Defining classical and quantum chaos through adiabatic transformations

We thank the referee for their careful reading of our manuscript and for their detailed questions. Below are our responses:

Figures 1, 3, and 4 are too sketchy. What does the color scale represent in Fig. 1, and what type of system are the authors considering? Additionally, the axes in Figs. 3 and 4 are not specified and should be clarified.

Per the referee's suggestion, we have added explicit labels to figures 3 and 4 which should address their concerns. In the case of Fig. 1, that figure is meant to be understood as a sketch, but we agree that our initial presentation of the figure was too sketchy. We have added clarifying comments which should address the referee's concerns.

In the numerical simulations of the various systems, could the authors provide some Poincaré surface of sections or more standard physical representations of the underlying classical dynamics and compare their results with standard tools of classical and quantum chaos?

This concern is similar to others raised in the first referee report; as such, we refer the referee to our answer there for more complete details. To briefly recapitulate, we have expanded the discussions of the introduction (Sec. 1) to better reflect the relationship between our probe, the fidelity susceptibility, and other well-known dynamical probes, particularly Lyapunov exponents. In addition, the new section 6.3 shows how Lyapunov exponents can be used to separate trajectories into regular and chaotic regions in a mixed phase space, revealing new features of the fidelity susceptibility. We believe that these Lyapunov exponents provide similar information to Poincaré maps while being somewhat more quantitative.

It is essential to clearly differentiate the new results of this paper from what was previously known. My understanding is that the novel contribution of this work lies in extending the concept of fidelity susceptibility to the classical chaotic regime and studying the classical-quantum correspondence for this observable in the semi-classical regime. Since the semi-classical regime of quantum chaos in systems with few degrees of freedom has been extensively explored, the authors should aim to relate their findings to more conventional tools and emphasize both the similarities and distinctions compared to existing results.

We have partially addressed these comments already, but we would like to return to the question of which results are novel in this work. We agree with the referee that our primary contribution is to establish the utility of the fidelity susceptibility, which is expressed through the long time response as a probe of chaos and ergodicity in classical systems (prior work has already shown this for quantum systems). However, it is also important to acknowledge that our conceptual understanding of classical chaos is rather different from (although wholly consistent with) the existing literature. In particular, our choice to focus on the fidelity susceptibility offers a unified perspective on both quantum and classical chaos and ergodicity. The standard classical perspective focused on Lyapunov exponents runs into severe difficulties as exemplified in our new discussion.

For example, we know that weakly non-integrable systems with small Lyapunov exponents (such as air) which allow turbulence are much more chaotic than strongly nonintegrable systems such as water. We also expanded Fig. 2 to highlight that the definition of chaos based on fidelity is extremely intuitive as it targets the long-time instability of time averaged probability distributions (quantum or classical) to small perturbations. We do not understand the mathematical connections (if any) between Lyapunov exponents and long-time instabilities. Likewise, we do not understand connections (if any) between short-time operator growth, recently conjectured to be a measure of quantum chaos, and long-time response. These remain very interesting unsolved questions. By now, we have several additional many-particle classical models, such as the central spin model and a perturbed Ishimori chain, where we see again that the fidelity approach to chaos works very well. Our colleagues at BU in the group of D. Campbell applied this approach to contrast the FPUT chain versus the integrable Toda lattice, and again results very convincingly show that the FPUT model is chaotic in the long time limit. Those results will be reported in separate works.

I believe the authors should temper their claim and instead state that fidelity susceptibility is a highly sensitive probe of chaos in both classical and quantum systems, as first demonstrated in Ref. [33] for quantum many-body isolated systems. This should not be presented as a new definition of chaos but rather as an interesting probe, new in the context of classical chaos.

I believe the authors’ central claim is overstated and presents a misleading interpretation of the physics at play, especially in the low-degree-of-freedom systems they consider. The transition to chaos in classical Hamiltonian systems with few degrees of freedom, as explored in this paper, is already well understood. The combination of the KAM and Poincaré-Birkhoff theorems ... describes a fractal structure of phase space, with regular islands embedded in chaotic seas. At weak integrability breaking, most trajectories remain regular, and the chaotic regions occupy a vanishingly small volume. If I understand Fig. 1 correctly, the authors characterize this regime as “maximal chaos,” but in my view, this regime should instead be described as maximally sensitive. In contrast, far from integrability, where a significant portion of the phase space becomes chaotic, this is what I would consider the truly maximally chaotic regime. What the authors term "maximal chaos" seems more appropriately labeled as a regime of maximal sensitivity, where small perturbations destroy rational tori and lead to the formation of hyperbolic and elliptic fixed points, with the former giving rise to chaos through homoclinic/heteroclinic tangles.

A solid understanding of the quantum regime has also been established, at least in the semiclassical limit the authors consider... In this regime, eigenstates can be classified as either regular or chaotic. Regular eigenstates are localized on regular tori, as per EBK quantization, while chaotic eigenstates spread across the chaotic sea. At weak integrability breaking, most eigenstates remain regular, with only a few chaotic states. I agree that some eigenstates exhibit heightened sensitivity in this regime... Yet, many eigenstates remain regular in this regime, which is why I would not call it maximally chaotic, but rather maximally sensitive.

The authors' focus on instability in both classical and quantum regimes is interesting, but it should not obscure the fact that much of the phase space remains regular rather than chaotic. The observable they consider does not appear to distinguish between regular islands and the chaotic sea, thus failing to reflect the intricate phase space structure. This is both a limitation and a potentially interesting feature of their observable.

We thank the referee for raising this important point. The main reason our response took so long is that we wanted to investigate this issue very carefully. The results are actually in line with our message and agree with the points raised by the referee. In response to these comments, we added the whole new section 6.3, updated Fig. 2, and added some relevant comments throughout the text.

Our work focuses primarily on the phase space averaged fidelity susceptibility, which is analogous to the phase space averaged Lyapunov exponent. Such measures do not see inhomogeneities in phase space. We believe it is very interesting to address the question of fluctuations of the fidelity susceptibility, which must be very strong in the mixed phase space regime. In quantum disordered systems these questions were addressed in Ref. https://arxiv.org/pdf/2009.04501 (see Fig. 6), where, in a similar chaotic but non-ergodic regime, the fluctuations of the fidelity susceptibility are very large. This is also consistent with what the referee wrote about quantum systems. To partially address this point in the current work, we analyzed the behavior of the spectral function and fidelity susceptibility in a regular part of the phase space and found that according to our criteria it is integrable, while in the chaotic part of phase space we found that the system is chaotic, as expected. When we do full phase space averaging of the fidelity susceptibility, the results are dominated by the chaotic phase space, which leads to a divergent response. In this respect, our approach to defining chaos through the fidelity susceptibility does not contradict the existing literature.

We thank the referee for his attentive reading of our manuscript and for his suggested revisions/questions. We will respond to the weaknesses he identified below:

>While the manuscript provides a comprehensive overview of the fidelity susceptibility and its relation to chaos, the connection between this measure and traditional notions of chaos (e.g., Lyapunov exponents) needs further elaboration. A more detailed discussion on how the proposed formalism aligns or diverges from classical chaos indicators would be beneficial.

We have added significant pedagogical discussion on these points to the introduction (Sec. 1), including an example which illustrates the fidelity susceptibility approach to chaos (see Fig. 2 and surrounding discussion). These discussions address the referee's concerns from a number of perspectives.

First, we point out that measures such as Lyapunov exponents have some conceptual inconsistencies: in particular, the existence of large Lyapunov exponents does not imply the existence of long-time and long-distance instabilities (``the butterfly effect''). This is to say nothing of the obvious difficulty of interpreting Lyapunov exponents in generic quantum mechanical systems. Second, the example treated in Fig. 2 provides an intuitive approach to diagnosing chaos that is equally applicable to both quantum and classical systems, circumventing the issues raised with Lyapunov exponents.


Beyond these high-level introductory points, we have performed additional calculations for the two-spin model studied throughout the remainder of the text. In particular, Sec. 6.3 of the revised manuscript introduces the concept of partial phase-space averaging: here, the idea is to separate trajectories into regular and chaotic domains. In Fig. 13 (b), we categorize trajectories into regular and chaotic domains based on their Lyapunov exponents, where regular trajectories are defined to have exponents which tend towards zero in the limit of long simulation times. Using this criteria, we also compute the spectral functions of trajectories corresponding to chaotic and regular regions (Fig. 13 (a)). The results of this analysis reveal that the fidelity susceptibility can be used to analyze chaos in different regimes of phase space; in particular, the presence of a mixed phase space or KAM region is not an obstacle to the application of our method. In combination with existing dynamical probes, our approach has the ability to separate chaos and ergodicity as dynamical concepts, as discussed throughout the manuscript.

>The manuscript mentions the slow convergence of quantum spectral functions to their classical counterparts near integrability. To highlight this convergence behavior explicitly, it would be useful to include more quantitative comparisons between quantum and classical results, possibly through additional figures or detailed discussions. Perhaps more details on the numerical methods should be provided, especially for the exact diagonalization and the treatment of finite-size effects.

Regarding the issue of numerical details, we refer the referee to Sec. 3.1, which provides an overview of our numerical methods in general. When specific calculations require us to deviate from the methods outlined in Sec. 3.1, we provide a complete discussion of our methods in the relevant section.

On the subject of convergence between quantum and classical data near integrability, we have significantly expanded the material in Sec. 6.2. In particular, we have performed additional calculations (see Fig. 12) which examine classical spectral functions as the integrability breaking parameter, $x$, is reduced. We find that as $x$ becomes small the classical results for the spectral function at low frequencies have very slow convergence to the asymptotic integrable limit. Similarly, at fixed small $x$ the low-frequency tail of the spectral function does not reach asymptotic behavior within the range of numerically accessible frequencies, indicating extremely slow convergence. We discuss the implications of these results in relation to our quantum data and estimate that convergence requires $S\sim 10^5$ even when $x$ is of order $1$. Conceptually, our results are qualitatively similar to observations in the context of Arnold diffusion for which we have added a corresponding reference.

>Identifying and characterizing the intermediate regime between integrable and ergodic behavior is intriguing. Additional examples or case studies illustrating the intermediate regime would enhance the reader's understanding.

We agree that this is an intriguing regime, and a number of works have pioneered efforts to explore it (see for example Refs. [33, 41] of our revised manuscript). It seems that the phenomena which can appear in this regime are quite diverse; in the case of our model, we see anomalously long relaxation times in the classical limit and anomalously slow recovery of classical data from quantum data. In addition, we have argued that our model does not exhibit thermalization for any choice of couplings (see Sec. 6.4). Further examinations of this regime are beyond the scope of the current work, but we hope the revised discussions of the introductory section will help sate the reader's curiosity.