Chaotic maps and flows: Exact Riemann-Siegel lookalike for spectral fluctuations

TL;DR

This paper derives an exact spectral correlation function for fully chaotic quantum maps and flows using a Riemann-Siegel lookalike, confirming predictions of random matrix theory and elucidating pseudo-orbit cancellations.

Contribution

It introduces a rigorous Riemann-Siegel lookalike approach to obtain exact spectral correlators for chaotic quantum systems with finite matrix dimension.

Findings

Exact two-point spectral density correlator for finite N

Recovery of Gaussian unitary ensemble results as N approaches infinity

Pseudo-orbit cancellations are explained by the Riemann-Siegel lookalike

Abstract

To treat the spectral statistics of quantum maps and flows that are fully chaotic classically, we use the rigorous Riemann-Siegel lookalike available for the spectral determinant of unitary time evolution operators $F$. Concentrating on dynamics without time reversal invariance we get the exact two-point correlator of the spectral density for finite dimension $N$ of the matrix representative of $F$, as phenomenologically given by random matrix theory. In the limit $N\to\infty$ the correlator of the Gaussian unitary ensemble is recovered. Previously conjectured cancellations of contributions of pseudo-orbits with periods beyond half the Heisenberg time are shown to be implied by the Riemann-Siegel lookalike.