in the Hamiltonian. First we explain how to account for next-nearest neighbor Hamiltonian terms in the context of fermionic PEPS algorithms based on simulating time evolution. Then we present benchmark calculations for the three models of fermions and compare our results against analytical, mean-field, and variational Monte Carlo results, respectively. Consistent with previous computations restricted to nearest neighbor Hamiltonians, we systematically obtain more accurate (or better converged) results for gapped phases than for gapless ones.