Classical and Quantum Dynamics in a Random Magnetic Field

TL;DR

This paper investigates the spectral properties of a quantum particle in a non-uniform magnetic field using supersymmetry, revealing universal Wigner-Dyson statistics and connecting quantum modes to classical irreversible dynamics.

Contribution

It introduces a novel analysis linking classical irreversible dynamics with quantum spectral statistics in a magnetic field using supersymmetry.

Findings

Classical density relaxation is governed by irreversible dynamics despite time-reversible kinetic equations.

Low-lying quantum modes correspond to classical eigenmodes, with a spectral gap separating higher modes.

Long-time quantum behavior exhibits universal Wigner-Dyson spectral statistics.

Abstract

Using the supersymmetry approach, we study spectral statistical properties of a two-dimensional quantum particle subject to a non-uniform magnetic field. We focus mainly on the problem of regularisation of the field theory. Our analysis begins with an investigation of the spectral properties of the purely classical evolution operator. We show that, although the kinetic equation is formally time-reversible, density relaxation is controlled by {\em irreversible} classical dynamics. In the case of a weak magnetic field, the effective kinetic operator corresponds to diffusion in the angle space, the diffusion constant being determined by the spectral resolution of the inhomogeneous magnetic field. Applying these results to the quantum problem, we demonstrate that the low-lying modes of the field theory are related to the eigenmodes of the irreversible classical dynamics, and the higher modes are separated from the zero mode by a gap associated with the lowest density relaxation rate. As a consequence, we find that the long-time properties of the system are characterised by universal Wigner-Dyson statistics. For a weak magnetic field, we obtain a description in terms of the quasi one-dimensional non-linear $\sigma$-model.