Random matrix universality in dynamical correlation functions at late times

Dear Editor,
we would like to thank both the referees for their valuable comments and appreciation of the results shown in the article. Following their suggestions and comments, we have edited certain parts of the manuscript. Below, we present a point-by-point response to all the queries of the referees.
Yours sincerely,
The authors

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%\vspace{1cm}We thank the referees for their comments which allowed us to better express the novelty of our work.

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\section{Referee 1}

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\subsection{Report}

We are glad the referee appreciates the clarity of the presentation and the accuracy of the analytical and numerical results. We also thank them for the insightful comments, which we believe helped us in improving the quality of our work. Below, we address the requested changes and remarks, and we hope these revisions make the manuscript suitable for publication.\\

1. We begin by addressing concerns about the relevance of observables whose averages do not depend on energy. \\
Firstly, let us give a number of instances in where such observables are found.
Observables with temperature or energy-independent average appear, for instance, in systems with symmetries and conservation laws (in the manuscript, we focus on a system with a continuous symmetry and conservation of total magnetization), in disordered systems at high temperature, or in Floquet systems. (See the reply to the other referee for more details). Following the referee's remarks, we have added these comments in the resubmitted version.\\
Secondly, let us remark that the literature on ETH usually focuses on energy-dependent observables.
In this case, when diagonal matrix elements depend on the energy, the plateau of correlations at long times would be {\it polynomially} small in the size of the system (see, for instance, Capizzi et al. arXiv:2405.16975). This would greatly hinder the observation of the ramp, which is instead \emph{exponentially} small with the size of the system.

Nonetheless, as we mention in the paper, there has been literature pointing out in some particular case the occurrence of the ramp in correlation functions, but the previous literature discussing similar features is a bit vague on some aspects (including when to expect the occurrence of the ramp and plateau and the precise quantity to look at), so we believe it is important to state clearly all the hypotheses and the predictions.

2. Concerning the experimental significance of the exponentially small ramp and plateau, we agree with the referee that this is a subtle effect and, for this reason, it is hard to measure it experimentally.
However, let us note that these exponentially small properties that we discuss here for correlation functions, such as the ramp and plateau, are shared by the spectral form factor (SFF), which is an object widely used to diagnose chaos in quantum systems. Even if the studies on the SFF are mainly numerical experiments, they are still quite relevant in the context of many-body quantum chaos.\\
We also mention that we do not hide the difficulty of observing this effect. On the contrary, we have pointed out that, unlike for the SFF, the strong fluctuations that characterize correlation functions hide the growing behavior of the ramp, making it visible only upon ensemble average. This is another feature that has not been discussed in the previous literature on the subject.

3. Concerning the temperature-dependent generalization, we agree with the referee that the previous sentence may have been misleading. We have clarified that one needs energy-independent observables also at finite temperatures. For this reason, we referred to the generalization as straightforward''. We hope that the revised version is now more clear.

\subsection{Requested changes}

$$\begin{itemize} \item If diagonal matrix elements are dependent on the energy, the plateau would be {\it polynomially} small in the size of the system (see, for instance, Capizzi et al. arXiv:2405.16975). This would greatly hinder the observation of the ramp. We have added this comment in the main text. \item At finite temperature, in the case considered here, we would still find an exponentially small plateau (in system size). On the other hand, in the presence of energy dependencies, polynomial scaling is found. We have added to the manuscript the correction due to this effect (see Eq. 36). \end{itemize}$$

\section{Referee 2}

\subsection{Report}

We thank the referee for the appreciation of our work and for providing valuable feedback that has guided us in refining the manuscript. In the following, we respond to the comments and outline the changes made, which we believe address all concerns and render the manuscript ready for publication.

\subsection{Requested changes}

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\begin{itemize}
\item
We thank the referee for bringing to our attention this reference, which is definitely relevant. We have added Joshi et al, PRX 12, 011018 (2022) after Eq. 18.
Note that in our case, the shift is due to the properties of the fluctuations of diagonal matrix elements, which vary in different universality classes.

\item We thank the referee for this comment, we have now added the following discussion in the manuscript:
An observable that is protected by symmetry or a conservation law does not show any energy dependency. In this paper, we focus on the following mechanism. We consider a disordered spin system that conserves the total magnetization $M^z=\sum_i \sigma_i^z$ and restricts ourselves to one of the sectors of the Hilbert space with fixed total magnetization $M\in \mathbb{Z}$. In the restricted Hilbert space, the thermal average of the total magnetization is, of course, $\langle M^z\rangle_\beta=M$. Then, we consider a given site $i$ and its local magnetization $\langle \sigma_i^z\rangle_\beta$, which fluctuates depending on the disorder realization. However, upon ensemble averaging, site-permutation symmetry is restored, and $\overline{\langle \sigma^z_i \rangle_{\beta}} = M/L$. Therefore, this quantity does not depend on temperature or energy. Note that this mechanism is not restricted to spin-1/2 systems but also applies to any particle system with a $U(1)$ symmetry that implies the conservation of the total number of particles.
Other examples of such observables could be drawn from disordered systems in their paramagnetic phase and at high energies where the ensemble average due to disorder implies the vanishing of expectation values of several quantities, at least in a certain range of temperatures (for instance, the (mixed) $p$-spin model in the transverse field).
Similar physics has also been observed in Floquet systems and our ETH arguments can be extended to such systems.
\item We have added to Fig. 5(c) an inset with a linear scale to illustrate the negative oscillations in an interval of time within the ramp.
\item We have replaced the confusing notation $\mathcal{C}$ by $\Gamma$.
\item The superscript/power 1 in Eqs. (27,29,31) was actually a power $-1$ but the minus sign gets mixed up with the overline in expressions such as $\overline{\rho(E)}^{-1}$ (the confusion is especially apparent when using the font used by Scipost). We have added an extra padding to fix this issue as in $\overline{\rho(E)}^{\,\,-1}$.
\end{itemize}

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