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Many-body chaos near a thermal phase transition
by Alexander Schuckert, Michael Knap
This is not the current version.
|As Contributors:||Michael Knap · Alexander Schuckert|
|Submitted by:||Schuckert, Alexander|
|Submitted to:||SciPost Physics|
|Subject area:||Condensed Matter Physics - Theory|
We study many-body chaos in a (2+1)D relativistic scalar field theory at high temperatures in the classical statistical approximation, which captures the quantum critical regime and the thermal phase transition from an ordered to a disordered phase. We evaluate out-of-time ordered correlation functions and find that the associated Lyapunov exponent increases linearly with temperature in the quantum critical regime, and approaches the non-interacting limit algebraically in terms of a fluctuation parameter. Chaos spreads ballistically in all regimes, also at the thermal phase transition, where the butterfly velocity is maximal. Our work contributes to the understanding of the relation between quantum and classical many-body chaos and our method can be applied to other field theories dominated by classical modes at long wavelengths.
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Anonymous Report 2 on 2019-6-24 Invited Report
This is a well-executed and well-written paper on a topic of great current interest. Almost all computations of OTOCs in quantum system involve some large-N limit, so it is welcome to see an analysis at finite N in a system in finite dimensions. The classical approximation should be qualitatively valid in the quantum-critical region. I would ask the authors to comment a bit more on issues related to other works:
(i) They observe a ballistic spread of chaos, unlike the diffusive behavior seen in random unitary networks. Do they have any comment on the reason for this difference ?
(ii) Why does 1+1 D KPZ behavior appear in a 2+1 D system ?
Anonymous Report 1 on 2019-6-14 Invited Report
Schuckert and Knap study OTOC growth in a relativistic scalar field theory close to a thermal phase transition using the classical statistical approximation. The properties of OTOCs in field theories and the connection between chaotic classical dynamics and the corresponding quantum dynamics are currently of great interest. The study is very carefully executed and well presented. The only weakness that one could identify is the lack of rigorous arguments as to why the classical statistical simulations faithfully capture the OTOC dynamics of the quantum system. As the authors also discuss, it would be desirable to have a more rigorous argument why the semi-classical method used here, gives faithful results for the quantum mechanical OTOC, or even a direct quantum calculation of it, is desirable but beyond the scope of this work. After discretization the model boils down to an array of classical coupled anharmonic (quartic) oscillators. In my opinion, even the classical dynamics of this system itself is interesting and the authors give good arguments why the classical statistical approximation should be justified.
In conclusion I recommend publication after minor revisions and clarifications (see comments below).
1) Formulation “chaos spreads ballistically” is found in several places. I find this a bit imprecise. What spreads is the OTOC, or the support of some Heisenberg operators. Chaos I would regard as a property of the dynamics itself which can be regular or chaotic but not as something that spreads in space.
2) The authors study a classical field theory with classical statistical methods. The argument is then always that this classical theory describes the low low-wavelength, long-distance properties of a corresponding quantum field theory well especially close to the phase transition. Has this conjecture been tested in any way? One argument for the classical description providing meaningful results for the OTOCs in the quantum model is that semi-classical methods have been shown recently to capture the behavior of the OTOC . Are the methods used in  equivalent to the ones used here for times smaller than the Ehrenfest time? What is the Ehrenfest time in the model studied here? Later the authors also consider values of G that are rather far away from the phase transition. Especially the quasi-particle peaks studied in Sec. 3 only exist sufficiently far away from the transition.
3) On the quantum versus classical field theory: Sometimes the authors speak of a classical statistical field theory. Although I know that this term is frequently used in the context of statistical mechanics of continuous systems it felt like mixing the term classical field theory with the method that is used to solve/sample it.
4) Appendix A: What is meant by “near” the finite temperature phase transition? Can this be quantified somehow?
5) Last sentence of Sec. 2: All the data shown is for a perturbation in the momentum field only. What exactly is meant by the statement that perturbing in phi gives “similar results” and “same chaotic properties”? Is the statement only valid for the local OTOC or is the equal spatial argument x a typo?
6) P. 6: “Fitting the line shape…” The peaks in Fig 1 do not seems to nicely fit a Breit-Wigner function for G>20. What region was fitted and what are the resulting confidence intervals of the fit?
7) I don’t quite understand in what sense eq. 15 is universal. It seems to me that both sides of the equation are general functions of t. What would a scaling collapse look like?
8) Exponential growth of the OTOC: From Fig. 3 it is not so convincing that the OTOC grows exponentially. It could also be something super-exponential according to the figure. Did the authors calculate longer times to verify the exponential behavior? I imagine that this is hard due to finite size effects. Also the method of obtaining errors for the fitted exponents seems questionable (cf Figure 4): Why would varying the size of the fit interval quantify the quality/confidence of the fit? It is interesting that the authors call this long-time behavior as usually the regime below the Ehrenfest time is considered short time.
In this context I was also wondering why the largest Lyapunov exponent should be the one that governs the OTOC growth. Shouldn’t it rather be the stability of phi with respect to the perturbation in pi which would be two specific phase space directions?
9) The authors mention on page 10 that the OTOC is expected to saturate around the Ehrenfest time. Usually saturation is expected on the Heisenberg time scale. Ref. 41 only argues that the OTOC saturates at times much larger than the Ehrenfest time as far as I understand. (see also e.g. arXiv:1812.09237 [quant-ph])
10) Appendix C “critical spectral function”: Is it correct that the individual trajectories are not time translation invariant?
11) Figure 9: Was the convergence also checked for the non-local OTOCs?
12) Word doubling “the the” on the bottom of p. 8.
13) Notation: Is could be confusing that lambda is used as the coupling constant as well as Lyapunov exponent and Lyapunov function.
14) In eq 28 and in the text above it the authors use “U” for the interaction parameters while it was defined a “g” before.
15) Eq 39: Should it be 0 instead of x in the denominator?