A universal bound on Quantum Chaos from Random Matrix Theory

TL;DR

This paper establishes a universal, model-independent bound on quantum chaos using Random Matrix Theory, linking spectral form factor to out-of-time-order correlators and providing precise limits for systems with varying degrees of freedom.

Contribution

It introduces a strict, model-independent bound on quantum chaos measures based on RMT principles, applicable to thermal systems with different degrees of freedom.

Findings

Derived bounds on spectral form factor for large and small systems.

Connected spectral form factor to out-of-time-order correlators as a measure of chaos.

Provided a mathematical derivation of the chaos bounds.

Abstract

In this article, using the principles of Random Matrix Theory (RMT), we give a measure of quantum chaos by quantifying Spectral From Factor (SFF) appearing from the computation of two-point Out of Time Order Correlation function (OTOC) expressed in terms of square of the commutator bracket of quantum operators which are separated in time. We also provide a strict model independent bound on the measure of quantum chaos, $-1/N(1-1/\pi)\leq {\bf SFF}\leq 0$ and $0\leq {\bf SFF}\leq 1/\pi N$, valid for thermal systems with a large and small number of degrees of freedom respectively. Based on the appropriate physical arguments we give a precise mathematical derivation to establish this alternative strict bound of quantum chaos.