Anisotropy-mediated reentrant localization

Submission summary

 As Contributors: Ivan Khaymovich Arxiv Link: https://arxiv.org/abs/2002.00013v3 (pdf) Date submitted: 2022-06-29 08:03 Submitted by: Khaymovich, Ivan Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Condensed Matter Physics - Computational Quantum Physics Approaches: Theoretical, Computational

Abstract

We consider a 2d dipolar system, $d=2$, with the generalized dipole-dipole interaction $\sim r^{-a}$, and the power $a$ controlled experimentally in trapped-ion or Rydberg-atom systems via their interaction with cavity modes. We focus on the dilute dipolar excitation case when the problem can be effectively considered as single-particle with the interaction providing long-range dipolar-like hopping. We show that the spatially homogeneous tilt $\beta$ of the dipoles giving rise to the anisotropic dipole exchange leads to the non-trivial reentrant localization beyond the locator expansion, $a<d$, unlike the models with random dipole orientation. The Anderson transitions are found to occur at the finite values of the tilt parameter $\beta = a$, $0<a<d$, and $\beta = a/(a-d/2)$, $d/2<a<d$, showing the robustness of the localization at small and large anisotropy values. Both extensive numerical calculations and analytical methods show power-law localized eigenstates in the bulk of the spectrum, obeying recently discovered duality $a\leftrightarrow 2d-a$ of their spatial decay rate, on the localized side of the transition, $a>a_{AT}$. This localization emerges due to the presence of the ergodic extended states at either spectral edge, which constitute a zero fraction of states in the thermodynamic limit, decaying though extremely slowly with the system size.

Current status:
Has been resubmitted

Dear Editor,

Hereby we resubmit our manuscript for consideration in SciPost Physics.
Following the recommendations of both referees, we have restructured the manuscript and implemented corrections.
Please see the list of changes below and the point-to-point replies to the referees in the previous version as comments.

Sincerely yours,
Xiaolong Deng, Alexander L. Burin, and Ivan M. Khaymovich

List of changes

1. We clarify the experimental feasibility of our results and added some additional examples from the former Appendix G.
a. Section 2 has been renamed to “Model and its symmetry”.
b. Section 3 has been renamed to “Overview of the numerical and analytical results”: now it contains Fig. 2 and the reference to all the results from Sec. 4 and 5.
c. Section 4 “Numerical results” contains the following subsections:
i. Sec. 4.1. “Finite-size flow of the ratio $r$-statistics”, consisting of Fig. 3 and the first part of the former Appendix A.1.
ii. Sec. 4.2 “Eigenstate properties: multifractal analysis and wave-function spatial decay”, consisting of the results for the spectrum of fractal dimensions $f(\alpha)$, fractal dimensions $D_q$, and the wave-function spatial decay: Figs. 4-7 and the former Appendix A.2.
iii. Sec. 4.3 “Wave-packet dynamics. Return probability” consists of Fig. 8 and the former Appendix B.2.
iv. Sec. 4.4 “Finite-size mobility edge and the fraction of ergodic states” consists of Figs. 9-10 and the second part of the former Appendix A.1 and Appendix A.3.
d. Section 5 “Analytical methods and results”, first, explains with the help of the new Fig. 11 the main idea of the localization in the spectral bulk due to the presence of the high-energy ergodic states. It contains the following subsections
i. Sec. 5.1 “Main idea of the renormalization group analysis”, including the former Appendix E, and
ii. Sec. 5.2 “Ioffe-Regel criterion for the fraction of high-energy ergodic states”, including the former Appendix F.
e. Appendix A.1 has been moved to Secs. 4.1, 4.4.
f. Appendix A.2 has been moved to Sec. 4.2.
g. Appendix A.3 has been moved to Sec. 4.4.
h. Appendix B.1 has been renamed to Appendix B.
i. Appendix B.2 has been moved o Sec. 4.3.
j. Appendix C (before C.1) has been renamed as the new Appendix A.
k. Appendices C.1-C.3 have been removed.
l. Appendix D has been renamed as the Appendix C.
m. Appendices E and F have been moved to the first and second subsections of Sec. 5.
n. Appendix G has been partly merged to the introduction.
3. We have separated the description of finite-size effects and the main crucial points of our numerical analysis.
4. We have unified the notations throughout the text.
5. We have clarified the sliding window average, used to produce the results in Fig. 2, and also the difference between energy-resolved r-statistics and the one averaged over the spectrum.
6. We have also clarified the main issue with the energy-resolved physical quantities and our focus on the bulk of the spectrum in the rest of the work, where the properties are homogeneous away from the critical points $\beta = a$.
7. We have discussed the subleading effects of the disorder amplitude $W$ with respect to the exponent $a$ of the power law and the anisotropy parameter $\beta$.
8. We have clarified our claim of ergodic (for $\beta=2>a=1.5$), critical ($\beta = a = 1.5$), and localized ($\beta=1<a=1.5$) properties in Fig. 2.
9. We have clarified the definition of d-dimensional PLRBM used for the comparison in Fig. 4.
10. We have improved the caption of Fig. 10 of the energy-resolved IPR sorted by increasing amplitude.
11. Fig. 1(a) has been replaced.
12. Fig. 3 has been combined from the former Figs. 3 and 6.
13. Fig. 9 has been combined from the former Fig. 7 and the inset to Fig. 3(b).
14. Fig. 11 has been added for the clarification of the analytical idea.
15. Several relevant references have been added [27, 47, 48] or updated [36].
16. Due to removing Appendices C.1-C.3 and G, the reference [65-74, 76-77] have also been removed.
17. Several clarifications, minor amendments, and corrections of typos have been done throughout the manuscript.

Submission & Refereeing History

Resubmission scipost_202208_00078v1 on 29 August 2022

Resubmission 2002.00013v3 on 29 June 2022
Submission 2002.00013v2 on 21 July 2021

Reports on this Submission

Anonymous Report 1 on 2022-8-18 (Invited Report)

• Cite as: Anonymous, Report on arXiv:2002.00013v3, delivered 2022-08-18, doi: 10.21468/SciPost.Report.5554

Strengths

1 - 2D Anderson transition for the effective long-range tunneling model coming from diluted tilted dipoles is studied apparently showing an interesting reentrant transition
2 - Extensive numerical data support the analysis

Weaknesses

Some numerically based conclusions are not convincing

Report

I believe the manuscript has been significantly improved with respect to the previous version. It contains significant new results on the rich 2D model, some of them not obvious at first glance. The paper combines pure numerical studies with renormalization group analysis. I hope that the authors may take into account the suggested changes as listed below.

Requested changes

1. Considering the gap ratio statistics the authors rejected my suggestion to make the more detailed energy dependent study arguing that the energy dependence shows only close to $\beta \approx a$ transition. While this is not documented in the figures, it is clearly stated in p.8 so I withdraw my suggestion. Still considering gap ratio statistics Fig.3 shows that finite size scaling of two crossings present in case (c) and (d) yields different $\beta_{AT}$ values. On the other hand, as mentioned in p.5 there is a symmetry relating $0<\beta<2$ interval with $\beta>2$. Is it not this symmetry which leads to the second crossing in panels (c) and (d)? Can one be more quantitative about the relation between the corresponding $\beta_{AT}$ values (1.45 versus 3.1 for panels g,h) ? May be we learn more about the consistency of the finite size scaling from such a comparison?
2. This referee is entirely lost with Fig.4. The caption and the horizontal axis says $D_2$ versus $a$. On the other hand the points both yellow and blue are denoted by $a=0$. Is it not a bit inconsistent? Caption says yellow squares are for $W=10$ while in the figure $W=20$ is indicated. Something is simply wrong here.
3. Fig.5 seems to show that three color lines for different values of $L$ yield the extrapolated black curve which is far far above finite size results. Is such an extrapolation not too courageous? What are the errors of this procedure?
4. The authors are asked to review and correct some formulae. In (17) the equality with Dirac delta function is of course wrong for $t>0$, also imaginary unit "i" is missing from exponents in (17) and (18).
5. Few places require editing. Eg. p. 3 "By tailor the optical forces" should be "Tailoring the optical forces"; Double dot ending para after (2) in p. 5 is not needed etc...

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

Author:  Ivan Khaymovich  on 2022-08-28  [id 2764]

(in reply to Report 1 on 2022-08-18)
Category:

We are very grateful to the Referee 1 for his/her detailed second review of our manuscript. Below we present the list of amendments resulting from the comments of the Referee. The revised version of the manuscript, with the changes marked by red font and yellow highlights, is attached to the reply.

Weaknesses - Some numerically based conclusions are not convincing.

Reply: We believe that in the revised version of the manuscript the numerically-based conclusions, consistent with the analytical predictions, are convincing for the Referee 1.

Report: I believe the manuscript has been significantly improved with respect to the previous version. It contains significant new results on the rich 2D model, some of them not obvious at first glance. The paper combines pure numerical studies with renormalization group analysis. I hope that the authors may take into account the suggested changes as listed below.

Reply: We thank the Referee 1 for high evaluation of our revised version and provide the reply below.

Requested changes 1. Considering the gap ratio statistics the authors rejected my suggestion to make the more detailed energy dependent study arguing that the energy dependence shows only close to $\beta = a$ transition. While this is not documented in the figures, it is clearly stated in p.8 so I withdraw my suggestion.

Reply: We thank the referee for this.

Requested changes : 1. (continuation) Still considering gap ratio statistics Fig.3 shows that finite size scaling of two crossings present in case (c) and (d) yields different $\beta_{AT}$ values. On the other hand, as mentioned in p.5 there is a symmetry relating $0<\beta<2$ interval with $\beta>2$. Is it not this symmetry which leads to the second crossing in panels (c) and (d)? Can one be more quantitative about the relation between the corresponding $\beta_{AT}$ values (1.45 versus 3.1 for panels g,h) ? May be we learn more about the consistency of the finite size scaling from such a comparison?

Reply: Yes, indeed, the pairs of crossings in panels (c) and (d) of Fig. 3 are related to each other by the exact symmetry (3) of the model: the fact that the critical exponents are the same in each of the pairs implicitly confirms this. In the revised version, in order to make the symmetry clear, we have added the clarifying phrase into the text: “Note that the pairs of crossings $\beta_{AT}$ in Fig. 3(c, d) are related to each other with respect to the symmetry (3) within the above mentioned error bar, while the critical exponents are just the same.”. In addition, we have added the error bars for $\beta_{AT}$ given by $\pm 0.05$ for $\beta<2$ and by $\pm 0.1$ for $\beta>2$.

Requested changes : 2. This referee is entirely lost with Fig.4. The caption and the horizontal axis says $D_2$ versus $a$. On the other hand the points both yellow and blue are denoted by $a=0$. Is it not a bit inconsistent? Caption says yellow squares are for $W=10$ while in the figure $W=20$ is indicated. Something is simply wrong here.

Reply: We thank the Referee 1 for pointing out these typos. In the revised version of the manuscript we have modified both the figure and the caption accordingly.

Requested changes: 3. Fig.5 seems to show that three color lines for different values of L yield the extrapolated black curve which is far far above finite size results. Is such an extrapolation not too courageous? What are the errors of this procedure?

Reply: Yes, indeed, Fig. 5 shows the extrapolation of the finite size spectrum of fractal dimensions $f(\alpha, L)$ to the infinite system size. As usual (see, e.g., [16, 17, 37]) with $f(\alpha)$ the main finite size correction of $f(\alpha, L)$ is given by the vertical shift (i.e. the weakly $L$-dependent prefactor in the probability distribution of $\alpha$, given by subleading corrections in (12)). As the shape of $f(\alpha, L)$ is practically the same for all the system sizes that we considered, we just get rid of the (unknown) prefactor by extrapolation (12). The deviations of the extrapolated value of the maximum of $f(\alpha)$ from $1$ gives an estimate for the error bar of this procedure.

Requested changes: 4. The authors are asked to review and correct some formulae. In (17) the equality with Dirac delta function is of course wrong for $t>0$, also imaginary unit "i" is missing from exponents in (17) and (18). 5. Few places require editing. Eg. p. 3 "By tailor the optical forces" should be "Tailoring the optical forces"; Double dot ending para after (2) in p. 5 is not needed etc…

Reply: We thank the Referee 1 for pointing out the misprints and correct them accordingly in the revised version.