Local correlation functions of arbitrary order for the Falicov-Kimball model

TL;DR

This paper derives a general analytical expression for local n-particle vertex functions in the Falicov-Kimball model, highlighting their increasing magnitude with particle number and their significant impact on self-energy corrections in diagrammatic DMFT extensions.

Contribution

It provides the first analytical formula for n-particle vertices in the Falicov-Kimball model, enabling analysis of higher-order correlations beyond the two-particle level.

Findings

Higher-order vertices increase in magnitude with particle number.

Three-particle vertex corrections are comparable to two-particle corrections in certain cases.

Higher-order vertices have a moderate effect on Feynman diagrams due to Green's function damping.

Abstract

Local n-particle vertex functions represent the crucial ingredient for all diagrammatic extensions of dynamical mean field theory (DMFT). Hitherto their application has been restricted -with a few exceptions- to the n=2-particle vertex while higher-order vertices have been neglected. In this paper we derive a general analytical expression for the n-particle (one-particle reducible) vertex of the Falicov-Kimball model (FKM). We observe that the magnitude of such vertex functions itself strongly increases with the particle-number n. On the other hand, their effect on generic Feynman diagrams remains rather moderate due to the damping effect of the Green's functions present in such diagrams. Nevertheless, they yield important contributions to the self-energy corrections calculated in diagrammatic extensions of DMFT as we explicitly demonstrate using the example of dual fermion (DF) calculations for the two-dimensional FKM at quarter filling of stationary f-electrons where corrections to the self-energy obtained from the three-particle vertex are indeed comparable in magnitude to corresponding corrections stemming from the two-particle vertex.