Classical Quantization of Local Hall Conductivity in 2D Ballistic Systems

TL;DR

This paper demonstrates that in classical 2D ballistic systems, the local Hall conductivity is quantized in units of a fundamental constant, with irregular dependence on magnetic field points in chaotic billiards, and discusses quantum smoothing effects.

Contribution

It establishes classical quantization of local Hall conductivity in 2D ballistic systems and explores its dependence on magnetic field points and quantum effects.

Findings

Hall conductivity is quantized in units of e^3/(2π^2 ħ c)

Quantization depends irregularly on magnetic field location in chaotic billiards

Quantum effects smooth the classical quantization

Abstract

We study the linear Hall response of 2D ballistic system on inhomogeneous magnetic field. We establish that in classical limit the Hall conductivity response on local magnetic field is quantized in units of $\alpha_H \equiv \frac{e^3}{2 \pi^2 \hbar c}$. The quantized value depends on the point where the field is applied, this dependence being irregular in chaotic billiards. The phenomenon allows for direct tracing of special electron trajectories that belong to fractal repellor. We discuss how quantum effects smooth the quantization.