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Timescales of quantum and classical chaotic spin models evolving toward equilibrium
by Fausto Borgonovi, Felix M Izrailev, Lea F Santos
Submission summary
Authors (as registered SciPost users):  Lea Santos 
Submission information  

Preprint Link:  scipost_202402_00033v2 (pdf) 
Date submitted:  20240601 03:40 
Submitted by:  Santos, Lea 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We investigate the quench dynamics of a onedimensional strongly chaotic lattice with $L$ interacting spins. By analyzing both the classical and quantum dynamics, we identify and elucidate the two mechanisms of the relaxation process of this systems: one arises from linear parametric instability and the other from nonlinearity. We demonstrate that the relaxation of the singleparticles energies (global quantity) and of the onsite magnetization (local observable) is primarily due to the first mechanism, referred to as linear chaos. Our analytical findings indicate that both quantities, in the classical and quantum domain, relax at the same timescale, which is independent of the system size. The physical explanation for this behavior lies in the fact that each spin is constrained to the surface of a threedimensional unit sphere, instead of filling the whole manydimensional phase space. We argue that observables with a welldefined classical limit should conform to this picture and exhibit a finite relaxation time in the thermodynamic limit. In contrast, the evolution of the participation ratio, which measures how the initial state spreads in the manybody Hilbert space and has no classical limit, indicates absence of relaxation in the thermodynamic limit.
Author indications on fulfilling journal expectations
 Provide a novel and synergetic link between different research areas.
 Open a new pathway in an existing or a new research direction, with clear potential for multipronged followup work
 Detail a groundbreaking theoretical/experimental/computational discovery
 Present a breakthrough on a previouslyidentified and longstanding research stumbling block
List of changes
The changes are detailed in the replies to the Referees.
Current status:
Reports on this Submission
Report
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 welldefined 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.
Recommendation
Publish (meets expectations and criteria for this Journal)
Author: Lea Santos on 20240726 [id 4658]
(in reply to Report 2 on 20240708)
We thank the Referee for the comment. Notice that, as we write in Sec.II.A, we use hbar= 1/\sqrt{S(S+1)}. If this was not clear, it should now be more evident with the added Eq.(4).
Our normalization is the usual way to study the classical limit. It guarantees that the classical and quantum widths of the energy shell coincide.
Notice also that the normalization of the second term with L is only needed when \nu<1, which is not our case.
Strengths
Authors have clarified a few points lacking in the previous manuscript
Report
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 v.s. nonlinear 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. Furthermore, the discussion of nonlinear 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 nonlinearity generates chaos?
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?
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.
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 loglog plot. If a dynamical feature can't be seen in the loglog 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 loglog plot as both regimes hold over many length and time scales.
In summary, I don't see diffusive like behavior in the energy spreading as claimed 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.

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. 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.
Requested changes
To make the manuscript clear, consistent and not misleading:
1) Please reduce emphasis on linear v.s. nonlinear chaos, or more deeply investigate nonlinear 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.
Recommendation:
My recommendation to the authors is to prepare a more thorough and complete analysis of the problem they are considering, and prepare the manuscript as a new submission. I believe the new submission would meet publication requirements if
1) misleading statements regarding the diffusive regimes are removed
2) a more comprehensive literature review is performed, and the problem they are investigate is more clearly motivated and stated.
3) a more thorough investigation into the difference between linear and nonlinear chaos is performed. In particular, how can the two mechanism be distinguished, and do they effect the relaxation times? I believe it would be helpful to investigate interesting limits of the model such as small J0 and nu\approx \infity where localization physics is expected to occur.
My recommendation to the editor is to reject the current submission.
Recommendation
Reject
Anonymous on 20240726 [id 4659]
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 nonlinear 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 timedependent 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 nonlinear 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 nonlinearity 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 singleparticles 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 loglog 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 loglog 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 semianalytically 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 yaxis 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 . nonlinear chaos, or more deeply investigate nonlinear 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.