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Multifractality and its role in anomalous transport in the disordered XXZ spin-chain
by David J. Luitz, Ivan M. Khaymovich, Yevgeny Bar Lev
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|Authors (as registered SciPost users):
|Yevgeny Bar Lev · Ivan Khaymovich · David J. Luitz
|Bar Lev, Yevgeny
The disordered XXZ model is a prototype model of the many-body localization transition (MBL). Despite numerous studies of this model, the available numerical evidence of multifractality of its eigenstates is not very conclusive due severe finite size effects. Moreover it is not clear if similarly to the case of single-particle physics, multifractal properties of the many-body eigenstates are related to anomalous transport, which is observed in this model. In this work, using a state-of-the-art, massively parallel, numerically exact method, we study systems of up to 24 spins and show that a large fraction of the delocalized phase flows towards ergodicity in the thermodynamic limit, while a region immediately preceding the MBL transition appears to be multifractal in this limit. We discuss the implication of our finding on the mechanism of subdiffusive transport.
Submission & Refereeing History
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Reports on this Submission
- Cite as: Anonymous, Report on arXiv:1909.06380v3, delivered 2020-02-27, doi: 10.21468/SciPost.Report.1543
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.
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:
$\bullet$ 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.
$\bullet$ 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.
$\bullet$ When authors conclude that mutlifractality $\neq$ anomalous transport they should be careful since (as they write themselves) mutifractality can be strongly basis dependent.
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.
- Cite as: Anonymous, Report on arXiv:1909.06380v3, delivered 2020-02-14, doi: 10.21468/SciPost.Report.1453
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.
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).
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. , cited in the summary; they are also ignored in the present manuscript.
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.
(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.
(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. ).