Integrability of the diffusion pole in the diagrammatic description of noninteracting electrons in a random potential

TL;DR

This paper examines the conditions under which the diffusion pole exists in disordered electron systems, concluding it can only be present in the metallic phase in dimensions greater than two due to integrability constraints.

Contribution

It derives a nonlinear integral equation for two-particle irreducible vertices and establishes the integrability condition for the diffusion pole in disordered systems.

Findings

Diffusion pole exists only if integrable.

The diffusion pole is restricted to metallic phases.

Existence depends on system dimensionality and phase.

Abstract

We discuss restrictions on the existence of the diffusion pole in the translationally invariant diagrammatic treatment of disordered electron systems. We use the Bethe-Salpeter equations for the two-particle vertex in the electron-hole and the electron-electron scattering channels and derive for systems with time reversal symmetry a nonlinear integral equation the two-particle irreducible vertices from both channels must obey. We use this equation to test the existence of the diffusion pole in the two-particle vertex. We find that a singularity of the diffusion pole can exist only if it is integrable, that is only in the metallic phase in dimensions $d>2$.