Symmetry Classes

TL;DR

This paper reviews Dyson's classification of random matrix ensembles based on symmetries and extends it to include chiral Dirac fermions and superconductors, linking symmetry classes to Riemannian symmetric spaces.

Contribution

It provides a modern perspective on Dyson's Threefold Way and introduces a minimal extension connecting symmetry classes to Riemannian symmetric spaces.

Findings

Dyson's Threefold Way is revisited and contextualized.

A minimal extension incorporates chiral Dirac fermions and superconductors.

Symmetry classes correspond to large families of Riemannian symmetric spaces.

Abstract

Physical systems exhibiting stochastic or chaotic behavior are often amenable to treatment by random matrix models. In deciding on a good choice of model, random matrix physics is constrained and guided by symmetry considerations. The notion of 'symmetry class' (not to be confused with 'universality class') expresses the relevance of symmetries as an organizational principle. Dyson, in his 1962 paper referred to as the Threefold Way, gave the prime classification of random matrix ensembles based on a quantum mechanical setting with symmetries. In this article we review Dyson's Threefold Way from a modern perspective. We then describe a minimal extension of Dyson's setting to incorporate the physics of chiral Dirac fermions and disordered superconductors. In this minimally extended setting, where Hilbert space is replaced by Fock space equipped with the anti-unitary operation of particle-hole conjugation, symmetry classes are in one-to-one correspondence with the large families of Riemannian symmetric spaces.