Defining classical and quantum chaos through adiabatic transformations

We thank the referees for their attentive reading of our previous submission. Their comments have led us to perform new computations which should relieve their most serious concerns (see figure 13 of our new submission and surrounding discussion). Our previous conclusions were based on a rather inaccurate and bold extrapolation of the spectral function to the $\omega\to 0$ limit. We have now performed an scaling analysis to locate the maximum of the fidelity-susceptibility, $\chi$. Our results indicate that the point of ``maximum chaos'' defined according to $\chi$ occurs exactly where one would expect, around $x\sim2.5$ in our notation, where $x$ is the integrability breaking parameter.


First, let us note that our primary motivation in this work is to test whether criteria for chaos proposed by us for quantum systems can be applied to classical models. Our results indicate that there are no contradictions in applying our criteria to classical models, and with the helpful suggestions of the referees, we have performed additional calculations to confirm this picture.


Let us also point out that traditional definitions of chaos in both quantum and classical systems have severe problems, which often lead to controversies in the literature. Consider the canonical example of chaotic dynamics, namely the Earth's atmosphere. This is a many-particle system consisting of air molecules that engage in collisions. Often, these dynamics are modeled classically, which is clearly accurate at long distances; however, any microscopic theory of atomic collisions necessarily invokes quantum mechanics. In this sense, one cannot define chaos without understanding how it manifests in both quantum and classical mechanics. Even if we assume that the dynamics are completely classical, it is neither practically nor computationally feasible to apply existing definitions of chaos to such a system, as it is impossible to prepare two copies of the atmosphere with nearly identical initial conditions. Moreover, such initial conditions are not even well-defined due to the quantum uncertainty principle (not to mention much larger statistical uncertainties). Hence trajectories will separate in space long before Lyapunov exponents can even be detected. Yet, it is clear that chaos exists in the atmosphere irrespective of approximations used to study the dynamics and is related to the development of long-time instabilities, such as tornadoes, which are clearly detectable in physical observables without any need to create two copies of the system or perform an echo.


Nevertheless, the scientific community has developed heuristics for identifying chaos in many-body systems. Returning to the example of Earth's atmosphere, we say that this system is chaotic because its long-time behavior exhibits instabilities. These long-time instabilities are naturally encoded in the time-average of trajectories in classical mechanics and correspond to quantum mechanical eigenstates. From this perspective, it is natural to associate the instability of eigenstates and/or time-averaged trajectories with chaotic dynamics. As explained in our manuscript, there is a natural quantity associated with these instabilities, the fidelity-susceptibility. Importantly, a growing body of work on both quantum and classical systems indicates that this definition of chaos is viable and the authors are not aware of any counterexamples (see for example https://arxiv.org/abs/2502.12046, https://journals.aps.org/prb/abstract/10.1103/PhysRevB.106.054208, https://arxiv.org/abs/2502.09711). In several of these references, the point of maximum chaos was found near the localization transition where long-time magnetization patterns show complex fractal structures. In this sense, the definition of maximal chaos is perfectly intuitive. We note that none of OTOCs, operator spreading, or level statistics exhibit any special features at the localization transition.



Now, let us consider the key concerns raised by the referees at a high level. The intuition they have appealed to, which is indeed relevant to our study, pertains to few-particle classical systems. As they have noted, there is substantial literature on this subject and a consensus has been reached that most of the phase space is regular near integrability. There is no contradiction between this observation and our arguments; we have corrected the text that led to these concerns. Following the referee's suggestion, we defined and analyzed the fidelity susceptibility for different trajectories and found that for regular trajectories $\chi$ is finite, while for chaotic trajectories it diverges as inverse waiting/cutoff time $\mu\to 0$. We also carefully analyzed the scaling of the maximum of $\chi$ and found that our previous statement was incorrect: the maximum occurs at intermediate integrability breaking strengths, as expected. On the other hand, in extended models without a mixed phase space, maximum chaos occurs at weak integrability breaking strength, which is again completely consistent with our everyday experience.



One measure of chaos, which is favored in parts of the literature and presumably by the referees, is the fraction of phase space which exhibits chaotic dynamics. This is perhaps a reasonable measure in few-particle systems, but it does not generalize to many-body systems and does not apply to quantum systems. We expect that the phase space of even a weakly non-integrable many-body system is dominated by chaotic domains. It is therefore desirable to find alternative measures of chaos which apply equally to few-particle and many-body systems; we propose that the fidelity-susceptibility is such a measure. Having said that, we of course understand that the fidelity-susceptibility is just a number characterizing a particular average over phase space. One can study more refined measures such as the fluctuations of $\chi$ (over different trajectories, different eigenstates, or over the center of mass time of two-point functions like in glasses), large deviations of $\chi$, the difference between typical and average fidelity susceptibility, or the full metric tensor. All of these studies are potentially very interesting and important. They have the potential to reveal additional details about the inhomogeneity of chaos, especially in non-thermalizing or slowly thermalizing regimes.



In consideration of this discussion, we have relaxed our terminology around ``maximal chaos'' throughout our new submission, either explicitly stating what we mean by ``maximum chaos" or simply replacing ``maximum chaos'' with maximum $\chi$. However, none of these considerations lead to any inconsistencies or modify our key idea that the phenomena at the heart of chaos are long-time instabilities.

We note that changes to the manuscript in this version are written in blue text for the referee's convenience.

-We have added a new section (6.3) which includes a numerical analysis of the maximally chaotic point (see Fig. 13 and surrounding discussion). This is meant to address the core of the referee's concerns, namely that the point of maximal chaos is not adjacent to the integrable regime (more specific discussion of this terminology is included in our responses to the referees).

-We have added discussions, particularly in the introduction and conclusion, which should be of pedagogical interest to readers.