Quantum conductance problems and the Jacobi ensemble

TL;DR

This paper analyzes the probability distributions of scattering matrix blocks in one-dimensional quantum transport problems, connecting them to the Jacobi random matrix ensemble through various mathematical methods.

Contribution

It introduces new calculations of singular value distributions for different symmetry classes of scattering matrices using three distinct mathematical approaches.

Findings

Derived singular value distributions for various symmetry classes of scattering matrices.

Connected scattering matrix properties to the Jacobi random matrix ensemble.

Applied three mathematical methods to obtain these distributions.

Abstract

In one dimensional transport problems the scattering matrix $S$ is decomposed into a block structure corresponding to reflection and transmission matrices at the two ends. For $S$ a random unitary matrix, the singular value probability distribution function of these blocks is calculated. The same is done when $S$ is constrained to be symmetric, or to be self dual quaternion real, or when $S$ has real elements, or has real quaternion elements. Three methods are used: metric forms; a variant of the Ingham-Seigel matrix integral; and a theorem specifying the Jacobi random matrix ensemble in terms of Wishart distributed matrices.