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Coherent forward scattering as a robust probe of multifractality in critical disordered media

by Maxime Martinez, Gabriel Lemarié, Bertrand Georgeot, Christian Miniatura, Olivier Giraud

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Olivier Giraud
Submission information
Preprint Link: scipost_202210_00061v1  (pdf)
Date accepted: 2022-12-22
Date submitted: 2022-10-11 16:54
Submitted by: Giraud, Olivier
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Quantum Physics
Approach: Theoretical


We study coherent forward scattering (CFS) in critical disordered systems, whose eigenstates are multifractals. We give general and simple arguments that make it possible to fully characterize the dynamics of the shape and height of the CFS peak. We show that the dynamics is governed by multifractal dimensions $D_1$ and $D_2$, which suggests that CFS could be used as an experimental probe for quantum multifractality. Our predictions are universal and numerically verified in three paradigmatic models of quantum multifractality: Power-law Random Banded Matrices (PRBM), the Ruijsenaars-Schneider ensembles (RS), and the three-dimensional kicked-rotor (3DKR). In the strong multifractal regime, we show analytically that these universal predictions exactly coincide with results from standard perturbation theory applied to the PRBM and RS models.

Published as SciPost Phys. 14, 057 (2023)

Reports on this Submission

Anonymous Report 2 on 2022-11-20 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202210_00061v1, delivered 2022-11-20, doi: 10.21468/SciPost.Report.6163


1-interesting details to previous publication
2-new simulation of system, with perspective for experimental realization
3-well written


1-main results earlier published


In their manuscript the authors extend previous studies of the coherent forward scattering peak in disordered single particle systems from the Anderson localized regime to the critical state at an Anderson localization transition. They identify two distinct dynamical regimes, applicable in the (non-commuting) limits of large systems sizes and long times, and give general arguments relating peak-height and -shape to multifractal properties of the critical system. In the infinite size system the forward peak saturates to the level compressibility (as previously conjectured in [44] and also demonstrated in [46]), while in the opposite limit (N finite, t \to \infty) the peak height is modified from random matrix prediction by a finite size correction that depends on the multifractal dimension D_2 of critical wave functions (as also previously reported in [46]). They verify predictions against three models for critical disordered systems, finding good agreement.

I think the manuscript reports on interesting work. It is well written, provides interesting details to their previous Letter, Ref. [46], and extends numerical studies in the latter to two additional models for the Anderson localization transition. Specifically, the detailed numerical analysis of the 3d kicked rotor seems interesting for possible future experiments. I, therefore, support publication of the manuscript in its present form. A minor remark is that I would appreciate if the authors could comment on the statistics of the forward peak? I.e. how do fluctuations of the peak height in the critical state compare to those in the Anderson localized regime?

  • validity: top
  • significance: high
  • originality: high
  • clarity: high
  • formatting: excellent
  • grammar: excellent

Report 1 by Vladimir Kravtsov on 2022-11-7 (Invited Report)

  • Cite as: Vladimir Kravtsov, Report on arXiv:scipost_202210_00061v1, delivered 2022-11-07, doi: 10.21468/SciPost.Report.6081


1- Very well written paper
2-Detailed derivation of main results
3-Comparison of numerics on three different models, both Hamiltonian (PLBRM) and Floquet (kick-rotor models), confirms the main results


1- significant overlap with the earlier short paper of the same authors Ref.[46]. For an expert the derivations of results in this paper is sufficient to understand their validity and significance. The present manuscript does not add much to understanding of the phenomenon.


This is a very comprehensive and well written paper, the extension of a short paper of the same authors Ref.[46]. Besides a detailed derivation of all the main results (presented partially in Ref.[46]) this paper contains not only the dynamics of the peak in CFS but also a new results about the form of the wings of the peak and, especially, much more numerics on three different models that confirms the analytical results.

In view of a great current interest to non-ergodicity of extended states in quantum disordered systems with and without interaction, this study (as well as Ref.[46]) is a significant progress in the field with a potential to observation of the effects in real experiments.

I do not have doubts about validity of the results and their significance. The only reservation I have is that the most interesting and fundamental findings are already communicated by authors in Ref.[46].

On the other hand, many of the important papers on the subject of non-ergodicity of extended states in disordered quantum systems are published in SciPost Physics. So, this paper may be a good addition to already existent collection in SciPost Physics. I think it will have an impact just because it is easy to read, also for non-specialists.

In view of this I recommend to publish the paper in its present form.

  • validity: top
  • significance: high
  • originality: good
  • clarity: top
  • formatting: excellent
  • grammar: excellent

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