SciPost Submission Page
Density of states correlations in Lévy Rosenzweig-Porter model via supersymmetry approach
by Elizaveta Safonova, Aleksey Lunkin, Mikhail Feigel'man
Submission summary
| Authors (as registered SciPost users): | Mikhail V. Feigel'man · Aleksey V. Lunkin · Elizaveta Safonova |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202509_00043v1 (pdf) |
| Date accepted: | Dec. 4, 2025 |
| Date submitted: | Sept. 25, 2025, 5:55 p.m. |
| Submitted by: | Elizaveta Safonova |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
|
| Approach: | Theoretical |
Abstract
We studied global density-of-states correlation function $R(\omega)$ for L\'evy-Rosenzweig-Porter random matrix ensemble in the non-ergodic extended phase. Using an extension of Efetov's supersymmetry approach we calculated $R(\omega)$ exactly in all relevant ranges of $\omega$. At relatively low $\omega \leq \Gamma$\, (with $\Gamma \gg \Delta$ being the effective miniband width) we found GUE-type oscillations with period of level spacing $\Delta$, decaying exponentially at the Thouless energy scale $E_{Th} = \sqrt{\Delta \Gamma/2\pi}$. At high energies $\omega \gg E_{Th}$ our results coincide with those obtained via cavity equation approach. Inverse of the effective miniband width, $1/\Gamma$, is shown to be given by the average of the local decay times over L\'evy distribution.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
Dear Editor,
please find below our detailed reply to all comments of the referees. We implemented all corrections requested by Referees #1 and #3, and also most part of corrections requested by the Referee #2. Hopefully, it is sufficient to accept the manuscript for publication.
Sincerely yours, E. Safonova, A. Lunkin and M. Feigel’man
Report 1 (text of the referee is in italic)
Remark: a seemingly important technical trick is presented in eq.(71). It is written that such a formula is justified for "smooth distributions". It would be helpful to explain more explicitly what means "smooth" in this context: what is the scale which ensures validity of this approximation?
We include explanation of this point after Eq. (72)
Report 2 (text of the referee is in italic) .
The statement “almost no exact theoretical results are available” is too strong; there exist some mathematical results that should be cited.
We cannot follow the above advice due to the absence of specific information from the Referee regarding “mathematical results”
The discussion of “correlations” is unclear. Several works (Mirlin et al., Roy et al.) on the role of correlations should be cited.
The term “correlations” is used in the manuscript in several different meanings; we are not sure what kind of correlations the Referee had in mind regarding his note above. Thus it is not clear to us what exactly we need to cite.
In addition, it is not obvious why this aspect is emphasized here since correlations of disorder are not included in the present model.
We discuss correlations between matrix elements of the Hamiltonian of many-body quantum system in due course of general introduction to the field of our research. Such introduction is necessary, to our opinion, in spite of the absence of such correlations in the specific model we study in the paper.
For example, the sentence “however, to study level correlations at not-so-large energies a more elaborated technique is needed” should be clarified: why does the cavity method fail in this regime?
Here quite different kind of correlations is discussed. Answer to the above question is provided in the beginning of page 3 in the updated text.
Some basic quantities, such as the Inverse Participation Ratio (IPR), are not defined.
Definition of IPR is included, see beginning of p.4 in the updated text.
In Sec. 2.2, references should be added to works explaining in more detail the supersymmetric techniques employed.
Relevant references are added.
The notation of supervectors overlaps with that of eigenstates, which makes the presentation confusing.
Eigen-vector’s notation ψi is replaced by notation Ψi
In Sec. 3, the authors should explain why the functional integral approach is needed here, and why a standard Hubbard–Stratonovich decoupling cannot be applied.
Explanation is added to the 1st paragraph in Section 3.
The derivations are presented in a very technical way. While the appendix gives details, the main text should highlight more clearly which aspects are non-trivial, and what physical insights are gained only thanks to this method.
Several additional explanations are added in various locations
The predictions should be more deeply discussed: what do they imply for return probability,
multifractality (is there a signature of D2?), and comparisons with Gaussian RP, log- normal RP or cavity methods
Discussion is added in the large paragraph in the end of page 14; it covers at least partially the above request of the referee. However, we cannot say anything about log-normal RP model (apart of providing citation to the relevant paper by Kravtsov and Khaymovich, 2021). We also note that “comparison to cavity method” was already made in the beginning of page 3.
Possible connections to Anderson or MBL transitions should be commented on.
There are no “possible connections” apart from those we already mentioned in the Introduction.
Figures are rare, hardly legible, and their nature is unclear (are these numerical simulations or analytic curves?). They should be made more reproducible, better captioned, and more extensively interpreted.
Number of figures is exactly the one we need to demonstrate our results. Additional explanations on the origin of curves shown in the plots are added in captions to all the figures. In addition, the size of Fig.3 is increased.
Report 3 (text of the referee is in italic)
1- Clarification of Eq. (43) similarity to Gaussian Hermitian RP ensemble: Provide additional commentary on why the two-point function behavior on intermediate energy scales (larger than mean level spacing, smaller than Thouless energy) coincides with the Gaussian Hermitian RP ensemble result, and possibly offer a physical explanation for this apparent convergence despite the differences in microscopic spectral structure between the Lévy and Gaussian RP models
Explanation is added in the new paragraph that is now the 2nd one in the Sec.5
2- Consider adding more explanatory text or intuitive descriptions for the most technical derivations; Ensure that physical motivations are clearly stated before diving into technical calculations;
Explanations are added to: 1) the beginning of Sec.3, 2) paragraph after Eq.(18), 3) paragraph after Eq.(25) 4) beginning of Sec.4.1, 5) paragraph above Eq. (36)
3- Conduct thorough proofreading to eliminate typos, misprints, formatting, and typographical errors throughout the text
Proofreading is conducted
List of changes
Current status:
Editorial decision:
For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)
Reports on this Submission
Report #3 by Anonymous (Referee 2) on 2025-10-25 (Invited Report)
The referee discloses that the following generative AI tools have been used in the preparation of this report:
Chatgpt used to correct English formulation
Strengths
-
One of the few analytical descriptions of the dynamical properties of non-ergodic delocalization, closely related to many-body localization—a still-debated phenomenon.
-
A non-trivial generalization of the supersymmetry approach to a random-matrix model with a broad, fat-tailed distribution.
Weaknesses
- Very technical presentation — dense and challenging to read, but offering profound and insightful content.
Report
Regarding the references I suggested adding in the Introduction, these include works by Imbrie and De Roeck, Huveneers et al. on mathematically rigorous results concerning MBL, several papers by Roy and Logan, and the recent work of Mirlin et al. [Phys. Rev. B 109, 214203 (2024)] on the effects of disorder correlations on the Anderson transition in random graphs. I agree with the authors that disorder correlations constitute an important aspect of MBL viewed from the perspective of Anderson localization in Fock space; however, since this point is not actually addressed in their paper, the discussion remains somewhat misleading.
Concerning the limitations of the cavity equation in describing the regimes discussed by the authors, I find the one-sentence explanation provided unsatisfactory. The authors seem to suggest that the cavity-method approach cannot capture finite-size effects (at least for dynamical properties). Yet, the cavity method does predict multifractal properties, which are inherently finite-size effects. I therefore encourage the authors to elaborate further on the limitations of this well-known and powerful approach, especially since they have employed both methods and are well positioned to discuss this issue in depth.
Finally, I still believe it would be valuable for the authors to list the main dynamical consequences that follow from their study and to comment on whether these results could help clarify certain aspects of the MBL problem.
These comments are intended as suggestions. The paper is already very interesting and, in my view, fully merits publication.
Requested changes
See report.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Strengths
1- The authors provide a detailed analytical calculation of the two-point density–density correlation function, a central and technically demanding quantity;
2- The topic is timely and relevant, as it connects with ongoing interest in fractal and nonergodic phases in disordered and complex systems;
Weaknesses
1- The analysis is technical and dense, which may limit accessibility to a broader physics audience not already familiar with the supersymmetric approach.
Report
I am satisfied with the authors’ responses and with the modifications they have made. The current version of the manuscript represents a clear improvement over the original submission.
I therefore recommend the paper for publication in SciPost Physics in its present form.
Recommendation
Publish (meets expectations and criteria for this Journal)
Strengths
Weaknesses
Report
Requested changes
None
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)