# Quantum Scrambling and State Dependence of the Butterfly Velocity

### Submission summary

 As Contributors: Xizhi Han · Sean Hartnoll Arxiv Link: https://arxiv.org/abs/1812.07598v2 Date submitted: 2019-03-27 Submitted by: Hartnoll, Sean Submitted to: SciPost Physics Domain(s): Theoretical Subject area: Quantum Physics

### Abstract

Operator growth in spatially local quantum many-body systems defines a scrambling velocity. We prove that this scrambling velocity bounds the state dependence (for example, the temperature dependence) of the out-of-time-ordered correlator in local lattice models. This amounts to a basic constraint on quantum chaos from locality. Thus, for example, while operators that do not grow with time can have an associated butterfly velocity, that velocity is necessarily temperature-independent. For scrambling operators, in contrast, the butterfly velocity shows a crossover from a microscopic high temperature value to a distinct value at temperatures below the energy gap.

###### Current status:
Editor-in-charge assigned

### Submission & Refereeing History

Submission 1812.07598v2 on 27 March 2019

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### Report

The growth of out-of-time-order correlators has been shown to be intimately connected to the scrambling of quantum information, whereby the information in the local degrees of freedom of a quantum many-body system is spread over its many degrees of freedom. These concepts are of great interest due to the connection between the fast-scramblers and black-holes, as well as the intimate connection between scrambling and the dynamics of entanglement entropy and thermalization. So far progress has been made on two separate fronts: in the large N or weak-coupling regimes, or a regime no immediately relevant to the original scrambling bounds, probing scrambling in quantum simulators starting with pure states.
Here Han and Hartnoll study the temperature dependence of the butterfly velocity in finite systems in 1D (both numerical and analytical results). I found this discussion quite interesting, and relatively novel. Their proof relies on a new quantity called the scrambling velocity which allows them to bound the rate of change of the butterfly velocity as a function of the temperature.

### Requested changes

I think the paper is suitable for publication however I would like the authors to address the following points:

1. I find the motivation for the scrambling velocity confusing. If $v_s$=0 then $v_B$ is temperature independent. The authors state that this is often true in non-interacting systems. However, I think this is only true for arbitrary state and operators, only in non-interacting systems. Is that correct? Since any unitary dynamics which generates entanglement should in principle lead to non-zero $v_s$.
2. In the case of non-interacting models where $v_s$ is non-zero but $v_B$ is independent of temperature can the authors provide some more intuition? I understand the quasi-particle picture however are there classes of spin models the authors suspect may present such characteristics?
3. The authors at the start of the paper focused on chaotic systems. Can they specifically define what they mean by this? That is, do they require exponential growth of the OTOC or saturating some bound for the Lyapunov exponent? later on they study an integrable system which is not chaotic so I was slightly confused by the initial focus on the chaotic systems.
4. Finally how important is the notion of temperature in the results? In many quantum simulation platforms the preparation of a pure state at a specific energy is accessible. Can the results be reformulated in terms of energy of the pure state?

• validity: high
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Many-body quantum dynamics and operator spreading are topics of much current interest. Most analytic treatments of these subjects appeal to various tractable limits in special models (such as large N and low temperature in the SYK model). Complementary approaches using "random-circuit" models have been used to make predictions about "strongly quantum" many-body systems away from large N/weak coupling limits, but these models are often random in time so that energy is not conserved and hence one must work with the equal-weight infinite temperature ensemble. As such, analytic results on strongly quantum systems at finite temperature are sorely lacking.

The paper by Han and Hartnoll attempts to address this gap by studying the temperature dependence of the butterfly velocity. They introduce a new quantity, the scrambling velocity, and use this to prove that the rate of change of the butterfly speed with temperature is upper bounded by the scrambling velocity. In other words, models with a zero scrambling velocity should have a temperature independent butterfly speed. This is an interesting result that is analytically derived and numerically verified, and I recommend publication. However, the paper would be strengthened if the following points could be addressed:

1. While the notion of the scrambling velocity and its relation to the state dependence of the butterfly speed is conceptually interesting, it would be useful to have some concrete examples of systems where $v_S$ is actually zero (besides the free-field example mentioned in the paper). While the introduction to the paper seems to be motivated by the question of scrambling in strongly quantum lattice models, it seems as though all such lattice models have non-zero scrambling velocity (?) As the authors discuss, even non-interacting systems with a quasiparticle description can have a non-zero $v_S$, and the temperature independence of $v_B$ in these systems has to then be separately argued for.

2. Following up on 1, do the authors see a way to generalize their result so that the free case directly follows instead of needing extra discussion? For example, perhaps the bound can be made for the slowest scrambling speed over the space of all operators? I'm also confused by real vs. momentum space. The OTOC is defined for local operators in real space, while quasiparticles of free theories generally live in momentum space. Won't it be the case that acting with a local operator creates quasiparticles of all momenta and hence gives a non-zero scrambling speed? I understand the semi-classical intuition that a quasiparticle with a (q,x) propagates, but does not grow, in space so that free systems should intuitively have $v_S=0$. However, I'm not sure if the definition for $v_S$ in (8) actually captures this.

3. The LHS of equation (9), $|\partial_\beta \lambda(v;\rho)|$, is positive. The RHS is $\frac{2h}{a}(v_s(v) - (\xi +\xi_{LR})\lambda(v)).$ Should there be an absolute value surrounding $\lambda(v)$ on the RHS? $\lambda(v)$ can be negative, in which case $|\partial_\beta \lambda(v;\rho)|$ wouldn't be upper bounded by $v_S$.

4. The authors use the words "quantum chaos" quite casually in the introduction, using it almost synonymously with scrambling or the growth of the OTOC. However, interacting integrable systems also scramble operators and show a growth in the OTOC, but are non-chaotic. Indeed, while there are well-defined measures of intermediate-to-late time chaos in MB systems (say as diagnosed by the spectral form factor for times greater than the Thouless time, or by nearest neighbor eigenenergy level repulsion), the connections of these measures of chaos to the early-time growth of the OTOC has not been conclusively established. It would be helpful for the authors to separate "scrambling" from "chaos" in their discussions.

5. On the last paragraph of page 2 which summarizes results, "velocity dependent Lyapunov exponent" is used before it is defined.

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