Replica approach to the generalized Rosenzweig-Porter model

1 - interesting and relevant problem in focus
2 - developed analytical approach to solve it and its prominent application to other physical quantities of interest

1 - some comparison with the previous literature is missing
2 - misprints

The authors consider a quite well-known Rosenzweig-Porter (RP) random-matrix model using the replica approach to calculate the averaged density of states (DOS) $\rho(\lambda)$ and the level compressibility $\chi(E)$ in the energy interval $[-E,E]$ for large, but finite matrix sizes $N\gg 1$.
This calculations open the possibility to understand the results for the level compressibility, conjectured in [23] and checked numerically in [31], showing a wide crossover region between Wigner-Dyson and Poisson results.

I have several questions and comments to the authors:
1) First of all, the calculation of the mean density of states $\rho(\lambda)$ is quite similar to
(a) the Itsykson-Zuber formula used in [36, 23] to calculate a spectral form factor,
as well as
(b) to the well-known coherent potential approximation which might be exact for the model.

(a) The first one allows to calculate a spectral form factor which has been shown in [23] to have a universal exponential decay after the rescaling of the energies, equivalent to $\delta=1$ in Sec. 3.5 of the current manuscript.
Please comment whether that universal result is related to the one, considered in the current manuscript, and whether one of them can be obtained from the other.

(b) The second approximation is rather well-known and used to calculate the density of states in disordered systems (see, e.g., Appendix A in [60] and the reference [73] therein).
Please comment whether the above coherent potential approximation is exact in the case of the RP model (at least it is known to be exact for the Cauchy distribution of the diagonal disorder, giving Eq. (50) straightforwardly).

2) As a follow up, I suggest the authors to make a clear distinction of the results for $\rho(\lambda)$ in their paper for all $\gamma$ to the results of [36, 37], where it was calcualted for $\gamma=2$ and, if possible, comment on the difference in methods, used for this calculation.
It seems that all the deviations from [36, 37] in the current manuscript are related to $\nu\neq 1$ and $\gamma=1$ (see the next item). Please comment on this explicitly in the manuscript.

3) In all the numerical calculations of finite $\eta$ parameter and the system size $N$, one should understand that the parameters $\nu$ and $\gamma$ are interchangeable.
Indeed, any deviations of $\gamma$ from $1$ used at finite $N$ (e.g., $\gamma = 1.1$ at $N=2000$ used in several figures) is equivalent to the redefinition of parameters $\nu$ and $\gamma$ by $\nu'=\nu N^{-(1-\gamma)/2}$ and $\gamma'=1$.
For the above parameters $N^{\gamma-1} \simeq 2.2$ and the range of changes of $\nu$ from $1$ to $10$ is much bigger.
As a result, all the results (e.g., the deviations of the density of states from a semicircle at $\gamma<1$ and from $p_a(\lambda)$ at $\gamma>1$) for finite $\eta$ and $N$ can be obtained in the thermodynamic limit at $\gamma=1$ and various $\nu$.
Please comment on this in the manuscript.

4) Another interesting issue, known for the RP model from a numerical paper [31], is that unlike a finite $E$-crossover in the level compressibility, a so-called power spectrum shows a rather sharp change of the behavior between the Wigner-Dyson and Poisson limits.
Moreover, the energy scale, separating two limits, which can be extracted quite accurately from the numerical simulations, shows the limit of $\delta=1$, considered in Sec. 3.5 of the current manuscript.
It would be very useful to have analytical predictions for the power spectrum. Please comment, whether the approach, used in the manuscript, can be applied for such a calculation.

5) Formula (13)=(125) for the level compressibility interpolates between Wigner-Dyson $\chi(y\ll 1)\ll 1$ and Poisson $\chi(y\gg 1)\simeq 1$ level statistics, however, the leading order in $y\ll 1$ is not given by $[\delta/E]\ln[E/\delta]$.
Please comment on why it is the case and how to restore this Wigner-Dyson limit for the RP model.

6) Another comment, related to the above formula (13)=(125) is that it explicitly uses $p_a(0)$, therefore it assumes the regular character of the distribution $p_a(a)$ at $a=0$. Please comment on the singular, but normalizable distributions of $a$.

7) Yet another interesting issue about the results for the RP model, related to the distribution $p_a(a)$ is mentioned in https://journals.aps.org/prb/abstract/10.1103/PhysRevB.106.094204:
The standard phase diagram of the RP with the ergodic, fractal, and localized phases at $\gamma<1$, $1<\gamma<2$, and $\gamma>2$ is tunable by the "dimensionality" of the distribution $p_a(a)$.
By "dimensionality" I mean the power $d$ with which the distance between sorted diagonal elements $a_n < a_{n+1}$ grows $a_{n+k} - a_n \propto k^d$.
Indeed, as considered in the above paper, for the Hermitian diagonal disorder case the "dimensionality" of $p_a(a)$ is $d=1$, while for the non-Hermitian case $d=2$.
The latter case of $d=2$ moves the localization transition from $\gamma=2$ to $\gamma=1$.
It might be interesting to use the analytical results of the current manuscript for non-integer values of $d$.

8) In the literature (see, e.g., https://doi.org/10.1016/j.aop.2017.12.009) the replica-symmetric consideration is associated with the ergodic phase, while the replica-symmetry broken state is associated with the multifractality.
As in the mathematic literature (see, e.g., [24]) the fractal phase of the RP model is called ergodic on the fractal, one may expect that the replica-symmetric solution describes the RP fractal phase well.
However, already a multifractal phase should be probably described by a replica-symmetry broken solution. Please comment on this naive logic in the manuscript.

The other comments are minor and related to the presentation of the work, therefore I have put them to the "Requested changes".

0. Please address the physical questions of the report above.

1. References:
- page 3, phrase "multifractal in the whole insulating phase [6-9]": please add also a couple of other references referring to this fact
https://doi.org/10.1103/PhysRevB.91.081103 - see Fig. 5
https://doi.org/10.1103/PhysRevB.96.104201 - see, e.g., Fig. 11
https://doi.org/10.21468/SciPostPhysCore.2.2.006 - see, e.g., Figs. 2-7
https://doi.org/10.1103/PhysRevB.97.214205 - see, e.g., Fig. 3

- page 3, last full paragraph, phrase "relovent methods [35, 36]": it might be useful for a general reader to mention the usage of Itsykson-Zuber formula which allows to calculate the spectral form factor in [36] for $\gamma = 2$ and in [23] for all $\gamma$ values.

- page 6, last paragraph of Sec. 1.1, phrase "and many other random matrix ensembles..." may also include the following examples:
$\beta$-ensemble https://doi.org/10.1103/PhysRevE.105.054121
Aubry-Andre model with $p$-wave superconductivity http://dx.doi.org/10.1103/PhysRevLett.110.176403, http://dx.doi.org/10.1103/PhysRevLett.110.146404 (and many others),
Aubry-Andre model with modulation of hopping http://dx.doi.org/10.1103/PhysRevB.91.014108
Flat-band systems with Aubry-Andre potential https://arxiv.org/abs/2205.02859 and https://arxiv.org/abs/2208.11930

Please also correct misprints, e.g.,
- page 5, just before Eq. (4), phrase "such energy band grows with the system size as $\Delta E \propto N^{1-\gamma}$" - as $\gamma>1$ one should replace "grows" by "decays".
- page 6, first paragraph, phrase "for $\gamma\geq 2$ this implies that the average escape time from site $i$ grows with $N$": as the escape time $\Delta t = \hbar/\Delta E$ it increases with $N$ for all $\gamma>1$. Please correct this.
- page 6, phrase "all the fractal dimensions $D_q$ are degenerate and equal to $D_{\gamma}$ for all $q$": here "for all $q$" should be replaced by "for all $q\geq 1/2$" or by "for all positive integer $q$".

2. The term "deformed Gaussian ensemble" is used in different sense in the literature (see, e.g., Eqs. (4-5) in http://dx.doi.org/10.1088/1751-8113/49/14/145005).
Please clarify in the manuscript, in which sense you use this term.

3. For the density of states with the Wigner distribution of the bare diagonal matrix elements there is no numerics in the manuscript.
Please consider the possibility to add the numerical comparison with the derived analytical results in Sec. 2.4.2.

4. The ansatz in Eq. (82) appears not very naturally. Please explain its origin in the manuscript.

5. There is a discrepancy between Secs. 3.4.1 and 3.5. Indeed, in the very end of Sec. 3.4.1, there is the claim that the position of the maximum "$E_{\max}(\eta)$ grows monotonically (and sublinearly) with $\eta$".
At the same time in Sec. 3.5 it is shown that the universal scaling appears at the linear scaling of $E$ with $\eta$, i.e. at $\delta=1$, while at $\delta<1$ the Poisson behavior (130) is valid.
Please comment on this discrepancy in the manuscript.

1 - The paper is extremely well written, with a fresh pedagogical style and many examples worked out explicitly
2 - The results are timely and novel, and the derivations are elegant
3 - The fact that the replica method can be cleverly employed to keep track of sub-leading large-N terms is very interesting, and provides a handy and powerful tool that can be probably used in many other problems

None

This paper was a true pleasure to read. It definitely meets all the criteria for acceptance, and meets (if not exceeds) the general expectation of clarity, correctness, and broad interest. The topic is timely, and the calculations are performed with competence, and with a pedagogical style that should appeal even to non-experts. I have no hesitation whatsoever in recommending that the paper be published in its current form, and I do not have many intelligent remarks that would improve such a top-class manuscript. My only very minor (and optional) remark is that perhaps the authors could add a comment about the 'unreasonable effectiveness' of eq. (63)-(64) in computing the level compressibility, in spite of the fact that sums of complex logarithms are not necessarily equal to complex logarithms of products - a curious fact that was analysed in detail in the paper "Index of a matrix, complex logarithms, and multidimensional Fresnel integrals [https://iopscience.iop.org/article/10.1088/1751-8121/abccf9] - and also noted on pag. 4 of ref. [72].

None

- The work is very detailed and well-written.
- The Rosenzweig-Porter model is of timely interest.

- The physical significance of the results is not immediately clear.

For a generalization of the Rosenzweig-Porter model, de authors perform an analytical study on the density of states, number of eigenvalues in an interval, and the spectral compressibility. Large-$N$ expressions for these quantities are obtained using an approach based on the replica method.

In the generalization of the Rosenzweig-Porter model under consideration, the elements of the deformatiom matrix ($A$) [Eq. (1)] are sampled from either a Cauchy or a Wigner (semicircle) distribution. For general distributions of these elements, next a scaling function for the level compressibility has been obtained.

The Rosenzweig-Porter model and its generalizations are of timely interest (see the 6th paragraph of the introduction). The interest has however been focused mainly on the fractal / multifractal properties of the eigenstates rather than on the eigenvalues. Driven by physical motivations, most of the generalizations proposed before are next concerned with non-standard distributions of the elements of the matrix $B$ [Eq. (1)] instead of the the elements of the deformation matrix. Because of this, the physical significance of this work is presumably limited.

The manuscript is well-written and discusses all calculations in great detail, making it accessible even for readers not familiar with the subject.

In my opinion this work meets all "general acceptance criteria". In its current form, however none of the "expectations" for SciPost Physics is clearly met. I tend to say that the expectation "opens a new pathway in an existing or a new research direction, with clear potential for multipronged follow-up work" could be fulfilled if the authors could argue that their results are of physical importance. Currently, I would recommend publication in SciPost Physics Core after (very) little revision.

- Page 4: The authors claim that "the GUE case can be analyzed with only minor changes of the calculations that we develop". I doubt whether this is true. The Edwards-Jones formula, an essential ingredient in the caluclations presented in this work, can for example not be trivially extended to complex-valued matrices (see e.g. Ref. [69]). This statement needs justification.

- Page 18: It might be helpful to add a brief comment on what is meant by "quenced" and "annealed". Although it is well-known in the field, to me it feels like a missing point since the rest of the work is essentially self-explanatory.

- Page 27: If I understand correctly, "at low energy" here means "in a small range around zero energy". As "low energy" can also be interpreted as "close to the ground-state energy", it might be worth re-considering the formulation.

- Page 30: In Eqs. (B.3) and (B.4), I think $i \le j$ should read $i < j$.