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.
Highlights
This paper is about operator growth: intuitively, given an operator such as ni which initially acts on only a finite number of sites, how much does time evolution scramble the information in ni? Put another way, how complicated is the operator ni(t)? To answer this question carefully, we introduce a new formalism, following [24]
The fact that the canonical operators of size s have an exponentially small length leads to a significant slowdown in the dynamics of our operator size
We studied several large N models with U(1) symmetry and showed that in the highly polarized sector with charge density n 1, the charged SYK model saturates our bound while the random dynamics including Brownian SYK model and random quantum automaton circuit do not
Summary
There is a conjectured universal “bound on chaos" [1] in many-body quantum systems: loosely speaking, a suitably defined out-of-time-ordered correlator (OTOC) at finite temperature is constrained to obey tr ρ[A(t), B]† ρ[A(t), B]. The dynamics must necessarily slow down and (1.2) (ignoring the precise 2π prefactor) is fixed by the Heisenberg uncertainty principle: ħh This is one manifestation of a conjectured “Planckian" bound on quantum dynamics and thermalization, whereby the fastest time scale (at least, of thermalization) in a low temperature quantum system is 1/T. The argument (1.3) is far from rigorous, and strictly speaking there are plenty of counter-examples to (1.1), e.g. in free fermion models [1, 12] Is it possible, at least under certain circumstances, to prove that quantum dynamics truly must slow down at low energy? This slowdown in effectively classical operator growth processes relative to quantum-coherent operator growth processes is reminiscent of the quadratic speed up of quantum walks over classical random walks [22, 23]
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