Probing semi-classical chaos in the spherical $p$-spin glass model

We thank the referee for her/his comments.

In what follows, we try to answer all the points in her/his report, in the same order they were raised:

1) As discussed in Phys. Rev. Lett. 85, 2589, the classical limit is appears when the dimensionless constant $\Gamma = \hbar^2/(M J) $ is sent to zero, irrespectively of $N$ and $E$. We clarified this point in the new version of the manuscript, at the outset of Section 2.

2) The Kolmogorov-Sinai entropy (density) depicted in Fig. 2 (right) corresponds to the average of the N positive Lyapunov exponents as defined in our manuscript. From this definition, we notice that the close alignment between this average and the maximum Lyapunov exponent, shown in Fig.1 (right), suggests that the entire set of exponents tends to be concentrated around a single value. While this observation hints at intriguing aspects of the Lyapunov exponents' distribution, we must acknowledge that the precise physical underpinning of this phenomenon remains elusive to us.

3) As pointed out by the referee, the extrapolation of $q_1$ at the finite $t_{max}$ may raise concerns, given that the curves are still evolving at that point. It is important to note that ergodicity breaking is indeed observed only at infinitely long times, with a specific sequence of limits: first, $t_w\to\infty$, and then $\tau\to\infty$ (as described by eq.(17) ). In contrast, for any fixed $t_w$, the correlation function invariably tends towards zero as $\tau$ approaches infinity (does not showing weak ergodicity breaking). While acknowledging this, we emphasize that our calculated values of $q_1$ and the ergodicity breaking energy $E_d$ are indeed estimates depending on the (finite) $t_{max}$. We note that even increasing $t_{max}$ by a factor of 4 does not lead to a significant alteration in our prediction for $E_d$ (as demonstrated in Appendix C), but do not exclude that that the value of $E_d$ we compute could be modified by taking a value of $t_{max}$ which is several order of magnitude above the ones we use here.
We also notice that, even at finite $t_{max}$, our estimation of the ergodicity breaking transition is valuable for a comparison with the predictions we obtain from the fidelity susceptibility. We notice that for a fixed $t_{max}$ both the correlation function and the fidelity, that we plot in Fig. 5, give the same estimates for the ergodicity breaking transition. These clarifications have been incorporated into the updated manuscript.

4) As suggested by Referee 2, our investigation is essentially focused into classical chaos, with quantum fluctuations serving to sample neighboring trajectories. This clarification has been incorporated into the revised manuscript. Notably, our findings can potentially extend beyond the small $\hbar$ regime using a Wigner function corresponding to a realistic micro-canonical or thermal state, as opposed to the one defined in equation (11). With this adjustment, one could quantitatively describe quantum fluctuations atop the micro-canonical manifold. Our reasoning is substantiated by the observation in SciPost Phys. 9, 021 (2020) that the Truncated Wigner Approximation (TWA) accurately reproduces quantum dynamics for operators, in the large-$N$ limit of spin-glass models. Accordingly, we expect the quantum Lyapunov exponent could be correctly reproduced by computing an out-of-time order correlator (OTOC) within TWA (instead of the quantity $d_J(t)$ defined in equation (12), which is equivalent to an OTOC only for $\hbar \to 0$). A similar rationale applies to the fidelity susceptibility, which can be computed via eq. (30) for both classical and quantum dynamics. Notably, it is worth highlighting that, to the best of our knowledge, the Lyapunov exponent and the fidelity susceptibility have yet to be tested within TWA for deep quantum regimes (non-zero $\hbar$ and finite $T$). Moreover, computing the exact thermal Wigner function for our studied model poses a formidable challenge. We have included this discussion in the revised manuscript.

5) We improved Appendix C clarifying more in details the steps leading to eq. (21) of the manuscript.

6) We fixed the typos noticed by the referee in the new version of the manuscript.

We thank the referee for his comments and feedback.

In the following, we try to address the points and criticism he raised:

1) Upon further consideration, we concur with the referee's observation regarding the term "semi-classical." As our results primarily pertain to the $\hbar\to 0$ limit, we have replaced the term with "classical" wherever appropriate.

2-a) As noted by the referee, the nomenclature "spin-glass" is historically rooted, as the model utilizes position and momentum operators. Specifically, the quantum variant of the p-spin spherical model was introduced in Phys. Rev. Lett. 85, 2589 as a toy model that mimics the glass transition observed in more realistic spin models. In particular, this model exhibits an equilibrium phase diagram analogous to the one of a randomly diluted quantum Ising magnet coupled to a transverse field, a system experimentally investigated in Phys. Rev. Lett. 67, 2076. We have incorporated these insights at the outset of Section 2.

2-b) We agree that our definition for the Wigner function is not meant to be rigorous. However, we expect that our approximation improves as $\hbar$ is sent to 0, as the initial condition are sampled from a configuration space which is closer and closer to the phase space where the center of the wave-packets lies (specifically, the Cartesian product between the sphere and its tangent space). While this approximation is useful for probing classical chaos, it is important to acknowledge that it does not necessarily reproduce the correct quantum fluctuations that one would retrieve from the exact Wigner function for the microcanonical state. We clarified this point in the new version of the manuscript, at the end of Section 2.

3) Possible indications of a bound on chaos in the p-spin glass have previously been explored in the study conducted by Bera et al. It was demonstrated therein that the Lyapunov exponent's value (computed using the regularized OTOC) consistently remains well below the Maldacena-Shankar-Stanford (MSS) bound, irrespective of temperature $T$ and $\hbar$. Nonetheless, it's noteworthy that the Lyapunov exponent they computed exhibits a linear dependence on $T$ at low temperatures and in the limit of $\hbar\to 0$. This shared feature aligns with the classical counterpart of the SYK model introduced in PhysRevB.100.155128. Unfortunately, in our framework we are not able to test such linear dependence for two reasons. The first one is that we are working at fixed $E$, an obstacle which we could in principle circumvent making use of the (non-trivial) correspondence between $E$ and $T$ in the spin-glass phase. Secondly, our classical annealing algorithm restricts us from reaching the classical ground state energy ($E_0$) due to entrapment within local metastable states upon crossing an energy threshold ($E_{th}>E_0$). Nonetheless, we observe that within the energy range we explored, the Lyapunov exponent closely aligns with values computed by Bera et al. at finite $T$. We have referenced the discussion on the MSS bound found in the work by Bera et al. in the updated version of the manuscript.

4) We acknowledge the distinction between fixed points in chaotic discrete-time maps and Hamiltonian continuous-time dynamics, as pointed out by the referee. What we meant in the manuscript is that one possible criterion for chaos lies in the presence of tangles (either homoclinic or heteroclinic) originating from one of more fixed points of the continuous-time dynamics, as better established for low-dimensional systems. Our idea was that a higher number of fixed points on a micro-canonical manifold in continuous-time dynamics leads to a possible proliferation of tangles, thus enhancing chaos. However, we recognize the vagueness of this idea and, in the revised manuscript, have refrained from further speculation on the connection between maximal complexity and maximal Lyapunov, focusing solely on their correlation.

We thank the referee for her/his comments.

In our paper, we aim to offer a physical rationale for the emergence of maximal chaos at intermediate energies in the p-spin glass model. While previous work by Bera et al. just highlighted a correlation between maximal chaos and a crossover in the correlation function, our contribution provides a potential explanation grounded in the rugged landscape's topology. Furthermore, our study introduces a pioneering exploration of the fidelity susceptibility's efficacy in detecting ergodicity breaking within classical systems. This approach aligns coherently with established tools used in spin-glass literature.

Considering these aspects, we believe our work offers valuable insights into the problem's physical nature while introducing innovative techniques for understanding ergodicity breaking in classical systems. As such, we think that our findings could warrant publication in SciPost Physics.

In the following, we address comments raised by the referee:

1 - We cannot perform a complete quantitative comparison between our results and those of Bera due to the differences in our calculations. Our approach considers finite energy density $E$, while Bera et al. work with a finite temperature $T$. Although $E$ and $T$ may have a one-to-one correspondence, exact collapse may not occur. Moreover, in the classical limit, the exponent computed by Bera et al. is obtained essentially from the log of the disorder averaged OTOC (see eq. S2.1 in Supp. Mat. of Phys. Rev. Lett. 128, 115302), while in our calculation we apply the log before averaging over the disorder (see eqs. (12) and (13) of our manuscript) to observe a cleaner linear ramp. A similar issue is discussed, from a more general perspective, also in Phys. Rev. Lett. 118, 086801. Despite these obstructions, we can roughly compare our numerical results for the Lyapunov exponent (Fig. 1 right) with Bera et al.'s $\hbar = 0.03$ case (Fig.3-(c) of Phys. Rev. Lett. 128, 115302), observing a similar maximum value for the Lyapunov exponent, of approximately 0.6 in both cases. From this estimation we can at least conclude that our results and the ones presented by Bera et al. do not contradict each other.
We added these considerations to the new version of the manuscript (Section 3).

2 - We acknowledge the importance of the cited references and we have included them in the new version of the manuscript.