Causality vs. Ward identity in disordered electron systems

TL;DR

This paper investigates the conflict between causality and Ward identity in disordered electron systems, showing that in the diffusive regime, the Ward identity cannot be fully satisfied without violating causality.

Contribution

It proves that assuming the diffusion pole and electron-hole symmetry leads to a partial violation of the Ward identity due to causality constraints.

Findings

Ward identity is partly violated in diffusive regime

Causality constrains the self-energy solutions

Physical explanation of the violation provided

Abstract

We address the problem of fulfilling consistency conditions in solutions for disordered noninteracting electrons. We prove that if we assume the existence of the diffusion pole in an electron-hole symmetric theory we cannot achieve a solution with a causal self-energy that would fully fit the Ward identity. Since the self-energy must be causal, we conclude that the Ward identity is partly violated in the diffusive transport regime of disordered electrons. We explain this violation in physical terms and discuss its consequences.