Multifractality and its role in anomalous transport in the disordered XXZ spin-chain

**Referee**

After studying the report of another referee and authors' reply I see two different approaches. The authors infer error
bars and conclusions based solely on data and on scaling relations for conventional multifractal systems. This is fair
approach given, that there exists no proper theory for MBL systems. At the same time, the referee keeps emphasizing
the severe nature of finite size effects and concludes that multifractality is a signature of the critical region. The
extremely large critical region is indeed supported by the growing body of work, however is not rigorously established
either.

**Our reply**
We thank the referee for the careful reading of our work, the other Referee reports and our replies. We also appreciate the constructive remarks of the Referee, which helped us to improve the presentation of our results.

**Referee**

Hence, I suggest that the authors and referee can meet each other "in between": In the v3 of their paper authors
already implemented a number of changes to acknowledge strong nature of finite size effects. I suggest them to do it
more explicitly, suggesting that a different interpretation is possible. In addition to the parts of the main text that
discuss deviation from the scalings, it would be nice to include such acknowledgment also in:

Introduction, right after sentence "Our analysis thus allows us to locate a region in the extended phase which appears
to be nonergodic." one can explain that it is not possible to rule out strong finite size effects citing refs 57-59.∙
In the discussion instead of writing "While we cannot rule out a slow residual flow to GOE..." authors can say that this
would provide an alternative explanation in lines with strong finite size effects suggested in Ref 57.∙

**Our reply**

We thank the referee for the suggestion and follow it in the new version of the manuscript.

**Referee**

When authors conclude that mutlifractality ≠ anomalous transport they should be careful since (as they write
themselves) mutifractality can be strongly basis dependent.
We have made a number of changes, which makes this point more clear, in the last paragraph of the discussion.

Finally, let me reiterate the basis for my recommendation to accept this work. In my view it does address an
important question using state of the art method. I do not know any other instances of the fractal spectrum studies
for many-body systems. Observation of fractal spectrum symmetry is also an interesting finding that calls for an
explanation. Provided authors stay more open on the interpretation of the numerically exact data, this work provides
a non-trivial addition to the literature.

**Our reply**

We thank the Referee for the recommendation to accept our work and hope that the current version is more open on the interpretations.

**Referee**

My answer to the authors’ reply I would like to keep brief. In essence, I am not satisfied with the reply because it does
not address the main point of my criticism: I do not believe that the multifractal scaling analysis presented in
this work gives exponents with an accuracy as claimed by the authors. Instead, the material presented in the
manuscript is, in my opinion, inconclusive. For instance, the summary makes a claim: “One of the central outcomes of
our study suggests that the previously observed anomalous dynamics is not related to multifractality of many-body
eigenstates”. In my mind, the claim has no clear support from the presented material.

**Our reply**

We thank the referee for the second evaluation of our work. The performed analysis meets the high standards of the Monte-Carlo and high energy physics communities. We have used a state of the art bootstrap (Monte Carlo resampling) analysis for all results shown in the manuscript, tracing back the error through the entire data analysis chain and removing correlations by blocking. We are therefore confident that the non-systematic error bars shown in the manuscript are reliable. As is always the case for multifractal studies the data is more reliable for smaller (in magnitude) moments, and our data is not different in this respect. We have strong evidence of ergodicity for smaller q-s, but the flow to ergodicity for larger q-s is less conclusive. A point, which we clearly discuss in the manuscript.

**Referee**

Furthermore, I consider the manuscript misleading because the dramatic impact of finite-size corrections is not
sufficiently accounted for in the paper’s conclusions. The authors say “To the best of our knowledge, the multifractal
spectrum of the disordered XXZ chain has not been studied before due to severe finite-size behavior […], which
hindered reliable extrapolation […].” What the authors don’t say is why to believe that their system sizes (up to L=24)
are significantly better than earlier ones (L=18,20,22).

**Our reply**

As we say in the text, we are not aware of previous studies of the multifractal spectrum of the XXZ model, therefore we are not sure what the Referee means when he asks us to compare our system sizes to the “earlier ones”. As can be seen from our results, there is a significant curving in the data even for the lower moments, for system size up-to L=16, 18, which is roughly the limit of exact diagonalization.
Of course one cannot exclude other sources of the finite-size effects, but in the extrapolation of multifractal measures (critical exponents tau_q and the spectrum of fractal dimensions f(alpha)) the Hilbert space dimensions plays the role of the extrapolating parameter (the finite size corrections are proportional to 1/ln N, but not 1/ln L). As the Hilbert space dimension spreads in our work from 252 to 2 704 156 (which makes 104 times difference) which is at the maximal range achieved with exact diagonalization methods so far. Going to L=24 from L=22, increases the Hilbert space by a factor of 4 compared to L=22, which is not a negligible increase. One can compare to similar efforts done for the 3D Anderson model for example.

**Referee**

Finally, I believe that the presentation of the status of the field MBL as offered in this manuscript is selective and
biased. Evidence indicating the importance of finite-size effects have been reported in other recent studies, e. g. by
the Princeton group, see below. As far as I can see, this evidence is ignored in the review authored by Luitz and Bar
Lev, Ref. [23], cited in the summary; they are also ignored in the present manuscript.

**Our reply**

While we are not sure why the Referee uses this stage to discuss his/her dissatisfaction with a different manuscript that two of us authored, we have included the discussion of finite-size effects in the XXZ model that the Referee mentions. We however feel that the relevance is tangential, since these studies discuss finite-size effects in the context of dynamical quantities and not multifractality, which are not necessarily related.

**Referee**

Summarizing, in my opinion the manuscript should not be published.

I repeat the observations underlying my recommendation: The authors have a spread in system sizes, L=10-24, i.e. by
only a factor of roughly two. The common belief – and also my own experience – is that usually a spread in system
sizes by (at least) two orders of magnitude is required in order to safely identify a power law and the corresponding
exponent in critical localization behavior. While this statement does not , in principle, exclude the possibility of
exceptions, in practice strong indications exist in the data presented in this (and earlier) work that exclude the
possibility that the disordered XXZ-model is of such an exceptional sort.
(1) Immediate evidence for this claim: Strong finite size effects are observed in the data, see e.g. Fig. 2 at W=2.2 and
2.6. In their data analysis the authors do not account for these effects; subleading corrections to scaling are not
included in the fitting. Therefore, the conclusions may be seriously impaired by the latter.

**Our reply**

It is likely that the experience of the Referee lies mostly with non-interacting systems, where indeed system sizes with a linear dimension of about L=200 were required to reliably perform the multifractal analysis [10.1103/PhysRevLett.105.046403]. This however corresponds to a Hilbert space dimension of L^3 =10^6. We would like to point out that the Hilbert space dimension of the L=24 system is 2,704,156, which is of the same order of magnitude used for the Anderson transition.

**Referee**

(2) Indirect evidence: (a) The present paper reports that typical and average values for exponents do not coincide.
Such a finding is hard to reconcile with conventional wisdom; it is, however, in my experience a typical indicator of
finite-size corrections that have not been accounted for, properly. (b) Similarly, f(\alpha) is reported to be non-
convex, see Fig. 6. Also this behavior is contradicting established knowledge and, usually, emphasizes the importance
of corrections to scaling.

**Our reply**

The common lore behind the concavity of f(alpha) comes from its calculation in terms of the inverse Legendre transform, which is able to reconstruct only the concave hull of f(alpha) from the critical exponents tau_q (see, e.g., Touchette Phys.Rep. 478, 1 (2009) for the review on the inverse Legendre transform). However this view is somehow dated, since already 5 years ago the first example of non-convex spectrum of fractal dimensions f(alpha) was found in Ref. [42] and rigorously proven mathematically in 2018 [P. von Soosten and S. Warzel, Letters in Mathematical Physics 109(4), 905 (2019)]. This example was followed by several other random-matrix [68] and many-body models [Micklitz et al PRL 123, 125701 (2019), Faoro et al. Annals of Physics 409, 167916 (2019)] showing similar behavior of f(alpha).
In this work (as well as in growing body of recent works) the spectrum of fractal dimensions has been calculated directly from the distribution function of the wave function coefficients approximated a histogram of wave function amplitudes with logarithmic bins alpha = -ln |psi|^2 / ln N.

Following from this non-concavity of f(alpha), the effective q where tau_q starts to deviate from its ergodic behavior tau_q = q-1 may correspond to the positive f(alpha) values at the smallest alpha where f(alpha) is still concave as around the maximum alpha = alpha_0.

**Referee**

(3) Circumstantial evidence: The authors agree in their response that the critical point in the model they investigated
is not well known. The value cited in the text, W_c\sim 3.7, has been questioned, e.g. by Devakul and Singh, who give
a much larger value W_c \approx 4.5 as a lower bound (PRL 2015). The issue is significant for the present case
because it demonstrates the importance of corrections to scaling; the asymptotic behavior in this model is just not
under computational control.

I add that subsequent analysis of Khemani, Sheng and Huse (PRL 2017) corroborates this conclusion. In fact, these
authors recommend explicitly to study MBL in quasiperiodic systems rather than in the random systems considered by
the present authors, because “the entanglement structure at the critical fixed points in RG studies [..] indicates that
the asymptotic disorder dominated regime in these random models might only be apparent in samples larger than
~100 spins [..], [..].” The importance of finite size effects has been emphasized again also by other authors and, in
particular, also for the range of disorder values considered in the present manuscript. (e.g. Weiner et al.; Doggen et al.
).

**Our reply**

We hardly see the relevance of the discussion of the critical disorder strength to our point. All the works in the literature agree that the critical disorder is at least W_c = 3.7, while some works mentioned by the Referee and by us in the previous reply claim W_c ~ 4.5 or higher. In the current work we deliberately stayed within the presumably ergodic phase, by choosing the disorder strength W<=3.0, which is far below all the estimates of the transition point. Moreover we suggest that our data can be consistent with multifractal behavior already at W = 2.6 which is at least by 30 - 40 % below the estimated critical disorder. As we mentioned above, we have added the discussion of finite-size effects to the beginning of the Model section.