Defining classical and quantum chaos through adiabatic transformations

The paper describes rather thoroughly the sensitivity of the fidelity susceptibility and of some related spectral function, and show that this sensitivity varies significantly at small omegas with the nature of the dynamics, being largest in the nearly integrable regime.

The central claim of the authors, that this sensitivity can be used as a new definition of chaos for both classical and quantum systems, is incompatible with the common definition of chaos and with the common consensus about the transition to chaos.

I fully agree with the point of view of referee 2, that the authors’ central claim is overstated and represents a misleading interpretation of the physics at play. I would add that, beyond this central claim, the introduction to the paper contains a number of statements that I would also consider somewhat misleading.

Probably, the crux of the problem I have with the position taken in this work appears already in the third sentence of the paper, when the authors state that "It is widely believed that chaos usually leads to ergodicity [....]". In fact, this is not a belief. There is a well-defined "hierarchy of chaos", according to which Anosov ==> Kolmogorov ==> Bernouilli ==> Mixing ==> Ergodic [cf. R. Bowen (1975), or Berkovitz et al. Stud. Hist. Philos. Sci. C 37 (2006)], and this is a theorem, not a "belief".

The confusion probably arises from the fact that the authors do not distinguish between the properties of a system and the properties of a given trajectory. Being ergodic, or mixing, or an Anosov system is a property of a dynamical system. Being "sensitive to initial conditions (or perturbations)", which is what the authors mean by "chaotic", is a property of a given trajectory. So when an integrable system is perturbed, "some" trajectories may of course become chaotic almost immediately, but this clearly does not make the whole system chaotic.

For classical dynamics, the "transition to chaos", i.e. how a system goes from integrability to chaos, is a subject that has been studied thoroughly since the seventies, from Berry's 1978 paper on "Regular and Irregular Motion", to his les Houches lecture in 1989, or the classic Lichtenberg and Lieberman book (1983) and Martin Gutzwiller's (1990). There is a strong consensus within a large community that the KAM regime, despite the presence of a few chaotic trajectories, should be considered at best (or worst) as weakly chaotic, and most of the time is described as nearly integrable. Suggesting, as the authors do, that this community has been wrong for half a century and that this regime corresponds to "maximal chaos" should be done with a bit of care and stronger justifications than the one provided here. In particular, this community is aware that the regime of strong chaos, corresponds, for some quantities, to a relative stability as rapid mixing may amount to averaging. These mechanisms are rather well understood, and in any case not at all in contradiction with the common definition of chaos.

On the quantum mechanical side of the problem, the situation is a bit more open. Indeed, after several years of debate, the quantum chaos community finally came to a consensus to define the academic field of quantum chaos as "the study of quantum systems whose classical analog is chaotic", rather than framing the question in terms of whether the quantum properties are intrinsically chaotic (in the sense of sensitivity to initial conditions or perturbations, etc.). While I personally would prefer to keep this point of view, I understand that the consensus here is not as strong as for the classical regime (and the community that has reached this consensus is somewhat smaller).

However, I would here also follow referee 2's point of view that since the definition of "maximal chaos" introduced by the authors applies equally to classical and quantum systems, and since it is not the correct concept in the classical case, it is not a correct concept in the quantum case either.

In conclusion, the paper contains interesting results on the sensitivity of eigenstates to perturbations, and shows that this sensitivity is maximal (for reasons we can clearly understand) in the nearly integrable regime. These results are presumably not as earth-shattering as the authors imply, but they clearly deserve to be eventually published. However the contextualization of this result is deeply misleading, and major changes still need to be made before this paper can appear in print.

1-Provide a better account of the transition to chaos, and in particular distinguish the properties of a system (or of the portion of phase space of a system) and the characteristics of a single trajectory.

In particular, the following sentences should be changed to distinguish between the existence of a small number of chaotic trajectories and properties of the global system (this is not an exhaustive list):

-- p2: " It is believed that, in general, motion becomes unstable or chaotic if the number of independent degrees of freedom describing a classical system is larger than the number of independent conservation laws."

-- p2: " When energy is the only conserved quantity, we expect that the classical motion of a single particle is both chaotic and ergodic in d > 1 spatial dimensions." (The fact that this statement is not true is corrected in the following sentence, but who really "expect" this today ?)

--p4:"There are plenty of examples, such as models described by KAM theory, where a system can be chaotic but non-ergodic and conversely." (A KAM system is never referred to as "chaotic", although some of its trajectories may)

--p15:"This model has been studied previously both in the quantum and classical limits [3,59] and is known to be integrable when the couplings satisfy
[...............................]
and chaotic otherwise. "
==> **chaotic** should be replaced by **non-integrable** here.

--caption of Fig1 :" The integrable and ergodic ETH phases are separated by a broad chaotic but non-ergodic regime characterized by maximal eigenstate sensitivity. "

--P23: " This model is integrable for x = 0 and x → ∞ (see Eq. (38)) and chaotic for any finite nonzero x"

2- p3: The ref [20], is, as its title ("regular and irregular wave-function") imply, not concerned with spectral statistics and the ref [21] is a study of "level clustering in the regular spectrum", so not really focused on chaotic systems. In any case neither [20] not [21] is particularly concerned with billiard. The discussion at the bottom of p3 (sentence starting by "Perhaps the most accepted definition of quantum chaos) is too imprecise and should be modified.

3- p3 : K(\tau) is a spectral statistics. The characteristic form of K(\tau) mentioned is thus just a consequence of the BGS conjecture, and not of ETH (which is an hypothesis about eigenfunctions).

4-p4 (top) : "both ETH and the BGS conjecture are really statement about ergodicity". This is not correct, even in system (or region of phase space) which are fully ergodic, deviation from the RMT can be found if mixing is not fast enough (see eg Bohigas et al prl 1990 & Phys Rep 1993).

5-p5: The paragraph "However it is intuitively clear that this effect is not related to Lyapunov instabilities [.....] will be minimal no matter how long we wait" is very confusing to me. I'm not a specialist in fluid mechanics, but my own intuition, which I think will be shared by many people, is that viscous fluids have small Lyapunov exponents, and non-viscous fluids have large Lyapunov exponents. The "intuition" mentioned by the authors is clearly not as universally shared as they assumed, and cannot be stated as fact without at least a reference to a paper where it is justified in detail.

Same thing for the sentence "The fact that maximal chaos defined in this way occurs at small integrability breaking perturbations agrees with our everyday intuition as we discussed above."

6-p7:" It is visually obvious that the motion in the top (bottom) panels is regular (chaotic). This fact can be quantified by analyzing the scaling of the distance between the two trajectories in time, which is a standard method for measuring chaoticity."
* What is "visually obvious" is that the right and left panels looks more different in the bottom line than in the top one. But is the statement that any time dependent function with a little bit of structure is necessarily associated with chaos ?
* Again, what is propagated here is only one trajectory. So we don't know if what is represented is a property of the system, or just of this particular trajectory.

7-P14: "Therefore the issue of connecting trajectories defined through short time expansions to chaos remains an open problem, both in quantum and classical systems."
I am not sure how much this question is open. Classical chaos is always defined as a long time property of a system (there is always a \lim_{\infty} involved in the definition). So it is not so much of a surprise that not a lot can be told about chaos and integrability through a short term expansion of the motion. And as for "quantum chaos", it is well recognized that RMT behavior is expected only on short energy scale, namely below the Thouless energy, and that everything beyond that energy scale (which correspond to short time dynamics) does not distinguish between chaotic or regular dynamics. Connecting short time dynamics to chaos seems to me more a dead end than an open problem.

8-P14 : section 3 : it would be good to see Poincaré sections of the dynamics, to get a sens of how much chaos is present for the various set of parameters discussed. In particular, showing a single trajectory as done in Fig. 3 provides very little information about the dynamics.

9-p19: " In each case, the low-frequency weight of Φ_ZZ drops off as ω → 0, consistent with our expectations for integrable systems."
==> can the authors specify where this expectation was formulated ?

10-p21: How is defined the "strength of integrability" ?

11-p21: Fig 7: I was not able to locate the definition of the "rescaled" fidelity.

12-p22: why is S=50 considered as "relatively small" ?

13-p24: scaling \omega_S = 1/S^{3/4} :
==> is it just that the reasoning according to which the mean level spacing ~ 1/S^2 is too simplistic (in the sens that this mean-level spacing might vary with S), or is it that the scale \omega_S eventually is not related to the mean level spacing ?

14-p25 Eq (43) : The factor C =3*10^{-5} is indeed "anomalously small". Can the author explain why they cannot calculate it analytically? Is it because they can only compute the scaling S*(x/\omega)^2, but not the full results, or because they have a prediction for the prefactor, but this prediction is off by a few orders of magnitude? In the first case they should explain what makes it impossible to get this prefactor (usually perturbative calculations are simple enough to get the full functional dependence). In the second case, such a large discrepancy may merit further discussion.

Ask for major revision

see my report

see my 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.

see my report

Ask for minor revision

The authors have answered all the questions and comments of the referees, and the manuscript has considerably improved. Therefore, I recommend publication.

Publish (easily meets expectations and criteria for this Journal; among top 50%)