Generalization of the Poisson kernel to the superconducting random-matrix ensembles

TL;DR

This paper extends the Poisson kernel to superconducting random-matrix ensembles, providing a new distribution for the scattering matrix in chaotic normal-superconducting systems with arbitrary coupling.

Contribution

It introduces a generalized Poisson kernel for nonstandard symmetry classes in superconducting systems, which cannot be derived from maximum entropy principles.

Findings

Derived the distribution of the scattering matrix for arbitrary coupling.

Generalized the Poisson kernel to Altland-Zirnbauer symmetry classes.

Calculated conductance distribution for a single-channel Andreev quantum dot.

Abstract

We calculate the distribution of the scattering matrix at the Fermi level for chaotic normal-superconducting systems for the case of arbitrary coupling of the scattering region to the scattering channels. The derivation is based on the assumption of uniformly distributed scattering matrices at ideal coupling, which holds in the absence of a gap in the quasiparticle excitation spectrum. The resulting distribution generalizes the Poisson kernel to the nonstandard symmetry classes introduced by Altland and Zirnbauer. We show that unlike the Poisson kernel, our result cannot be obtained by combining the maximum entropy principle with the analyticity-ergodicity constraint. As a simple application, we calculate the distribution of the conductance for a single-channel chaotic Andreev quantum dot in a magnetic field.