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Universality in Anderson localization on random graphs with varying connectivity

by Piotr Sierant, Maciej Lewenstein, Antonello Scardicchio

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Submission summary

Authors (as registered SciPost users): Piotr Sierant
Submission information
Preprint Link:  (pdf)
Date accepted: 2023-05-24
Date submitted: 2023-03-16 11:01
Submitted by: Sierant, Piotr
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
Approaches: Theoretical, Computational


We perform a thorough and complete analysis of the Anderson localization transition on several models of random graphs with regular and random connectivity. The unprecedented precision and abundance of our exact diagonalization data (both spectra and eigenstates), together with new finite size scaling and statistical analysis of the graph ensembles, unveils a universal behavior which is described by two simple, integer, scaling exponents. A by-product of such analysis is a reconciliation of the tension between the results of perturbation theory coming from strong disorder and earlier numerical works, which seemed to suggest that there should be a non-ergodic region above a given value of disorder $W_{E}$ which is strictly less than the Anderson localization critical disorder $W_C$, and that of other works which suggest that there is no such region. We find that, although no separate $W_{E}$ exists from $W_C$, the length scale at which fully developed ergodicity is found diverges like $|W-W_C|^{-1}$, while the critical length over which delocalization develops is $\sim |W-W_C|^{-1/2}$. The separation of these two scales at the critical point allows for a true non-ergodic, delocalized region. In addition, by looking at eigenstates and studying leading and sub-leading terms in system size-dependence of participation entropies, we show that the former contain information about the non-ergodicity volume which becomes non-trivial already deep in the delocalized regime. We also discuss the quantitative similarities between the Anderson transition on random graphs and many-body localization transition.

List of changes

In the revised manuscript we have added clarifications in Sec. 4.2 and 4.3 and an Appendix B which contains further details on the finite size scaling procedures.

Published as SciPost Phys. 15, 045 (2023)

Reports on this Submission

Anonymous Report 2 on 2023-4-28 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2205.14614v3, delivered 2023-04-28, doi: 10.21468/SciPost.Report.7113


The work is detailed and well-presented, with state-of-the-art numerical calculations.


1. nu=1/2 possibility, which the authors now claim is entirely possible, must be highlighted in the main text. By this, I imply that the authors relegate that discussion to the appendix, which is unfair. It would be essential to bring it back into the main text.

2. I am afraid I still have to disagree with the author's insistence that this is an MBL-like transition. I reiterate that massive correlations and dense connectivity make these models different from an MBL problem. It is easy to acknowledge that.


In general, the other possibility of nu~1 is intriguing, given that there was almost a kind of agreement with nu~1/2 from previous numerical studies and some analytical calculations. Therefore, I support the publication of the current manuscript.

I also agree with the previous referee's comment about the abstract. The words like "complete analysis" and "unprecedented precision and abundance" are probably overstated. Numerical studies always leave room for another conclusion. I think having an abstract with a small amount of skepticism would serve better for the community of Anderson transitions with a "finite connectivity" tree-like structure.

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

Report 1 by Gabriel Lemarié on 2023-4-21 (Invited Report)

  • Cite as: Gabriel Lemarié, Report on arXiv:2205.14614v3, delivered 2023-04-21, doi: 10.21468/SciPost.Report.7084


Strengths are the same as in my previous report.


Weaknesses are listed in my new report.


Before providing my opinion on the authors' response to my previous report, I would like to highlight several points that I believe are essential to consider. First, it is important to note that my judgment may not be entirely impartial as I am actively involved in the ongoing debate regarding the critical properties of the Anderson transition. Additionally, I believe that the article has merit beyond the discussion of critical properties and, therefore, should be published regardless of the outcome of this dispute. Lastly, given the scientific dispute regarding the critical properties, it is crucial that all arguments can be freely defended, providing another reason to publish the manuscript.

Following this preamble, I would like to express my appreciation for the authors' response to my report, which I found to be interesting and supported by new analyses. However, I must state that I do not find the modification of the manuscript satisfactory from my perspective. The authors demonstrate in their answer that they are unable to distinguish which of the two scenarios, critical exponent 1 or critical exponent 1/2, is more compatible with their observable <r>. Therefore, there is considerable uncertainty between $\nu=1$ and $\nu=1/2$, which is not well reflected in the manuscript. I believe that the authors should acknowledge this uncertainty and explicitly state that their data are compatible with an exponent of 1/2, as predicted by Fyodorov-Tikhonov-Mirlin and the finite-size scaling analyses of Tikhonov-Mirlin, Biroli-Tarzia, and Garcia-Mata et al. However, the authors should also mention that another analysis is possible, which provides $\nu=1$, which is an interesting conclusion as it offers a nice explanation of recent observations about non-ergodic delocalization. Unfortunately, the authors relegate this critical discussion to an appendix, and it does not appear clearly in the abstract, introduction, or conclusion. I, therefore, suggest that the authors present their indecision more clearly in their manuscript, taking the necessary precautions to avoid any potential bias.

I would like to make a few comments about the new version of the manuscript and reply of the authors by order of importance:

- I have first a comment about the finite-size scaling approach proposed by the authors. In general, we introduce irrelevant corrections as a way to describe data at too small system sizes. Moreover, it should be stressed that irrelevant corrections are usually terribly difficult to characterize: one needs extremely precise data with a very large variation of system size.

In the manuscript, the authors propose that the critical behavior is $<r>=<r>_P$ and that at $W_c$ the variations of $<r>-<r>_P$ on L are irrelevant corrections. However, the behavior they assume for the irrelevant corrections, $<r> \sim \overline{L}^{-2}$, does not work for small system sizes. So the irrelevant corrections they introduce do not allow them to describe the critical properties at small system sizes. In their finite-size scaling analysis, they crucially exclude small system sizes on their fits. 

On the contrary, the other scaling hypothesis proposed in Garcia-Mata et al. [81, 131] does not assume irrelevant corrections, and we consider $<r>-<r>_P\sim L^{-\omega}$ with $\omega$ between 1.5 and 2 as the critical behavior. Doing that, we are able to describe data for all system sizes (see my previous report and the appendix B of the new manuscript).

When the authors say `` data collapses will a posteriori confirm our assumptions'', they should clearly say that it is the case only for the largest system sizes. The collapse shows systematic and important deviations with Eq. (10) close to the transition at small system sizes.Moreover, since the systematic deviations originate from their assumption that $ <r>-<r>_P\sim \overline{L}^(-2)$ at criticality, I think it would be interesting to adopt the same finite-size scaling analysis as in Fig. 8, but with $<r>-<r>(W_c)$ as a function of $(W-W_c)/W_c * L^{1/\nu}$ with $<r>(W_c)$ the critical data (fully numerical, no assumption about it) and $\nu$ to be fitted. What is the value of $\nu$ one would obtain that way? Is the collapse of the data better when one considers all system sizes?
- Below Eq.~(18), when the authors comment on the work [81], I think it would be fair to say that \textit{all} the data of the authors are compatible with a critical exponent $\nu=1/2$ if we take into account nonlinear corrections to the scaling parameter $\xi$ (see my previous report).

Imposing $\xi= A |W-W_c|^{-\nu}$, A a constant, with a scaling function $F(L/\xi)$, the authors find with the scaling approach of [81] $\nu=0.64-0.67$ and deviations to scaling quite close to the transition. But using nonlinear corrections, for example $\xi = A  |W-W_c|^{-\nu} + B |W-W_c|^{-2\nu}$, one finds that $\nu=0.5$ is compatible with the data and that one can describe the data for all system sizes and quite far from the transition point.

Now why nonlinear corrections to the scaling parameter are important? Because the system sizes considered here are small. The behavior  $\xi = A |W-W_c|^{-\nu}$ is valid only in the close vicinity of the transition. If you have huge system sizes, you can have $L\gg \xi$ with however W close to $W_c$. However, if you have only small system sizes, you need to go far from $W_c$ to have $L>>\xi$ and far from $W_c$ there are nonlinear corrections.

-Appendix B: ``$\nu=1$ is the exponent of the transition coming from the localized region which is undoubtedly correct''. I disagree. In the papers [81] and [131], we have shown that there are two critical exponents in the localized regime, $\nu=1$ and $\nu_\perp=1/2$, $\nu=1$ which controls averaged observables and $\nu_\perp=1/2$ which controls typical observables. Importantly, average or typical does not relate to the effects of rare disorder configurations but rare branches (in a single disorder configuration) which are known to play an important role in the related problem of directed polymers.The $\nu=1$ critical exponent found in ``Bethe lattice works'' originates from the fact that all branches are considered, hence averaged quantities are considered (even if one takes the average of the logarithm of the Imaginary part of the Green's function in the pool method).

The Bethe lattice works have not described the quantity <r>. We claim <r> is a typical quantity (associated with a critical exponent 1/2). The question raised by the value of the critical exponent in the localized phase is therefore related to the question of <r> being an average or typical quantity. The authors have beautiful data in the localized regime; they find as in Garcia-Mata et al. an exponential decay as a function of $L$ towards $<r>_P$, and they propose a scaling assumption in the localized regime which is identical to our scaling assumption. My previous report has shown that their data are compatible with $\nu=1/2$ also in the localized regime. So why do they still claim that $\nu=1$ for <r> in the localized regime?

- Below Eq. (18): ``Their suggested corrections are necessary to recover the behavior for $W_T\sim 1/L$''. I disagree. What I said in the previous report, as shown in Garcia-Mata et al, is that the delocalized regime is characterized by a correlation volume $\Lambda$, which (very) close to the transition should vary exponentially as $\Lambda \sim c_1 \exp[c_2 (W_c-W)^{-1/2}]$ (as in the theory of Fyodorov-Tikhonov-Mirlin) where $c_1$ and $c_2$ are constants. In particular, the crossover of <r> to the ergodic value $<r>_{GOE}$ is controlled not by a linear scaling, ratio of $L/\xi$, but by a volumic scaling, ratio of $N/\Lambda$ where N is the number of sites in the system. The two scalings are distinct in infinite dimensionality. Now the form of $\Lambda$, which again is true only very close to the transition (see in particular Tikhonov-Mirlin, 2019), suggests the form Eq.~(38), not $W_T\sim 1/L$. As I said in my previous report, this should not be taken too seriously as, for such small system sizes, reaching the ergodic regime means being very far from $W_c$ where we have nonlinear corrections, etc... But the fact is that the data of the authors are not incompatible with Eq.~(38) (I think this is a pure coincidence).

-Below Eq. (18): ``the lengthscale $\xi_2 \sim |W-W_c|^{-1}$ determines the ergodic behavior''.And Section 4.3: "This behavior... is the best one compatible with... $W_c$." I disagree. The authors show in their appendix that the ``lengthscale'' (I claim it is a volume) which controls the crossover to the ergodic regime can be equally well fitted by Eq.~(38) which does not seem to indicate a critical exponent 1. Once again, the authors should at least say that different behaviors are compatible with their data and that they propose $\xi_2 \sim |W-W_c|^{-1}$.

I believe one cannot draw a firm conclusion about the critical exponent by looking at data in the asymptotic ergodic regime because for the small system sizes considered, they are far from $W_c$ and may show important nonlinear corrections.

-Below Eq. (18): ``We feel that our assumption, which does not separate the behavior at the critical point from thet in the delocalized region is preferable''.  And Section 4.3: I also disagree with the phrase "the only way these two behaviors...". I disagree. The data show systematic deviations with respect to the scaling assumption Eq. (10) far from $W_c$. To be able to show that the behavior  $\xi_2 \sim |W-W_c|^{-1}$ comes from the scaling behavior at criticality, one would need to show that the data are compatible with the scaling assumption (without any nonlinear correction) up to the ergodic regime, which is not the case as recognized by the authors. 

-Abstract: I believe that the phrases "We perform a... complete analysis of the Anderson ..." and "The unprecedented precision and abundance" are exaggerated. While the authors present interesting results, the study is not entirely comprehensive since they only consider the critical properties of one observable, <r>, and only in the delocalized regime. Previous studies have presented similar precision and abundance.

Requested changes

See my report

  • validity: ok
  • significance: good
  • originality: good
  • clarity: ok
  • formatting: excellent
  • grammar: excellent

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