Semiclassical approach to discrete symmetries in quantum chaos

TL;DR

This paper uses semiclassical methods to analyze spectral correlations in quantum chaotic systems with discrete symmetries, revealing how symmetry types influence statistical behavior consistent with random matrix theory predictions.

Contribution

It extends semiclassical techniques to systems with discrete symmetries, showing how different irreducible representations affect spectral statistics and aligning results with RMT.

Findings

Real representations yield GOE statistics

Complex representations yield GUE statistics

No correlations between non-degenerate subspectra

Abstract

We use semiclassical methods to evaluate the spectral two-point correlation function of quantum chaotic systems with discrete geometrical symmetries. The energy spectra of these systems can be divided into subspectra that are associated to irreducible representations of the corresponding symmetry group. We show that for (spinless) time reversal invariant systems the statistics inside these subspectra depend on the type of irreducible representation. For real representations the spectral statistics agree with those of the Gaussian Orthogonal Ensemble (GOE) of Random Matrix Theory (RMT), whereas complex representations correspond to the Gaussian Unitary Ensemble (GUE). For systems without time reversal invariance all subspectra show GUE statistics. There are no correlations between non-degenerate subspectra. Our techniques generalize recent developments in the semiclassical approach to quantum chaos allowing one to obtain full agreement with the two-point correlation function predicted by RMT, including oscillatory contributions.