Periodic-orbit theory of universal level correlations in quantum chaos

TL;DR

This paper uses semiclassical periodic-orbit theory to demonstrate universal spectral correlations in quantum chaos, deriving the full spectral form factor for all times and symmetry classes.

Contribution

It extends previous work by establishing the complete two-point correlator and spectral form factor for quantum systems with chaotic classical limits, including all time regimes.

Findings

Universal behavior of level correlations confirmed

Spectral form factor derived for all times

Applicable to systems with and without time-reversal symmetry

Abstract

Using Gutzwiller's semiclassical periodic-orbit theory we demonstrate universal behaviour of the two-point correlator of the density of levels for quantum systems whose classical limit is fully chaotic. We go beyond previous work in establishing the full correlator such that its Fourier transform, the spectral form factor, is determined for all times, below and above the Heisenberg time. We cover dynamics with and without time reversal invariance (from the orthogonal and unitary symmetry classes). A key step in our reasoning is to sum the periodic-orbit expansion in terms of a matrix integral, like the one known from the sigma model of random-matrix theory.