Out-of-equilibrium Eigenstate Thermalization Hypothesis

Referee 1

1) We thank the referee for this point.
For the correlator $\langle A(t)B\rangle$ similar considerations lead to $F_{AB}^{(n)}(\omega)=f_A(\omega)f_{B}(-\omega)g_{AB}(\omega)$ with $g_{AB}(\omega_{ij}) = \overline{R_{ij}^A R^B_{ji}}$. This was mentioned, in different terms, already in Ref.[Phys. Rev. Lett. 125, 050603 (2020)] and Ref.[Phys. Rev. Lett. 129, 170603 (2022)]. We introduced a footnote on page 3 to highlight this point. Since our main focus is the study of post-quenched dynamics, we have decided not to discuss these cross-correlations in detail.

2) The Referee is right, $\Psi_{ij}$ being of rank-1 implies additional higher-order relations, for instance $\tilde{R}_{ij}\tilde{R}_{jk} \tilde{R}_{ki} = (1+\tilde{R}_{ii})(1+\tilde{R}_{jj})(1+\tilde{R}_{kk})$ with $i\neq j\neq k$ and so on. It seems that they all descend from (14a) and (14b).
At order $3$, using first (14a) and then (14b), we find
$$\begin{equation} \tilde{R}_{ij}\tilde{R}_{jk} \tilde{R}_{ki} = \tilde{R}_{ik} (1+ \tilde{R}_{jj} )\tilde{R}_{ki} = (1+\tilde{R}_{ii})(1+\tilde{R}_{jj})(1+\tilde{R}_{kk}). \end{equation}$$
Similarly at order $4$, using twice (14a) and then (14b).
$$\begin{equation} \tilde{R}_{ij}\tilde{R}_{jk} \tilde{R}_{kl}\tilde{R}_{li} = \tilde{R}_{ik} (1+ \tilde{R}_{jj} )\tilde{R}_{ki}(1+\tilde{R}_{ll} ) = (1+\tilde{R}_{ii})(1+\tilde{R}_{jj})(1+\tilde{R}_{kk})(1+\tilde{R}_{ll}) \end{equation}$$
We expanded Appendix A.2 to mention these constraints there.

3) We thank the referee for the remark on multi-time correlation functions, which we believe is very interesting.
To properly capture such correlation functions, our ansatz has to be extended. In the revised version, we discuss this in section 2.4. Regarding the title, indeed our ansatz is useful beyond out-of-equilibrium context, but we wanted to emphasize the dynamical aspect, not captured by conventional ETH.

4) Let us consider a state which is a small perturbation of an eigenstate, $| \psi \rangle= e^{i \epsilon B}|E_n\rangle$. By expanding for small $\epsilon$ one gets:
$$|\psi\rangle = \frac{1}{\sqrt{1+\epsilon^2 \langle E_n| B^2|E_n\rangle}} \left( |E_n\rangle + i \epsilon B|E_n\rangle \right)$$
so that $\langle \psi| A(t) | \psi\rangle \simeq \langle E_n | A |E_n\rangle + i \epsilon \langle E_n| [ A(t) , B ] |E_n\rangle$.
We readily find
$$\begin{equation} \Psi_{ij} \simeq \delta_{i,n}\delta_{j,n}- i\epsilon [B_{in} \delta_{j,n} -B_{nj} \delta_{i,n} ] = \delta_{i,n}\delta_{j,n}- i\epsilon e^{- S(E_n)/2} [f_B(\omega_{in}) R_{in}^B \delta_{j,n} -f_B(\omega_{jn}) R_{nj}^B \delta_{i,n}]. \end{equation}$$
Tensor $\Psi_{ij}$ for this state doesn't have a smooth diagonal component, it is thus beyond the scope of our ansatz.
This is because the perturbation is applied to an eigenstate. If instead the parturbation is applied to the type of states that we consider, the ansatz is stable. We comment on this in section 2.4.
Generalization of ansatz to include the states proposed by the referee would be an interesting task for the future.

5) From the numerical results we have, it seems that the out-of-equilibrium ETH'' ansatz has the same finite-size effects as the standard one. In fact, the convergence with the system size $N$ in Fig.~3 of the manuscript is very similar to the one we obtain for the matrix elements $\overline{A_{ij}A_{ji}}$. We wanted to include a figure to show better this but it is not possible in the text version that we use here.

6) We thank the referee for the careful reading and have corrected the typo.

Referee 2

1) We thank the referee for careful reading; we have removed the reference to the supplementary material, which was unnecessary.

2) We have extended the caption of Fig.1 to add a description of panel (b).

3) To first approximation, the probability distribution of the off-diagonal matrix elements follows that of the product of two uncorrelated Gaussian random variables, resulting in a distribution described by the modified Bessel function of the second kind. We have added a discussion of this point in section 2.4 of the manuscript, and show corresponding numerical results in Fig.~2 in the revised version.

4) We thank the referee for pointing out a set of confusing notations. We have address that in the revised version.

5) We thank the referee for the careful reading. We have corrected the typos.