SciPost Submission Page
Signatures of a critical point in the many-body localization transition
by Ángel L. Corps, Rafael A. Molina, and Armando Relaño
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Ángel L. Corps · Rafael Molina |
| Submission information | |
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| Preprint Link: | scipost_202010_00019v2 (pdf) |
| Date accepted: | 2021-05-06 |
| Date submitted: | 2021-02-15 09:16 |
| Submitted by: | L. Corps, Ángel |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
Disordered interacting spin chains that undergo a many-body localization transition are characterized by two limiting behaviors where the dynamics are chaotic and integrable. However, the transition region between them is not fully understood yet. We propose here a possible finite-size precursor of a critical point that shows a typical finite-size scaling and distinguishes between two different dynamical phases. The kurtosis excess of the diagonal fluctuations of the full one-dimensional momentum distribution from its microcanonical average is maximum at this singular point in the paradigmatic disordered $J_1$-$J_2$ model. For system sizes accessible to exact diagonalization, both the position and the size of this maximum scale linearly with the system size. Furthermore, we show that this singular point is found at the same disorder strength at which the Thouless and the Heisenberg energies coincide. Below this point, the spectral statistics follow the universal random matrix behavior up to the Thouless energy. Above it, no traces of chaotic behavior remain, and the spectral statistics are well described by a generalized semi-Poissonian model, eventually leading to the integrable Poissonian behavior. We provide, thus, an integrated scenario for the many-body localization transition, conjecturing that the critical point in the thermodynamic limit, if it exists, should be given by this value of disorder strength.
Author comments upon resubmission
Thank you for sending us the Referees' comments. We have revised our manuscript accordingly. Below you can find the response to each of the Referees.
Yours sincerely,
The Authors
List of changes
We have integrated the comments/suggestions of the referees with our work. Here we list them.
(A) Major changes
A1. The abstract has been rewritten to clarify the novelty of our manuscript (in particular the word `unambiguous' has been removed).
A2. Added Section 3.3., where a finite-size scaling of the kurtosis excess $\gamma_{2}(\overline{\Delta}_{n})$ is presented. Added Fig. 3 consisting of three panels.
A3. Previous Fig. 7 has been modified and the new Fig. 8 is now located in Section 5. This figure now consists of two panels, and it contains finite size-scaling considerations for both thermalization and spectral statistics, which offers an integrated scenario. \\
(B) Other changes
B1. Added several references.
B2. In general, some parts of the text involving well-known previous results have been either simplified or entirely removed (the reader is thus simply referred to original references) as they could have influenced the perception of novelty.
B3. The meaning of colored points in previous Fig. 6 (now Fig. 7) has been clarified.
B4. We have stated more explicitly that the first claim in Section 5 is a conjecture.
B5. Several clarifications and explanations have been added throughout.
Published as SciPost Phys. 10, 107 (2021)
Reports on this Submission
Report #2 by Anonymous (Referee 1) on 2021-3-30 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202010_00019v2, delivered 2021-03-30, doi: 10.21468/SciPost.Report.2742
Report
The authors have made a considerable effort to rewrite the manuscript, to improve the presentation, and to address the criticism from the referees raised in the first round. In my opinion the authors did a good job and have sharpened the message of their analysis.
The main achievements of their work are given below:
- They perform a thorough analysis of both the kurtosis excess of the observable matrix elements and the long-range spectral statistics via the averaged power spectrum.
- They establish the connection between the peak of the kurtosis excess and the breakdown of the RMT description, and identify this point as a possible candidate for the transition point in the thermodynamic limit.
- These results, combined together, constitute a new result in the field of widely studied disordered spin-1/2 chains.
For these reasons, I recommend the manuscript to be published in SciPost Phys as is.
Report #1 by Anonymous (Referee 2) on 2021-3-18 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202010_00019v2, delivered 2021-03-18, doi: 10.21468/SciPost.Report.2714
Report
The authors have improved their manuscript and clarified in their text and response what they consider to be the main advances of their work. They have also clarified that strong statements made in the first version are conjectures inspired by their data rather than definite conclusions.
There are still a couple of points in their response and updated manuscript that I am a bit unsure about. First, there is no sense, as far as I can see, that $\omega_c(L)$, obtained as the maximum of the kurtosis excess, is a $\textit{singular point}$. It is just a feature of finite size data and there is nothing singular about it. In particular, since it clearly and strongly flows with system size, the authors at one point write that they can not strictly conclude if it is reflecting an underlying critical point or just a finite-size crossover, in which it would not even be singular in the thermodynamic limit. In my mind it is confusing to talk about $\omega_c(L)$ as a singular point when what it really is is just a way of estimating the critical disorder strength from finite size data. It may be a nice way of doing this, and it does correlate well with estimates of when the Thouless energy becomes equal to the Heisenberg energy, but I do not see that it has otherwise a higher status than some of the other ways of estimating the critical disorder strength in finite size systems.
It is true that this way is better than using crossings in level spacing statistics (or r ratios), but in my mind it is well understood that these crossing are not very accurate. In their response to my first report the authors claim that the kurtosis excess is special since it gives a peak at (or close) to the transition. On page 20 in their manuscript they also compare this with entropies stating that they change monotonically with disorder strength, citing several references including Ref. [37] (Kjäll et al). This is not fully accurate since for example in Ref. [37] the fluctuations of the entanglement entropy are shown to peak at the transition and the maximum of the variance could be used in the very same way as here to estimate the critical disorder strength (Ref. [37] instead used finite-size scaling collapse to estimate the critical disorder strength). So the kurtosis excess is not the first or only quantity to have the features the authors claim is novel. I think this is OK. It doesn't really affect the potential value of this work but I think it is not useful to write and discuss it in a way where it is made to seem like it reveals more than it actually does.
The above are comments on the presentation and the discussion and interpretation of the data. I think it is exaggerating a bit what can be read from the data, and how it positively compares with other data. I don't think this is needed (or useful) as some of the data is actually quite nice. I like figures 5 and 6 for example, and the way of analysing this and the physics that comes from it, is a useful addition to the literature. I think it is consistent with what is already known, and even if there isn't really anything fundamentally new the work provides a nice way of analysing data.
One last comment: in the authors reply they state that the concept of Thouless energy becoming the Heisenberg energy has only recently entered the discussion. This is not strictly true. Arguably this goes back to Thouless' early work, albeit in the Anderson transition. But even in the context of MBL it goes back further, and it was for example discussed by Serbyn, Papić and Abanin in Phys. Rev. B 96, 104201 (2017), which is not cited.
In any case, if the authors tidy up these last presentation issues (see requested changes ) I think it deserves publication. I am undecided if I think it belongs more to the SciPost Physics of SciPost Physics Core, so I'll live it up to the editorial college to decide in case the editor-in-charge goes forward with publication. If it does pass for SciPost Physics, it would maybe be in expectation 3. (Open a new pathway in an existing or a new research direction, with clear potential for multipronged follow-up work;), but for SciPost Physics Core it satisfies 2. (Detail one or more new research results significantly advancing current knowledge and understanding of the field.)
Requested changes
1-Clarify in what sense $\omega_c(L)$ is a singular point or, more likely, if it is not a singular point, rephrase the discussion of it as being a finite-size estimator of the critical value for the disorder strength.
2-If the maximum of the kurtosis excess $\omega_c(L)$ is fundamentally different signature as a precursor to the MBL transition than, say the maximum in the variance of the eigenstate entanglement entropy, clarify this difference and contrast with quantities that behave in a similar way (instead of only contrasting it with quantities that do not). Note that I used the word fundamentally different since of course it is different in the details of the quantities that are being looked at.