# Many-body quantum dynamics slows down at low density

### Submission summary

 As Contributors: Andrew Lucas Arxiv Link: https://arxiv.org/abs/2007.10352v3 (pdf) Date accepted: 2020-11-10 Date submitted: 2020-10-15 02:47 Submitted by: Lucas, Andrew Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Quantum Physics Approach: Theoretical

### 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.

Published as SciPost Phys. 9, 071 (2020)

Dear editor/referees,

We thank you for the very positive reports. We have made the minor requested changes to our manuscript and believe it is now ready for publication.

Sincerely,
Xiao, Yingfei, Andy

### List of changes

Response to the first referee:

We have fixed the typos that were pointed out, and modified the discussion above (1.4) to clarify what we meant by finite density at infinite T as an “analogue” of finite T: both correspond to dynamics in a very small part of Hilbert space.

Response to the second referee:

We added a new paragraph to the introduction (containing equations (1.6) and (1.7)) which explains why (1.5) is a universal bound at finite density.

Another tiny change:

In our discussion of the Schwinger-Keldysh formalism, we changed "world index" to "contour index" since the latter appears more customary in the literature.

### Submission & Refereeing History

Resubmission 2007.10352v3 on 15 October 2020
Submission 2007.10352v2 on 28 July 2020

## Reports on this Submission

### Report

I thank the authors for answering my question, and now recommend publication.

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### Report

The author's have added clarifications as I had requested. I am satisfied and recommend for publication.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -