Functional integration method for 1D localization, multipoint correlators and persistent current in mesoscopic ring at arbitrary magnetic fields

TL;DR

This paper develops an exact functional integration approach for analyzing 1D localization, multipoint correlators, and persistent current in mesoscopic rings under arbitrary magnetic fields, avoiding perturbative methods.

Contribution

It introduces a novel exact functional integral method for 1D localization problems, enabling calculation of physical quantities without perturbation theory.

Findings

Derived closed-form functional representations for physical quantities.

Calculated the mean square dispersion of localized wave functions.

Evaluated the persistent current expectation value for arbitrary magnetic flux.

Abstract

Starting from the Abrikosov-Ryzhkin formulation of the 1D random potential problem I find closed functional representations for various physical quantities. These functional integrals are calculated exactly without the use of any perturbative expansions. The expressions for the multipoint densities correlators are obtained. Then I evaluate the mean square dispersion of the size of localized wave functions. As a physical application of the method, I find the expectation value of the persistent current in mesoscopic ring with arbitrary magnetic flux $\Phi$. (For small $\Phi$ this problem has been solved by O.Dorokhov). The case when the random potential has finite correlation length is considered too.