On systems of maximal quantum chaos

TL;DR

This paper explores the unique features of maximally chaotic quantum systems, emphasizing their hydrodynamic nature, suppression of certain exponential growths, and implications for operator scrambling, supported by SYK and holographic models.

Contribution

It provides evidence that maximally chaotic systems are governed by a hydrodynamic effective field theory and identifies a key signature of maximal chaos related to operator growth suppression.

Findings

Hydrodynamic effective field theory describes maximally chaotic systems.

Suppression of exponential growth in commutator squares is a hallmark of maximal chaos.

Operator scrambling differs fundamentally in maximally chaotic systems.

Abstract

A remarkable feature of chaos in many-body quantum systems is the existence of a bound on the quantum Lyapunov exponent. An important question is to understand what is special about maximally chaotic systems which saturate this bound. Here we provide further evidence for the `hydrodynamic' origin of chaos in such systems, and discuss hallmarks of maximally chaotic systems. We first provide evidence that a hydrodynamic effective field theory of chaos we previously proposed should be understood as a theory of maximally chaotic systems. We then emphasize and make explicit a signature of maximal chaos which was only implicit in prior literature, namely the suppression of exponential growth in commutator squares of generic few-body operators. We provide a general argument for this suppression within our chaos effective field theory, and illustrate it using SYK models and holographic systems. We speculate that this suppression indicates that the nature of operator scrambling in maximally chaotic systems is fundamentally different to scrambling in non-maximally chaotic systems. We also discuss a simplest scenario for the existence of a maximally chaotic regime at sufficiently large distances even for non-maximally chaotic systems.