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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_00052v2
Date submitted: 2020-07-26 10:25
Submitted by: Kuno, Yoshihito
Submitted to: SciPost Physics
Discipline: Physics
Subject area: Condensed Matter Physics - Theory
Approaches: Theoretical, Computational

Abstract

We study a canonical many-body-localized (MBL) system with power-law correlated disorders: $s=\frac{1}{2}$ spin chain in a random magnetic field. The power-law-correlated disorder can control the critical regime of finite systems between the MBL and thermal (ergodic) phases by varying its exponent, and it let us investigate the MBL transitions in detail. Static-eigenstate and dynamic properties of MBL are studied by numerical methods for systems with various long-range correlations. By using energy-resolved distribution of the localization length (DLL) obtained for the non-interacting and interacting cases, we show that certain novel universality exists for the MBL transitions. The static-eigenstate as well as dynamic MBL are investigated by the DLL to find essential properties of the phase transition out of MBL.

Current status:
Editor-in-charge assigned


Author comments upon resubmission

Dear Editors,

We read the referees’ reports very carefully, and have revised the manuscript.
The revision is major. We have performed additional calculations of new physical quantities, and obtained novel and interesting results for many-body localization (MBL) and its phase transition. We have also performed large system-size calculations to clarify the phase diagram for non-interacting case, etc. to improve the reliability of the observations in the previous manuscript. However, we think that newly obtained findings are quite important and warrant publication. Therefore, the major points of the revised manuscript are different from those of the previous manuscript. We carefully added the new findings in the revised manuscript and try to respond as much as possible to informative comments from the referees. None the less, we would like to submit the new manuscript as a revised version of the previous one as it studies the same model. In the following, we first summarize a list of changes made, and then responses to the referees’ comments and recommendations.

1. For the non-interaction case, we have performed large system-size calculation, i.e., the system size L=2000. (Section 4).
2. To observe the phase diagram of the non-interacting system, we calculated the energy-resolved distribution of the localization length (DLL). The DLL gives a useful insight for investigation of localization properties of system. By the data of the DLL and other physical quantities, we identified the location of the crossover regime between the MBL and thermal states. (Section 4)
3. For the interaction systems, we have done the calculations of the standard deviation of the entanglement entropy (SDEE). With calculations of level-spacing ratio (LSR) and SDEE, we identify the parameter region corresponding to the critical regime between the MBL and thermal states. (Section 5.2)
4. Besides the strength of the disorder, the present system has another parameter, which controls the phase diagram of the MBL and thermal states: the exponent of the correlation of the disorder. This means that there exists a phase-boundary line of the MBL state, and therefore, we can study `universality’ of the phase transition out of the MBL state. To obtain new insight on MBL, we calculated the DLL in the central parts of the critical regimes and in the vicinity of the transition points for multiple values of the parameters. Then, we compared the DLLs obtained for different parameters to get important novel observations. (Section 5.2)
5. Dynamical properties of the systems are also understood from the view point of the DLL. (Section 6)
6. Title of the manuscript has been changed.
7. References have been added.
8. Grammatic errors have been corrected.

To the second referee

Thank you very much for your careful reading the manuscript and useful comments and suggestions. The followings are responses to your comments and recommendations.

1. For the non-interacting cases, we performed the large-scale calculation up to L=2000. The number of disorder realization is 500-600. In the previous manuscript, we said N_d= 2^3, but this is a typo, and the actual number is 2\times 10^3.
Although we have obtained rather reliable results for the PR and SSEE, we judge that it is not easy for our numerical resources to obtain a phase diagram in the thermodynamic limit. Therefore, we have omitted the phase diagram for non-interacting case, and only conclude that critical regime seems to exist in the vicinity W=2.5 for finite system size. To corroborate this conclusion, we calculated energy-resolved distribution of localization length (DLL), which gives detailed information of the many-body quantum states. (Fig.8 and 9.) As we mentioned in the above, the DLL plays an important role in the present work. (Section 4)

2. For the experimental realization of disorders with the long-range correlations, for example, in cold atom experiments, the disorder is generated by a speckle laser light. Actually, the disorder potential has somewhat real-space correlation, but controllable power-law correlated disorder has not yet been implemented in the experiments.

3. For the interacting case studied in Sec.5, we employ the multi-fractal analysis (MFA), and identify the ETH (thermal), critical and MBL regimes by using the MFA and LSR. We emphasize there that the obtained phase diagram is only for a finite system size. It is not easy to obtain the phase diagram in the thermodynamic limit even though the long-range correlation of disorders enhances the critical regime between the ETH and MBL states. Section 5.2 is new and devoted to the study of the critical regime and the phase transition out of the MBL state. We mostly focus on the cases with \gamma=0.2 and 2.0. By calculating SDEE, we identify the critical regime between the ETH and MBL states. (Fig.7.) From the SDEE and DLLs, we obtain very important observations concerning to MBL, including universality of the phase transition and reminiscence of the local-bit. We think that the new findings in Sec. 5 warrant publication in SciPost, even though the present work cannot identify a definite phase diagram in the thermodynamic limit.
4. At the end of section 2, we clarify the physical meaning of the power-law correlation of the disorder. Power-law correlation is a superposition of short-range correlations and is qualitatively different from a single-short correlation with adjusted strength. However, for the interacting case with J_z=1, calculations of some physical quantities, such as SDEE and DLL at critical points, exhibit incredible coincidence for different parameters. We think that this coincidence must depend on the filling factor (magnetization) of the system. Anyway, this is a future problem.
5. The scaling functions are explicitly shown in Fig.5. We employ the symmetric scaling. More elaborated analysis is a future work.
6. (For 6 and 7 of the referee’s comments.) Thank you very much for comments on the technical details. We would like to keep the manuscript as its present form for no improvements were found.
7. Typos have been corrected.
8. We do not have a clear answer for the problem of the finite-size effect.
9. We have corrected 10^{-4} to 10^{-3}. Anyway, this deviation is too small to be effective.
10. Thank you very much for your comments on the grammatic errors. We have blushed up English.
11. References have been added.


To the first referee

Thank you very much for careful reading the manuscript and useful comments. The followings are responses to your comments and recommendations.

1. As explained in the above, we determined the locations of transition points by using the LSR and SDEE. (Fig. 7.)
2. We examined the fitting without imposing the constraint a<1, and found the same results.
3. In Fig. 5 (a), the fitting line deviates from the numerical data. However, $W=2.0$ is located well inside of the ETH phase, and therefore the constant seems irrelevant. The other cases of (b) and (c) exhibit the satisfactory fitting between the line and data.

Concerning to the referee’s broader questions:
4. In the revised manuscript, we explicitly show how the long-range correlation broadens the critical regime (Fig.4), but we judge that if the critical regime survives in the thermodynamic limit is a difficult problem. Instead, we clarify the novel nature of MBL by using the DLL and SDEE (Section 5). Then, we think that the present work warrants publication in SicPost.
5. At the end of Sec. 7, we refer to the recent work on MBL from the view point of the Fock space. This work obviously can be applied to the present model. This is an interesting future work.

List of changes

1. For the non-interaction case, we have performed large system-size calculation, i.e., the system size L=2000. (Section 4).
2. To observe the phase diagram of the non-interacting system, we calculated the energy-resolved distribution of the localization length (DLL). The DLL gives a useful insight for investigation of localization properties of system. By the data of the DLL and other physical quantities, we identified the location of the crossover regime between the MBL and thermal states. (Section 4)
3. For the interaction systems, we have done the calculations of the standard deviation of the entanglement entropy (SDEE). With calculations of level-spacing ratio (LSR) and SDEE, we identify the parameter region corresponding to the critical regime between the MBL and thermal states. (Section 5.2)
4. Besides the strength of the disorder, the present system has another parameter, which controls the phase diagram of the MBL and thermal states: the exponent of the correlation of the disorder. This means that there exists a phase-boundary line of the MBL state, and therefore, we can study `universality’ of the phase transition out of the MBL state. To obtain new insight on MBL, we calculated the DLL in the central parts of the critical regimes and in the vicinity of the transition points for multiple values of the parameters. Then, we compared the DLLs obtained for different parameters to get important novel observations. (Section 5.2)
5. Dynamical properties of the systems are also understood from the view point of the DLL. (Section 6)
6. Title of the manuscript has been changed.
7. References have been added.
8. Grammatic errors have been corrected.

Submission & Refereeing History

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Resubmission scipost_202004_00052v2 on 26 July 2020

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