A Supersymmetry approach to billiards with randomly distributed scatterers

TL;DR

This paper derives an explicit analytical expression for the density of states in chaotic billiards with randomly distributed scatterers, revealing a phase transition in the strong coupling limit when scatterers exceed a certain number.

Contribution

It introduces a supersymmetry method to analytically compute the density of states in billiards with random scatterers, including a novel phase transition analysis.

Findings

Explicit formula for density of states with broken time-reversal symmetry

Discontinuous change in density of states at strong coupling when scatterers exceed matrix size

Analytical approach applicable to chaotic billiards with impurities

Abstract

The density of states for a chaotic billiard with randomly distributed point-like scatterers is calculated, doubly averaged over the positions of the impurities and the shape of the billiard. Truncating the billiard Hamiltonian to a N x N matrix, an explicit analytic expression is obtained for the case of broken time-reversal symmetry, depending on rank N of the matrix, number L of scatterers, and strength of the scattering potential. In the strong coupling limit a discontinuous change is observed in the density of states as soon as L exceeds N.