Hybridization of wave functions in one-dimensional localization

TL;DR

This paper refines the hybridization approach to describe wave function correlations in one-dimensional localized systems, providing quantitatively precise asymptotic expressions that align with known exact results.

Contribution

It develops a method to derive exact asymptotic expressions for wave function correlations in 1D localization, extending the hybridization model with log-normal distribution assumptions.

Findings

Accurately describes correlation functions at the Mott length scale

Reproduces known exact results within the considered frequency orders

Provides asymptotic behavior of correlation functions in different 1D systems

Abstract

A quantum particle can be localized in a disordered potential, the effect known as Anderson localization. In such a system, correlations of wave functions at very close energies may be described, due to Mott, in terms of a hybridization of localized states. We revisit this hybridization description and show that it may be used to obtain quantitatively exact expressions for some asymptotic features of correlation functions, if the tails of the wave functions and the hybridization matrix elements are assumed to have log-normal distributions typical for localization effects. Specifically, we consider three types of one-dimensional systems: a strictly one-dimensional wire and two quasi-one-dimensional wires with unitary and orthogonal symmetries. In each of these models, we consider two types of correlation functions: the correlations of the density of states at close energies and the dynamic response function at low frequencies. For each of those correlation functions, within our method, we calculate three asymptotic features: the behavior at the logarithmically large "Mott length scale", the low-frequency limit at length scale between the localization length and the Mott length scale, and the leading correction in frequency to this limit. In the several cases, where exact results are available, our method reproduces them within the precision of the orders in frequency considered.