Local Density of States Correlations in the Lévy-Rosenzweig-Porter random matrix ensemble

Response to the 2nd Referee

We would like to thank the Referee for the report. We have implemented all the requested changes and added comments around Eq. (9) to address request 4. The misprints mentioned in points (1) and (2) of the report have been corrected. Additionally, we have included the short-time asymptotics on Fig. 3, left. We believe that the analytical arguments from Ref. [15] are only valid in the short-time regime (\Gamma t << 1). However, this short-time behavior has a slope similar to the line approximating the intermediate-time behavior, which is clearly visible in the updated plot.

Response to the 1st Referee

We are thankful to the Referee for the very detailed report. Below, we provide the list of changes we made following the Referee's suggestions. 1. The discussion has been expanded around Eqs. (2) and (3), to make it more clear that we are concerned with the case of h_0=const and N->inf. In this case, the model is always in the delocalized phase and most importantly, the physical quantities like DOS fluctuations or IPR of the eigenstates behave regularly in the standard N->inf delta->0 limit. 1.1 \eta was replaced to \delta 2. We have added a comment at the end of Appendix С regarding the influence of correlations that arise for sufficiently large values of the real part of the self-energy. Our analysis focused on the bulk of the spectrum; therefore, this observation does not affect our conclusions. 3. The missing values for \mu and \delta have been added, see Fig. 1.
We believe that the statement "the system stays well enough inside the fractal phase" is a result of confusion between different formulations of the problem (in particular, the scaling of h_0 with N). In our choice of scaling (constant h_0 at N->inf) the system remains in the delocalized phase. The question of whether the finite-size behavior of the system at particular choice of parameters (i. e . whether its properties at finite N resemble those of delocalized or fractal phase) can not be decided based on the scaling arguments; our numerics illustrate that at the values of N we choose, the system remains well inside the delocalized regime. We extend the discussion of the scaling in the section “Model” and added a reminder to the introductory part of the section “Numerics”. 4. We believe this comment is closely related to the previous one. As we have mentioned above, the data shown on Fig. 1, illustrates the convergence of the numerical results to the semi-analytical solution in the limit of p=N \to \infty. The perfect agreement between theory and numerics in Fig. 2 and Fig. 3 demonstrates that finite-size effects are negligible and, therefore, not considered. 5. 5.1 Question: «What is the precise mathematical meaning of the “semi-classical approximation”?» Answer: The “semi-classical approximation” is not exactly a “mathematical” term, but rather a standard way to refer to a particular approximation of neglecting the self-energy in computing the integrals in the like of Eq. (8). We expanded a comment after Eq. (8) to make it more clear. Question: «In other words, what does «≈» used after Eq. (8) mean specifically?» Answer: There is no anymore “≈” after Eq. (8). Question: «How extensive is the range of \xi around \epsilon that contributes to the integral?» Answer: The self-energy in Eq. (8) can be neglected for \epsilon - \xi - R \sim \gamma << R \sim h_0^2/W << \xi, \epsilon \sim W. 5.2 We adjusted our usage of the term “self-averaging” so that when it is used it indeed implies that fluctuations of the “self-averaging” quantity are small. In some cases, see for example discussion after Eq. (18), we don’t need the “self-averaging” property but instead rely only on the law of large numbers, which does not necessarily require finite variance of a considered random variables (see, for example, [https://terrytao.wordpress.com/2008/06/18/the-strong-law-of-large-numbers/]), even in its strong form. 5.3 The assumption is N \gg 1 (explicitly added) , analogous to the standard form of the central limit theorem. Our results, like the Eqs. (20) and (22) are valid for the typical values of the variables (in the bulk of the distribution), not in the tails. 5.4 We expanded the comment on how exactly semiclassical approximation helps in deriving Eq. (27) in the discussion above the equation itself. 5.5 We defined the meaning of the term “basis state” in the discussion above Eq. (36). 5.6 We have made it more clear in the new version of the text that the term “self-averaging” refers not to \gamma but to the return probability R (see the text above Eq. (38)). The return probability R is self-averaging because it is a sum of a large number of random contributions, limited to the range [0, 1]. 5.7 Since there are no parameters in this approximation, we refer to it as “not fully controllable”. For the same reason, it would be hard to provide applicability conditions for it and we use it as a hint only (in particular, it fails to capture the coefficient in the time-dependence correctly). 6. We explain the meaning of the quantity Gamma_0^(sc) in the discussion after Eq. (45). Indeed, this notation was not properly introduced there; this is fixed in the new version (see the end of the paragraph after Eq. (45)).

1. The discussion of scaling has been expanded in the "Model" and "Numerics" sections.
2. The explanation of the transition between Eq. 18 and Eq. 19 has been updated.
3. Applicability conditions have been added (above Eq. 27 and in the introduction of Appendix A.2).
4. Figure 3 (left) has been updated.
5. The introductory part of Appendix C has been revised, with an additional comment added at the end.
6. The clarity and readability of the text have been improved.
7. Misprints have been corrected.