Out-of-equilibrium Eigenstate Thermalization Hypothesis

In this paper an extension of the eigenstate thermalisation hypothesis is formulated. The extension makes it possible to discuss relaxation after a quantum quench in chaotic many-body quantum systems. It does so by formulating an Ansatz for matrix elements (in the basis of eigenstates of the Hamiltonian for the system) of a projector onto an initial state, and for matrix elements of an observable in the same basis. The Ansatz is statistical, and correlations between the two sets of matrix elements are a central part of it. Various statistical checks of the Ansatz are given and it is tested against numerics.

The work constitutes an important development in the area of chaotic many-body dynamics and is likely to become a standard reference in the field. For these reasons, it easily satisfies SciPost's second acceptance criterion, of "opening a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work".

The paper is generally clearly written and will be easy to follow for others working in the field.

I recommend publication after the authors have considered the following points:

[1] There are references in the text to [44] Supplemental Material. Is this in fact the appendices? If so, please correct the reference. I could not find any other material.

[2] The caption for Fig 1 refers to panel (a) but not panel (b). Although panel (b) is mentioned in the main text, I think the figure caption should be extended.

[3] Can the data in Fig 1(b) be used to test the probabilty distribution of $\tilde{R}_{ij}$? If so, I suggest that this would be a useful addition to the paper.

[4] I found the notation of Eq 37 hard to decipher. It is written that $Z(\beta)$ generalises $Z(0)$ defined in (6) but: (i) the notation $Z(0)$ does not appear in (6) and (ii) if I set $\beta=0$ in (37) I appear to get $Z(0)=1$. I suggest this should be clarified.

[5] I noticed two typos:
(a) in the text above (8) I think $R_{ij}$ should be $\tilde{R}_{ij}$;
(b) in (18) the sign in the exponential is probably wrong.

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The paper is certainly interesting and perhaps even overdue as since early days of ETH there were many discussions of relation between ETH and relaxation of observables. It was also realized that the key are the correlations between the matrix elements of observables and the overlaps of the initial state with the eigenstates but to my knowledge this is the first serious work in this direction.

I think the paper is well written and well organized. In addition it contains various numerical and analytical tests of consistency of results so there is no question in my mind that it fully deserves publication.

Below I want to list some questions and suggestions, which are optional, and the authors might (or might not) want to address them.

1. The authors effectively study cross-correlations between the matrix elements of the i) projector operator to the initial state $P=|\Psi\rangle\langle \Psi|$, which, as they note, is the rank one operator and has special properties, and ii) the matrix elements of some observable $A$. I think it would be nice first to formulate similar cross-correlation function between two observables say $A$ and $B$ and define the analogue of the function $f(\omega) g^\ast (\omega)$. When $A=B$ this would be $|f(\omega)|^2$, which is studied a lot in the literature, but generally $f\neq g$. After formulating this cross-correlation for arbitrary operators it would be easier to understand the subtleties peculiar to $ B=P$.

2. From Eq. (5) it follows that $|\Psi_{ij}|^2=\Psi_{ii}\Psi_{jj}$ , which seems to suggest that there are roughly $D^2$ independent phases in $\Psi_{ij}$ but only $D$ independent amplitudes. I wonder if this constraint plays any role in defining $\tilde R_{ij}$. Perhaps this is what allows the authors to derive Eq. (13), but maybe the authors could say in words what is important. I clearly can extend Eq. (13) to add products with more terms and more constraints on $\tilde R$. Do they follow automatically from (13) or are there more constraints?
3. Related to the previous question. Can I use this machinery to study objects like $\langle \psi | A(t) A(t') |\psi\rangle$ or one must introduce more functions like in higher odder ETH the authors developed. Maybe a general comment of how one can use the formalism for practical calculations. By the way, I am not sure that "out of equilibrium ETH" is a good title. ETH is still equilibrium to my taste as the authors use equilibrium eigenstates, just as I wrote, the important point is that one of the operators is the projector operator. If we e.g. use a projector to a mixed state, for example a pure polarized state for 10 sites in the middle and the identity matrix or some thermal state for the rest. For which length of the partial projector the suggested ETH will become equilibrium then?
4, I wonder if the authors can comment on how their formalism connects with the Kubo linear response when the state $|\Psi\rangle$ is a perturbed eigenstate of the Hamiltonian with say the operator $\epsilon A$ or more generally $\epsilon B$. In this case $\langle \Psi| A(t) |\Psi\rangle$ should reduce to a standard integral of a two-point function and hence be expressed through the standard ETH. This question connects to my question 1.

5. In Fig. 2 there seem to be a good convergence to the thermodynamic limit at large frequencies as perhaps expected. Can the authors make some claims about say short time relaxation. Does this convergence differ qualitatively from convergence of the cross-correlation (or the spectral function) of normal observables? Does the new ETH formalism allow them to reduce finite size effects?

6. The easiest one :). There is a typo right below Eq. (28).

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