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Anomalous mobility edges in one-dimensional quasiperiodic models
by Tong Liu, Xu Xia, Stefano Longhi, Laurent Sanchez-Palencia
This is not the current version.
|As Contributors:||Laurent Sanchez-Palencia|
|Date submitted:||2021-08-18 17:46|
|Submitted by:||Sanchez-Palencia, Laurent|
|Submitted to:||SciPost Physics|
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.
Author comments upon resubmission
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."
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".
List of changes
- "bands" changes into "energy intervals"
- Discussion about the effect of the phase enhanced and shifted to end of paragraph below Eq. (6)
- Added an appendix (new Apppendix B)
Submission & Refereeing History
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Reports on this Submission
Anonymous Report 2 on 2021-10-27 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202108_00049v1, delivered 2021-10-27, doi: 10.21468/SciPost.Report.3743
I am happy with the changes the authors made and would recommend the manuscript be published as it is. Regarding Fig. 4, I understand that in a localized phase, disorder average with respect to the global phase of potential usually would not lead to qualitatively change. However, in an extensive or thermalized phase, some observables (for example the level spacing statistics) are sensitive to the potential phase, and only the average value gives the meaningful prediction. In a critical phase, however, to the best of my knowledge, this is still open. I believe Fig. 4 nicely show that IPR is not sensitive to the potential phase even in a critical phase.
No further changes are required.
Anonymous Report 1 on 2021-10-12 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202108_00049v1, delivered 2021-10-12, doi: 10.21468/SciPost.Report.3652
1. The manuscript provides a thorough analysis of anomalous mobility edges (AME) in lattice models with unbounded quasi-periodic potentials. Through extensive numerics, the authors show that these potentials turn the energy states of the free energy band [-2J,2J] from delocalized to critical (multifractal), while adding new localized states outside such band.
2. The authors show the existence of AME in a modified SSH model with quasi-periodic tunneling rates.
3. Implementation of the studied model with cold atoms or photonic lattices is discussed.
I can't see.
The manuscript present interesting new results on the physics of anomalous mobility edges (AME), which are energies separating localized from critical (multifractal) states in the context of unbounded quasi-periodic lattices.
The obtained results are quite interesting and I can recommend publication in this journal. Here are some points that the authors could consider to improve the presentation:
1. In the finite-size scaling analysis for the exponent beta (Fig.2a, Fig.3c, Fig.5) the authors observed a linear dependence of beta when plotted as a function of 1/m. Since the system size L=Fm, where Fm is the m-th Fibonacci number, one finds m \propto log(L) for large enough m. The finite-size correction to the exponent beta is therefore of order 1/log(L). Is this behavior general and expected for second order phase transitions or is it specific to the model considered ?
2. A comment to Fig.4. To my understanding, the results shown do not depend on the value of the phase because the irrational parameter alpha used in the calculation is approximated by
the ratio of consecutive Fibonacci numbers. This, together with the periodic boundary conditions, makes the choice of the phase irrelevant and therefore disorder averaging can be avoided.
3. From the Appendix A it seems that, if the disorder potential v(x) is unbounded and quasi-periodic, the Lyapunov exponent is independent of the specific form of v(x), as the final result coincides with the clean limit case. Is this statement correct?
1. In Fig5: which value of a is used and what is the corresponding value of Em
2. The authors mentioned a "multifractal theorem". What are they referring to exactly ?