Asymptotic limit of high spatial dimensions and thermodynamic consistence

TL;DR

This paper investigates the thermodynamic consistency and $\Phi$-derivability of high-dimensional limits in quantum itinerant models, revealing that vertex functions retain momentum dependence and vertex corrections may not vanish, affecting mean-field theory validity.

Contribution

It demonstrates that in high spatial dimensions, vertex functions retain momentum dependence and mean-field theory's $\Phi$-derivability depends on external sources.

Findings

Vertex functions retain momentum dependence in high dimensions.

Vertex corrections to conductivity do not vanish in the mean-field limit.

Mean-field theory is $\Phi$-derivable only with specific external sources.

Abstract

The question of thermodynamic consistence and $\Phi$-derivability of the asymptotic limit of high spatial dimensions for quantum itinerant models is addressed. It is shown that although the irreducible $n$-particle Green functions are local, reducible vertex functions retain different momentum dependence. As a consequence, the vertex corrections to conductivity do not generally vanish in the mean-field limit. The mean-field theory is a $\Phi$-derivable approximation only if regular nonlocal or anomalous local external sources are admitted.