Quantum echo dynamics in the Sherrington-Kirkpatrick model

Dear Editor,

thank you for handling our submission. We apologize for the delay in this resubmission. We are pleased to thank the referees for recommending the publication of the paper on SciPost Physics. In this resubmission, we believe to have addressed the final comments and suggestions raised by Referee 2.

Yours sincerely,

Silvia Pappalardi, Anatoli Polkovnikov, and Alessandro Silva

Reply to the Report of Prof. Jalabert:

We thank the referee for the comments and careful reading. Here, we reply to the corresponding points, specifying the changes included in the resubmitted manuscript.

The referee writes:

In answering the point 1 of my original report, the authors claim that the echo observable $\mu(t)$ can be referred to as OTOC, like “all the time-dependent correlation functions which have an unusual time-ordering”. I was not questioning the “legal right” that the authors claim to have in choosing such nomenclature. But it’s just not common sense to induce such confusion by using a name that means something different for almost all practitioners. If the authors were to define the product of an arbitrary even or an odd number of operators at various times, would they also call this object OTOC? They dubbed “square commutator” what everybody else calls OTOC. But such a choice is also questionable within their line of thought, since “square commutator” does not carry the notion of time, and could eventually refer to the square of any pair of commutators (not necessarily at different times).

Our response:

Following the suggestion of the referee, we have changed the sentences where we refer to the echo $\mu(t)$ as the OTOC, but we kept the one in the introduction where we say contains an OTOC. In fact, while we agree about the fact that all practitioners define $\langle \hat B(t) \hat A(0) \hat B(t) \hat A(0)\rangle$ as the OTOC, we still believe that objects as $\langle \hat B(t) \hat A(0) \hat B(t) \rangle$ lie in the same class, sharing the unusual time-ordering and a similar classical limit. More precisely, OTOC are defined as multi-point and multi-time correlation functions (more or equal than three body) which cannot be represented on a single Keldysh contour, following the work by Alainer, Faoro, Ioffe, Annals of Physics (2016). They are characterized by an unusual time-ordering which prevents them from appearing in standard causal response functions. In order to be more accurate, we have added this definition in the introduction of the revised manuscript. Furthermore, for the particular initial state we are considering, i.e. $\hat A|\psi_0 \rangle = \alpha_0 |{\psi_0}\rangle$, the OTOC appearing in the echo $\mu(t)$ is equal to the ``standard'' OTOCs, by a constant factor, i.e.

To conclude, we would like to reiterate that our motivation for choosing the echo $\mu(t)$ is very similar to what discussed by Boris Fine and collaborators in Refs.[23-24]. There, they propose the echo as a simple and easy way to measure observables and encode irreversible dynamics via some particular OTOC.

The referee writes:

Concerning my points 2 and 3, the authors state in the manuscript and in the response that “whenever the classical limit is chaotic” … the square commutator, and then $\mu(t)$, are expected to grow exponentially in time because they encode “the square of the derivatives of the classical trajectory to respect to the initial conditions”. However, a quantum system might have a classical analogue, while the derivatives of the classical trajectory with respect to the initial condition might not be exponential because the dynamics are not fully chaotic. Therefore, having a classical analogue is not a sufficient condition for observing an intermediate time-window of exponential growth. Moreover, it is not obvious to me the identification made between a system having a semi-classical analogue and the existence of a classical limit for the “square commutator” (the latter based on the validity of the saddle point mean-field approximation ensured by the large N-limit). Some spin $1/2$ chains are shown to fulfil the second definition, but none of them fulfils the first one.

Our response: We thank the referee for raising this point. More precisely, we meant that every time the classical limit is well defined, all quantum observables (including the echo $\mu(t)$) shall display semi-classical dynamics before $t_{\text{Ehr}}$. At this point, if the classical limit is sufficiently chaotic, i.e. the derivatives of the trajectory to respect the initial condition immediately grow exponentially fast, then also the quantum echo observable should reproduce such exponential growth. In the example of the SK model under consideration, the classical limit is fully chaotic as witnessed by the exponential growth of the echo within TWA. Hence, one could predict that the quantum $\mu(t)$ should grow exponentially fast in time before $t_{\text{Ehr}}$. Following the comment of the referee, in the revised manuscript, we have specified chaotic when referring to the semi-classical limit as a condition for the exponential growth.

Let us now briefly comment on the second part of the referee's observation. As the referee points out, typically quantum spin 1/2 chains with short-range interactions do not have a classical limit, either semiclassical correspondence of the square-commutator. However, in the case we are studying, the validity of the semiclassical limit is ensured by the presence of long-range interactions and in particular of the large $N$ limit. Even if this is not an "obvious'' classical limit (like it would be for large $S$ spin chains), anyways the large $N$ limit is enough for having a semi-classical description and semi-classical dynamics. Physically, this follows from the fact that each spin is subject to a slowly changing magnetic field generated by interactions with many other spins. So effectively each spin evolves in an external field, for which the semiclassical description is exact. This semiclassical behaviour is manifested both in the dynamics of local observables (Figure 1 and 2) and in more complicated observables, like the echo (Figure 4). This has been shown numerically in Section 5 and analytically in appendix B.

The referee writes:

The tutorials about the Bopp formalism, included in Sec. 5 of the manuscript, in Appendix A, and in the response to Referee I, could be indeed helpful. But, other than the correction later implemented by one of the authors, I notice that in the paragraph before Eq. (23), $\alpha$ and $\beta$ are not boson operators, but complex phase-space variables, that in Ref. [70] “Vacational” has to be changed to “Variational”, and that Ref. [71] deals with the non-linear -model.

Our response: We would like to thank the referee for pointing out these mistakes. We have corrected them in the resubmitted version of the manuscript.

- We added the definition of OTOC in the introduction as a "multi-point and multi-time correlation functions which cannot be represented on a single Keldysh contour".
- We have specified *chaotic* when referring to the semi-classical limit as a condition for the exponential growth.
- We have corrected the typos in the references [70-71].
- We implemented the corrections to the Bopp formalism.