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Many-body chaos and anomalous diffusion across thermal phase transitions in two dimensions

by Sibaram Ruidas, Sumilan Banerjee

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Authors (as registered SciPost users): Sumilan Banerjee · Sibaram Ruidas
Submission information
Preprint Link: scipost_202105_00019v1  (pdf)
Date submitted: 2021-05-14 17:08
Submitted by: Banerjee, Sumilan
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
Approaches: Theoretical, Computational


Chaos is an important characterization of classical dynamical systems. How is chaos linked to the long-time dynamics of collective modes across phases and phase transitions? We address this by studying chaos across Ising and Kosterlitz-Thouless transitions in classical XXZ model. We show that spatio-temporal chaotic properties have crossovers across the transitions and distinct temperature dependence in the high and low-temperature phases which show normal and anomalous diffusions, respectively. Our results also provide new insights into the dynamics of interacting quantum systems in the semiclassical limit.

Current status:
Has been resubmitted

Author comments upon resubmission

Dear Editor,

We are resubmitting our paper which was reviewed by two referees. We are delighted to receive broadly positive reviews from both the referees. We thank the referees for recommending our paper for publication after some revision. We append a detailed response to the comments and queries made by the reviewers. We have modified the manuscript and reorganised the figures to make the paper more followable. We think that the modifications made in the revised manuscript following referees' comments have significantly improved our paper. We hope that our paper will now be accepted for publication in SciPost.

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Thanking you and with our best regards, Yours sincerely, Sibaram Ruidas and Sumilan Banerjee

Response to Referee Report

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Response to Referee Report

We are grateful to both the referees for careful reading of our manuscript and many important and useful questions, comments and suggestions. We also thank the referees for recommending publication in SciPost. Following the comments and suggestions by the referees we have reorganized and modified the figures, added additional discussions and analysis. Due to these modifications, the manuscript is now much improved in our opinion. Below we address referees’ comments point by point.

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We are grateful to both the referees for careful reading of our manuscript and many important and useful questions, comments and suggestions. We also thank the referees for recommending publication in SciPost. Following the comments and suggestions by the referees we have reorganized and modified the figures, added additional discussions and analysis. Due to these modifications, the manuscript is now much improved in our opinion. Below we address referees’ comments point by point.

Anonymous Report 1 on 2021-2-13 Invited Report

We thank the referee for several crucial comments, for recommending our paper for publica- tion and finding our study to be “a natural and necessary addition to the literature exploring the connection of many body chaos and transport”. Comment 1: The authors consider two different measures of the butterfly effect in classical spin system, the decorrelator D(x,t) and ⟨δSr(t)^2⟩. All the reported results of the parame- ters such as Lyapunov exponent, t0 and butterfly velocity are for the decorrelator, while it is claimed that the results are the same for the other quantity as well. I think this comparison warrants more evidence, if not in the main text, then as a section in the appendix. Also, why these quantities differ is not immediately clear from the text, and a little more exposition would be nice. Since the claim is that these quantities differ in O(ε^2), and the data to be fitted are ∼ 10−8 → 10−1 while ε ∼ 10−4, there should be more clarity on how well the quantities derived from these two quantities match up.

Response: We thank the referee for the above comment. We agree that our discussion on this in the manuscript was somewhat confusing. Initially the quantities D(r,t) and ⟨δSr(t)^2⟩ differ in O(ε^2) only at r = 0, i.e. the former starts from 0 and the latter from ε^2, due to their definitions and the special choice of the perturbation. We have clarified this point in the main paper below Eq.3 and also have added a separate subsection (B.1) in the appendix. Due to the initial difference, the two quantities differ quantitatively at later times, but they give rise to the same λ_L and v_B as we show in Fig.13 in the modified manuscript for one particular temperature in the easy-plane case as an example. We have verified that D(x, t) and ⟨δSr(t)^2⟩ give the same λL and vB for other parameter values.

Comment 2: The generalized Lyapunov exponent D(x,t) is fitted to a ballistic form ∼ exp[λ_Lt(1−(x/v_B(t−t_0))^ν)], it is claimed that ν ∼ 2 for all the cases, but there is no evidence shown for this, and it would be nice to see the goodness of the fit, especially as a function of Temperature. This is relevant, because in local quantum models, it has been shown that the generalized Lyapunov exponent is possibly not ballistic, but has a diffusively broadening wavefront, exp[λ_Lt(1−(x/v_B(t−t_0)))^{1+p}], with p = 1 definitively shown for random circuit models. From the locus of λ_L(x, t) = 0, there doesn’t seem to be broadening at these timescales in the model considered here, but it will be interesting the study this near the thermal phase transition, where, as the authors point out, chaotic behaviour could manifest similarly for classical and quantum case. I also think that the lack of wavefront broadening in their study is a significant finding and should be highlighted more.

Response: We are grateful to the referee for raising this important point. Following referee’s suggestion, we have re-analyzed our data for λ_L(x,t) in much more detail and done fitting with goodness of fit estimates for the easy-plane anisotropy. We have estimated the range of applicability of a velocity-dependent Lyapunov exponent ansatz, namely the range over which the data for λ_L(x,t) for different t could be collapsed into a single curve as a function of v = x/(t − t_0). Theses analysis are reported in a new section (C) in the appendix. In Appendix C, we estimate the exponent ν along with its errors and goodness of fit. We show temperature dependence of ν across the KT and Ising transitions (Fig.14). As the plot suggests, ν ≈ 2.0 at high temperature and decreases to ∼ 1.0 at low temperature. This result does not affect the conclusion of the ballistic propagation of chaos front. The range over which λ_L(x,t) can be collapsed as a function of v around v = v_B shrinks with temperature. As a result, the goodness of fit is reasonable only for T >0.6. As the referee suggested, we also tried to fit the collapse regime with λ_L(1−x/(t−t_0)v_B)^{1+p} along with its goodness of fit. Motivated from Ref.[34], this functional form is valid for p>= 0 only outside the lightcone (v ≥ vB). We find p ≃ 0 at high and moderate temperature with a healthy goodness of fit (P ) value (0.2 <=P <= 0.5) which suggests that there is no broadening in the chaos front. At lower the temperatures, collapse regime outside the lightcone shrinks and we do not get a good fit. We highlight the lack of wavefront broadening in our study in the overview of the results in Section 3 in the modified manuscript.

Comment 3: For the easy plane case (KT transition), it is suggested that there is an anomalous subdiffusion for spin transport in the low temperature phase. However, as the authors also point out, the dynamical exponent α has an upward drift with time (Fig. 12 in appendix), so it is not clear whether this persists in the long-time limit. The authors claim that in these timescales, there is already evidence of ballistic chaotic behaviour, which would suggest that the mechanisms of dynamical spin transport and chaos are different. However, there is an onset time-scale for chaos, t_0 as denoted by the authors, which also seem to diverge at low temperatures. This would suggest a possible conflict, and the non-exponential behaviour of the decorrelator could coincide with subdiffusion at these timescales. Is this a possibility?

Response: We thank the referee for bringing up another very relevant point. We have now tried to clarify this issue in Section 4.3. The transient subdiffusive regime in Fig.16(a) (Fig.12 in the previous manuscript) persists at least over a temporal range 20<t <80. Indeed, this regime partially overlaps with the non-exponential growth regime t < t_0. However, the maximum value of t_0 attained at the lowest temperature studied here is ∼ 35 [Fig.4(a)]. But the transient subdiffusive regime extends beyond t_0 and continues deep inside the light cone. Thus the sub-diffusive regime also overlaps with the exponential growth and the ballistic spread of chaos over the system sizes considered here.

Comment 4: Fig.1 seems to have disconnected figures lumped together, which is disorienting in the First read. Fig 1(a) is perfect for the introduction, but it was not clear if the reader is supposed to pay attention to 1(b), (c), (d), (e) while going through the introduction. I think it would improve the presentation of the paper if the 1(b)-(e) is separated from the summary phase diagram and presented along with the technical results for chaos.

Response: As suggested by the referee, we have separated the Fig.1 into two different figures, Fig.1 and Fig.2.

Comment 4: There are some issues with labelling in the figures. The authors consider two anisotropies - ∆ = 0.3 and ∆ = 1.2 - it will make the reader’s life much easier to label each relevant figure with this so that they are not disoriented. Furthermore, the legends in the plot should be labelled for clarity – for example, in 2(c), the different plots are for temper- atures, so that should appear in the label in the figures. Same for 1(d), 2(a), 2(b), 3, 4(a), 5(a), 7, 9, 10 and 11.

Response: We thank the referee for the useful suggestions for improving the quality of presentation of the figures. We have made all the changes as suggested – the anisotropy value is explicitly mentioned in all the figures to avoid confusion and the legends also have been marked with the relevant parameter.

Anonymous Report 2 on 2021-3-10 Invited Report

We thank the referee for very useful remarks, for recommending our paper for publication and finding that “the manuscript provides a thorough and careful analysis of a set of questions” that will be “of interest to the wider community interested in many-body chaos in the quan- tum setting and its connection to the semiclassical limit”.

Comment 1: While the numerical analysis appears to be sound, I think the manuscript would benefit from a clearer discussion of how their results relate to the existing large-N com- putations discussed in Ref. [18] of the present manuscript (and to some extent, Ref. [41] of the quantum O(N) model studied within a different large-N setup). I realize that the authors have made some brief passing remarks, but I think the readers would benefit from a discussion of the key features that the earlier large-N calculations have clearly missed, which perhaps the present calculation captures better (For instance, I understand that the KT physics is much better cap- tured here). The overall discussion is somewhat confusing and a bit haphazard, in my opinion.

Response: We thank the referee for the suggestion to incorporate a discussion on how our results relate to earlier O(N) model calculations. We have included a detailed discussion on this issue at the end of the Section 2. As we have elaborated, the crucial difference between the model and dynamics studied in our work and the dynamics of O(1) model in Ref.18 and O(N) model for large N in Ref.41 lies in the dynamical universality classes. The latter and the associated dynamical critical exponent have important consequences for the spread of chaos or the butterfly speed at the phase transitions, and in the KT phase for easy-plane anisotropy. Also, related with this aspect is the existence of and additional conserved quantity, namely z component of the magnetization, in the model considered in our work. The additional con servation largely dictates spin transport in our case, unlike the O(N) models in Ref.14,41.

Comment 2: Similarly, I am not sure to what extent the authors actually provide a ra tionalization of the temperature-dependence and the natural scale associated with the butterly velocity (vB) away from the high-temperature limit (especially near the transition). Unlike in the quantum case, where theories with a well-defined ”z” (dynamical exponent) show a temper- ature dependence that follows naturally, how does one understand the temperature dependence of v_B in the present setting?

Response: We thank the referee for bringing into our attention this important point. Similar to the case of quantum critical point (Ref.53 in the modified manuscript), the dynamics studied in our work has well-defined dynamical exponents z at the Ising transitions for the easy-axis case and in the KT phase and at the KT transition for the easy-plane case. We discuss these in Section 2 in the revised manuscript. Based on these dynamical scalings, one can write down dynamical scaling laws for OTOC and v_B for the critical points and critical phases, as we discuss in Section 4.2 and Appendix C. These scaling laws enable us to obtain temperature and system size dependence of v_B away from the high-temperature limit. However, as we discuss in the modified manuscript, verification of these scaling laws in some of the parameter regimes is beyond the scope of our study, and finite system size and time accessed in our present work.

Minor Commnts:

Comment 1: On page 4, the authors write: ”In contrast, short-range quantum models with Finite local Hilbert space,....”. I know that this has often been believed to be true, but I know of at least one concrete ”counterexample”, i.e. a model with short-range interactions, a finite lo- cal Hilbert and without an obvious semiclassical limit involving random unitary circuits, which shows exponential growth of the OTOC over an extended period of time: arXiv:2009.10104. (The paper also explicitly presents the appropriate criterion that helps identify the regime where the exponential regime is well isolated.) For the sake of completeness, I think this should at least be cited.

Response: We thank the referee for the comment. We have modified the quoted sentence, cited the paper mentioned by the referee and added a discussion on the work in the context of our study.

Comment 2: On page 5, the authors while referring to the interplay of operator spread- ing in random unitary circuits and diffusion, make the remark that ”these toy models are non chaotic”. I am assuming that the authors have a very specific diagnostic in mind, related to the exponential growth of the OTOCs here? Otherwise, I think these models are still very much ”chaotic” (though it might be hard to use many of the conventional metrics of chaos for these models). The authors should probably be more precise in their remark and also specify appropriate caveats.

Response: We thank the referee for the above remark. Our reference to the random unitary circuit models as “non chaotic” was indeed based on the absence of exponential growth of OTOC in these models. We agree that these models are very much chaotic based on other diagnostics and we have added this caveat to the relevant sentence in Section 2.

List of changes

1. We have reorganized the figures and modified figure legends to improve the presentation.
2. Two new appendices (Appendix C and D) and a new subsection (B.1) in Appenidx B have been added following referees’ suggestions and comments.
3. We have incorporated additional discussions in the main text on several points raised by the referees and modified the text in a few places in response to referees’ comments.
4. We have added to two figure panels in the Appendix.
5. Several new references have been added along with the modified text and new sections.

Submission & Refereeing History

Resubmission 2007.12708v3 on 2 October 2021

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Resubmission scipost_202105_00019v1 on 14 May 2021

Reports on this Submission

Anonymous Report 1 on 2021-6-12 (Invited Report)


The authors have responded to all the questions raised in my report diligently and made the requisite changes. I recommend the article for publication.

Requested changes

1. A minor point - some references are not up to date - it will be good to review the references and make sure that the latest references are used. One that I noticed was Ref 42 which is now published in Phys Rev B (

  • validity: top
  • significance: high
  • originality: good
  • clarity: high
  • formatting: good
  • grammar: perfect

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