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A microscopic theory of Anderson localization of electrons in random lattices

Václav Janiš

TL;DR

This work develops a fully microscopic, conserving theory of Anderson localization for electrons in random lattices by combining parquet-based two-particle self-consistency with a local mean-field foundation (CPA). The approach distinguishes electron-hole and electron-electron scattering channels, using high-dimensional and Ward-identity-consistent constructions to connect microscopic vertices to macroscopic observables, and reveals localization as a quantum bound state of an electron and a hole. A local approximation to the two-particle irreducible vertex yields a mean-field-type critical behavior in $d>2$, including a bifurcation of the irreducible vertex and a bound-state interpretation of the localized phase, with a divergence of the characteristic scale $A(E;oldsymbol{ extomega})$ signaling the transition. The framework shows that Anderson localization cannot be captured by static, linear-response quantities alone; the diffusion pole, dynamical conductivity, and the two-particle response function must be treated self-consistently to describe both metallic diffusion and the localized bound-state regime.

Abstract

The existence of Anderson localization, characterized by vanishing diffusion due to strong randomness, 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 upon a recent publication \cite{Janis:2025ab} and present a microscopic theory of disordered electrons covering both the metallic phase with extended Bloch waves and the localized phase where the 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-particle and two-particle Green functions that satisfy the necessary conservation laws and causality requirements in the whole 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 properties in both metallic and localized phases, including critical behavior at the Anderson localization transition.

A microscopic theory of Anderson localization of electrons in random lattices

TL;DR

This work develops a fully microscopic, conserving theory of Anderson localization for electrons in random lattices by combining parquet-based two-particle self-consistency with a local mean-field foundation (CPA). The approach distinguishes electron-hole and electron-electron scattering channels, using high-dimensional and Ward-identity-consistent constructions to connect microscopic vertices to macroscopic observables, and reveals localization as a quantum bound state of an electron and a hole. A local approximation to the two-particle irreducible vertex yields a mean-field-type critical behavior in , including a bifurcation of the irreducible vertex and a bound-state interpretation of the localized phase, with a divergence of the characteristic scale signaling the transition. The framework shows that Anderson localization cannot be captured by static, linear-response quantities alone; the diffusion pole, dynamical conductivity, and the two-particle response function must be treated self-consistently to describe both metallic diffusion and the localized bound-state regime.

Abstract

The existence of Anderson localization, characterized by vanishing diffusion due to strong randomness, 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 upon a recent publication \cite{Janis:2025ab} and present a microscopic theory of disordered electrons covering both the metallic phase with extended Bloch waves and the localized phase where the 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-particle and two-particle Green functions that satisfy the necessary conservation laws and causality requirements in the whole 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 properties in both metallic and localized phases, including critical behavior at the Anderson localization transition.

Paper Structure

This paper contains 17 sections, 87 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model and fundamental relations
  3. Hamiltonian and averaged resolvent
  4. Two-particle vertex and electron-hole correlation function
  5. Ward identities, diffusion pole, and generalized Einstein relation
  6. Perturbation expansion
  7. Coherent potential approximation
  8. Beyond the local mean field
  9. Parquet equations and two-particle self-consistency
  10. High-dimensional approximation
  11. Mean-field critical behavior in dimensions $d>2$
  12. Anderson localization
  13. Dynamical vertex, conserving vertex, and the Ward identity
  14. Local approximation
  15. Metallic phase
  16. ...and 2 more sections

Figures (2)

  • Figure 1: Diagrammatic representation of the Bethe-Salpeter equation for multiple scattering in the electron-hole channel (diffuson). The two-particle vertices, characterized by wave vectors $\mathbf k,\mathbf k', \mathbf q$, are connected with off-diagonal one-electron propagators, and the double-primed wave vectors are summed over the Brillouin zone.
  • Figure 2: Diagrammatic representation of the Bethe-Salpeter equation for multiple scattering in the electron-electron channel (cooperon). The two-particle vertices in this channel are characterized by wave vectors $\mathbf k,\mathbf k',\mathbf Q,$. It is connected with the electron-hole channel by inverting one of the electron lines.