Distribution of local density of states in disordered metallic samples: logarithmically normal asymptotics

TL;DR

This paper investigates the asymptotic behavior of the local density of states distribution in disordered metallic samples across different dimensions, revealing logarithmically normal asymptotics in quasi-1D and 2D cases, and a different form in 3D.

Contribution

It provides a detailed analysis of LDOS distribution asymptotics using supersymmetric sigma-models, confirming results with exact solutions and previous calculations, and extends understanding to 3D systems.

Findings

LDOS distribution has logarithmically normal asymptotics in quasi-1D and 2D samples.

Exact solutions confirm the asymptotics in quasi-1D.

In 3D, the asymptotics differ, following a different exponential form.

Abstract

Asymptotical behavior of the distribution function of local density of states (LDOS) in disordered metallic samples is studied with making use of the supersymmetric $\sigma$--model approach, in combination with the saddle--point method. The LDOS distribution is found to have the logarithmically normal asymptotics for quasi--1D and 2D sample geometry. In the case of a quasi--1D sample, the result is confirmed by the exact solution. In 2D case a perfect agreement with an earlier renormalization group calculation is found. In 3D the found asymptotics is of somewhat different type: $P(\rho)\sim \exp(-\mbox{const}\,|\ln^3\rho|)$.