Energetic and dynamic properties of a quantum particle in a spatially random magnetic field with constant correlations along one direction

TL;DR

This paper investigates the spectral and transport properties of a quantum particle in a plane under a unidirectionally constant magnetic field, demonstrating conditions for ballistic motion and localization, with implications for random magnetic field models.

Contribution

It provides rigorous spectral and transport analysis for quantum particles in a unidirectionally constant magnetic field, including random cases, extending previous results by Iwatsuka and Mueller.

Findings

Spectrum is almost surely absolutely continuous under certain conditions.

Quantum motion is ballistic along the magnetic field direction.

Dynamical localization occurs perpendicular to the magnetic field.

Abstract

We consider an electrically charged particle on the Euclidean plane subjected to a perpendicular magnetic field which depends only on one of the two Cartesian co-ordinates. For such a ``unidirectionally constant'' magnetic field (UMF), which otherwise may be random or not, we prove certain spectral and transport properties associated with the corresponding one-particle Schroedinger operator (without scalar potential) by analysing its ``energy-band structure''. In particular, for an ergodic random UMF we provide conditions which ensure that the operator's entire spectrum is almost surely absolutely continuous. This implies that, along the direction in which the random UMF is constant, the quantum-mechanical motion is almost surely ballistic, while in the perpendicular direction in the plane one has dynamical localisation. The conditions are verified, for example, for Gaussian and Poissonian random UMF's with non-zero mean-values. These results may be viewed as ``random analogues'' of results first obtained by A. Iwatsuka [Publ. RIMS, Kyoto Univ. 21 (1985) 385] and (non-rigorously) by J. E. Mueller [Phys. Rev. Lett. 68 (1992) 385].