Survival probability in Generalized Rosenzweig-Porter random matrix ensemble

1- synergy of analytical and numerical approaches
2- very careful numerical analysis
3- simple interpretation of numerical results in analytical terms
4- very high expertise of the authors

1- relevance to MBL seems superficial in the present version of the manuscript; the authors should elaborate on this point (see report)
2- some definitions in Sec. 3 should be made more transparent
3- some numerical observations require more intuitive explanations

The manuscript addresses numerically and analytically dynamics in the generalized Rosenzweig-Potter model (GRPM) which is a toy-model for studying non-ergodic multifractal phase. The time evolution of the survival probability $R(t)$ defined through a certain projection on a small fraction of eigenstates was investigated for ergodic, multifractal, and localized phases, and the fractal exponent was extracted from the behavior of $R(t)$. The paper presents a very thorough numerical analysis of the model, which is very timely and important in view of the hot debates in the community on the existence of delocalized non-ergodic phases in physical models. The analytical consideration based on the simple Ansatz (18) is by and large consistent with the numerical observations.

Before recommending this interesting work for publication, I would like to clarify the following points:

1. As a major motivation for this study, the authors mention the possibility of a subdiffusive behavior in the delocalized phase in the problem of many-body localization (MBL). I certainly agree with the authors that this is indeed an important motivation, but it is not immediately clear to me whether the analysis of the GRPM sheds more light on that issue. First, it would be nice to be sure that subdiffusion requires non-ergodicity and non-ergodicity always manifests itself in many-body problems through a subdiffusive behavior. Second, in contrast to the GRPM (where the existence of nontrivial exponents is kind of put in by hand in the formulation of the problem -- through the statistics of matrix elements), the subdiffusive behavior in the conventional setting for the MBL problem (with short-range interactions) is an emergent phenomenon. Therefore, I believe that it would be very beneficial for the readers if the authors elaborated on the potential relevance of their results to MBL in the discussion section. Currently, the connection between the two problems appears to be rather superficial, in my view. A a side remark, I would like to propose the authors to add some recent reference in connection to the subdiffusive behavior in the MBL problem, e.g., arXiv:1603.06588, arXiv:1807.05051, arXiv:1809.02137, arXiv:1809.02894, and, perhaps, some other works that contributed to the debate on whether subdiffusion is a transient or a true thermodynamic phenomenon. I also think that it would be useful to mention earlier works where the survival probability was studied in a many-body setting in connection to the relaxation in quantum dots (Altshuler-Gefen-Kamenev-Levitov model), in particular,
P.G. Silvestrov, Phys. Rev. B 64, 113309 (2001).

2. The observed behavior of $R(t)$ seems to contradict the above-mentioned potential relation between the GRPM and the subdiffusive behavior in interacting models. Indeed, it turns out that $R(t)$ in the ergodic phase of the GRPM decays slower (as $1/t^2$) than in the multifractal phase (where the decay is exponential). For a subdiffusive behavior, one would naively expect the opposite -- because of the increased return probability characteristic of subdiffusion. The counter-intuitive faster decay in the non-ergodic phase requires a simple explanation.

3. It is not quite clear to me whether the $1/t^2$ decay of $R(t)$ in the ergodic phase is not an artifact of the choice of the initial state which is produced by projecting on the window of eigenstates with hard boundaries. If instead of the hard cutoff in the definition of $\hat{P}_f$ one uses a soft window of the same width determined by $f$ (this can be achieved by introducing decaying coefficients $c_n$ in the definition of $\hat{P}_f$), would not this eliminate the peculiar features observed for the hard window? The argument behind this is quite simple: the Fourier transform of a box is different from the Fourier transform of, say, a Gaussian. Of course, the study with the chosen form of $\hat{P}_f$ is interesting and instructive on its own; the question is rather about universality of the findings in the manuscript.

4. Definitions in Sec. 3 require clarification. Indeed, there two types of labels in the problem: $r$ for the sites, and $n$ for the eigenenergies. Quite unexpectedly, Eq. (8) uses the site label $r$ for the energy $\varepsilon_r$. The relation only briefly mentioned above Eq. (8) and is rather confusing (... with its maximum occuring at $r=n$). What is $\varepsilon_r$? As far as I see, the formal "algorithm" is as follows: (i) label sites by $r$; (ii) solve for for the eigenenergies and eigenfunctions -- this gives an order set of $\varepsilon_n$ with the corresponding $\psi_n(r)$; (iii) for each $n$ starting with $n=1$ find $r_m(n)$ such that $|\psi_n(r_m)|^2$ is maximum out of all $|\psi_n(r)|^2$; (iv) based on this relabel the site $r_m(n)=n$ everywhere; (v) repeat for the next $n$. This should completely relabel all the sites compared to the original labeling used for finding the eigenenergies. I believe this should be clearly explained. In this regard, while such relabeling is natural for localized states, where the eigenfunctions are dominated by single site and hence there is a clear correspondence between eigenenergy index and the site label, it is rather tricky to associate some single site to a given eigenstate $\psi_n$ -- this is the reason for my confusion.

In conclusion, the manuscript can be reconsidered for publication after the authors have considered the above points. I strongly believe that the authors would find my remarks, questions, and suggestions useful for further improving their paper.

1- update references to the subdiffusion in the many-body setting
2- further elaborate on the relevance of the GRPM to MBL
3- discuss universality of the results with respect to the form of $\hat{P}_f$
4- clarify definitions and labeling used in Sec. 3

(see report for details)

1- very interesting

2- very clear

no weakness

The GRP model is known as the simplest matrix model where
a Non-Ergodic Extended phase with multifractal properties
exists besides the Localized phase and the Ergodic phase.
Here the authors study the dynamical properties of the model,
and obtain that the survival probability R(t) can be used as a clear signature
to distinguish the three phases.
The various regimes and approximations are clearly explained,
with well chosen figures displaying the corresponding numerical data.
This work is definitely very interesting within the field of random matrix models,
but also relevant for the field of Many-Body-Localization as explained in the Introduction.

As a consequence, I strongly recommend the publication of this manuscript.

no change are required

See attached report

See attached report

See attached report

See attached report