Diagrammatic content of the DMFT for the Holstein polaron problem in finite dimensions

TL;DR

This paper analyzes the diagrammatic content of DMFT in the Holstein polaron problem, highlighting its limitations in finite dimensions due to the neglect of inter-site vertex corrections.

Contribution

It clarifies the diagrammatic basis of DMFT for the Holstein polaron and discusses its limitations in finite dimensions, especially regarding polaron spreading.

Findings

DMFT sums a specific class of local diagrams in the Holstein polaron problem.

In finite dimensions, DMFT cannot accurately describe polaron spreading.

Vertex correction limitations of DMFT persist at finite densities and with phonon dispersion.

Abstract

In the context of the Holstein polaron problem it is shown that the dynamical mean field theory (DMFT) corresponds to the summation of a special class of local diagrams in the skeleton expansion of the self-energy. In the real space representation, these local diagrams are characterized by the absence of vertex corrections involving phonons at different lattice sites. Such corrections vanish in the limit of infinite dimensions, for which the DMFT provides the exact solution of the Holstein polaron problem. However, for finite dimensional systems the accuracy of the DMFT is limited. In particular, it cannot describe correctly the adiabatic spreading of the polaron over multiple lattice sites. Arguments are given that the DMFT limitations on vertex corrections found for the Holstein polaron problem persist for finite electron densities and arbitrary phonon dispersion.