Localization in a rough billiard: A sigma model formulation

TL;DR

This paper models quantum particle localization in a rough billiard using a sigma model, revealing analytical localization length tied to classical diffusion, advancing understanding of quantum chaos in boundary-rough systems.

Contribution

It introduces a sigma model formulation for quantum dynamics in rough billiards, providing analytical results for localization length based on classical diffusion.

Findings

Localization length equals classical diffusion constant

Analytical expression for localization in rough billiards

Quantum dynamics described by supersymmetric sigma model

Abstract

We consider the quantum dynamics of a particle in a weakly rough billiard. The Floquet operator for reflection at the boundary is obtained as a unitary band matrix. The resulting dynamics in angular momentum space can be treated in the framework of the one-dimensional supersymmetric nonlinear sigma model. We find analytically localization and the corresponding localization length $\xi=D_{cl}$ where $D_{cl}$ is the classical diffusion constant due to boundary scattering.