Level statistics inside the vortex of a superconductor and symplectic random matrix theory in an external source

TL;DR

This paper models the energy level statistics inside a superconductor vortex core affected by impurities using symplectic random matrix theory, providing explicit formulas for the density of states and correlations.

Contribution

It introduces a symplectic random matrix ensemble with an external source to describe vortex core energy levels in superconductors, extending previous unitary ensemble work.

Findings

Derived explicit formulas for density of states.

Described crossover from clean to dirty limits.

Generalized previous random matrix models.

Abstract

In the core of the vortex of a superconductor, energy levels appear inside the gap. We discuss here through a random matrix approach how these levels are broadened by impurities. It is first shown that the level statistics is governed by an ensemble consisting of a symplectic random potential added to a non-random matrix. A generalization of previous work on the unitary ensemble in the presence of an external source (which relied on the Itzykson-Zuber integral) is discussed for this symplectic case through the formalism introduced by Harish-Chandra and Duistermaat-Heckman. This leads to explicit formulae for the density of states and for the correlation functions, which describe the cross-over from the clean to the dirty limits.