Dynamical Detection of Level Repulsion in the One-Particle Aubry-André Model
Round 1
Reviewer 1 Report
The manuscript presented is a variation on the work listed as [31]. While in [31] random disorder for finite systems is investigated, the present manuscript deals with quasi-periodic potential presenting very similar concepts. Thus while the results are new is the strict sense and publishable, the real credit for originality goes to [31].
I believe that before publication the manuscript has to be modified/improved in several aspects.
The manuscript considers one dimensional (1D) single particles physics within the Aubry-Andre tight binding model. Systems in 1D are strictly integrable. There is no quantum chaos involved (as quantum chaos is a notion describing behavior of quantum systems chaitic in the classical limit). Therefore the introduction is inadequate. Especially for the volume dedicated to Shmuel Fishman one should be precise. Most of the references are redundant and should be replaced by a careful references to well studies Aubry-Andre model.
The interesting observation of the paper, after [31], is that finite systems give false signs of delocalization when the localization length is comparable to the system size. Still the claim that statistics resembling GOE, GUE and GSE is obtained by varying the system size is grossly over advertized.
It is well known that one can create "exact" arbitrarry level statistics even for 1D system that is, by definitio, integrable. I refer the authors to papers of Da Hsuan Feng , e.g.Hua Wu, D. W. L. Sprung, Da Hsuan Feng, and Michel Vallières Phys. Rev. E 47, 4063 (and references therein) where it is shown that one can generatate exactly GOE spectrum in a 1D integrable system. See also Quantum nonintegrability in finite systems, Physics Reports, 1995 which should be a valuable reference.
Authors show that Aubry-Andre model for finite sizes allows for observation of Poisson, GOE-like, GUE-like, GSE-like, picket fence like spectra. The real catch-word is "like" - what does it mean? The manuscript touches two properties - level repulsion exponent beta (obtained after unfolding levels) and the averaged gap ratio rbar that does not require unfolding.
Unfolding procedure is not described. As shown by Garcia-Garcia (2001) one may get any statistics by a careless unfolding. How the unfolding is done in this paper? There is a single parameter ksi (linked to hbar and the system size) which characterizes the obtained spectra (according to authors). Still the values of beta and rbar obtained seem not to lead to he same property. to claim that GxE-like spectra are obtained a necessary condition is that for a given ksi both rbar and beta correspond to say GOE. Is it so? Or beta=1 is obtained for say ksi_1 and rbar=0.53 (as appropriate for GOE) for a different ksi_2? If this is the case - then it is clear that the obtained spectra have little in common with gaussian random matrices and the agreement (e.g. beta=1) is a coincidence resulting from the continuity of beta while changing ksi. The figures are not at all helpful in this matter. Fig.1a and b rather suggest that characteristic for GxE ensembles values of beta occur at different ksi than values for rbar. The table of values would be much better than the plots. Still, at the level of plots, the discrepancy seems large pointing towards the conclusion that statement like "GOE-lke" or "GSE-like" are misleading. While may be level repulsion parameter beta coincides - other spectral properties are markedly different. The correlation hole should be properly defined. While the authors discuss this feature in several papers it would be good if a red arrow points towards this festure in the plots with a proper explanation. Precisely what the experiment should observe? I believe that the string of reference to other studies of Aubry-Andre model is needed. A casual search of the net reveals; Nilanjan Roy and Auditya Sharma; Phys. Rev. B 100, 195143 which is both relevant and includes many relevant references. In view of these - how the authors overcome the gaps problem in unfolding the spectra? (to link first and the last point of this report)Author Response
We thank the Referee for carefully reading our paper and for the very pertinent remarks, which led to a significantly improved new version of our manuscript. All comments were all taken into account in the new version of the paper. He/She is right that our work is an extension to our paper in Phys. Rev. E 100, 022142 (2019). The value of this new work is to deal with an experimental model of great current interest and with system sizes that are experimentally accessible. The studies shown in this new work could certainly be tested in experiments with cold atoms.
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REFEREE: The manuscript considers one dimensional (1D) single particles physics within the Aubry-Andre tight binding model. Systems in 1D are strictly integrable. There is no quantum chaos involved (as quantum chaos is a notion describing behavior of quantum systems chaitic in the classical limit). Therefore the introduction is inadequate. Especially for the volume dedicated to Shmuel Fishman one should be precise. Most of the references are redundant and should be replaced by a careful references to well studies Aubry-Andre model.
The interesting observation of the paper, after [31], is that finite systems give false signs of delocalization when the localization length is comparable to the system size. Still the claim that statistics resembling GOE, GUE and GSE is obtained by varying the system size is grossly over advertized.
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AUTHORS: The Referee is right that the 1D noninteracting Aubry-Andre model is integrable. This is exactly why we explicitly wrote in the second paragraph of the introduction:
``This is a finite-size effect, not a signature of chaos, but it can still be used as a way to demonstrate how the properties of the spectrum get manifested in the dynamics of realistic quantum systems.''
We now extend the sentence as
``This is a finite-size effect, not a signature of chaos. Wigner-Dyson distributions in these non-chaotic 1D models emerge when the localization length is larger than the system size. However, these models can still be used as a way to demonstrate how the properties of the spectrum get manifested in the dynamics of realistic quantum systems.''
This new sentence should give no false impression of any association with quantum chaos.
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REFEREE: It is well known that one can create "exact" arbitrarry level statistics even for 1D system that is, by definitio, integrable. I refer the authors to papers of Da Hsuan Feng , e.g.Hua Wu, D. W. L. Sprung, Da Hsuan Feng, and Michel Vallières Phys. Rev. E 47, 4063 (and references therein) where it is shown that one can generatate exactly GOE spectrum in a 1D integrable system. See also Quantum nonintegrability in finite systems, Physics Reports, 1995 which should be a valuable reference.
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AUTHORS: We agree with the Referee and thank him/her for the references, which we were not aware of. Relano et al have also created an integrable Hamiltonian with GOE statistics. We now cite all of these references at the end of Sec.2.1.1
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REFEREE: Authors show that Aubry-Andre model for finite sizes allows for observation of Poisson, GOE-like, GUE-like, GSE-like, picket fence like spectra. The real catch-word is "like" - what does it mean?
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AUTHORS: We are careful to use the word ``like'', whenever we talk about our results for the Aubry-Andre model to make it clear that the model is not chaotic, but shows GOE-like, GUE-like and GSE-like distributions due to finite size effects. Notice that we explicitly write in the end of Sec.2.1.1 that
``It is important to emphasize that the different level spacing distributions obtained with the model are not linked with the symmetries of the Hamiltonian. The Hamiltonian matrix used here is real and symmetric...''
We have now added one more paragraph at the end of this section, which finishes as:
``they emerge due to finite size effects.''
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REFEREE: Unfolding procedure is not described. As shown by Garcia-Garcia (2001) one may get any statistics by a careless unfolding. How the unfolding is done in this paper?
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AUTHORS: We are well familiar with the unfolding procedure and have done it in numerous papers. Notice that we have a work published with Garcia-Garcia as well. We now describe the unfolding procedure in the beginning of Sec.2.1. However, it should be clear that the purpose of this work is to show the effects of level statistics in the dynamics, not to provide a thorough analysis of the details of the distributions. Our purpose is to show that level repulsion can be detected experimentally by studying the evolution of an experimental quantum model and an experimental quantity.
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REFEREE: There is a single parameter ksi (linked to hbar and the system size) which characterizes the obtained spectra (according to authors). Still the values of beta and rbar obtained seem not to lead to he same property. to claim that GxE-like spectra are obtained a necessary condition is that for a given ksi both rbar and beta correspond to say GOE. Is it so? Or beta=1 is obtained for say ksi_1 and rbar=0.53 (as appropriate for GOE) for a different ksi_2? If this is the case - then it is clear that the obtained spectra have little in common with gaussian random matrices and the agreement (e.g. beta=1) is a coincidence resulting from the continuity of beta while changing ksi. The figures are not at all helpful in this matter. Fig.1a and b rather suggest that characteristic for GxE ensembles values of beta occur at different ksi than values for rbar. The table of values would be much better than the plots. Still, at the level of plots, the discrepancy seems large pointing towards the conclusion that statement like "GOE-lke" or "GSE-like" are misleading. While may be level repulsion parameter beta coincides - other spectral properties are markedly different.
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AUTHORS: This is an interesting point, which we had actually thought about exploring further, but not in this work. While both \beta and \tilde{r} capture short-range correlations, they are not exactly the same quantities. We are not sure that they should exactly coincide in our non-chaotic model. However, a more systematic study is worth doing taken various different models into account, including true chaotic models. We substitute our sentence
``it is evident that there is not an exact one-to-one correspondence between the two.''
to
``it is evident that there is not an exact one-to-one correspondence between the two, but a more systematic comparison of the two quantities together with a careful unfolding procedure is worth doing. In this case, various different models should be taken into account, including true chaotic models.''
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REFEREE: The correlation hole should be properly defined. While the authors discuss this feature in several papers it would be good if a red arrow points towards this festure in the plots with a proper explanation. Precisely what the experiment should observe?
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AUTHORS: We explain already in the Introduction what the correlation hole is and also cited there various references. In Sec.3, we now extend the paragraph starting as ``The main result of Fig.~\ref{fig:SP} is ...'' to make it more evident what the correlation hole is in Fig.2. It reads:
``The main result of Fig.~\ref{fig:SP} is the fact that for experimental sizes (few dozens of sites), the correlation hole emerges at times ($t<10^2$) and values of the number operator ($n_{1,L/2} (t)>10^{-2}$) that are experimentally reachable. The correlation hole is the dip below the saturation point of the dynamics. In all panels of Fig.~\ref{fig:SP}, the saturation of the dynamics is marked with a horizontal dashed line for the smallest and the largest system sizes. The correlation hole corresponds to the values of the numerical curves that are below this dashed line. The difference between saturation and minimum of the hole is most evident for the GSE-like spectrum in Fig.~\ref{fig:SP}~(e).''
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REFEREE: I believe that the string of reference to other studies of Aubry-Andre model is needed. A casual search of the net reveals; Nilanjan Roy and Auditya Sharma; Phys. Rev. B 100, 195143 which is both relevant and includes many relevant references. In view of these - how the authors overcome the gaps problem in unfolding the spectra? (to link first and the last point of this report)
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AUTHORS: We added this reference and an early one from Harper, although the most relevant reference is the already cited 1980 paper by Aubry and Andre.
Reviewer 2 Report
The authors propose a way to detect correlations in the spectrum of the Hamiltonian by looking on the decay of the density, which is a quantity which is easily observable in cold-atoms experiments, unlike other measures of chaos, such as entanglement or OTOCs.
The work presents a detailed analysis of this claim for a particular noninteracting model (the Aubry-André model), which is also accesible in experiment. The work is clearly written, and addresses a timely subject.
My main reservation is that the study is limited to noninteracting systems, which can be easily simulated on classical computers (as the authors successfully do). In fact for system sizes much larger that are possible in cold atoms experiments. While the results of this work probably apply also to interacting systems, the times where the "correlation hole" will be visible becomes exponential in system size, making it impractical for cold atoms experiments, of about L>20 sites, which are currently limited to about (T~3000/J). Interacting systems with L<30 sites, can again be easily studied on a classical computer.
A minor point that I have is regarding notation. I feel that first quantization notation will be more clear in the context of this work, since the authors work in the limit of one particle (there is no finite particle density).
With this said, the work presents an interesting idea of how chaos emerges in simple physical observbles, and therefore deserves publication.
Author Response
We thank the Referee for carefully reading our work and for the very positive review, which made us very happy. We reply to his/her comments below.
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REFEREE:
My main reservation is that the study is limited to noninteracting systems, which can be easily simulated on classical computers (as the authors successfully do). In fact for system sizes much larger that are possible in cold atoms experiments. While the results of this work probably apply also to interacting systems, the times where the "correlation hole" will be visible becomes exponential in system size, making it impractical for cold atoms experiments, of about L>20 sites, which are currently limited to about (T~3000/J). Interacting systems with L<30 sites, can again be easily studied on a classical computer.
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AUTHORS:
The correlation hole is certainly seen in interacting models as well. This is what we have shown in various other papers that we wrote about disordered interacting chaotic models and even clean interacting chaotic models. The issue with these models is exactly what the Referee said, it takes very long times for the hole to be seen and this happens at tiny values. This is the main point of our work, the fact that we can still detect the hole, if we focus on simple noninteracting models, as the experimental Aubry-Andre model.
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REFEREE:
A minor point that I have is regarding notation. I feel that first quantization notation will be more clear in the context of this work, since the authors work in the limit of one particle (there is no finite particle density).
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AUTHORS:
We chose to keep the notation as it is, since this is the same used in new works about the Aubry-Andre model, such as the now cited Phys. Rev. B 100, 195143 (2019).
Round 2
Reviewer 1 Report
The authors responded positively to the remarks and have taken them, at least partially, into account. While a study of the relation between different parameters (ksi_1 and rbar) would be valuable, I agree that this would form a significant extension of the present work. I recommend publication in the present form.
