Classical Quantization of Local Hall Conductivity in 2D Ballistic Systems

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.