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Effects of power-law correlated disorders in a many-body localized XXZ spin chain

by Takahiro Orito, Yoshihito Kuno, Ikuo Ichinose

Submission summary

As Contributors: Ikuo Ichinose · Yoshihito Kuno
Preprint link: scipost_202004_00052v1
Date submitted: 2020-04-30
Submitted by: Kuno, Yoshihito
Submitted to: SciPost Physics
Discipline: Physics
Subject area: Condensed Matter Physics - Theory
Approaches: Theoretical, Computational

Abstract

Effects of correlations in disorders in MBL systems have not been clarified so far. We study effects of power-law-correlated disorders in a typical many-body-localized (MBL) system: $s=\frac{1}{2}$ spin chain in a random magnetic field. We numerically investigate the localization properties of the system rather in detail. The power-law-correlated disorders induce localization properties essentially different from those of a uniform white-noise disorder, as we show for Anderson localization as well as MBL. Notably, we find that the power-law-correlated disorders tend to enhance a critical phase. In addition, we also systematically investigate entanglement properties.

Current status:
Editor-in-charge assigned



Reports on this Submission

Anonymous Report 2 on 2020-5-28 Invited Report

Strengths

1. The manuscript raises an interesting problem of influence of controlled correlations in disorder on many-body localization transition.

2. The results are presented in a logical manner.

Weaknesses

1. Numerical data are obtained for limited system sizes and averaged over relatively small number of disorder realizations. I consider this a significant weakness as the work is purely numerical.

2. It is hard to be convinced about the validity of the claim about the critical regime that separates the ETH and MBL phases. Presented results do not show significant differences between the correlated disorder analysed by the authors and the standard uncorrelated disorder.

3. Manuscript is not well written, there are numerous grammar errors that force the reader to guess what is meant by the authors.

Report

The manuscript by T. Orito, Y. Kuno, and I. Ichinose considers a problem of many-body localization transition in quantum spin chains with power-law correlated disorder. The problem is original and interesting. The power-law correlated disorder can be regarded as an intermediate step between the standard cases of uncorrelated uniform disorder and quasiperiodic potential with correlations of infinite range. For that reason, I believe that the manuscript has potential to be published. However, the authors should convincingly address the following points:

1. System sizes considered in Sec. 4 that deals with the non-interacting system are $L\approx100-200$. While this is sufficient to qualitatively illustrate the reasoning of the authors, such system sizes do not allow for convincing quantitative analysis of the data. A specific, quantitative criteria that allow to distinguish between the delocalized, critical and localized regimes should be proposed. The present criterion for the critical phase ("the PR exhibits sharp increase for small L, but it tends to saturate as L is getting large") is very loose. What is meant by "tends to saturate"? If PR saturates, then the state is localized. Perhaps some sub-linear growth of PR with L is meant? The problem is well illustrated by the curve with $\gamma=1.8$ in Fig. 3 (a): in my opinion there is a tendency for saturation of PR for this data, yet the authors believe that the system is delocalized for those parameters. Similar applies to $\gamma=0.3$ data in Fig. 3 (c): does the PR really saturate or maybe it shows some slow increase with L?
This demonstrates why larger system sizes and more quantitative criterion is necessary to really distinguish the phases. This should be done and is, to my understanding, numerically feasible - the size of the matrix is $L\times L$ so going to system sizes of few thousands seems feasible. I also do not understand why the number of disorder realizations is so shockingly small, e.g. $N_d=2^3=8$(!). Much larger number of disorder realizations (few hundreds or even few thousands) would lead to data of much better quality. Similar points about system size and number of disorder realizations apply to results for SSEE.

2. Is there any experimental relevance of the power-law correlated disorder? It is written in the text: "Such a control of the correlations in the disorders is feasible in recent real experiments on cold atoms". Is this really the case? Can authors be more specific about experimental protocol to create such correlated disorder or quote a specific experiment where such disorder was implemented?

3. The authors distinguish for $J_z>0$ (analogously to the non-interacting case) three regimes: ergodic, critical and MBL for $J_z>0$. To differentiate between the phases they analyse system size scaling of participation entropy $S_2$, level spacing ratio (LSR) and entanglement entropy of eigenstates (EE). The authors suggest that the critical regime persists to large system sizes, in contrast to the uncorrelated disorder case as claimed in Phys. Rev. X 7, 021013 (2017).
However, the presented data for LSR and EE look very similarly to the uncorrelated case: there is a strong increase of disorder strength at which LSR deviates from value for GOE and also a visible shift of crossing points to larger disorder strengths with increasing system size. It is thus hard to be convinced about the claim regarding the critical regime. Since this is one of the important results of the papers, authors should address this point extensively. A more quantitative analysis of the ergodic, critical and MBL regimes along the lines of 2005.09534 could be helpful to clear the matter. Another thing is that the results would be more convincing if larger system sizes were considered. Reaching system sizes $L=20$ is standard nowadays with shift-invert method, see e.g. SciPost Phys. 5, 045 (2018).

4. The authors claim that the "critical phase is strongly affected by the properties of the long-range correlations". While the rate of logarithmic growth of entanglement indeed depends on the value of $\gamma$, an analogous behavior is observed for uncorrelated disorder, see e.g. Phys. Rev. B 93, 060201(R) (2016). Is the rate of growth of EE really sensitive to the value of $\gamma$ or is it just an "effective strength" of disorder that is tuned by $\gamma$?

5. The authors estimate a critical exponent $\nu$ by using finite-size scaling with respect to $W_c$. It is not clear what finite scaling is meant. If the standard finite-size scaling analysis analogous to Phys. Rev. B 91, 081103 (2015) (ref. [37]) is meant, then it should be noted that recent works Phys. Rev. Lett. 123, 180601 (2019), arXiv:2004.02861 suggest that symmetric scaling as in Ref. [37] should be replaced by an asymmetric scaling different on the two sides of the transition.

6. Instead of rescaling the entanglement entropy shown in Fig. 6 (b) by Page value $E_T$, the authors could use value for spin chains with conserved total $S^z$ given in Phys. Rev. Lett. 119, 220603 (2017). This probably will reduce the finite size effects of $EE$ data.

7. Data presented in Fig. 5 (a) would be better fitted if $a^P>1$. Perhaps such a behaviour of $S_2$ is analogous to a super-linear growth of entanglement entropy observed in certain parameter regime in arXiv:2005.09534?

8. $S_0$ in the labels of axes in Fig. 5 is equal to $\ln(D)$, is that correct? Should there be $S_q$ instead of $S_0$ eq. (13)?

9. The finite size effects at MBL transition have been a subject of recent debate triggered by work arXiv:1905.06345 (see e.g. arXiv:1911.04501, arXiv:1911.06221 or arXiv:1911.07882). I wonder whether the finite size effects are enhanced or reduced by the presence of the correlations in disorder.

10. The discrepancy between the correlation function $C(l)$ for numerically generated disorder shown in Fig. 1 and the desired power-law correlations becomes significant at large $l$. For instance, for $\gamma=0.8$ it is certainly much bigger than $\mathcal O(10^{-4})$ as the authors state in the text. Correction or appropriate comment is needed there.

Requested changes

1. The points 1-10 of the report should be addressed.

2. There exists a vast literature on the topic of MBL. The manuscript has only 41 references. In my opinion the authors should extend the list of references to give a better comparison of their results with results obtained for uncorrelated disorder. One example is work SciPost Phys. Core 2, 006 (2020) which is certainly relevant to the multifractal analysis performed by the authors.

3. Referee code of conduct states that I should " assess the level of clarity of the manuscript, as well as its general formatting and level of grammar". My general feeling is that the manuscript is not written well and some work should be devoted to improving the formatting and grammar. Examples of sentences of phrases that probably need to be fixed or clarified are listed below:

- "In particular, effects of disorders with power-law correlations are focused, which are feasible in recent experiments."

-"Notably, the global phase diagram clearly exhibits the enhancement of critical regimes, in which various physical quantities related to localization exhibit behavior neither extended nor genuine localized states."

- "Section V is devoted for study on the interacting case."

- "We numerically investigate the localization properties of the system rather in detail"

-"In the experimental side"

- "Although the disorders of Eq. (3) slightly deviate from the genuine power-law correlation $l^{-\gamma}$, but it is a good approximation for studying localization."

- "In this paper, we shall investigate how this type of disorder affect the phase diagram of the spin model of Eq. (1)"

- "of energy eigen states"

- "Concerning to the critical regime at the intermediate values of W"

- "In fact, the previous works on some correlated disorders have gotten the same conclusions with ours [17],"

- "The direct EE value is too subtle to quantify the difference between the critical regime and delocalized and localized phase"

- "Our results of the system-size scaling of the PR are shown in Fig. 3 (a)-(c), in which three data correspond to the typical delocalized, critical and localized states, respectively."

- "For the delocalized phase (Fig. 3 (a)), the PR apparently exhibits linear increase proportional to L, which is nothing but the behavior of the delocalized phase."

-"As shown in Fig. 4 (a) and (b), the values of $a^p$ and $b^p$ have a possibility to capture the ETH-critical transition as far as our numerical system sizes."

-"the time evolution of the state accompanies a change of the EE caused by its dephasing"

- "the increase of the EE is very slow, i.e., and $EE \propto log(t)$, which is a maker of the MBL state."

  • validity: ok
  • significance: good
  • originality: high
  • clarity: good
  • formatting: reasonable
  • grammar: below threshold

Anonymous Report 1 on 2020-5-27 Invited Report

Strengths

1. The study of correlated disorder in the MBL context is original.
2. The paper is well-organized.

Weaknesses

1. The main claim, "the power-law-correlated disorders tend to enhance a critical phase", should be clarified.
2. The scaling analysis should be discussed more in depth (this is related to the previous point).

Report

This paper studies the XXZ spin-1/2 chain in the presence of a disordered external field. This model is known to host ETH and MBL phases in a number of cases. Here, the authors focus on the case of power-law correlated disorder, exploring the phase diagram in the (disorder strength $W$, correlation decay exponent $\gamma$) plane. This is an interesting, and to my knowledge new setting.

The authors report a critical field $W_c$ which increases slightly with decreasing $\gamma$. Most interestingly, they report a broadening of a "critical region" as $\gamma$ is decreased. To locate the critical region, the authors fit the participation entropy to the form $S = a S_0 + b \ln S_0$. The point where $b$ is minimal is the left boundary of the region, while the right boundary is the crossing point of the gap ratio. I have several critical comments at that stage:
1. I find it dangerous to define a physical region using two different observables. Indeed, finite-size effects play out differently for different observables, and as a result, they yield non-identical estimates of $W_c$. Can the authors comment on the stability of this region at the thermodynamic limit? An option is to look at how the extend of the region varies as a function of system size.
2. It seems to me that the authors impose $a \leq 1$ in the fits. I think this constraint biases the estimation of the parameter $b$, and I would be curious to see what happens it is relaxed.
3. The fit of Fig. 5 (a) could visibly be improved by taking into account a constant term
\[
S = a S_0 + b \ln S_0 + c.
\]
The constant term plays an important role for ground states [1,2]. For the MBL problem, Ref. [3] argued that the constant (physically related to a non-ergodicity volume) diverges at the transition. Therefore, it cannot be neglected close to $W_c$, i.e. where the authors observe a critical region. Is a minimum in $b$ still observed when the constant term is taken into account?

Overall, I believe this study is interesting, in particular for its claim of the existence of a "critical" region of some kind. However, I think the status of the critical region needs to be clarified by addressing the points listed above. I finish with some broader (and perhaps not answerable!) questions:
1. The XXZ Hamiltonian can be recast as a single-particle tight-binding problem on the configuration graph. The on-site energies are random and correlated [4], even when real-space variables $\eta_j$ are not. How is this correlation affected by the addition of real-space power-law correlations? Is the form similar to one of those which were investigated in [4], or is it different?
2. The Aubry-André potential is in some sense even more strongly correlated than the one studied here, since its correlation does not decay with distance. However, the AA model does not seem to host a particularly large critical region. Can the authors comment on that, perhaps in connection with Griffiths effects?

[1] Shannon and entanglement entropies of one- and two-dimensional critical wave functions
[2] Universal Behavior beyond Multifractality in Quantum Many-Body Systems
[3] Multifractal scalings across the many-body localization transition
[4] Many-body localization due to correlated disorder in Fock space

Requested changes

Major changes
* Addressing the 3 points listed in the report.

Minor changes
* I could spot a few grammatical problems, such as "Section V is devoted for study on the interacting case" which I would rephrase as "Section V is devoted to the study of the interacting case". Addressing those could improve the readability.
* Define $S_0$ in the text

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

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