Many-body quantum dynamics slows down at low density

TL;DR

This paper investigates how quantum many-body systems with conserved charge exhibit slowed dynamics at low density, providing new bounds on chaos indicators and analyzing models like SYK and automaton circuits.

Contribution

It introduces a density-dependent operator size and derives bounds on Lyapunov exponents, connecting these to models like SYK and quantum circuits.

Findings

Operator growth slows algebraically at low density.

New bounds on Lyapunov exponents stronger than Lieb-Robinson bounds.

Density dependence of chaos saturates in the charged SYK model.

Abstract

We study quantum many-body systems with a global U(1) conservation law, focusing on a theory of $N$ interacting fermions with charge conservation, or $N$ interacting spins with one conserved component of total spin. We define an effective operator size at finite chemical potential through suitably regularized out-of-time-ordered correlation functions. The growth rate of this density-dependent operator size vanishes algebraically with charge density; hence we obtain new bounds on Lyapunov exponents and butterfly velocities in charged systems at a given density, which are parametrically stronger than any Lieb-Robinson bound. We argue that the density dependence of our bound on the Lyapunov exponent is saturated in the charged Sachdev-Ye-Kitaev model. We also study random automaton quantum circuits and Brownian Sachdev-Ye-Kitaev models, each of which exhibit a different density dependence for the Lyapunov exponent, and explain the discrepancy. We propose that our results are a cartoon for understanding Planckian-limited energy-conserving dynamics at finite temperature.