Ballistic electron motion in a random magnetic field

TL;DR

This paper develops a new derivation of the non-linear sigma-model for electron motion in a 2D random magnetic field, avoiding traditional approximations, and explores localization behavior at different length scales.

Contribution

It introduces a novel derivation method for the sigma-model in a random magnetic field, applicable down to the electron wavelength, and compares it to models with random potential.

Findings

The derived sigma-model differs from the random potential model.

Averaging over fluctuations yields the standard sigma-model and localization behavior.

Abstract

Using a new scheme of the derivation of the non-linear $\sigma$-model we consider the electron motion in a random magnetic field (RMF) in two dimensions. The derivation is based on writing quasiclassical equations and representing their solutions in terms of a functional integral over supermatrices $Q$ with the constraint $Q^2=1$. Contrary to the standard scheme, neither singling out slow modes nor saddle-point approximation are used. The $\sigma$-model obtained is applicable at the length scale down to the electron wavelength. We show that this model differs from the model with a random potential (RP).However, after averaging over fluctuations in the Lyapunov region the standard $\sigma$-model is obtained leading to the conventional localization behavior.