Proof of phase separation in the binary-alloy problem: the one-dimensional spinless Falicov-Kimball model

TL;DR

This paper demonstrates phase separation in the one-dimensional Falicov-Kimball model's ground states, showing a transition from periodic to mixed configurations depending on ion density and coupling strength.

Contribution

It provides a detailed analysis of phase separation phenomena in the Falicov-Kimball model, revealing a phase transition driven by coupling and ion density.

Findings

Phase separation occurs when ion density is far from 1/2.

Ground states transition from periodic to mixed configurations.

A phase transition is identified as a function of coupling strength.

Abstract

The ground states of the one-dimensional Falicov-Kimball model are investigated in the small-coupling limit, using nearly degenerate perturbation theory. For rational electron and ion densities, respectively equal to $\frac{p}{q}$, $\frac{p_i}{q}$, with $p$ relatively prime to $q$ and $\frac{p_i}{q}$ close enough to $\frac{1}{2}$, we find that in the ground state the ion configuration has period $q$. The situation is analogous to the Peierls instability where the usual arguments predict a period-$q$ state that produces a gap at the Fermi level and is insulating. However for $\frac{p_i}{q}$ far enough from $\frac{1}{2}$, this phase becomes unstable against phase separation. The ground state is a mixture of a period-$q$ ionic configuration and an empty (or full) configuration, where both configurations have the same electron density to leading order. Combining these new results with those previously obtained for strong coupling, it follows that a phase transition occurs in the ground state, as a function of the coupling, for ion densities far enough from $\frac{1}{2}$.