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Anomalous mobility edges in one-dimensional quasiperiodic models
by Tong Liu, Xu Xia, Stefano Longhi, Laurent Sanchez-Palencia
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Submission summary
| Authors (as registered SciPost users): | Laurent Sanchez-Palencia |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2105.04591v1 (pdf) |
| Date submitted: | 2021-06-03 14:41 |
| Submitted by: | Sanchez-Palencia, Laurent |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Mobility edges, separating localized from extended states, are known to arise in the single-particle energy spectrum of disordered systems in dimension strictly higher than two and certain quasiperiodic models in one dimension. Here we unveil a different class of mobility edges, dubbed anomalous mobility edges, that separate bands of localized states from bands of critical states in diagonal and off-diagonal quasiperiodic models. We first introduce an exactly solvable quasi-periodic diagonal model and analytically demonstrate the existence of anomalous mobility edges. Moreover, numerical multifractal analysis of the corresponding wave functions confirms the emergence of a finite band of critical states. We then extend the sudy to a quasiperiodic off-diagonal Su-Schrieffer-Heeger model and show numerical evidence of anomalous mobility edges. We finally discuss possible experimental realizations of quasi-periodic models hosting anomalous mobility edges. These results shed new light on the localization and critical properties of low-dimensional systems with aperiodic order.
Current status:
Reports on this Submission
Report #1 by Anonymous (Referee 3) on 2021-7-26 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2105.04591v1, delivered 2021-07-26, doi: 10.21468/SciPost.Report.3294
Report
This is a very good study on the mobility edge in some 1D quasiperiodic models. The numerical investigation seems solid, and the results are very interesting.
The authors first analytically prove that for a specific model, only eigenstates with eigenenergies in between [-2,2] are non-localized. Since no extended state is allowed for unbounded potential, they further conclude that these eigenstates should be critical. Numerical calculations confirm their conclusion. They have also numerically investigated some other models, and generalize their conclusion.
I think this is an interesting work that definitely deserves to be published. I only have some minor suggestions for the authors (see the requested changes box).
Requested changes
(1) The authors claim that, in a sentence below Eq. (1), that the phase "theta" is irrelevant. This is, however, usually not entirely true in the investigation of localization physics in quasiperiodic models. One usually need to take the average of "\theta" to mimic the randomness of a genetic disorder. Although this average is usually more important in the extensive phase than the localized phase. I however wonder has this \theta dependency has been considered and investigated. I can see in Eq. (3) that the analytical analysis to some degree includes this average by the integration over \theta, so the question is more concerning on the numerical analysis.
(2) The energy gap is usually an important observable for the studies of localization. The gap ratio usually can give us information on whether the eigenenergies follows Gaussian Orthogonal Ensemble distribution or Poisson distribution. This is an important signature of an extensive/localized spectrum. For completeness, I suggest the authors carry out this analysis.
(3) This is a very minor problem, but I am a bit confused about the choice of their word "band". In the introduction, the third paragraph, the authors claim that "... can give rise to a full band of critical states ...". In condensed matter physics, a "band" usually refers to a set of eigenenergies that are close to each other, with a relatively large gap between the different bands. In Figure 1(b), one can see that the spectrum consists of multiple bands, and these bands can sometimes cross the mobility edge. Therefore, I think the term "band" might be a bit misleading. That being said, this seems to me only a problem of choice of word.
Author: Laurent Sanchez-Palencia on 2021-08-18 [id 1689]
(in reply to Report 1 on 2021-07-26)Dear Editor,
We thank your for forwarding the report of the Referee. We are grateful to the latter for his/her very positive report on our work. The Referee has made a couple of suggestions, which we answer below We also briefly indicate the corresponding changes we made on the manuscript and resubmit a revised version of the manuscript.
Referee's comment 1 : "The authors claim that, in a sentence below Eq. (1), that the phase "theta" is irrelevant. This is, however, usually not entirely true in the investigation of localization physics in quasiperiodic models. One usually need to take the average of "\theta" to mimic the randomness of a genetic disorder. Although this average is usually more important in the extensive phase than the localized phase. I however wonder has this \theta dependency has been considered and investigated. I can see in Eq. (3) that the analytical analysis to some degree includes this average by the integration over \theta, so the question is more concerning on the numerical analysis."
Answer : It is known from the extensive mathematical literature on the quasi-Mathieu operator (Aubry-André model) that the spectrum and localization properties of the model do not depend on the phase \theta when the incommensurate ratio \alpha is Diophantine. In our case, we use the inverse golden ratio, which is indeed a Diophantine number. Therefore, we believe that the phase is essentially irrelevant, excluding the zero-measure set of values of \theta leading to a diverging potential.
Nevertheless, we agree with the Referee that --to the best of our knowledge-- there is no rigorous proof of this conjecture for our model. To clarify this point, we have run new calculations with different values of \theta. We indeed found that the spectrum and the localization properties are unchanged, see new appendix B in the resubmitted manuscript.
Referee's comment 2 : "The energy gap is usually an important observable for the studies of localization. The gap ratio usually can give us information on whether the eigenenergies follows Gaussian Orthogonal Ensemble distribution or Poisson distribution. This is an important signature of an extensive/localized spectrum. For completeness, I suggest the authors carry out this analysis."
Answer :
The level spacing statistics is indeed a rather popular approach to determine the localization properties of disordered or quasi-periodic systems. In fact, there exists a variety of methods to determine the localization properties. In our work, we already use three of them: (i) Exact calculation of the Lyapunov exponent (for the diagonal model), (ii) scaling of the IPR, and (iii) multifractal analysis. Although we agree with the Referee that a study of the level spacing statistics could be performed, we think it goes beyond the scope of our study and it would not provide us with additional information about the onset of anomalous mobility edges in our models.
Referee's comment 3 : "This is a very minor problem, but I am a bit confused about the choice of their word "band". In the introduction, the third paragraph, the authors claim that "... can give rise to a full band of critical states ...". In condensed matter physics, a "band" usually refers to a set of eigenenergies that are close to each other, with a relatively large gap between the different bands. In Figure 1(b), one can see that the spectrum consists of multiple bands, and these bands can sometimes cross the mobility edge. Therefore, I think the term "band" might be a bit misleading. That being said, this seems to me only a problem of choice of word."
We thank the Referee for pointing out this possible confusing choice of a word. In the revised version of the manuscript, we now use the more neutral word "energy interval" instead of "band".