Power spectrum and form factor in random diagonal matrices and integrable billiards

TL;DR

This paper investigates how truncations in random diagonal matrices affect the power spectrum and form factor, revealing that traditional assumptions linking them break down in truncated spectra, with implications for integrable quantum systems.

Contribution

It provides a nonperturbative analysis of spectral statistics in truncated RDMs and shows their relevance to bounded quantum systems with integrable classical dynamics.

Findings

Power spectrum behavior deviates from form factor predictions in truncated spectra.

Truncated RDM spectra better describe bounded integrable quantum systems.

Numerical simulations support the theoretical analysis.

Abstract

Triggered by a controversy surrounding a universal behaviour of the power spectrum in quantum systems exhibiting regular classical dynamics, we focus on a model of random diagonal matrices (RDM), often associated with the Poisson spectral universality class, and examine how the power spectrum and the form factor get affected by two-sided truncations of RDM spectra. Having developed a nonperturbative description of both statistics, we perform their detailed asymptotic analysis to demonstrate explicitly how a traditional assumption (lying at the heart of the controversy) -- that the power spectrum is merely determined by the spectral form factor -- breaks down for truncated spectra. This observation has important consequences as we further argue that bounded quantum systems with integrable classical dynamics are described by heavily truncated rather than complete RDM spectra. High-precision numerical simulations of semicircular and irrational rectangular billiards lend independent support to these conclusions.