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Quantum echo dynamics in the SherringtonKirkpatrick model
by Silvia Pappalardi, Anatoli Polkovnikov, Alessandro Silva
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Authors (as registered SciPost users):  Silvia Pappalardi · Anatoli Polkovnikov · Alessandro Silva 
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Preprint Link:  https://arxiv.org/abs/1910.04769v1 (pdf) 
Date submitted:  20191015 02:00 
Submitted by:  Pappalardi, Silvia 
Submitted to:  SciPost Physics 
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Academic field:  Physics 
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Approach:  Theoretical 
Abstract
Understanding the footprints of chaos in quantummanybody systems has been under debate for a long time. In this work, we study the echo dynamics of the SherringtonKirkpatrick (SK) model with transverse field under effective time reversal. We investigate numerically its quantum and semiclassical dynamics. We explore how chaotic manybody quantum physics can lead to exponential divergence of the echo {of} observables and we show that it is a result of three requirements: i) the collective nature of the observable, ii) a properly chosen initial state and iii) the existence of a welldefined semiclassical (large$N$) limit. Under these conditions, the echo grows exponentially up to the Ehrenfest time, which scales logarithmically with the number of spins $N$. In this regime, the echo is well described by the semiclassical (truncated Wigner) approximation. We also discuss a shortrange version of the SK model, where the Ehrenfest time does not depend on $N$ and the quantum echo shows only polynomial growth. Our findings provide new insights on scrambling and echo dynamics and how to observe them experimentally.
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Report 2 by Rodolfo Jalabert on 2019125 (Invited Report)
 Cite as: Rodolfo Jalabert, Report on arXiv:1910.04769v1, delivered 20191205, doi: 10.21468/SciPost.Report.1378
Report
Report on the manuscript “Quantum echo dynamics in the SherringtonKirkpatrick model”
by S. Pappalardi, A. Polkovnikov, and A. Silva
The present manuscript concerns the echo dynamics of quantum operators for the SherringtonKirkpatrick spin model with transverse magnetic field. Numerical and analytical approaches are used to investigate the conditions for the existence of an exponential growth of the echo observables. The subjects of quantum echoes and outoforder correlators (OTOCs) treated in this manuscript are timely and the results obtained are interesting within the present theoretical efforts to understand the scrambling of quantum information.
A number of questions need to be addressed in order to assess the significance of the proposed contribution.
1) While the second moment of the echo operator $A_{\epsilon}(t)A$ corresponds in lowest order in $\epsilon$ to the square commutator (or OTOC), the studied quantity $\mu(t)$ differs, in general, from the OTOC. Even in the case in which the initial state is an eigenstate of the operator $A$, the quantity $\mu(t)$ differs from the OTOC. It is true that in this case both quantities have as a contribution the maximally crossed term $AB(t)AB(t)$. However, in the general case, the term arising from the matrix element of $B(t)A^{2}B(t)$ is present in the double commutator, but not in $\mu(t)$. It is only for unitary operators that the maximally crossed term suffices to describe the OTOC. Therefore, the case of the local operators treated in this work would be comprised, since the onsite spin operators are at the same time Hermitian and unitary, but the nonlocal collective observables discussed in this work are not unitary.
The difference between the OTOC and $\mu(t)$ is not unimportant since the terms that make up for their difference have been shown to be crucial in obtaining the exponential growth of the former as the difference between parametrically larger components.
While the authors expressed that the properties of the distribution of the echo operator are left out for future work, it should be clearly indicated in this manuscript that $\mu(t)$ is a different object than the OTOC. In addition, the authors should try to establish a connection with previous results for $\mu(t)$ in the case that they exist in spin chains or in other kind of systems.
2) The possibility of observing an exponential growth of $\mu(t)$ is linked to the existence of a sufficiently long timewindow between the initial (perturbative) powerlaw growth and the longtime saturation. And such a condition is shown to put constraints on the nature of the initial state and of the observable. This line of thought is supported by the numerical simulations and the implementation of the truncated Wigner approximation on the SherringtonKirkpatrick model for the chosen parameters. However, in a general case, no argument is given on why in the intermediate timewindow the growth of $\mu$ should be necessarily exponential. This is a central issue of current interest since the behavior of the OTOC in the nonperturbative intermediate regime, as well as for longtimes, are not simply dictated by the integrable or nonintegrable character of the system, especially in the case where the classical analog is not welldefined.
3) The large Nlimit is presented as a semiclassical one, determining one of the three requirements needed to obtain the exponential divergence of the echo observables. In addition, the largeN limit is linked with the improvement of the truncated Wigner approximation. However, the proper semiclassical limit of a spin system is that in which the number of possible spin components goes to infinity, while in the present work the spin 1/2 is kept throughout. The necessity of having a large N appears as a condition for enlarging the intermediate timewindow, and it is thus unrelated to a semiclassical approximation. While, in interacting manybody systems, 1/N sometimes plays an analogous role than that of $\hbar$ for a onebody system, the usage of semiclassical limit as synonymous of large N is not justified in the present model.
4) The chosen model, in its longrange version, shares with the SachdevYeKitaev model the feature of having a nonlocal (global) interaction, and both of them exhibit the exponential growth of the echo observables. It would be important to signal the relevant differences between the two models and why a nonlocal interaction seems to be a crucial ingredient in obtaining the abovediscussed behavior.
5) A cleaner presentation of the manuscript is needed. Some of the figure captions refer to “bottom” and “top” panels while describing horizontal layouts (i.e; Fig.1), while the symmetrical confusion also appears (i.e. Fig.4). Often the ordinate axislabel is missing. Misspellings and repeated words are to be corrected.
Addressing the abovepresented questions and deepening the discussion of the physics behind the numerical findings should increase significance of this interesting manuscript.
Author: Silvia Pappalardi on 20200217 [id 738]
(in reply to Report 2 by Rodolfo Jalabert on 20191205)
We are grateful to the Referee for her/his careful reading of the manuscript and for pointing out the potentially broad impact of our results.
A new version manuscript has been already resubmitted on the Arxiv, where we have made few changes according to the Referees comments. However, we attach here a copy where the most relevant changes are marked in blue.
In the following, we provide detailed answers to the Referee's specific comments.
Question (1):
We thank the referee for this comment. Since a similar issue has been raised also by the Referee 1, we have added few lines in the introduction to the new version of the manuscript, improved the discussion in Section 2 and included a new small appendix with the details of the classical limit of the echo observables.
First of all, let us specify the general meaning that one can associate to OTOC. Even if in recent literature the name OTOC is usually related to correlators as $\langle \hat A \hat B(t) \hat A \hat B(t) \rangle$, ``out of timeordered correlators'' OTOC can be referred to all the timedependent correlation functions which have an unusual timeordering. Namely, they describe correlation functions which cannot be computed with conventional techniques that assume casual time evolution and therefore need to be evaluated via extended Keldysh contours, like the one developed in [Aleiner, Faoro, Ioffe, Annals of Physics 2016]. In this sense, the echo observable $\mu(t)$ can be refereed as OTOC, since it contains terms as $\langle \hat B(t) \hat A \hat B(t) \rangle$, see Eq.(4).
In the second place, in the semiclassical limit, every kind of OTOC contains quadratic terms in the derivatives of the initial conditions, as was already known by Larkin and Ovchinnikov. This is true also for the echo $\mu(t)$. This statement can be shown by the use of timedependent Bopp operators, a standard formalism of phasespace methods which allows computing timedependent correlation functions, by representing quantum operators as phasespace variables. This is extensively discussed in the Appendix of [Schmitt, Sels, Kehrein and Polkovnikov, PRB 2019] for a semiclassical limit of the SYK. Anyhow, in order to be as selfconsistent as possible, we have added a new Appendix to the manuscript, with the derivation of the semiclassical limit of $\mu(t)$, based on Bopp operators. We show for bosonic and spin operators, that the semiclassical echo observable contains quadratic terms in the derivatives of the classical trajectory to respect to the initial conditions, hence it encodes the Lyapunov exponent, exactly as the square commutator. For further details on this derivation, we refer to the response (1) to the Report 1, where we have reported the main steps of the proof.
In conclusion, we acknowledge that the square commutator $c(t) =  \langle [\hat B(t), \hat A]^2 \rangle$ contains a term $\langle B(t)A^2 B(t)\rangle$, which is missed by $\mu(t)$. For this reason, we added a sentence in Section 2. Since we find this very interesting, we would like to ask if the Referee could kindly point us the reference where ``the terms that make up for their difference have been shown to be crucial in obtaining the exponential growth of the former as the difference between parametrically larger components.''
Question (2):
We thank the referee for this comment since this point is indeed a nontrivial statement and it is exactly one of the purposes of the paper. If we believe that there are no other initial time scales, our claim can be restated as follows: If there exist a timeinterval $t^\ast\leq t \leq t_{\text{Ehr}}$, where $t_{\text{Ehr}}$ diverges in the thermodynamic limit, then in the same interval the $\mu(t)$ should be semiclassical. Therefore it behaves like the square of the derivatives to respect to the initial conditions, see also the Refs.[1618] of the manuscript and the new Appendix A. When the latter grows exponentially with a rate given by the classical (generalized) Lyapunov exponent, then  in that time interval  also the quantum $\mu(t)$ should grow exponentially with the same rate. On the other hand, when the system does not have a semiclassical analogue, i.e. $t_{\text{Ehr}}\sim t^*$ does not depend on the system size, there is no timeregime where the quantum $\mu(t)$ would behave semiclassically. Therefore there is no reason in principle to have exponential growth. This is typically the case of shortrange interacting lattice systems, which do not have a classical analogue and for which it has been proved that the squarecommutator grows at most exponentially fast [Kukulian et al.]. This statement is firmly supported by our numerics on longrange (semiclassical) and shortrange (fullyquantum) models.
Question (3):
We thank the Referee for this comment since our previous version did not provide the underlying argument to our statements. We have added a discussion and a new appendix in order to better justify our claims. The justification can be summarized as follows. TWA can be proved as the saddle point of the Keldysh action in a pathintegral formulation of the timeevolutions. Within this framework, representation of initial conditions through the Wigner function is exact, while the approximation occurs at the level of the timeevolution, which is solved by the saddlepoint approximation expanding in small quantum fluctuations. Even if the present work deals with spins $1/2$, this alltoall interacting model has a meanfield character and the large $N$limit ensures the validity of the saddle point approximation, as confirmed by the numerics. More formally (see Appendix B), one can estimate the corrections to the saddlepoint solution, which are found to scale as $1/N$.
Question (4):
We thank the referee for highlighting this point, since the presence of nonlocal interactions, shared by the SYK and SK models, is indeed crucial to have a semiclassical limit and hence to observe the exponential growth of the OTOC. Even if in the previous version we already pointed out that the models share several analogies, we have now added a sentence to signal that the SYK and SK differ in many respects. For example, they have different equilibrium phasediagrams, which inevitably lead to differences in the dynamics. This is particularly evident at low temperatures, where the SK model possesses a glassy phase below a critical transverse field, i.e. see [Andreanov and Muller, PRL 2012]. On the other hand, such a phasetransition is absent in the SYK model. Anyhow, as pointed out by the Referee, the two models share the feature of having nonlocal alltoall random interaction. Indeed, a semiclassical analysis was performed by one of us on the SYK using the TWA on fermionic bilinears, see [Schmitt, Sels, Kehrein and Polkovnikov, PRB 2019], where the authors found similar results concerning the exponential growth of the echo. In that paper, the role of the shortrange interactions was not addressed, and the analysis presented in the present work shows how the nonlocality of the interactions affects the classical limit (even on a system of spins $1/2$) and therefore the exponential growth. Also in that paper there was no analysis of the observables and the echo operator and the choice, which satisfies the criteria we derived here was rather coincidental. We believe that finding these conditions is another important new result of the present work.
Question (5):
We changed the figure's captions and added the ordinate axis where missing. We made an effort to improve the presentation, correcting misspellings and rewriting some sections.
Anonymous Report 1 on 20191129 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1910.04769v1, delivered 20191129, doi: 10.21468/SciPost.Report.1347
Strengths
The authors address timely a subject of interest to a broad range of communities
Weaknesses
The article is not very clear, and makes some far fetched statements.
Report
The manuscript addresses a subject which is of current interest in a wide range of communities. The authors study the evolution of an echo defined for observables. This quantity is related to the original proposals of the Loschmidt echo which tried to characterize quantum chaos and irreversibility in quantum systems by the decay of the overlap between a state evolved with a Hamiltonian and another state evolved with (almost) the same (slightly perturbed) Hamiltonian. The subject has developed a lot in the last few years but using a different approach, which is characterizing quantum chaos by how information is scrambled over time. This is quantified by the (exponential) growth (or decay, depending on how it is defined) of the outoftime ordered correlators. There has been an increase of interest since the deduction of an upper bound (essentially temperature) for the growth rate in certain maximally chaotic systems, that could possibly be related to black hole dynamics.
In such a context the authors provide some results which could be worthy of interest, but which in my opinion suffer from some problems.
I’ll first point out what I find very interesting. The authors point out the known difficulty in many body systems to find the alleged exponential growth of the OTOC. The authors make an interesting analysis showing that the problem is essentially due to two things: the type of initial state, and the type of observable. In order to have a large enough window to accommodate for an exponential growth the initial state should be extensively far from the saturation value, and at least one of the observables should be nonlocal.
These results are analytical for the quantity $\mu$ of Eq (2) which they define as echo. It does not depend on the model.
After the analytical results the authors proceed to show numerical results for the SherringtonKirkpatrick model, which is a system of spins with infinite range interaction that has a glass transition. In the numerical results the authors show first that the exact dynamics of $\mu(t)$ do indeed behave as predicted and then they use the semiclassical Truncated Wigner approximation (TWA) to try to show that they can extract the exponential behavior from the model (in the suitable initial state +observable combination) and in addition they show that this is not possible for limited range interaction.
The results (though not ground breaking) could be interesting enough to be published in light of the present interest of the subject in one and manybody complex systems.
However there are many concerns which I enumerate below which should be addressed before it can be accepted.
(1) I think the authors try to force the relation between $\mu$ and OTOC. Indeed the square commutator — at order $epsilon^2$ — appears in the second moment of $\mu$ but that is no reason to speak interchangeably of the behavior of one quantity and the other, especially of exponential growths (or decays, or whatever). I think this things should be clearly stated somewhere, or everywhere. If there is indeed a onetoone relation to the growth of the OTOC, please show it. I’m not saying $\mu$ is not a good quantity to characterize scrambling, it is just not the most widely used, if it is indeed one, then prove it. I think for completeness a better explanation could be included in the manuscript.
(2) The explanation and justification of the TWA is rather scarce, I was not familiar with it and found it rather difficult to follow in the text. Since the subject appeals to a broad audience, then effort could (maybe should) be made so that the article is as self contained as possible.
(3) I think I missed the point of using the TWA altogether. Then again it might be me, but it could have been better explained. In Fig 5, if no TWA is plot, I wouldn’t bet on an exponential growth “window”. The TWA does indeed show an exponential growth, but after the Ehrenfest time. So, paradoxically, the exponential regime through the semiclassical approximation could be obtained when the validity of the approximation breaks (?).
These are my major concerns which prevent me from recommending publication. I have some other minor comments.
In the introduction, the reference to Peres in the question of irreversibility [14] is correct, however when giving credit about Loschmidt echo, i guess the Jalabert & Pastawski (PRL 86,2490 (2001)) should be cited, and maybe also due to its impact Jacquod, Silvestrov & Beenakker (PRE, 64, 055203R).
In page 3 the list (Refs 2731) of works where the Lyapunov exponent has been found (or exponential sensitivity) seems incomplete.
Im not sure about something in Eq. (1). They say $\hat{A}$ evolves first with $\hat{H}$, then a rapid rotation is applied ( $\epsilon\hat{B}$) and then it evolves with the reversed Hamiltonian $\hat{H}$. The first forward evolution of $\hat{A}$ would be
\[
\hat{A}(t)=e^{i\hat{H}t}\hat{A}e^{i\hat{H}t}
\]
but in Eq(1) the signs are not like this.
Saying Eq (2) “contains OTOC” is far fetched.
Page 10, The “TWA is known to asymptotically describe quantum echoes at short times”, please provide a reference.
The “TWA for the SK model” section is not very clear.
Section 6: “Let us now turn to the dynamics of the echo and hence OTOC…” This direct relation is not established clearly.
Requested changes
(stated in the report)
Author: Silvia Pappalardi on 20200217 [id 739]
(in reply to Report 2 by Rodolfo Jalabert on 20191205)Question (1):
Question (2):
Question (3):
Question (4):
Question (5):
Summary of changes:  We rewrote the introduction in a more cohesive manner.  We added a small comparison between the SK and SYK models.  We added a new appendix containing the derivation of the semiclassical limit of the echo observable and the square commutator.  We added a new appendix containing the derivation of TWA and its validity for the SK model.  We rewrote the introduction to TWA in Section 5.  We changed the caption of the pictures and added ordinate labels when missing. In Fig.4, we added an exponential function to guide the reader's eyes to the thermodynamic limit.  We improved the discussion of the numerical findings in Section 6.
Attachment:
ScraSkSemiClass_SciPost_v7_sp_compressed.pdf
Rodolfo Jalabert on 20200227 [id 747]
(in reply to Silvia Pappalardi on 20200217 [id 739])Second report on the manuscript “Quantum echo dynamics in the SherringtonKirkpatrick model”
by S. Pappalardi, A. Polkovnikov, and A. Silva
I appreciate the improvements made in the presentation and discussion of the manuscript, and I think that it is now publishable in SciPost Physics.
I have nevertheless a few comments and optional suggestions.
In answering the point 1) of my original report, the authors claim that the echo observable $\mu(t)$ can be refereed to as OTOC, like “all the timedependent correlation functions which have an unusual timeordering”. I was not questioning the “legal right” that the authors claim to have in choosing such nomenclature. But it’s just not common sense to induce such a confusion by using a name that means something different for almost all practitioners. If the authors were to define the product of an arbitrary even or odd number of operators at various times, would they also call this object OTOC? They dubbed “square commutator” what everybody else calls OTOC. But such a choice is also questionable within their line of thought, since “square commutator” does not carry the notion of time, and could eventually refer to the square of any pair of commutators (not necessarily at different times).
The authors ask if the “Referee could kindly point” … “the reference where …". Such a discussion can be found in Ref. [18] of the latest version of the manuscript, as well as in some of its references, i.e. Phys. Rev. Lett. 121, 210601 (2018). However, please notice that nor in my original report, nor thought this answer, I am requesting to increase the citation list with rather wellknown concepts.
Concerning my points 2) and 3), the authors state in the manuscript and in the response that “whenever the classical limit is chaotic” … the square commutator, and then $\mu(t)$, are expected to grow exponentially in time because they encode “the square of the derivatives of the classical trajectory to respect to the initial conditions”. However, a quantum system might have a classical analog, while the derivatives of the classical trajectory with respect to the initial condition might not be exponential because the dynamics is not fully chaotic. Therefore, having a classical analog is not a sufficient condition for observing an intermediate timewindow of exponential growth. Moreover, it is not obvious to me the identification made between a system having a semiclassical analogue and the existence of a classical limit for the “square commutator” (the latter based on the validity of the saddle point meanfield approximation ensured by the large Nlimit). Some spin ½ chains are shown to fulfill the second definition, but none of them fulfils the first one.
The tutorials about the Bopp formalism, included in Sec. 5 of the manuscript, in Appendix A, and in the response to Referee I, could be indeed helpful. But, other than the correction later implemented by one of the authors, I notice that in the paragraph before Eq. (23), $\alpha$ and $\beta$ are not boson operators, but complex phasespace variables, that in Ref. [70] “Vacational” has to be changed to “Variational”, and that Ref. [71] deals with the nonlinear $\sigma$model.