Conservation laws in disordered electron systems: Thermodynamic limit and configurational averaging

TL;DR

This paper examines the challenge of extending probability conservation laws from individual disordered electron systems to their configurational averages in the thermodynamic limit, highlighting issues caused by localized states and diffusion poles.

Contribution

It clarifies the limitations of norm conservation in averaged disordered systems and discusses the mathematical and physical implications of localized states and diffusion poles.

Findings

Norm conservation holds in individual realizations but not in averaged systems.

Localized states break translational invariance, complicating averaging.

Diffusion poles cause conflicts with causality and analyticity in the theory.

Abstract

We discuss conservation of probability in noniteracting disordered electron systems. We argue that although the norm of the electron wave function is conserved in individual realizations of the random potential, we cannot extend this conservation law easily to configurationally averaged systems in the thermodynamic limit. A direct generalization of the norm conservation to averaged functions is hindered by the existence of localized states breaking translational invariance. Such states are elusive to the description with periodic Bloch waves. Mathematically this difficulty is manifested through the diffusion pole in the electron-hole irreducible vertex. The pole leads to a clash with analyticity of the self-energy, reflecting causality of the theory, when norm conservation is enforced by the Ward identity between one- and two-particle averaged Green functions.