On the Spectral Form Factor for Random Matrices

Abstract

In the physics literature the spectral form factor (SFF), the squared Fourier transform of the empirical eigenvalue density, is the most common tool to test universality for disordered quantum systems, yet previous mathematical results have been restricted only to two exactly solvable models [Forrester 2020]. We rigorously prove the physics prediction on SFF up to an intermediate time scale for a large class of random matrices using a robust method, the multi-resolvent local laws. Beyond Wigner matrices we also consider the monoparametric ensemble and prove that universality of SFF can already be triggered by a single random parameter, extending the recently proven Wigner-Dyson universality [Cipolloni, Erdős, Schröder 2021] to some larger spectral scales. Remarkably, extensive numerics indicates that our formulas correctly predict the SFF in the entire slope-dip-ramp regime, as customarily called in physics.

arXiv comment33 pages, 3 figures

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• Single-Particle Universality of the Many-Body Spectral Form Factor
• Dip-ramp-plateau for Dyson Brownian motion from the identity on $U(N)$
• Higher-order gap ratios of singular values in open quantum systems
• Spectral Form Factors of Topological Phases
• No-resonance conditions, random matrices, and quantum chaotic models
• Statistics of the Random Matrix Spectral Form Factor
• Disorder-free localisation in continuous-time quantum walks : Role of symmetries
• Eigenstate Thermalization in 1+1-Dimensional SU(2) Lattice Gauge Theory Coupled with Dynamical Fermions
• Comparative Study of Indicators of Chaos in the Closed and Open Dicke Model