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Probing semi-classical chaos in the spherical $p$-spin glass model

by Lorenzo Correale, Anatoli Polkovnikov, Marco Schirò, Alessandro Silva

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Submission summary

Authors (as registered SciPost users): Lorenzo Correale · Marco Schirò · Alessandro Silva
Submission information
Preprint Link:  (pdf)
Date submitted: 2023-03-30 12:23
Submitted by: Correale, Lorenzo
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Statistical and Soft Matter Physics
Approaches: Theoretical, Computational


We study semiclassically the dynamics of a quantum $p$-spin glass model starting from initial states defined in microcanonical shells. We compute different chaos estimators, such as the Lyapunov exponent and the Kolmogorov-Sinai entropy, and find a marked maximum as a function of the energy of the initial state. By studying the relaxation dynamics and the properties of the energy landscape we show that the maximal chaos emerges in correspondence with the fastest spin relaxation and the maximum complexity, thus suggesting a qualitative picture where chaos emerges as the trajectories are scattered over the exponentially many saddles of the underlying landscape. We also observe hints of ergodicity breaking at low energies, indicated by the correlation function and a maximum of the fidelity susceptibility.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 3 on 2023-7-13 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2303.15393v1, delivered 2023-07-12, doi: 10.21468/SciPost.Report.7497


1. This is a timely topic. And, the authors offer a good grasp of the field and its many nuances. The questions posed by the authors and their approach are well formulated.

2. The analytical approach is sound and rather simple to follow.


1. Nature of several approximations are not clarified.

2. Distinction between classical, semi-classical, and non-classical can be further elucidated.


This paper studies the chaos in the spherical p-spin glass model. The authors use a semiclassical approach to compute several indicators of chaos. A strong aspect of this work is to discuss such indicators and show that they follow the same qualitative behavior. I would recommend for publication once the authors address the following comments. That said, the editors should give a higher priority to the other referees as I am not an expert in all aspects of this work.

- Perhaps a naive question is what controls the strength of hbar? Taking hbar to be a dimensionful constant, what does it mean for hbar being small or large? Is this a statement about large N or E?
- Beside the qualitative similarity between Fig. 1 and 2 (both right panels), what can be said about their quantitative relationship. Is it just a coincidence that they take more or less the same values?
- The long-time limit in Fig. 3 is taken by the time average in [39,40]. This doesn't seem completely justified as the curves are still evolving at t=40. Does it make sense to extrapolate to longer times to extract q_1?
- Can the authors provide a brief discussion on what conclusions of their work will be altered beyond the semiclassical limit (hbar -> 0).
- They are quite a few assumptions in the steps (in Appendix W) leading to Eq. (21). Further clarification would be helpful to the reader.

There are quite a few typos in the text. Couple of examples:

- I believe the last equality in eq. (41) should involve the derivative with respect to sigma_i (0).
- There is a factor of i missing in front of v in the definition of G(z).
- ...

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

Author:  Lorenzo Correale  on 2023-09-01  [id 3943]

(in reply to Report 3 on 2023-07-13)
answer to question
reply to objection
pointer to related literature

We thank the referee for her/his comments.

In what follows, we try to answer all the points in her/his report, in the same order they were raised:

1) As discussed in Phys. Rev. Lett. 85, 2589, the classical limit is appears when the dimensionless constant $\Gamma = \hbar^2/(M J) $ is sent to zero, irrespectively of $N$ and $E$. We clarified this point in the new version of the manuscript, at the outset of Section 2.

2) The Kolmogorov-Sinai entropy (density) depicted in Fig. 2 (right) corresponds to the average of the N positive Lyapunov exponents as defined in our manuscript. From this definition, we notice that the close alignment between this average and the maximum Lyapunov exponent, shown in Fig.1 (right), suggests that the entire set of exponents tends to be concentrated around a single value. While this observation hints at intriguing aspects of the Lyapunov exponents' distribution, we must acknowledge that the precise physical underpinning of this phenomenon remains elusive to us.

3) As pointed out by the referee, the extrapolation of $q_1$ at the finite $t_{max}$ may raise concerns, given that the curves are still evolving at that point. It is important to note that ergodicity breaking is indeed observed only at infinitely long times, with a specific sequence of limits: first, $t_w\to\infty$, and then $\tau\to\infty$ (as described by eq.(17) ). In contrast, for any fixed $t_w$, the correlation function invariably tends towards zero as $\tau$ approaches infinity (does not showing weak ergodicity breaking). While acknowledging this, we emphasize that our calculated values of $q_1$ and the ergodicity breaking energy $E_d$ are indeed estimates depending on the (finite) $t_{max}$. We note that even increasing $t_{max}$ by a factor of 4 does not lead to a significant alteration in our prediction for $E_d$ (as demonstrated in Appendix C), but do not exclude that that the value of $E_d$ we compute could be modified by taking a value of $t_{max}$ which is several order of magnitude above the ones we use here.
We also notice that, even at finite $t_{max}$, our estimation of the ergodicity breaking transition is valuable for a comparison with the predictions we obtain from the fidelity susceptibility. We notice that for a fixed $t_{max}$ both the correlation function and the fidelity, that we plot in Fig. 5, give the same estimates for the ergodicity breaking transition. These clarifications have been incorporated into the updated manuscript.

4) As suggested by Referee 2, our investigation is essentially focused into classical chaos, with quantum fluctuations serving to sample neighboring trajectories. This clarification has been incorporated into the revised manuscript. Notably, our findings can potentially extend beyond the small $\hbar$ regime using a Wigner function corresponding to a realistic micro-canonical or thermal state, as opposed to the one defined in equation (11). With this adjustment, one could quantitatively describe quantum fluctuations atop the micro-canonical manifold. Our reasoning is substantiated by the observation in SciPost Phys. 9, 021 (2020) that the Truncated Wigner Approximation (TWA) accurately reproduces quantum dynamics for operators, in the large-$N$ limit of spin-glass models. Accordingly, we expect the quantum Lyapunov exponent could be correctly reproduced by computing an out-of-time order correlator (OTOC) within TWA (instead of the quantity $d_J(t)$ defined in equation (12), which is equivalent to an OTOC only for $\hbar \to 0$). A similar rationale applies to the fidelity susceptibility, which can be computed via eq. (30) for both classical and quantum dynamics. Notably, it is worth highlighting that, to the best of our knowledge, the Lyapunov exponent and the fidelity susceptibility have yet to be tested within TWA for deep quantum regimes (non-zero $\hbar$ and finite $T$). Moreover, computing the exact thermal Wigner function for our studied model poses a formidable challenge. We have included this discussion in the revised manuscript.

5) We improved Appendix C clarifying more in details the steps leading to eq. (21) of the manuscript.

6) We fixed the typos noticed by the referee in the new version of the manuscript.

Report 2 by Juan-Diego Urbina on 2023-7-12 (Invited Report)

  • Cite as: Juan-Diego Urbina, Report on arXiv:2303.15393v1, delivered 2023-07-12, doi: 10.21468/SciPost.Report.7459


1-large scale numerical calculations, supplemented with technical intuition to get cleaner results.

2- good choice of the vast literature, focusing on the present aspects of interest.

3- an effort to physically interpret the results and clarify the connections established, and to justify the strong assumptions behind the numerical calculations


1- mixed use of terminology that is already standard, not easy to follow for a reader with expertise in quantum chaos and/or semiclassical methods

2- too many approximations and assumptions (surely well justified) involved in the numerical calculations, lack of a benchmark simulation (maybe with some small system) to check them.

3- some small level of conceptual confusion about the role of fixed points in classical and quantum chaos.


Dear Editor,

This is my report on the paper "Probing semi-classical chaos in the spherical p-spin glass model" by Correale et al.

First of all I want to apologize for the delay in submitting my report, as it took considerable efforts to go through the large amount of results, considerations and approximations involved.

I find the paper very well written, with a complete set of relevant and timely) references and appendixes clarifying several technical aspects. The scientific content is clearly sound, relevant and original, so I am happy to recommend the paper for publication, after the following questions and considerations are satisfactorily addressed.

1) I disagree with the usage of the term "semi-classical", because the actual computations presented in the paper are purely classical. This comment is part of my little personal attempt to avoid the use of well established and accepted quantum chaos terminology in a way that makes confusing for a person of this community to understand what the authors are actually doing. I simply don't see what is "semi-" in this context of a purely classical approach.

2) as the whole analysis is based on the truncated Wigner approximation, clarifying to what extent this way of defining a classical limit is rigorous is important in my opinion. Here two aspects appear relevant to me that seem to be very natural and obvious to the authors, but are actually quite delicate from a semiclassical perspective. a) the quantum degrees of freedom are represented by spin operators and the symplectic structure that emerges in the classical limit by no means is obtained by supplementing them with conjugated momentum operators as in eq. 1. To put dramatically, for N=1 this Hamiltonian has nothing to do with the usual quantization of a large spin degree of freedom. My guess is that this is simply the way one defines spin glasses, but the connection with the precise notion of spin and specially with its classical limit is then obscure for a non-expert. b) in the same venue, the construction of the Wigner function in systems with compact configuration space is an old and delicate problem, and certainly the definition used by the authors can not be rigorous (for a discussion and proposal see Fischer et al 2013 New J. Phys. 15 063004).

3) Any signature of a bound on chaos? In particular, one should consider what type of indicator provides the classical counterpart of the quantum Lypaunov exponent defined through the regularized OTOC.

4) Although much tantalizing, I found the claim of the authors-connecting the mechanism of stochastic chaos due to scattering against an exponentially large number of potential saddles with the well established mechanism where proliferation of fixed points is a signature of chaotic motion- a bit misleading. The nature of the saddles is completely different in the two theories, as unstable fixed points in the Hamiltonian case actually represent unstable periodic orbits (fixed points of the Poincare map) rather than actual fixed points of the dynamics.

Looking forward to hear further comments from the authors, and to correspondingly give my final recommendation for publication in SciPost.

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

Author:  Lorenzo Correale  on 2023-09-01  [id 3942]

(in reply to Report 2 by Juan-Diego Urbina on 2023-07-12)

We thank the referee for his comments and feedback.

In the following, we try to address the points and criticism he raised:

1) Upon further consideration, we concur with the referee's observation regarding the term "semi-classical." As our results primarily pertain to the $\hbar\to 0$ limit, we have replaced the term with "classical" wherever appropriate.

2-a) As noted by the referee, the nomenclature "spin-glass" is historically rooted, as the model utilizes position and momentum operators. Specifically, the quantum variant of the p-spin spherical model was introduced in Phys. Rev. Lett. 85, 2589 as a toy model that mimics the glass transition observed in more realistic spin models. In particular, this model exhibits an equilibrium phase diagram analogous to the one of a randomly diluted quantum Ising magnet coupled to a transverse field, a system experimentally investigated in Phys. Rev. Lett. 67, 2076. We have incorporated these insights at the outset of Section 2.

2-b) We agree that our definition for the Wigner function is not meant to be rigorous. However, we expect that our approximation improves as $\hbar$ is sent to 0, as the initial condition are sampled from a configuration space which is closer and closer to the phase space where the center of the wave-packets lies (specifically, the Cartesian product between the sphere and its tangent space). While this approximation is useful for probing classical chaos, it is important to acknowledge that it does not necessarily reproduce the correct quantum fluctuations that one would retrieve from the exact Wigner function for the microcanonical state. We clarified this point in the new version of the manuscript, at the end of Section 2.

3) Possible indications of a bound on chaos in the p-spin glass have previously been explored in the study conducted by Bera et al. It was demonstrated therein that the Lyapunov exponent's value (computed using the regularized OTOC) consistently remains well below the Maldacena-Shankar-Stanford (MSS) bound, irrespective of temperature $T$ and $\hbar$. Nonetheless, it's noteworthy that the Lyapunov exponent they computed exhibits a linear dependence on $T$ at low temperatures and in the limit of $\hbar\to 0$. This shared feature aligns with the classical counterpart of the SYK model introduced in PhysRevB.100.155128. Unfortunately, in our framework we are not able to test such linear dependence for two reasons. The first one is that we are working at fixed $E$, an obstacle which we could in principle circumvent making use of the (non-trivial) correspondence between $E$ and $T$ in the spin-glass phase. Secondly, our classical annealing algorithm restricts us from reaching the classical ground state energy ($E_0$) due to entrapment within local metastable states upon crossing an energy threshold ($E_{th}>E_0$). Nonetheless, we observe that within the energy range we explored, the Lyapunov exponent closely aligns with values computed by Bera et al. at finite $T$. We have referenced the discussion on the MSS bound found in the work by Bera et al. in the updated version of the manuscript.

4) We acknowledge the distinction between fixed points in chaotic discrete-time maps and Hamiltonian continuous-time dynamics, as pointed out by the referee. What we meant in the manuscript is that one possible criterion for chaos lies in the presence of tangles (either homoclinic or heteroclinic) originating from one of more fixed points of the continuous-time dynamics, as better established for low-dimensional systems. Our idea was that a higher number of fixed points on a micro-canonical manifold in continuous-time dynamics leads to a possible proliferation of tangles, thus enhancing chaos. However, we recognize the vagueness of this idea and, in the revised manuscript, have refrained from further speculation on the connection between maximal complexity and maximal Lyapunov, focusing solely on their correlation.

Anonymous Report 1 on 2023-7-5 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2303.15393v1, delivered 2023-07-04, doi: 10.21468/SciPost.Report.7449


The authors present a semiclassical (small hbar) analysis of chaos and relaxation dynamics in the p-spin spherical glass model. The main motivation is coming from recent results on the quantum many-body chaos of this model obtained in the large-N limit by Bera et al.

The semiclassical limit is taken using the truncated Wigner approximation, and the chaos estimators used are the Lyapunov exponent and the Kolmogorov-Sinai entropy. Relaxation dynamics (and its obstruction due to weak ergodicity breaking) is also studied using correlation functions and fidelity susceptibility. Spin relaxation and chaos are fastest around zero energy. A physical picture in terms of semiclassical trajectories is provided.

The paper is very clear and the results will be interesting for the quantum chaos and spin glass community. This work is an application of a combination of novel tools of semiclassical chaos and relaxation dynamics to a specific spin glass model. Although this is certainly a useful addition to the subfield, I don’t see any novel insight or surprising result beyond what was already reported in Bera et al which would warrant publication in SciPost Physics.
I would thus recommend publication in SciPost Physics Core, provided my comments below are addressed.

1- Did the authors try a quantitative comparison of their Lyapunov exponent with the hbar going to zero limit of Bera et al? Since they were obtained using different techniques, it would be nice to see a quantitative match.
2- There are a few missing references about studies of chaos in related models motivated by similar connections to many-body quantum chaos:
a. Phys. Rev. B 103, 174302
b. PhysRevB.100.155128

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Author:  Lorenzo Correale  on 2023-08-31  [id 3940]

(in reply to Report 1 on 2023-07-05)
answer to question

We thank the referee for her/his comments.

In our paper, we aim to offer a physical rationale for the emergence of maximal chaos at intermediate energies in the p-spin glass model. While previous work by Bera et al. just highlighted a correlation between maximal chaos and a crossover in the correlation function, our contribution provides a potential explanation grounded in the rugged landscape's topology. Furthermore, our study introduces a pioneering exploration of the fidelity susceptibility's efficacy in detecting ergodicity breaking within classical systems. This approach aligns coherently with established tools used in spin-glass literature.

Considering these aspects, we believe our work offers valuable insights into the problem's physical nature while introducing innovative techniques for understanding ergodicity breaking in classical systems. As such, we think that our findings could warrant publication in SciPost Physics.

In the following, we address comments raised by the referee:

1 - We cannot perform a complete quantitative comparison between our results and those of Bera due to the differences in our calculations. Our approach considers finite energy density $E$, while Bera et al. work with a finite temperature $T$. Although $E$ and $T$ may have a one-to-one correspondence, exact collapse may not occur. Moreover, in the classical limit, the exponent computed by Bera et al. is obtained essentially from the log of the disorder averaged OTOC (see eq. S2.1 in Supp. Mat. of Phys. Rev. Lett. 128, 115302), while in our calculation we apply the log before averaging over the disorder (see eqs. (12) and (13) of our manuscript) to observe a cleaner linear ramp. A similar issue is discussed, from a more general perspective, also in Phys. Rev. Lett. 118, 086801. Despite these obstructions, we can roughly compare our numerical results for the Lyapunov exponent (Fig. 1 right) with Bera et al.'s $\hbar = 0.03$ case (Fig.3-(c) of Phys. Rev. Lett. 128, 115302), observing a similar maximum value for the Lyapunov exponent, of approximately 0.6 in both cases. From this estimation we can at least conclude that our results and the ones presented by Bera et al. do not contradict each other.
We added these considerations to the new version of the manuscript (Section 3).

2 - We acknowledge the importance of the cited references and we have included them in the new version of the manuscript.

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