SciPost Submission Page
Defining classical and quantum chaos through adiabatic transformations
by Cedric Lim, Kirill Matirko, Hyeongjin Kim, Anatoli Polkovnikov, Michael O. Flynn
Submission summary
| Authors (as registered SciPost users): | Michael Flynn |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202412_00029v1 (pdf) |
| Date submitted: | 2024-12-17 09:45 |
| Submitted by: | Flynn, Michael |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
We propose a formalism which defines chaos in both quantum and classical systems in an equivalent manner by means of \textit{adiabatic transformations}. The complexity of adiabatic transformations which preserve classical time-averaged trajectories (quantum eigenstates) in response to Hamiltonian deformations serves as a measure of chaos. This complexity is quantified by the (properly regularized) fidelity susceptibility. Physically this measure quantifies long time instabilities of physical observables due to small changes in the Hamiltonian of the system. Our exposition clearly showcases the common structures underlying quantum and classical chaos and allows us to distinguish integrable, chaotic but non-thermalizing, and ergodic/mixing regimes. We apply the fidelity susceptibility to a model of two coupled spins and demonstrate that it successfully predicts the universal onset of chaos, both for finite spin $S$ and in the classical limit $S\to\infty$. Interestingly, we find that finite $S$ effects are anomalously large close to integrability.
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 multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
List of changes
First, we note that changes to the manuscript in this version are written in blue text for the referee's convenience.
-Significantly expanded conceptual/pedagogical discussions of the introduction, see in particular Fig. 2 and surrounding discussion.
-Expanded the content and discussion of physics near integrability, see Fig. 12.
-Added a new section (6.3) which explains how phase space averages can be broken down into trajectories over regular or chaotic regions.
Current status:
Reports on this Submission
Strengths
see my report
Weaknesses
see my report
Report
In their revised manuscript and response to my previous report, the authors have clarified several points. However, they have not modified their main claim regarding the necessity to redefine chaos, which I still find too strong.
To summarize my initial concerns, which remain unaddressed: the authors make a bold claim that chaos in both quantum and classical systems should be redefined based on eigenstate sensitivity, quantified by fidelity susceptibility. Unlike conventional measures such as the Lyapunov exponent, which captures sensitivity to initial conditions, the authors focus on the response of eigenstates or time-averaged trajectories (e.g., KAM tori in phase space) to changes in a Hamiltonian parameter. They find that this sensitivity, when averaged over phase space, is maximal at weak integrability breaking—stronger than in the fully chaotic (ergodic) regime.
This is not surprising if fidelity susceptibility is primarily detecting bifurcations, such as the breaking of KAM tori as an integrability-breaking parameter is varied. By the Poincaré-Birkhoff theorem, rational tori will be broken, while KAM theorem ensures that sufficiently irrational tori persist. Thus, while the weak integrability-breaking regime is indeed fragile to parameter variations, it is misleading to call this “maximal chaos,” as most of the phase space remains regular.
Rather, the fact that the authors’ observable suggests "maximal chaos" in this regime should be seen as evidence that it is not a reliable indicator of chaos. If the authors instead framed their findings as showing that weak integrability breaking/mixed dynamics leads to maximal parameter sensitivity—without redefining chaos—I would have no objection to publication. However, given the well-established understanding of Hamiltonian chaos in low-dimensional systems, I cannot support a redefinition of chaos based on the authors' findings.
As I previously requested, the authors should better connect their results to established literature in both classical and quantum chaos and provide a careful phase-space analysis of their observable. Given that they focus on low-dimensional systems, this should be feasible.
Requested changes
see my report
Recommendation
Ask for minor revision
Report #1 by Tigran Sedrakyan (Referee 1) on 2025-2-5 (Invited Report)
Report
The authors have answered all the questions and comments of the referees, and the manuscript has considerably improved. Therefore, I recommend publication.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)