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arXiv:cond-mat/9811299

Falicov-Kimball Models: A Partial Review of the Ground States Problem

Gruber, Christian

Published Nov 20, 1998• 5 citations
arXivHTMLPDF

Abstract

In this review we present a biased review of the ground state properties of the Falicov-Kimball models in 1,2 and infinite dimensions, considering either fermions or hard-core bosons. In particular we want to show the very rich structure that these models exhibit and to point out the analogies and differences associated with the statistic of the quantum particles and the nature of the lattice (bipartite or not). The flux phase problem is also discussed.

DOI
10.48550/arxiv.cond-mat/9811299

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In this review we present a biased review of the ground state properties of the Falicov-Kimball models in 1,2 and infinite dimensions, considering either fermions or hard-core bosons. In particular we want to show the very rich structure that these models exhibit and to point out the analogies and differences associated with the statistic of the quantum particles and the nature of the lattice (bipartite or not). The flux phase problem is also discussed.
The Falicov-Kimball model is a lattice model of itinerant spinless fermions ("electrons") interacting by an on-site potential with classical particles ("ions"). We continue the investigations of the crystalline ground states that appear for various filling of electrons and ions, for large coupling. We investigate the model for square as well as triangular lattices. New ground states are found and the effects of a magnetic flux on the structure of the phase diagram is studied. The flux phase problem where one has to find the optimal flux configurations and the nuclei configurations is also solved in some cases. Finaly we consider a model where the fermions are replaced by hard-core bosons. This model also has crystalline ground states. Therefore their existence does not require the Pauli principle, but only the on-site hard-core constraint for the itinerant particles.
We consider the one-dimensional Falicov-Kimball lattice model as well as a continuum model, where the attractive potential between the electrons and the nuclei is a \ensuremath{\delta} function, with an equal number N of classical ``nuclei'' and fermion ``electrons.'' We find the exact leading behavior of the ground-state energy as a function of U for U>Uc{\mathit{U}}_{\mathit{c}}Uc​. The error term is rigorously estimated. For fixed N the system forms ``atoms'' which have an effective repulsion between them. From this we argue that when N becomes infinite the ground-state configurations of the nuclei should be periodic. The low-temperature structure of the model is also discussed.
The two-state spinless Falicov-Kimball model on a one-dimensional lattice is studied by means of well-controlled numerical procedures. Restricted phase diagrams in the grand-canonical ensemble and at zero temperature are constructed. The evolution of these phase diagrams, as the interaction parameter U is varied, along with the band structures corresponding to configurations of localized particles (ions) and densities of mobile particles (electrons), is monitored. The changes observed enable us to draw conclusions for the complete phase diagram and to make a qualitative plot of such a diagram.
We consider the Falicov-Kimball model in the case of an equal number of classical "nuclei" and quantum "electrons" with Fermi statistics. In one dimension we find an exact formula for the leading behavior of the ground-state energy as a function of the attractive potential U between electrons and nuclei. For U > U c the system forms "atoms" which have an effective repulsion between them. Similar results hold for a continuum model provided there is a sufficiently large hard core between the nuclei. Some low-temperature properties are also discussed.
The ground-state phase diagram of two-dimensional Falicov-Kimball model with nearest-neighbour and next-nearest-neighbour hoppings (characterized by t,t' constants, respectively) has been studied in perturbative regime, i.e. in the case when on-site Coulomb interaction constant U is much larger than t,t' ones. The fourth-order phase diagram exhibits a rich structure, more complicated than for the ordinary Falicov-Kimball model. Possible experimental implications of the presence of nnn term are shortly discussed.