Microscopic theory of Anderson localization

Abstract: 
Anderson formulated a simple problem of the propagation of a quantum particle in random lattices. He found that diffusion may vanish when disorder is strong. The disclosed vanishing of diffusion sparked a tremendous effort to understand this phenomenon. The vanishing of diffusion became a paradigm of quantum critical behavior.  Two techniques have been used to investigate the problem. One is the classical statistical physics of the critical behavior in the localized state. The other is the quantum theory of particle scattering on lattice impurities in the metallic state. The former approach features a diverging scale, localization length, whereas the latter employs a vanishing scale, static conductivity.  Both methods use effective models, and neither explains the microscopic origin of Anderson localization. 

A microscopic theory of Anderson localization must be built on the quantum theory of diffusion. I will present a way to reach the Anderson localization transition with a diverging scale beyond the mean-field theory of electronic structure of random metallic alloys, the coherent-potential approximation. The core of the approach is a renormalized diagrammatic expansion for the two-particle vertex functions. A nonlinear equation for the electron-hole irreducible vertex is derived from the parquet equations. A new timescale beyond the diffusion equation is introduced that diverges at the localization transition. The new solution beyond this critical point is shown to represent Anderson localization. An expression for the complex dynamic conductivity is obtained in concord with applicable conservation laws. It is found that the width of the central peak of the dynamic conductivity vanishes at the transition, not necessarily its height, static conductivity. The localized state is a quantum bound state formed between the propagating particle and the hole left behind, which splits from the scattering states described by extended Bloch waves.