Ballistic electron motion in a random magnetic field

Abstract

Using a scheme of the derivation of the nonlinear $\sigma$ model we consider the electron motion in a random magnetic field 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 $Q^2=1.\mathrm{}$ Contrary to the standard scheme, neither singling out slow modes nor saddle-point approximations 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. However, after averaging over fluctuations in the Lyapunov region the standard $\sigma$ model is obtained leading to the conventional localization behavior.