Anderson localization: A disorder-induced quantum bound state

TL;DR

This paper investigates Anderson localization by analyzing how electrons become localized due to disorder, identifying a quantum bound state formation that causes the transition, and describing the transition through divergence in a new time scale.

Contribution

It introduces a novel approach using Green functions and parquet equations to describe the localization transition via quantum bound states in high dimensions.

Findings

Divergence in a new time scale signals the localization transition.

The static diffusion constant's width, not height, vanishes at the transition.

Localized quantum bound states cannot be described by traditional wave equations.

Abstract

Electrons at the Fermi energy may lose their ability to propagate to long distances in certain random media. We use Green functions and solve parquet equations for the non-local electron-hole vertex in high spatial dimensions to describe the vanishing of diffusion in Anderson localization. It is caused by forming a quantum bound state between the diffusing particle and the hole left behind. Divergence in a new time scale proportional to the electrical polarizability signals the Anderson localization transition. Consequently, the height of the peak of the dynamical conductivity at zero frequency, the static diffusion constant, is not pushed to zero at the localization transition but rather its width. Spatially localized quantum bound states in the localized phase cannot be described by the continuity and wave equations in the Hilbert space of Bloch waves.