Crystalline Spectral Form Factors

TL;DR

This paper explores the crystalline-like behavior of the spectral form factor in highly repulsive quantum systems, deriving theoretical models and demonstrating this behavior through circuit and matrix ensemble simulations.

Contribution

It introduces a novel analysis of spectral form factors showing crystalline behavior in strongly repulsive eigenvalue systems, combining Coulomb gas models, permutation circuits, and random matrices.

Findings

Derived the Debye-Waller factor for SFF oscillation suppression

Estimated singularities at multiples of the Heisenberg time

Reproduced crystalline behavior using circuit and matrix models

Abstract

We investigate crystalline-like behavior of the spectral form factor (SFF) in unitary quantum systems with extremely strong eigenvalue repulsion. Using a low-temperature Coulomb gas as a model of repulsive eigenvalues, we derive the Debye-Waller factor suppressing periodic oscillations of the SFF and estimate the order of its singularities at multiples of the Heisenberg time. We also reproduce this crystalline-like behavior using perturbed permutation circuits and random matrix ensembles associated with Lax matrices. Our results lay a foundation for future studies of quantum systems that exhibit intermediate level statistics between standard random matrix ensembles and permutation circuits.