Random matrix universality in dynamical correlation functions at late times

1. It is shown (under some assumptions) that dynamical correlation functions in chaotic many-body quantum systems show a ramp and plateau structure at late times, similar to what is found in the spectral form factor.

2. As indicated in the manuscript, the connection between dynamical correlation functions and the spectral form factor has been discussed in the recent literature from other perspectives than the one given here, so the result is not completely fresh.

This paper applies the eigenstate thermalisation hypothesis (ETH) to a discussion of the late-time behavior of dynamical correlation functions in chaotic many-body quantum systems. The main result is that the dynamical correlation function shows a ramp and plateau structure like the spectral form factor. More precisely, the two are related after ensemble-averaging by a (specified) linear transformation. This result is derived and tested in three ways: from random matrix theory, from ETH, and from numerics on some long-range spin glass models.

On balance, I agree with the authors' assessment that the paper: provides a novel and synergetic link between different research areas, and opens a new pathway in an existing or a new research direction. However, given prior work on the topic using approaches other than ETH, I don't think that the level of novelty or potential are as high as for some SciPost publiations.

1. In addition to papers cited in the manuscript that discuss related ideas [7, 21-24, 25 and 26], the authors should cite and discuss Joshi et al, PRX 12, 011018 (2022). This paper is about the Partial Spectral Form Factor [PSFF] not correlation functions, but since the PSFF is given by an average over all correlation functions of operators with a given support, the results are very relevant to the current manuscript. While there are important differences [the PRX is about Floquet systems, while the manuscript is about Hamiltonian systems, and the PRX does not use ETH, which is the main point of the manuscript], there is also overlap in the random matrix results, especially in the idea of a shifted, connected correlation function [compare Eqns 17 and 18 of the manuscript with Eq 5 of the PRX]. Without intending to diminish the significance of the present manuscript, I think it would help readers in the field if this link were set out clearly.

2. I think it would be useful to add more discussion of a central assumption made at the bottom of page 9 (an observable has an average that does not depend on energy density ...). For instance, it would help to give one of two examples of systems with this property at the top of page 10.

3. In the discussion of Fig 5(c), it is asserted that ${\cal C}$ has negative oscillations but these cannot be seen in the present figure with a log axis. I think it would be useful to add an inset on a linear scale to illustrate this point.

4. I find the notation using ${\cal C}$ and $C$ to indicate related but distinct quantities [see e.g. Eq. 17] rather confusing. I suggest that the authors should use symbols that are easier to distinguish.

5. I don't see the reason for the superscript/power 1 on the rhs of Eqns 28 and 29. I suggest removing it or explaining the reason for this notation.

Publish (meets expectations and criteria for this Journal)

1. The results are clearly and concisely presented.
2. The analytical calculations are very detailed, and extensive numerical calculations have been carried out to illustrate the ramp and plateau in correlation functions.

1. The setting that the authors focus on, namely observables whose diagonal matrix elements do not depend on energy, is very restricted. In Hamiltonian systems, these matrix elements generically depend on energy.
2. The ramp and plateau in autocorrelation functions are exponentially small in the system size, so it is not clear that these features are observable in an experiment taking less than exponential time.

Thank you for sending this manuscript. The authors study the behavior of autocorrelation functions of local observables in chaotic many-body quantum systems. They explore a late-time ramp and plateau in the time-dependence of these autocorrelation functions. Focusing on situations where the matrix elements of the observable are described by random matrix theory, and so where the time dependence is controlled by spectral statistics, it is shown that the well-known ramp and plateau in the spectral form factor imply similar features in autocorrelation functions. Understanding when such a feature should arise in autocorrelation functions is an interesting question. The ramp and plateau behavior was identified for Sachdev-Ye-Kitaev models in Ref. [7] and in chaotic Floquet systems in Ref. [25]; in the latter setting the analysis is simplified by the fact that the diagonal matrix elements of the observable do not depend on (quasi)energy.

Here, the authors also focus on the setting where diagonal matrix elements do not depend on the energy, although this situation is atypical for Hamiltonian systems. A key aim of the present work is to justify when the ramp and plateau arise, but by focusing on such a restricted setting it is not clear whether that aim has been achieved. A basic question is the degree by which diagonal matrix elements can vary while still preserving a ramp and plateau. For example, if the diagonal matrix elements vary, on average, by an amount of order unity from one end of the spectrum to the other, what happens to the ramp and plateau? A simple model in the spirit of ETH would be e.g. diagonal elements which vary linearly with energy.

The ramp and plateau in autocorrelation functions have amplitudes that are exponentially small in system size, and so it is difficult to judge their significance. Within the framework of systems described by the ETH, are there any situations where the ramp and plateau become observable (in an experiment which does not require exponential time)?

Above Eq. (30), the authors write ‘we assume for simplicity that the system is put at infinite temperature (beta=0) since using a finite temperature is a straightforward generalization’. Although it may be straightforward in the situation where the diagonal matrix elements do not depend on energy, this assumption is masking the complexity of the problem in Hamiltonian systems. It would be useful to see explicitly how e.g. Eq. (45) is modified at finite beta, at least for a simple model where the diagonal matrix elements have a simple (but nontrivial) energy dependence.

Due to the issues above, for now I will not recommend publication in SciPost. However, a revised version, which addresses when the ramp and plateau are expected (beyond the one setting of observables whose diagonal matrix elements do not depend on energy), could be suitable for publication.

1. Provide additional details indicating when the ramp and plateau are expected. How are these features of autocorrelation functions suppressed if diagonal matrix elements depend on energy?
2. Elaborate on the amplitude of the ramp and plateau, and how this amplitude depends on temperature. When are these features not exponentially small in system size?

Ask for major revision