SciPost Submission Page
Probing semi-classical chaos in the spherical $p$-spin glass model
by Lorenzo Correale, Anatoli Polkovnikov, Marco Schirò, Alessandro Silva
This is not the latest submitted version.
This Submission thread is now published as
|Authors (as registered SciPost users):
|Lorenzo Correale · Marco Schirò · Alessandro Silva
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.
Submission & Refereeing History
You are currently on this page
Reports on this Submission
- 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).
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.
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.
- 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