Level dynamics and the ten-fold way

TL;DR

This paper studies how eigenvalues of Hamiltonians evolve over parameters in symmetric spaces, identifying the geometric and statistical structures that govern their dynamics.

Contribution

It introduces a framework for analyzing level dynamics on symmetric spaces, including the reduced manifold, Poisson structure, and eigenvalue density calculation.

Findings

Identified the reduced manifold for eigenvalue dynamics.

Established the Poisson structure for Hamiltonian evolution.

Derived the eigenvalue density on the reduced space.

Abstract

We investigate the parameter dynamics of eigenvalues of Hamiltonians ('level dynamics') defined on symmetric spaces relevant for condensed matter and particle physics. In particular we: 1) identify appropriate reduced manifold on which the motion takes place, 2) identify the correct Poisson structure ensuring the Hamiltonian character of the reduced dynamics, 3) determine the canonical measure on the reduced space, 4) calculate the resulting eigenvalue density.