GENERALIZED CIRCULAR ENSEMBLE OF SCATTERING MATRICES FOR A CHAOTIC CAVITY WITH NON-IDEAL LEADS

TL;DR

This paper introduces a generalized circular ensemble for scattering matrices in chaotic quantum cavities with non-ideal leads, providing a microscopic foundation for the Poisson kernel distribution in such systems.

Contribution

It derives a Lorentzian ensemble for the Hamiltonian and connects it to a generalized circular ensemble for the scattering matrix, extending Dyson's ensemble to non-ideal coupling scenarios.

Findings

The Lorentzian ensemble matches microscopic theory for disordered metals.

The derived P(S) distribution generalizes Dyson's circular ensemble.

The work justifies the Poisson kernel from a microscopic perspective.

Abstract

We consider the problem of the statistics of the scattering matrix S of a chaotic cavity (quantum dot), which is coupled to the outside world by non-ideal leads containing N scattering channels. The Hamiltonian H of the quantum dot is assumed to be an M x N hermitian matrix with probability distribution P(H) ~ det[lambda^2 + (H - epsilon)^2]^[-(beta M + 2- beta)/2], where lambda and epsilon are arbitrary coefficients and beta = 1,2,4 depending on the presence or absence of time-reversal and spin-rotation symmetry. We show that this ``Lorentzian ensemble'' agrees with microscopic theory for an ensemble of disordered metal particles in the limit M -> infinity, and that for any M >= N it implies P(S) ~ |det(1 - \bar S^{\dagger} S)|^[-(beta M + 2 - beta)], where \bar S is the ensemble average of S. This ``Poisson kernel'' generalizes Dyson's circular ensemble to the case \bar S \neq 0 and was previously obtained from a maximum entropy approach. The present work gives a microscopic justification for the case that the chaotic motion in the quantum dot is due to impurity scattering.