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Localization and fractality in disordered Russian Doll model
by Vedant Motamarri, Alexander S. Gorsky, Ivan M. Khaymovich
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Submission summary
Authors (as registered SciPost users):  Ivan Khaymovich · Vedant Motamarri 
Submission information  

Preprint Link:  https://arxiv.org/abs/2112.05066v2 (pdf) 
Date submitted:  20220603 06:18 
Submitted by:  Khaymovich, Ivan 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
Motivated by the interplay of Bethe Ansatz integrability and localization in a Richardson model of superconductivity, we consider a timereversal symmetry breaking deformation of this model, known as a Russian Doll Model (RDM) by implementing diagonal onsite disorder. The localization and ergodicitybreaking properties of a singleparticle spectrum are analyzed within a largeenergy renormalization group (RG) over the momentumspace spectrum. Based on the above RG, we derive an effective Hamiltonian of the model, discover a fractal phase of nonergodic delocalized states, with the fractal dimension different from the one of a paradigmatic RosenzweigPorter model, and explain it both in terms of the developed RG equations and matrixinversion trick.
Author comments upon resubmission
Thank you for communicating to us the Referee report on our manuscript entitled "Localization and fractality in disordered Russian Doll model". We would like to resubmit the article for further consideration for SciPost Physics.
We would like to thank the Referee for taking the time to have a careful read of our manuscript and for the report. The Referee communicated a really positive assessment of our manuscript, and she/he mentions that our conclusions "appear sound, and the adapted RG method employed here is novel enough to warrant publication in SciPost."
We provide a detailed reply to the Referee critique and attach it to the Referee report. We include the full reports in bold text, and comment in normal font to all the points indicating also the related changes made in the manuscript. We have also attached the revised manuscript with the performed changes highlighted in red to the reply to the referee report, so that the they are easier to spot.
Sincerely yours,
Vedant Motamarri, Alexander S. Gorsky, and Ivan M. Khaymovich
List of changes
 The references [25, 34, 36, 4047] have been added. The other references have been shifted accordingly.
 We have slightly changed the accents in the introduction and in the abstract.
 We have added a brief discussion of the powerlawlike localization and its relation to the frozen multifractal states and Chalker scaling in the end of Sec. 2.
 We have slightly reformulate this discussion after Eq. (8) in order to emphasize the finitesize crossover at $\sin \theta \sim W N^{1+\gamma/2}$.
 We have added a discussion of manybody sectors of the models after the previous item discussion.
 In addition, we have added the numerical results on the spectrum of fractal dimensions $f(\alpha)$, on the level $r$statistics across the spectrum, and on the wavefunction decay in Figs. 46 as well as the corresponding analytical calculations (3537).
Current status:
Reports on this Submission
Report #1 by Anonymous (Referee 2) on 202281 (Contributed Report)
 Cite as: Anonymous, Report on arXiv:2112.05066v2, delivered 20220801, doi: 10.21468/SciPost.Report.5475
Strengths
The main strength is convincing renormalization group treatment of the model by two different methods, which well agrees with outcome of the exact diagonalization
Weaknesses
Weakneses are mainly stylistic: the English is a bit lame to the extent that the precise meaning of some places becomes vague.
Report
I like the content of this work  a carefully executed renormalization in an alltoall "Russian Doll'' model with diagonal disorder and broken timereversal symmetry. Evidence for accuracy of the RG is overwhelming: two complementary methods give the same effective Hamiltonian which eventually can be treated (at least, for establishing the multifractality properties of eigenfunctions) largely to the same extent as the paradigmatic PorterRosenzweig model.
Unfortunately exposition is frequently vague, and I believe poor usage of English grammar contributes to vaguesness. Though I provide below a few suggestions for improvement, I would further recommend the authors to ask someone with a good command of English grammar to edit the paper throughout.
Requested changes
The research quality is high, but the exposition needs a certain editing before acceptance. I give below a few suggestions for improvement
(1) top of page 7:
hopping between plane waves
>
hopping between states in the plane wave basis
(2) after eq.(24): the above approximation works leading to ...
and the difference between ... and ... at most of the order
>
the above approximation works WELL leading to ... the difference between ... and ... and IS at most of the order
(3) top of page 11:
and inverse this matrix
>
and inverTS this matrix.
(4) with onesided divergence
>
with one sided unbounded growth
(5) where Ep<p >>1 are large and positive Ep<p∗ > 0
>
where Ep<p >>1 are large and positive.
(5) page 13:
"with a bandwidth b"  note that b is not defined in eq. 34 (I suppose it W/\Gamma?)
footnote 4 " there are some investigations which might have multifractal wave functions''  it sounds nonsensical ...
please try to reformulate.
(6) eq.(40): note that such a symmetry was originally discovered in:
AD Mirlin, YV Fyodorov, A Mildenberger, F Evers
Exact relations between multifractal exponents at the Anderson transition
Physical review letters 97 (4), 046803 (2006)
Please cite the original reference, not only the review [37].
(7) In conclusions: " we ... confirm the subleading character of the
approximations'' > we confirm that approximations we used are valid to the leading order, and all subsequent corrections are subleading (and write explicitly in which parameter)
Author: Ivan Khaymovich on 20220829 [id 2770]
(in reply to Report 1 on 20220801)** Report 1 of the Referee **
** Report **
** I like the content of this work  a carefully executed renormalization in an alltoall "Russian Doll'' model with diagonal disorder and broken timereversal symmetry. Evidence for accuracy of the RG is overwhelming: two complementary methods give the same effective Hamiltonian which eventually can be treated (at least, for establishing the multifractality properties of eigenfunctions) largely to the same extent as the paradigmatic PorterRosenzweig model. **
** Unfortunately exposition is frequently vague, and I believe poor usage of English grammar contributes to vagueness. Though I provide below a few suggestions for improvement, I would further recommend the authors to ask someone with a good command of English grammar to edit the paper throughout. **
We thank the referee for the careful reading of the manuscript. \textit{As per the referee's recommendation, we have carried out a thorough editing of the paper to make the exposition clearer. Below, we list down corrections made as per the referee's specific comments:}
** Requested changes **
** The research quality is high, but the exposition needs a certain editing before acceptance. I give below a few suggestions for improvement **
** (1) top of page 7:
hopping between plane waves
>
hopping between states in the plane wave basis **
We have replaced the phrase by "Indeed, the disorder term ... in the momentumspace basis... plays a role of the scattering between plane waves (or hopping),..."
** (2) after eq.(24): the above approximation works leading to ...
and the difference between ... and ... at most of the order
>
the above approximation works WELL leading to ... the difference between ... and ... and IS at most of the order **
We have followed the recommendation of the referee.
** (3) top of page 11:
and inverse this matrix
>
and inverTS this matrix. **
We have followed the recommendation of the referee.
** (4) with onesided divergence
>
with one sided unbounded growth **
In the correspondent place and in other ones we have replaced spectral divergence by the spectral unbounded growth.
** (5) where $E_{p<p^*} \gg 1$ are large and positive $E_{p<p^*} > 0$
>
where $E_{p<p^*} \gg 1$ are large and positive. **
We have followed the recommendation of the referee.
** (5) page 13:
"with a bandwidth b"  note that b is not defined in eq. 34 (I suppose it $W/\Gamma$?) **
We have followed the recommendation of the referee and added the definition of the effective parameter $b$.
** footnote 4 " there are some investigations which might have multifractal wave functions''  it sounds nonsensical ...
please try to reformulate. **
We have rephrased the footnote 4 according to the referee's comment.
** (6) eq.(40): note that such a symmetry was originally discovered in:
AD Mirlin, YV Fyodorov, A Mildenberger, F Evers
Exact relations between multifractal exponents at the Anderson transition
Physical review letters 97 (4), 046803 (2006)
Please cite the original reference, not only the review [37]. **
We have added the reference to the above paper and mentioned that the corresponding symmetry was originally discovered there.
** (7) In conclusions: " we ... confirm the subleading character of the
approximations'' > we confirm that approximations we used are valid to the leading order, and all subsequent corrections are subleading (and write explicitly in which parameter) **
We have reformulated the above phrase in the conclusions accordingly.