Timescales of quantum and classical chaotic spin models evolving toward equilibrium

I am submitting this report on behalf of an invited referee who has had some trouble with the SciPost interface. This referee was invited following the second reports of the two initial referees, and should be read after these. The referee refers to themself as "Referee 3."

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Dear editor, dear authors,

this is my report on this paper. I must apologize for the delay in submitting my opinion, due to a mixture of personal, professional reasons as well to problems in exploring the huge amount of information appearing in the web interface of SciPost.

I will divide this report in two pieces. First I will offer my opinion on how well the authors addressed the very substantive and relevant comments and criticisms of the first referee. Then I will comment on my own, and unfortunately new, fundamental criticisms to the conceptual aspects of the work specially related with the role, definition and usefulness of the classical limit within the so-called QCC program invoked in the paper, very much in line with the badly addressed only question of the second referee.

I will not go on describing the goals and claims again, as it has been done to the extreme in the previous communications. Instead, I will start with the second round of comments from the first referee and give my opinion about the relevance of the questions and the quality of the author's response.

Part I: on the answers to the second report

Response to referee 1:

The first aspect is that while the limit of linear chaos (small J0) is useful for finding the steady state energy spread, I don't see how it is necessary for understanding the system size independent relaxation time for the classical observables

Authors: The question puzzles us, because we dedicated the whole Sec.V to this explanation. It is written there: “Therefore, taking the property of linear chaos of our model into account, it becomes clear that increasing the system size L simply implies adding more harmonics in the expression of the driving force Fk and the time-dependent frequency Ωk(t)... This is why increasing L does not affect significantly the chaotic dynamics of single spins in their motion on the unit sphere.” In case our explanation was not sufficiently explicit, we now repeated the same idea with the new paragraph at the end of Sec.V: “In short, at time t>0, due to the interaction between the Sx and Sy components in Eqs.(7) all spins get involved in the dynamics independently of the system size. By increasing L, we simply increase the number of components in the frequency and force of each oscillator in Eqs.(9)-(10). This does not significantly affect the parametric instability responsible for the chaotic motion of Szk in Eq.(8).

Referee 3: I agree that, within the context of the "linear chaos" mechanism proposed by the authors, one must expect a relaxation that is roughly independent on system size IN THE CLASSICAL MODEL. I have very fundamental doubts of the possibility of extracting any useful information from this classical model about the quantum system when L goes to the thermodynamic limit, and therefore I consider this aspect of the work only partially addressed. The authors have shown certain dismissive tone when this question was raised in both previous reports, and this adds to my negative impression about how seriously the authors take the comments of the first referee.

Referee: Furthermore, the discussion of non-linear chaos in the paper is still very limited. How is this relaxation mechanism different from linear chaos, beyond the fact that the model is in a parameter regime in which the linear perturbation theory fails? Is there a specific mechanism, or is the statement just the general rule of thumb that non-linearity generates chaos?

Authors: The dominant mechanism of the motion is linear chaos. The nonlinear mechanism of chaos emerges in the second order of perturbation theory and does not influence the instability of the motion if compared to the parametric instability.

Refreee 3: Again, I found that this answer open more questions than the one it attempts to answer. At the classical level the emergence of chaos is simply not a perturbative effect, as the authors surely know. Invoking orders in perturbation theory here is something that requires a very careful analysis, starting with a systematic comparison between the solution of the classical equations of motion to show that indeed the perturbative arguments actually describe what happens. I agree with the second referee in that the concept of classical linear and non-linear chaos simply lack any conceptual or numerical support beyond providing an attractive way to think about many-body systems. My most fundamental concern is explined in the second section of the report.

Referee: Relatedly, the authors claim they identify two relaxtion mechanism (linear and non linear chaos), but then later say the linear mechanism has been discussed since the 60s. Which parts of the current work are the authors contributions?

Authors: Every part of the work is our contribution. All results are new: linear chaos being the main source of chaos in 1D strongly interacting spin models, analytical and numerical results for the relaxation timescales, etc. Both mechanisms, linear and nonlinear chaos, are well known in classical mechanics, but the role of the parametric instability in spin models was ignored so far.

Referee 3: When comparing the present work when the cited references I do agree with the second referee. At some point in the communications, the authors claim that the classical and quantum calculations are to be taken as independent contributions, but then the paper is about QCC as a way to estimate quantum results very hard to obtain numerically by means of nice classical considerations. This boils down, again, to the validity of their classical limit, something that I am not sure at all.

Referee: In summary, I am not convinced of the authors claim to have identified " two mechanisms of the relaxation process of this systems: one arises from linear parametric instability and the other from nonlinearity". In particular, I couldn't find in the manuscript a place where the two relaxations mechanisms are distinguished from each other.

Authors: As written in the abstract, we identified two mechanisms and then demonstrated that the relaxation of the single-particles energies and of the onsite magnetization is primarily due to linear chaos. It is strange that the referee cannot understand this point.

Referee 3: Leaving aside the unprofessional final comment, I simply think that the distinction between the claimed two mechanisms, an interesting observation in classical mechanics, is irrelevant for the quantum results. There is simply no evidence that the difference between linear and nonlinear chaos is relevant for the quantum results (this was stressed before by the referee, but then the authors claimed below that one must consider both classical and quantum picture and the estimates the former provide, and then we completed a circular argument).

Referee: --Diffusive Regime-- I don't see evidence of the diffusive regime. Diffusive behavior \Delta E^2 \propto t, shown in Fig 3 occurs for approximately 1 to two periods 1/J0. I wouldn't call this a regime. Typically regimes of dynamics last over multiple time scales . I don't see how universal features of dynamics can be identified within one or two time steps . Furthermore, in the paper, the authors claim the diffusive regime is shown in Fig 1, not Fig 3. In Fig 1, I see no evidence of a diffusive like behavior. As the authors state in their reply, you cannot see this behavior in the log-log plot. If a dynamical feature can't be seen in the log- log plot, I wouldn't call it a regime. In contrast, the crossover from ballistic to diffusive regimes discussed in particle transport is observable on a log-log plot as both regimes hold over many length and time scales. In summary, I don't see diffusive like behavior in the energy spreading asclaimed by the authors. Perhaps it would be more clear to refer to the time scale at which ballistic energy spreading ends, and the time scale at which saturation occurs.

Authors: First, the Referee needs to understand that the timescales are derived semi-analytically and then confirmed numerically. These timescales depend differently on the model parameters. Our analysis is not a purely numerical. Paying attention only to numerical data and ignoring the theory behind the results is, unfortunately, a common attitude nowadays. Second, Fig.1 and Fig.3 are exactly the same, but in a different scale. We wrote that diffusion is evident in Fig.3b, because different system sizes are considered there. Notice that the variance in the y-axis is renormalized by the length. This means that Delta E_0^2 exhibits a linear increase of two orders of magnitude in the range 0<J_0t<2.

Referee 3: The argument from the authors would be perfectly valid if their classical limit would be the leading order of some sort of semiclassical expansion. This is NOT the case. To make this point bluntly, the claim of the authors is that they explore the region where there is QCC, but then invike the QCC to justify and interpret their quantum results!! An attempt to check the QCC cannot be used to interpret quantum results from the classical picture. This is fundamental issue with the paper, that actually has its origins in the (unanswered) question raised by both refrees on how one actually defined the classical limit.



Response to referee 2

Referee: We thank the authors for answering our questions and improving the manuscript. We think the manuscript is good for publication. We accept the author’s explanation in terms of level spacing for our question #1 but wondering if our understanding could be more useful here. To clarify the question, what we have in mind is that as the spin S increases, the H_0 term in Eq (2) increases with S while the V term in Eq (3) increases with S^2. Thus, in our original comment we said “It is not clear that the interaction contribution to the Hamiltonian [Eq. (3)] provides a well-defined limit of large S, in the sense it is not extensive with spin length. If we normalize the interaction amplitude as J0/\sqrt{S(S+1)}”. Since the authors said after Eq. (36) “This means that for PR(t), the thermodynamic and the semiclassical limit of Eq. (32) lead to opposite conclusions, a result that requires further analysis.” We think it maybe a way to explain this question by using the proper normalization.

Referee 3: The fact that the limits lead to different results points to a very flawed definition of classical limit. I track the problem with the fact that large total (individual) spin S is a perfectly reasonable (and in fact rigorous) way to obtain a classical limit (in the sense of the stationarity of the action appearing in the path integral representation of the propagator, for example) in FEW PARTICLE SYSTEMS. When dealing with many-body systems, specially when addressing issues like the dependence with the system size and the thermodynamic limit, this is by no means the only, leave alone the best, way to define the classical limit. The naive argument used by the authors concerning the increasing density of energy levels already indicates that instead of S, one can perfectly define a classical limit by large L. sadly, at this point, any usefulness of the particular classical limit chosen by the authors goes out of the window and the work cannot be used beyond the claim that in certain regimes certain classical limit behaves similarly to the quantum system.



Part II

Now I will briefly mention my problems with the conceptual foundations of this work. I guess by now the authors that I strongly disagree with the definition and usefulness of the classical limit, and I am not ready to recommend the paper unless this issue is properly addressed. As mentioned before, there are at least two asymptotic regimes where one expect the emergence of classical descriptions in quantum systems, namely large S, large L or certain combination of the two. Although this is a very very hard problem, lack of justification on what is the appropriate "classical" description makes the present paper completely circular. So, can the authors actually shown that their classical limit emerges from some quantum considerations, specially in the limit of large L>>1? If so, I am ready to accept the results of the classical calculations as zeroth order limit of a well defined semiclassical expansion. Otherwise, the agreement between classical and quantum is nothing more than an observation that cannot be used to apply the analytical findings on the classical side to the quantum one.

Finally, there is a very problematic issue with, again, the very definition of linear vs non-linear chaos. Assume for a moment that we are dealing with an interacting but INTEGRABLE spin chain (many known examples). Note that the method of the authors to write the equations of motion as a set of non-linear parametric oscillators can be equally applied in this case. According to the logic of the authors, then, EVERY integrable spin chain will display, at least, linear chaos. This is of course a pretty ridiculous consequence of the claims made in the paper that simply makes the whole approach impossible to trust.

Although I am pretty sure that, together with the fact that the concerns of the first referee still stand in my opinion, this last argument gives a pretty much mortal blow to any work based on this parametric decomposition of the dynamics, I will be willing to read an improved version of the manuscript. In its present form, however, I do not recommend publication in SciPost.

Referee: Dear Authors and Editors, Upon review, I found that the authors have significantly improved the quality of their manuscript. The abstract is improved and the introduction is clear. Issues regarding the initial state, and Lyapunov exponent are resolved and the discussion of linear vs non-linear chaos is clarified. I still find two aspects of the manuscript misleading and must be addressed before the paper could be published. --Linear vs non linear chaos-- The first aspect is that while the limit of linear chaos (small J0) is useful for finding the steady state energy spread, I don't see how it is necessary for understanding the system size independent relaxation time for the classical observables --- Authors: The question puzzles us, because we dedicated the whole Sec.V to this explanation. It is written there: “Therefore, taking the property of linear chaos of our model into account, it becomes clear that increasing the system size $L$ simply implies adding more harmonics in the expression of the driving force $F_k$ and the time-dependent frequency $\Omega_k(t)$... This is why increasing $L$ does not affect significantly the chaotic dynamics of single spins in their motion on the unit sphere.”

In case our explanation was not sufficiently explicit, we now repeated the same idea with the new paragraph at the end of Sec.V: “In short, at time $t>0$, due to the interaction between the $S_x$ and $S_y$ components in Eqs.(7) all spins get involved in the dynamics independently of the system size. By increasing $L$, we simply increase the number of components in the frequency and force of each oscillator in Eqs.(9)-(10). This does not significantly affect the parametric instability responsible for the chaotic motion of $S_k^z$ in Eq.(8).

Referee: Furthermore, the discussion of non-linear chaos in the paper is still very limited. How is this relaxation mechanism different from linear chaos, beyond the fact that the model is in a parameter regime in which the linear perturbation theory fails? Is there a specific mechanism, or is the statement just the general rule of thumb that non-linearity generates chaos? --- Authors: The dominant mechanism of the motion is linear chaos. The nonlinear mechanism of chaos emerges in the second order of perturbation theory and does not influence the instability of the motion if compared to the parametric instability.

Referee: Relatedly, the authors claim they identify two relaxation mechanism (linear and non linear chaos), but then later say the linear mechanism has been discussed since the 60s. Which parts of the current work are the authors contributions? --- Authors: Every part of the work is our contribution. All results are new: linear chaos being the main source of chaos in 1D strongly interacting spin models, analytical and numerical results for the relaxation timescales, etc. Both mechanisms, linear and nonlinear chaos, are well known in classical mechanics, but the role of the parametric instability in spin models was ignored so far.

Referee: In summary, I am not convinced of the authors claim to have identified " two mechanisms of the relaxation process of this systems: one arises from linear parametric instability and the other from nonlinearity". In particular, I couldn't find in the manuscript a place where the two relaxations mechanisms are distinguished from each other. --- Authors: As written in the abstract, we identified two mechanisms and then demonstrated that the relaxation of the single-particles energies and of the onsite magnetization is primarily due to linear chaos. It is strange that the referee cannot understand this point.

Again: There are two classical mechanisms of Hamiltonian chaos, one due to the overlap of nonlinear resonances and the other due to parametric instability. As we explained in the paper, in our spin model, parametric instability emerges in the first order of perturbation theory in the interaction between spins, but the overlap of nonlinear resonances appears in the second order of perturbation theory. Thus, the instability of motion is mainly due to parametric instability and not to the nonlinear terms. This finding was confirmed numerically.

Referee: --Diffusive Regime-- I don't see evidence of the diffusive regime. Diffusive behavior \Delta E^2 \propto t, shown in Fig 3 occurs for approximately 1 to two periods 1/J0. I wouldn't call this a regime. Typically regimes of dynamics last over multiple time scales . I don't see how universal features of dynamics can be identified within one or two time steps . Furthermore, in the paper, the authors claim the diffusive regime is shown in Fig 1, not Fig 3. In Fig 1, I see no evidence of a diffusive like behavior. As the authors state in their reply, you cannot see this behavior in the log-log plot. If a dynamical feature can't be seen in the log- log plot, I wouldn't call it a regime. In contrast, the crossover from ballistic to diffusive regimes discussed in particle transport is observable on a log-log plot as both regimes hold over many length and time scales. In summary, I don't see diffusive like behavior in the energy spreading asclaimed by the authors. Perhaps it would be more clear to refer to the time scale at which ballistic energy spreading ends, and the time scale at which saturation occurs. --- Authors: First, the Referee needs to understand that the timescales are derived semi-analytically and then confirmed numerically. These timescales depend differently on the model parameters. Our analysis is not a purely numerical. Paying attention only to numerical data and ignoring the theory behind the results is, unfortunately, a common attitude nowadays. Second, Fig.1 and Fig.3 are exactly the same, but in a different scale. We wrote that diffusion is evident in Fig.3b, because different system sizes are considered there. Notice that the variance in the y-axis is renormalized by the length. This means that Delta E_0^2 exhibits a linear increase of two orders of magnitude in the range 0<J_0t<2.

Referee: Besides the above inconsistencies, I believe the paper is lacking in motivation and context. There is a large amount of research investigating classical approximations to quantum systems which is not acknowledged or addressed. In the context of spin models the relevant article is https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.011022 Another systematic investigation is performed here: https://www.sciencedirect.com/science/article/abs/pii/S0003491618301647?via%3Dihub The large body of literature surrounding these techniques often focuses on thermalization and relaxation times similar to the authors. How does the authors manuscript fit within the context of these works? Both of the above articles detail a systematic investigation into the correspondence between quantum and classical dynamics. --- Authors: It is unclear why the Referee decided only in the second round of the review process that the above references are relevant for our work. We cited them, but the Referee should understand that we are NOT “investigating classical approximations to quantum systems.” Our new approach is NOT an approximation, it does not employ any semiclassical approximation. Our results are exact. In the quantum case, we consider exact diagonalization and in the classical case, we integrate the classical equation of motion.

Referee: In particular, the second article shows that in systems that show localization, classical analogs fail to capture the dynamics at long times . Wouldn't similar localization physics occur in the authors model in the local limit (nu=\infty) and in the regime at small J0? This seems to be a particularly interesting limit to investigate: This is the regime where the parametric oscillator description is supposed to hold, and it's one where relaxation time scales may be dramatically different depending on potential localization physics . The manuscript could be substantially improved if this limit was systematically investigated. --- Authors: It does not make sense to study the limit of J0 small or nu=\infty. In both cases the classical chaos is weak or absent, therefore there is no diffusion, no thermalization in these limits. The whole paper is about chaotic systems. How could the Referee have missed this? Our approach works well when both the quantum model and its classical counterpart are strongly chaotic. This is crucial for the correspondence between the properties of both global and local observables we studied in the manuscript.

Referee: Requested changes To make the manuscript clear, consistent and not misleading: 1) Please reduce emphasis on linear vs . non-linear chaos, or more deeply investigate non-linear chaos and the parametric oscillator. 2) Please remove claims of a diffusive regime, or diffusive like behavior To improve motivation and context: 1) Improve literature review, and better motivate existing work --- Authors: We cited the references. As explained above, there is no reason to make any other change.