Spatial structure of anomalously localized states in disordered conductors

TL;DR

This paper investigates the spatial structure of anomalously localized states in disordered conductors using the sigma-model approach, transfer matrix method, and saddle-point approximation, revealing their properties across different dimensions and quantities.

Contribution

It provides a detailed analysis of ALS structures in various dimensions and for different physical quantities, highlighting their role in distribution tails and the effectiveness of approximation methods.

Findings

Exact solutions for ALS in quasi-1D systems

Saddle-point approximation accurately describes ALS intensity

ALS structures vary depending on the physical quantity considered

Abstract

The spatial structure of wave functions of anomalously localized states (ALS) in disordered conductors is studied in the framework of the $\sigma$--model approach. These states are responsible for slowly decaying tails of various distribution functions. In the quasi-one-dimensional case, properties of ALS governing the asymptotic form of the distribution of eigenfunction amplitudes are investigated with the use of the transfer matrix method, which yields an exact solution to the problem. Comparison of the results with those obtained in the saddle-point approximation to the problem shows that the saddle-point configuration correctly describes the smoothed intensity of an ALS. On this basis, the properties of ALS in higher spatial dimensions are considered. We study also the ALS responsible for the asymptotic behavior of distribution functions of other quantities, such as relaxation time, local and global density of state. It is found that the structure of an ALS may be different, depending on the specific quantity, for which it constitutes an optimal fluctuation. Relations between various procedures of selection of ALS, and between asymptotics of corresponding distribution functions, are discussed.