Lower bound for the segregation energy in the Falicov-Kimball model

TL;DR

This paper derives a lower bound for the ground state energy in the Falicov-Kimball model, aiding the proof of fermion segregation conjecture at large interactions, with explicit boundary terms in two dimensions.

Contribution

It provides a novel explicit lower bound for the model's energy, including boundary effects, crucial for understanding fermion segregation at intermediate densities.

Findings

Derived a lower bound involving bulk and boundary terms.

Explicit boundary coefficient obtained for density 1/2 in 2D.

Method adaptable for all densities with modifications.

Abstract

In this work, a lower bound for the ground state energy of the Falicov-Kimball model for intermediate densities is derived. The explicit derivation is important in the proof of the conjecture of segregation of the two kinds of fermions in the Falicov-Kimball model, for sufficiently large interactions. This bound is given by a bulk term, plus a term proportional to the boundary of the region devoid of classical particles. A detailed proof is presented for density n=1/2, where the coefficient 10^(-13) is obtained for the boundary term, in two dimensions. With suitable modifications the method can also be used to obtain a coefficient for all densities.