Ballistic electron motion in a random magnetic field

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) and, for certain distributions of the magnetic field, is unstable with respect to the formation of vortex-like singularities that are strongly localized in space. Apparently, in this case properties of the system are quite unusual. The result is essentially non-perturbative and cannot be obtained diagrammatically. For other distributions of the RMF, averaging over fluctuations in the Lyapunov region can be carried out and the standard $\sigma$-model is obtained leading to the conventional localization behavior.