Semiclassical Theory for Universality in Quantum Chaos with Symmetry Crossover

TL;DR

This paper develops a semiclassical framework to understand spectral statistics in quantum chaotic systems with symmetry crossover, aligning with random matrix theory predictions and connecting with nonlinear sigma models.

Contribution

It introduces a semiclassical approach to analyze symmetry crossovers in quantum chaos, bridging periodic-orbit theory with random matrix and sigma model formalisms.

Findings

Explicit second-order semiclassical calculations match RMT predictions.

Formal higher-order terms are derived and presented.

Diagrammatic NLS model expansion is linked to periodic orbit theory.

Abstract

We address the quantum-classical correspondence for chaotic systems with a crossover between symmetry classes. We consider the energy level statistics of a classically chaotic system in a weak magnetic field. The generating function of spectral correlations is calculated by using the semiclassical periodic-orbit theory. An explicit calculation up to the second order, including the non-oscillatory and oscillatory terms, agrees with the prediction of random matrix theory. Formal expressions of the higher order terms are also presented. The nonlinear sigma (NLS) model of random matrix theory, in the variant of the Bosonic replica trick, is also analyzed for the crossover between the Gaussian orthogonal ensemble and Gaussian unitary ensemble. The diagrammatic expansion of the NLS model is interpreted in terms of the periodic orbit theory.