Defining classical and quantum chaos through adiabatic transformations

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This work proposes a new definition of chaos, aimed at applying equally to both classical and quantum systems. Rather than defining chaos for example via the Lyapunov exponent in classical systems and OTOCs or ETH in quantum systems, the authors suggest characterizing it through the sensitivity of Hamiltonian eigenstates to small perturbations. While this notion may initially seem quantum in nature, the authors demonstrate that this is not the case; they define a classical fidelity susceptibility to probe the emergence of chaos in the weak integrability-breaking regime.

This study contributes to the growing interest in quantum chaos in closed or isolated many-body systems with a large number of degrees of freedom, particularly its relation to thermalization and out-of-equilibrium dynamics. Key tools and concepts such as the Eigenstate Thermalization Hypothesis (ETH) and Out-of-Time-Ordered Correlators (OTOCs) have emerged in this field. The current work builds upon previous studies by some of the present authors, notably Ref. [54], which offers a detailed and pedagogical description of fidelity susceptibility in both classical and quantum systems, and Ref. [33], which shows that quantum chaos in many-body systems can be sensitively probed via adiabatic deformations of eigenstates.

Following an introduction that motivates the authors' claim for a new definition of chaos—distinguishing it from short-time diagnostics like OTOCs or Lyapunov exponents, and from ergodicity measures such as ETH and Random Matrix Theory (RMT)—the authors provide a review of their analytical construction of fidelity susceptibility in both classical and quantum systems. They then study a two-spin model across different regimes, ranging from near-integrability to far from integrability.

I have a significant reservation regarding this paper, which prevents me from recommending it for publication at this time.

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, as summarized in Ott's book and Berry's lecture notes “Regular and Irregular Motion,” 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 (large-spin limit), as detailed in papers like [Bohigas, O., Tomsovic, S., & Ullmo, D. (1993). Manifestations of classical phase space structures in quantum mechanics. Physics Reports, 223(2), 43-133] and in the books by Gutzwiller and Haake. 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, such as in the phenomenon of chaos-assisted tunneling [Tomsovic, S., & Ullmo, D. (1994). Chaos-assisted tunneling. Physical Review E, 50(1), 145], where certain regular states near a chaotic sea couple resonantly with chaotic states, leading to large variations in the tunneling rate between regular islands when a parameter of the system $\lambda$ is vaied. 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.

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.

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.

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?

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.

Ask for major revision

1. The manuscript tackles an important and timely topic in theoretical physics, providing a unified approach to understanding chaos in both classical and quantum domains.

2. The use of fidelity susceptibility as a measure of chaos is innovative and offers new insights into distinguishing between integrable, chaotic but non-thermalizing, and ergodic regimes.

3. The paper is well-organized, and the arguments are generally clear and supported by thorough theoretical and numerical analysis.

1. 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.

2. 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.

3. 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.

The manuscript addresses a significant and complex topic with a novel approach that has the potential to advance our understanding of chaos in classical and quantum systems. Given that the authors address the points raised in this report, I would recommend the manuscript for publication in SciPost Physics.

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