Emergent fractal phase in energy stratified random models

Dear Editor,

We are grateful to both referees for the high evaluation of our paper and especially to referee 2 for thorough careful reading of the manuscript. His/her critical remarks have allowed us to significantly improve the presentation of our work.

In the revised version of our manuscript, we address all the points mentioned by the referee 2 (as the referee 1 has accepted our manuscript “as is”).

The point-to-point reply to the referee 2 is given below.

Sincerely yours, Anton G. Kutlin and Ivan M. Khaymovich.

Referee 2 report: Strengths 1. Useful technical results in a new family of models, interpolating between well known canonical models. 2. New analytic technique presented. 3. Writing is well paced, and with an explanatory style. Weaknesses 1. Results are not clearly organised/communicated, in particular no intuitive understanding of the results is given.

Reply: We thank the referee for the high evaluation of our work and for highlighting its results and hope that the revised version clearly represents our results, including the intuitive understanding of them.

Referee 2 report: The manuscript is well paced, and adopts a useful pedagogic style. However, despite this, it is unnecessarily difficult to follow, primarily due to (i) the lack of explicit definitions of the quantities of interest, and (ii) the lack of explanation of the physics underlying results.

Reply: In the revised version of the manuscript we have defined the quantity of interest explicitly and added the physical explanation of our results.

Referee 2 report: Major recommended change:

Organisation of results: My main concern with the paper is that the main results can be understood with simple physical intuitive arguments which are not clearly presented. I recommend that such short arguments should be included early in the paper, before delving into technical analysis, as they help guide the reader, and make the results accessible. In particular to readers who may be less interested in the details of the technical machinery which the authors develop.

I am sure the authors are aware of these simple arguments, but in the current version of the manuscript they are not highlighted in an accessible way.

Specifically, I refer to the fact the model has two regimes, which are distinguished by the hierarchy of the bandwidth of T ( = N^0) and the bandwidth of V ( = N^{1 - \gamma/2 - \beta / 2} ). In each of these regimes the fractal dimension can be found according to simple arguments (specifically: Energy stratification, and Fermi's golden rule).

Reply: We agree with the point of the referee about intuitive explanation of our results, however, there are a couple of warnings which are worth to mention: 1) A blind usage of the Fermi’s golden rule for the system (including already the Richardson’s model) may lead to the wrong conclusion of the ergodicity in the entire region gamma<1. This is in some sense similar to the point based on [34, 35], when the divergence of the locator expansion series is associated to the ergodic delocalization of wave functions. The examples of [56-61] explicitly show that for the correlated hopping terms both these simple and physically intuitive approaches might fail.

2) As the follow-up of the previous argument, strictly speaking, for the correlated models one cannot use Fermi's golden rule even in the regime \beta+\gamma>2 of the considered models. Moreover, even the known methods like the cavity equations or Dyson Brownian motion are not able to overcome this problem. This forced us to develop our own method which takes into account all such correlations explicitly.

Of course, result-wise we completely agree with the Referee that the above simple intuitive arguments work for the considered range of models, but it cannot keep us away from possible errors in more involved correlated models.

The list of main changes (numbers are according to the minor changes suggested by the referee):

0. (corresponds to the requested major changes)
We added an explanation of the results to the end of Sec. 2 before their technical derivation.
In addition, we have clarified the description of the model in the corresponding section in the pictorial way by adding a new figure.

1. We defined the fractal dimension D explicitly via the inverse participation ratio (IPR) in Eq. (8) on p 6.
In addition, in Eq. (13) of Sec. 3 we related the above fractal dimension to the local resolvent (Green’s function) which we calculate in the main part of the manuscript.

2., 3., 4. We corrected grammar and typos in the text and formulas.

5. We removed footnote 9 and slightly rewrote the corresponding discussion.
We thank the referee for pointing out that our example was incorrect: the orthogonality definitely matters, however this does not affect our general result.
Besides, in the qualitative explanation of the results we added a footnote 4 clarifying the mechanism of how orthogonality leads to D(\beta)>=\beta.

6. Finally, regarding your comment about using T for a potential and V for a kinetic term: it was not deliberate. We have changed the notation of the diagonal energy term T to H_0 in order to match the notations with the bunch of old works on the Rosenzweig-Porter and [16].