Equivalence of the Falicov-Kimball and Brandt-Mielsch forms for the free energy of the infinite-dimensional Falicov-Kimball model

TL;DR

This paper proves the explicit equivalence between two different mathematical formulations of the free energy in the Falicov-Kimball model within infinite dimensions, confirming their numerical consistency.

Contribution

It provides a detailed derivation demonstrating the equivalence of the real-axis and imaginary-axis free energy formulas for the Falicov-Kimball model.

Findings

The two formulas are numerically equal.

Explicit derivation of their equivalence is provided.

Confirms consistency of different theoretical approaches.

Abstract

Falicov and Kimball proposed a real-axis form for the free energy of the Falicov-Kimball model that was modified for the coherent potential approximation by Plischke. Brandt and Mielsch proposed an imaginary-axis form for the free energy of the dynamical mean field theory solution of the Falicov-Kimball model. It has long been known that these two formulae are numerically equal to each other; an explicit derivation showing this equivalence is presented here.