Anderson localization from the replica formalism

TL;DR

This paper investigates Anderson localization in quasi-one-dimensional disordered wires using the replica sigma-model, deriving exact transmission eigenvalue densities for certain symmetry classes with a semiclassical approach.

Contribution

It introduces a semiclassical method within the replica formalism to compute exact transmission eigenvalue densities in disordered wires, extending previous supersymmetry techniques.

Findings

Exact density of transmission eigenvalues for class CI superconducting wires.

Approximate density function for class A unitary systems, excluding large transmission tail.

Demonstrates the applicability of semiclassical replica methods to localization problems.

Abstract

We study Anderson localization in quasi--one--dimensional disordered wires within the framework of the replica $\sigma$--model. Applying a semiclassical approach (geodesic action plus Gaussian fluctuations) recently introduced within the context of supersymmetry by Lamacraft, Simons and Zirnbauer \cite{LSZ}, we compute the {\em exact} density of transmission matrix eigenvalues of superconducting wires (of symmetry class $C$I.) For the unitary class of metallic systems (class $A$) we are able to obtain the density function, save for its large transmission tail.