Crossovers between superconducting symmetry classes

TL;DR

This paper investigates how the symmetry class of a small metallic grain coupled to superconductors changes under different magnetic fields and couplings, using random matrix theory and supersymmetric sigma-models to derive exact density of states expressions.

Contribution

It provides the first exact analytical results for the density of states during symmetry class crossovers in superconducting systems.

Findings

Derived exact density of states expressions for symmetry class crossovers.

Identified how magnetic field and coupling influence the symmetry class.

Mapped the spectrum behavior across different symmetry regimes.

Abstract

We study the average density of states in a small metallic grain coupled to two superconductors with the phase difference $\pi$, in a magnetic field. The spectrum of the low-energy excitations in the grain is described by the random matrix theory whose symmetry depends on the magnetic field strength and coupling to the superconductors. In the limiting cases, a pure superconducting symmetry class is realized. For intermediate magnetic fields or couplings to the superconductors, the system experiences a crossover between different symmetry classes. With the help of the supersymmetric sigma-model we derive the exact expressions for the average density of states in the crossovers between the symmetry classes A-C and CI-C.