A microscopic theory of Anderson localization of electrons in random lattices

TL;DR

This paper develops a microscopic theory for Anderson localization of electrons in disordered lattices, unifying metallic and localized phases and providing a framework for analyzing critical behavior at the mobility edge.

Contribution

It introduces a comprehensive microscopic model that describes both diffusive and localized electron regimes, including a mean-field approximation for two-particle interactions.

Findings

Derived a local approximation for two-particle irreducible vertices.

Analyzed dynamic properties and critical behavior at the mobility edge.

Identified a new instability line for electron-hole correlations in the metallic phase.

Abstract

The existence of Anderson localization, characterized by vanishing diffusion due to strong disorder, has been demonstrated in numerous ways. A systematic approach based on the Anderson quantum model of the Fermi gas in random lattices that can describe both diffusive and localized regimes has not yet been fully established. We build on a recent publication \cite{Janis:2025ab} and present a microscopic theory of disordered electrons that covers both the metallic phase with extended Bloch waves and the localized phase, where a propagating particle forms a quantum bound state with the hole left behind at the origin. The general theory provides a framework for constructing controlled approximations to one- and two-particle Green functions that satisfy the necessary conservation laws and causality requirements across the full range of disorder strength. It is used explicitly to derive a local, mean-field-like approximation for the two-particle irreducible vertices, enabling quantitative analysis of the solution's dynamic properties in both metallic and localized phases, including critical behavior at the mobility edge. A new instability line for the dynamical electron-hole correlation function of the metallic phase is introduced.