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Timescales of quantum and classical chaotic spin models evolving toward equilibrium
by Fausto Borgonovi, Felix M Izrailev, Lea F Santos
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Authors (as registered SciPost users):  Lea Santos 
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Preprint Link:  scipost_202402_00033v1 (pdf) 
Date submitted:  20240222 19:01 
Submitted by:  Santos, Lea 
Submitted to:  SciPost Physics 
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Academic field:  Physics 
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Approach:  Theoretical 
Abstract
We investigate the quantum and classical quench dynamics of a onedimensional strongly chaotic lattice with $L$ interacting spins. By analyzing the classical dynamics, we identify and elucidate the two mechanisms of the relaxation process of these 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 conservation of the $L$ spin angular momenta. 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.
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In this manuscript, the authors studied the postquench dynamics for a chaotic onedimensional lattice spin model. They investigated the “quantumclassical correspondence” of this model and numerically showed that its classical version captures the time scales of quantum dynamics in the chaotic regime they focused on. Then, they used this correspondence to understand the relaxation dynamics especially in the quantum case. One focus of their discussion is the relaxation of a global observable (single particle energy spread), where they find the ballistic and diffusion timescales are both independent of system size. They also analyzed the system size independent diffusion timescale with a local observable (Sz for a local spin). In the last portion of the paper, they study the relaxation of participation ratio, which they claim is an inherently quantum metric that has a diverging relaxation timescale in the thermodynamic limit. Overall, the manuscript is well written and contains both numerical and semianalytical explanations to support their arguments. It should be of interest to readers working in the broad field of outofequilibrium quantum many body systems. We recommend publication after the authors address the following questions/comments.
1. 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)}, the \sqrt{S(S+1)} in the denominator of Eq. (36) now cancels, and the contrasting behaviour of \tau_N with system size L in the thermodynamic limit vs spin length S in the classical limit is removed. Can the authors comment on that?
2. In Fig.3 (b), it seems with the increase of L, the curves shift to left. But L=200 seems to shift in the opposite direction compared to L=100, can the author comment on that?
3. Is there a factor of 1/9 missing in the first line of Eq.(13)?
4. In Fig. 4, can the authors comment why Delta_Estat seems to go to zero in the integrable limit J0=0? It is not clear from their prior analysis, which focused on the ergodic case.
5. In the explanation of Fig. 5, the authors mention that the Gaussian should have a variance proportional to Dt. Can the authors explain what is the t in this expression? Right now, it may be confusing to the reader that this variance changes with time.
6. In Fig.10, it seems the L=4 data in panel (b) have more significant noises compare to (a), can the authors comment on that?
7. There is a typo B=0=1 at the caption of Fig2.
8. In Eq. (16), it appears the first term should be B0^2 instead of 1
9. The abbreviation ‘QCC’ is used in the Sec I but defined in Sec VII.
10. Should the “participation ratio” at the end of last paragraph of Sec VII be “inverse participation ratio”? It seems that is the quantity discussed in that reference.
11. A general comment, can the authors specify what is the S used in each plot?
Author: Lea Santos on 20240625 [id 4586]
(in reply to Report 2 on 20240405)
REFEREE:
In this manuscript, the authors studied the postquench dynamics for a chaotic onedimensional lattice spin model. They investigated the “quantumclassical correspondence” of this model and numerically showed that its classical version captures the time scales of quantum dynamics in the chaotic regime they focused on. Then, they used this correspondence to understand the relaxation dynamics especially in the quantum case. One focus of their discussion is the relaxation of a global observable (single particle energy spread), where they find the ballistic and diffusion timescales are both independent of system size. They also analyzed the system size independent diffusion timescale with a local observable (Sz for a local spin). In the last portion of the paper, they study the relaxation of participation ratio, which they claim is an inherently quantum metric that has a diverging relaxation timescale in the thermodynamic limit. Overall, the manuscript is well written and contains both numerical and semianalytical explanations to support their arguments. It should be of interest to readers working in the broad field of outofequilibrium quantum many body systems. We recommend publication after the authors address the following questions/comments.
AUTHORS:
We thank the Referee for the pertinent comments that were all addressed in the new version of the paper.
REFEREE:
1. 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)}, the \sqrt{S(S+1)} in the denominator of Eq. (36) now cancels, and the contrasting behaviour of \tau_N with system size L in the thermodynamic limit vs spin length S in the classical limit is removed. Can the authors comment on that?
AUTHORS:
We do not see why the Hamiltonian should be extensive in the spin number (not length). The spin length is fixed and the semiclassical limit is obtained by letting the distance between the quantized levels of the angular momentum go to zero, as usually done in spin systems.
REFEREE:
2. In Fig.3 (b), it seems with the increase of L, the curves shift to left. But L=200 seems to shift in the opposite direction compared to L=100, can the author comment on that?
AUTHORS:
Our main interest in this figure is the initial slope, which is common to all sizes (especially to the large sizes L=50, 100, 200). But after that, as the curves move to saturation, there are fluctuations.
REFEREE:
3. Is there a factor of 1/9 missing in the first line of Eq.(13)?
AUTHORS:
Yes, thank you.
REFEREE:
4. In Fig. 4, can the authors comment why Delta_Estat seems to go to zero in the integrable limit J0=0? It is not clear from their prior analysis, which focused on the ergodic case.
AUTHORS:
We added the following explanation to the text: "The fitting function is compatible with the fact that there is no energy spreading in the absence of interaction (J_0 = 0)".
REFEREE:
5. In the explanation of Fig. 5, the authors mention that the Gaussian should have a variance proportional to Dt. Can the authors explain what is the t in this expression? Right now, it may be confusing to the reader that this variance changes with time.
AUTHORS:
We improved the explanation by adding two sentences in the two paragraphs below Eq.27.
REFEREE:
6. In Fig.10, it seems the L=4 data in panel (b) have more significant noises compare to (a), can the authors comment on that?
AUTHORS:
The curve for L=4 fluctuates more, because the Hilbert space is smaller.
REFEREE:
7. There is a typo B=0=1 at the caption of Fig2.
AUTHORS:
Thank you.
REFEREE:
8. In Eq. (16), it appears the first term should be B0^2 instead of 1
AUTHORS:
Yes. Thank you.
REFEREE:
9. The abbreviation ‘QCC’ is used in the Sec I but defined in Sec VII.
AUTHORS:
Thank you. It was modified.
REFEREE:
10. Should the "participation ratio" at the end of last paragraph of Sec VII be "inverse participation ratio"? It seems that is the quantity discussed in that reference.
AUTHORS:
This is the correct definition. PR =1/sum Ck^4 while IPR =sum Ck^4, even though this has been incorrectly swapped in some papers, even by some of us!
REFEREE:
11. A general comment, can the authors specify what is the S used in each plot?
AUTHORS:
It was specified in the new version of the paper.
Author: Lea Santos on 20240625 [id 4587]
(in reply to Report 3 on 20240410)REFEREE:
I have carefully reviewed the submission titled "Timescales of quantum and classical chaotic spin models evolving toward equilibrium," in which the authors investigate the high temperature dynamics of the 1D longrange Ising model. The study explores the relaxation of local and total magnetization, as well as the inverse participation ratio. The authors introduce a diffusive model and a driven parametric oscillator to explain the relaxation of magnetization.
AUTHORS:
 We did not introduced a diffusive model. We studied exactly the quantum and classical Hamiltonians.
 We did not introduce a parametric oscillator; it is defined by the Hamiltonian.
REFEREE:
Next, they use a classical estimation of the energy uncertainty to explain the growth of the inverse participation ratio.
AUTHORS:
We did not use a classical estimation of the energy uncertainty to explain the growth of the inverse participation ratio. Perhaps by "energy uncertainty", the referee means the width of the LDoS.
REFEREE:
While the topic of the submission is of interest, I regret to note that the manuscript is not suitable for publication due to several significant issues with its presentation and argumentation. The writing quality is below the standard expected for publication, and many of the claims made in the abstract are not adequately supported within the body of the paper.
Having reviewed the authors' previous works, I am aware of their capability to produce highquality research. Therefore, I believe that this submission does not reflect their usual standards of excellence. I do not believe that the manuscript meets the high standards of SciPost, and I recommend that it be revised extensively or withdrawn from further consideration. I have detailed my specific concerns below.
AUTHORS:
Our work is of high quality and the results are timely and novel. The points raised by the Referee are addressed below.
REFEREE:
Lack of scientific rigor The ideas of linear v.s. non linear chaos is poorly developed. A major claim of the paper is that chaos has two mechanisms for relaxation of the magnetization: linear and nonlinear chaos. There are many issues with claim:
1) The claim is not made precise and it is not clear what linear chaos is. Eq 7 shows the equations of motion can be written as a parametric oscillator, while section V claims some of the dynamics can be captured by using equation 7 where the parameters of the parametric oscillator are given by the solution of the perturbed, nonchaotic hamiltonian. Imprecise claims are made about changing the random frequency distribution for the linear and parametric drive, but no clear way to distinguish linear from nonlinear chaos is ever given.
AUTHORS:
 The difference between linear and nonlinear chaos is widely discussed in the literature (see e.g. Ref.[40]). Our paper is not about this difference.
 Eq.7 shows that the motion of Sz is nonlinear only in the second order of perturbation of J0, that's why chaos is linear. This is explained already in the abstract: "linear parametric instability...referred to as linear chaos." See also Sec.V.
REFEREE:
2) As far as I could find, there is no where in the paper where “nonlinear” chaos is claimed. The authors appear to use section V to describe all relaxation processes. It would appear the abstract and section V are in contradiction.
AUTHORS:
The abstract properly says that "the relaxation of the singleparticles energies (global quantity) and of the onsite magnetization (local observable) is primarily due to ... linear chaos" and this is corroborated with the results in the main text.
REFEREE:
3) The model parametric oscillator they use does not appear to describe the dynamics well as L increases. The only results in this regard is shown in Fig 9) The parametric oscillator appears to only match for short times for L=21) While for L=51 it the parametric oscillator relaxes faster. Perhaps at L=100 it is even worse.
AUTHORS:
Fig.9 confirms that the dynamics is well described by linear chaos.
REFEREE:
4) No investigation of the parametric oscillator model is performed. Does it match for all J? Does the behavior of the parametric oscillator change with J and L? How about the distribution of frequencies in the linear and parametric drive? Does the parametric drive or the linear drive dominate? When is it necessary to include the nonlinear feedback?
AUTHORS:
The motion of Sz is defined by the parametric oscillator, which is nonlinear only in the second order of perturbation of J0) We do not study all of the properties of the parametric oscillator, because they are not relevant for the main goal of this work, which is to estimate the relaxation timescale.
REFEREE:
L spin angular momenta? The abstract claims the physical explanation for the relaxation time scales lies in the conservation of the L spin angular momenta. I didn’t find any information about this in the text. The effect of breaking the conservation law is not investigated.
AUTHORS:
 It is explained below Eq.6)  It does not make sense to talk about "breaking" the conservation of the angular momentum of each spin; this is a property of spin models.  The referee may have misunderstood our sentence. We now write explicitly in the abstract: "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."
REFEREE:
Energy diffusion is not observed The authors claim that after the ballistic energy spreading, the model under goes diffusive energy spreading with $DeltaE^2 \propto t$. A significant portion of the paper is directed to identifying the time scale at which the crossover from Baltic to diffusive spreading occurs and the time at which the equilibrium occurs after diffusion. However, Fig. 2 shows that the $DeltaE^2 \propto t$ scaling only matches numerical results at a single time. The yellow line in Fig 2b, that represents the diffusive spreading,looks like a tangent line. This is in contrast to the ballistic curve shown in purple which matches the dynamics for a sufficient time. As size increases the situation does not improve, Fig 2a shows that the diffusive lines remain poor descriptions of the energy relaxation.
AUTHORS:
 Fig.2 gives the global results of the dynamics, starting from the ballistic behavior up to saturation. Fig.2 is in the loglog scale, which makes possible the visualization of the ballistic behavior and the transition from ballistic to diffusion.
 To actually see the diffusion timescale, one needs to go to the linear plot in Fig.3) This figure shows that diffusion happens approximately between J0)t~0)1 and J0)t~1)5)
REFEREE:
Lyapunov exponent claims appear false In section III D, it is claimed "the Lyapunov tie tau_lambda as a function of is comparable to that of J0 is comparable to that of tau_d". I don’t see this. Fig 6 shows tau_lambda saturating to a fixed value, while tau_d appears to approach zero.
AUTHORS:
 It does not make sense to talk about Lyapunov time or diffusion time as J0 goes to infinity. In this limit, there is no diffusion and no chaos.
 Our claim that tau_lambda and tau_d are of the same order is restricted to values of J0 where diffusion is observed. This is now better explained in the text.
REFEREE:
Misc 1) The classical initial state is not justified. Why take S_x and S_y random? I think references to work on DTWA are missing here.
AUTHORS:
The classical initial state needs to be as close as possible to the quantum initial state, as originally explained below Eq.11 (now transferred to the end of Sec.II.B). The quantum initial state has fixed values for the spins in the zdirection, so clearly <Sx> = <Sy> =0) To build a classical initial state with zero average Sx and Sy, we need to take them random.
REFEREE:
Poor presentation of results One of the most challenging aspects of reviewing this paper was grappling with the significant issues related to its writing quality. The manuscript exhibits instances of inappropriate language and an abundance of typographical errors, which significantly hindered the reading experience. A few are list as follows. Definition of QCC: The acronym QCC (quantum classical correspondence) is heavily used in the introduction but is only defined at the end of the paper. This definition should be provided earlier to aid reader understanding, especially since it differs from the more common interpretation involving hbar > 0)
AUTHORS:
QCC is now defined in the introduction, as it should be.
REFEREE:
Reliance on Unexplained Results: The introduction heavily relies on results from Ref. 20, which are not explained in the paper. It's essential to either provide sufficient explanation or minimize dependence on external references to ensure the paper stands on its own.
AUTHORS:
Everything that is needed from Ref.19 for this paper is explained after Eq.6)
REFEREE:
Undefined Concepts: The Lyapunov exponent, lambda_+, is mentioned but not defined. Providing a clear definition would enhance the reader's understanding of the concept. "Projection of H onto H_0" This usually means $\tr[H H_0]$, but I don’t think that is how the authors are using it.
AUTHORS:
 The Lyapunov exponent lambda_+ is associated with the motion of a single spin, as now explained.
 The "projection" concerns the classical LDoS and is not giving by the trace suggested by the referee. The procedure was first introduced in 1998 and is explained in detailed in Ref.19)
REFEREE:
Random Discussion: The model section includes random discussion of the KolmogorovSinai entropy, which seems out of place and should be either expanded upon or removed.
AUTHORS:
It was removed.
REFEREE:
Missing Definitions and Clarity: The initial state in Eq. 11 is not defined, which could lead to confusion for readers. Additionally, the term "quantum and classical model" in Eqs. 1 and 2 is unclear and needs to be rephrased for clarity. Finally, the term “single particle energies” is not defined, and it is not clear what this refers to.
AUTHORS:
 The quantum initial state was defined in the paragraph of Eq.4) Now we explicitly write in italic "Quantum initial state". The classical initial condition was explained after Eq.11 and is now transferred to the end of Sec.II.B, where we explicitly write in italic "Classical initial conditions".
 Eqs. 1 and 2 are general. Whether S_k are operators or not is distinguished between Sec.II.A and Sec.II.B.
 The energies of the single particles are explicitly given in Eq.10)
REFEREE:
Poor Terminology: Replace "nonhomogenous" with "inhomogeneous" for accuracy.
AUTHORS:
Replaced.
REFEREE:
Poor Sentence Structure: The sentence "Specifically, ergodicity means …" at the end of section III is a runon sentence and should be revised for clarity and readability.
AUTHORS:
The sentence is in Sec.II and it was improved.
REFEREE:
Confusing Figure Contents: What distinguishes Fig. 5a and 5b is not clearly explained. Providing a clearer description or labeling would enhance the figure's effectiveness. Clarify the meaning of "dim" in Fig. 10 to improve reader understanding.
AUTHORS:
 Fig. 5a and 5b are the same, but the latter is in semilogscale, as now stressed in the caption.
 "dim" was defined before Eq.31 and it is now explained also in the caption of Fig.10)
REFEREE:
Confusing Sentences: The sentence “Let us first consider a very small interaction …” below Fig.8 is confusing and contains a redundancy. It should be revised for clarity. Also the sentence above Eq 21)
AUTHORS:
 We rephrased the sentence below Fig.8)  We do not know which sentence above Eq.21 the Referee is referring to.
REFEREE:
Poor Figure Labels and References: Explain the significance of the marks (circle, cross, diamond, etc.) in Fig. 12 to help readers interpret the figure accurately. Ensure that Fig. 12 correctly references Equations 19 and 21 for accuracy. Specify the meaning of sigma in Fig 10a
AUTHORS:
 The symbols are explained in the legend.
 Sigma is explained in the text and now also in the caption of Fig.10)
 The numbers of the equations in the legend of Fig.12 were changed.
REFEREE:
Unprofessional Language: 1) Referring to the participation ratio as the "quantum observable with no classical limit" is unclear and could be stated more precisely.
AUTHORS:
It is already explained below Eq.28: "This quantity is purely quantum, because it measures the effective number of manybody basis states..."
REFEREE:
2) The phrase “We decided to include it in this paper despite its lack of classical limit” could be rephrased to remove any perceived passiveaggressive tone. Consider stating the rationale for including the concept more objectively.
AUTHORS:
We didn't understand the comment.
REFEREE:
3) Phrases like “To keep the quantumclassical description as close as possible”, “To find the time $\tau_b$ at which .., we need to find the velocity …”, and “Combining these results one gets …” are more appropriate for lecture notes and educational material, then professional articles.
AUTHORS:
We didn't understand the comment.