Segregation in the Falicov-Kimball model

TL;DR

This paper demonstrates that in the Falicov-Kimball model, particles tend to segregate at zero temperature when away from half-filling, with mathematical bounds showing how domain shape influences kinetic energy minimization.

Contribution

It provides new mathematical bounds on eigenvalues of the Laplace operator and applies them to show segregation phenomena in the Falicov-Kimball model.

Findings

Particles segregate at zero temperature away from half-filling.

Domain shape affects kinetic energy through boundary terms.

Single compact domain minimizes kinetic energy for itinerant particles.

Abstract

The Falicov-Kimball model is a simple quantum lattice model that describes light and heavy electrons interacting with an on-site repulsion; alternatively, it is a model of itinerant electrons and fixed nuclei. It can be seen as a simplification of the Hubbard model; by neglecting the kinetic (hopping) energy of the spin up particles, one gets the Falicov-Kimball model. We show that away from half-filling, i.e. if the sum of the densities of both kinds of particles differs from 1, the particles segregate at zero temperature and for large enough repulsion. In the language of the Hubbard model, this means creating two regions with a positive and a negative magnetization. Our key mathematical results are lower and upper bounds for the sum of the lowest eigenvalues of the discrete Laplace operator in an arbitrary domain, with Dirichlet boundary conditions. The lower bound consists of a bulk term, independent of the shape of the domain, and of a term proportional to the boundary. Therefore, one lowers the kinetic energy of the itinerant particles by choosing a domain with a small boundary. For the Falicov-Kimball model, this corresponds to having a single `compact' domain that has no heavy particles.