Nonperturbative theory of power spectrum in complex systems

TL;DR

This paper develops a nonperturbative, universal theory for the power spectrum of complex quantum systems with stationary level spacings, applicable beyond random matrix models, and connects it to Painlevé transcendents.

Contribution

It introduces a nonperturbative formalism for the power spectrum in complex systems with stationary level spacings, including a universal formula involving Painlevé transcendents.

Findings

Exact universal power spectrum formula for large matrices

Application to quantum chaotic systems with broken time-reversal symmetry

Conjecture of a new mathematical identity involving Painlevé transcendents

Abstract

The power spectrum analysis of spectral fluctuations in complex wave and quantum systems has emerged as a useful tool for studying their internal dynamics. In this paper, we formulate a nonperturbative theory of the power spectrum for complex systems whose eigenspectra -- not necessarily of the random-matrix-theory (RMT) type -- posses stationary level spacings. Motivated by potential applications in quantum chaology, we apply our formalism to calculate the power spectrum in a tuned circular ensemble of random $N \times N$ unitary matrices. In the limit of infinite-dimensional matrices, the exact solution produces a universal, parameter-free formula for the power spectrum, expressed in terms of a fifth Painlev\'e transcendent. The prediction is expected to hold universally, at not too low frequencies, for a variety of quantum systems with completely chaotic classical dynamics and broken time-reversal symmetry. On the mathematical side, our study brings forward a conjecture for a double integral identity involving a fifth Painlev\'e transcendent.