Quantum diffusion in a random potential: A consistent perturbation theory

TL;DR

This paper develops a consistent perturbation theory for quantum diffusion in disordered systems, ensuring conservation laws via Ward identities and deriving key properties like the diffusion pole and diffusion constant.

Contribution

It establishes a method to enforce Ward identities in perturbation theory, leading to a fully conserving and consistent description of quantum diffusion.

Findings

Derived the low-energy asymptotics of the two-particle vertex

Provided an exact representation of the diffusion pole and diffusion constant

Analyzed vertex corrections due to maximally-crossed diagrams for weak localization

Abstract

We scrutinize the diagrammatic perturbation theory of noninteracting electrons in a random potential with the aim to accomplish a consistent comprehensive theory of quantum diffusion. Ward identity between the one-electron self-energy and the two-particle irreducible vertex is generally not guaranteed in the perturbation theory with only elastic scatterings. We show how the Ward identity can be established in practical approximations and how the functions from the perturbation expansion should be used to obtain a fully consistent conserving theory. We derive the low-energy asymptotics of the conserving full two-particle vertex from which we find an exact representation of the diffusion pole and of the static diffusion constant in terms of Green functions of the perturbation expansion. We illustrate the construction on the leading vertex corrections to the mean-field diffusion due to maximally-crossed diagrams responsible for weak localization.