Quantization of edge currents for continuous magnetic operators

TL;DR

This paper proves a quantization theorem for edge currents in magnetic Hamiltonians on a half-plane, extending the concept of edge channels to disordered systems and providing Gaussian bounds on the heat kernel.

Contribution

It introduces a quantization theorem for edge currents in magnetic operators with disorder and establishes Gaussian bounds on the heat kernel and its derivatives.

Findings

Quantization of edge currents in disordered magnetic systems.

Extension of edge channel concept to systems with randomness.

Gaussian bounds on heat kernel and covariant derivatives.

Abstract

For a magnetic Hamiltonian on a half-plane given as the sum of the Landau operator with Dirichlet boundary conditions and a random potential, a quantization theorem for the edge currents is proven. This shows that the concept of edge channels also makes sense in presence of disorder. Moreover, Gaussian bounds on the heat kernel and its covariant derivatives are obtained.