Many-body delocalization dynamics in long Aubry-André quasiperiodic chains

Submission summary

 As Contributors: Elmer Doggen Arxiv Link: https://arxiv.org/abs/1901.06971v1 Date submitted: 2019-01-23 Submitted by: Doggen, Elmer Submitted to: SciPost Physics Domain(s): Theor. & Comp. Subject area: Condensed Matter Physics - Theory

Abstract

We theoretically study quench dynamics in an interacting one-dimensional spin chain with a quasi-periodic on-site field, also known as the interacting Aubry-Andr\'e model of many-body localization. Using the time-dependent variational principle we assess the late-time behaviour for chains up to $L = 50$. The choice of periodicity of the quasi-periodic potential influences the dynamics, which is explained in terms of properties of the non-interacting problem. Furthermore, the decay of antiferromagnetic order is faster than a power law, and finite-size effects on the value of the critical disorder are weaker than in the purely random case.

Current status:
Editor-in-charge assigned

Submission & Refereeing History

Submission 1901.06971v1 on 23 January 2019

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Strengths

1- Application of TDVP to the quasiperiodic (QP) interacting system

Weaknesses

1- Proper convergence of the method with time-step and bond-dimension is not established
2- Conclusion about the drift of the dynamical exponent is drawn from a regime were the data doesn't appear to be converged
3- The study doesn't bring much new information compared to literature.
4- Poor presentation, which doesn't allow to read the results conveniently

Report

The authors numerically study the dynamics in a quasiperiodic (QP) interacting spin chain, which shows a MBL like transition as a function of the strength of the QP potential.

In their introduction the authors claim:

"The aim of the present work is to investigate the differences between purely random
and quasi-periodic disorder at system sizes inaccessible to exact diagonalization, using the
newly developed numerical technique of the time-dependent variational principle as applied
to matrix product states"

This statement is however misleading, since dynamics in QP interacting systems was studied using MPS based method in both [37] (L=100-200) and 10.1073/pnas.1800589115, for systems of up to 800 sites (cf, to L=50 that the authors use in this work). In fact one of the accents of 10.1073/pnas.1800589115 is on finite-size effects and comparison to the disordered case.

However, my main concern is not even with the originality of the work, but with the validity of its results. The authors use a very low bond dimension (xi<64, compared to xi=1000-2000, many time used in TEBD), and their method (1-site TDVP) doesn't allow them to estimate the discarded weight. Moreover, the comparison between the two shown bond-dimensions are done on a logarithmic scale and for different "disorder" realizations, which doesn't allow to estimate the time until which the results are reliable.

From Fig 1 it is obvious that this time is not longer than t~100 (which is also probably a conservative estimate). This can also seen from the comparison to TEBD, which the authors provide in their reply. Although the graph is also on a log-log scale, for W=4, the divergence with TEBD occurs already at t<100, where TEBD results appear to converge, as far as one ca n judge from this plot. While one can see that TEBD saturates the bond dimension 256 already at times t=20, the actually meaningful information is the accumulated discarded weight, which the authors don't provide.

As on can see from Fig 3 for t<100 and W=4, the drift in the dynamical exponent, which is one of the main results of the work is not significant, and in any case probably lies within the error bars (which are not shown for some reason). For W>5 the oscillations in the exponent do not allow to extract a coherent message, though the convergence issues I raised above, are presumably better there.

Another issue, which is not discussed in the paper is the convergence with respect to the time-step. Following a request by the first referee the authors do specify the time-step dt=0.1, in their reply, but don't provide any evidence that this time-step is sufficient to go to t=300 and W=8 for example.

In view of the above I agree with the report of the first referee that the conclusions of this work are most probably based on not converged data, and are therefore doubtful. I therefore do not think that the paper stands by the standards of SciPost.

Requested changes

See report

• validity: low
• significance: low
• originality: low
• clarity: high
• formatting: reasonable
• grammar: excellent

Strengths

1. Application of time-dependent variational technique with matrix product states to the problem of MBL with quasi-periodic disorder
2. Introducing distribution of spin densities as a possible experimental observable.

Weaknesses

1. Results seem not to be converged - that makes the claims of faster than power-law decay of imbalance doubtful.
2. The effect found and discussed, the dependence of the system behavior on periodicity of quasi-periodic potential is not new.

Report

The manuscript addresses many-body localization (MBL) in quasi-periodic on-site disorder in isotropic Heisenberg chain model - a paradigmatic model used for MBL studies. While the manuscript lists some of earlier works on this very topic [20,23,26] it misses, however, at least one early study PRB 87, 134202 (2013). Those earlier studies used typically exact diagonalization combined with the finite size scaling to estimate the transition to MBL. The present work considers experimentally accessible observables such as imbalance and distribution of spin densities whose time dynamics is evaluated using time-dependent variational principle with matrix product states. The authors report as their main finding (abstract and conclusion) the strong dependence of the system properties on the periodicity of the quasi-periodic potential which they trace to properties of the non-interacting problem. They stress that the decay of the anti-ferromagnetic order is faster than a power law (I believe on the delocalized side of the MBL transition).

The field of MBL is quite well established by now so making a significant contribution in the field is not as straightforward as say 5 years ago. In my opinion the work presented while providing some incremental value to the field, certainly does not meet the Scipost Physics criterion: "Articles provide in-depth, detailed reports of groundbreaking research within one or more subject areas."

The main claim, a strong dependence of the system dynamics on the periodicity of the quasi-periodic potential is a bit trivial and not new. It is explained more than 10 years ago in detail in V. Guearrera et al. New. J. Phys. 9, 107 (2007). In fact the on-site energy difference distribution (plotted in Fig.2 right panel of the manuscript) may be easily found analytically as proportional to $1/\sqrt{1-x^2}$ distribution with the amplitude of the effective disorder $W\sin(\pi\Phi)$ as it follows from that paper. Similar strong dependence on the quasi-periodicity was found in an interaction free problem by Major et al. Phys. Rev. A 98 053633 (2018).

While the application of the variational principle MPS approach to time dynamics seems to be an original contribution, this has been already reported in [22] for a purely random case. So one can hardly give credit for that. Especially, as time dependent dynamics using MPS (with e.g. TEBD) for MBL has been studied in a routine way, see [8], also
J. H. Bardarson, F. Pollmann, and J. E. Moore, Phys. Rev. Lett. 109, 017202 (2012)
P. Sierant, D. Delande, and J. Zakrzewski,Phys. Rev. A 95, 021601(R) (2017)
for different implementations.

The variational approach might have some advantage for long times due to the inherent energy conservation, the problem deserves a separate careful study in the MBL regime and close to it - comparisons as those in 1901.05824 do not suggest a clear advantage of the variational approach.

There is a strong believe that the system studied shows the mobility edge - the transition to MBL depends on energy (see Luitz et al Phys. Rev. B 91, 081103 (2015)). The present study considers a single initial anti-ferromagnetic state whose energy depends strongly on the disorder realization. This may affect the value of the critical disorder obtained. The question of the mobility edge is entirely ignored in the manuscript.

Apart from these general remarks I believe that the authors should (in some future submission) consider the following remarks
1. Results for $\chi=32$ and $\chi=64$ seem unconverged (Fig. 1a and Fig.1b). Why not show convergent results for sufficiently large $\chi$?
2. The present Fig.1 is unreadable - it requires huge enlarging to distinguish different curves.
3. Let me mention that standard TEBD results with $\chi=250$ do not agree with those reported in Fig.1.
4. Why the authors do not provide information that would allow a reader to reproduce their results? What was the time step in time-dependent integration? What was the accuracy criterion (none - just $\chi$ was defined is one of the options)?
5. The faster than power-law decay of the imbalance on the delocalized side, the main conclusion of the manuscript, is deduced from not converged results.
6. The claim for the critical disorder to be $W_c=5$ is a rough estimate only. The text suggest error bar of the order of 0.5. Can the authors do better with the converged results?
7. Authors claim a weak dependence on the system size for quasi-periodic potential in the introduction but they do not substantiate this claim further - showing just a single curve for $L=16$. How $W_c$ depends on the system size?

Requested changes

see above

• validity: low
• significance: low
• originality: ok
• clarity: ok
• formatting: good
• grammar: perfect

Author Elmer Doggen on 2019-03-07 (in reply to Report 1 on 2019-02-15)
Category:

Following a possibility provided by SciPost, we provide a preliminary response to the Referee's comments (Report 1). We will finalize the response and resubmit the paper, once the Editor-in-Charge invites us to do so upon the end of the refereeing round.

We thank the Referee for a detailed report. The referee's comments motivated us to improve the rigour of our results with some additional numerical computations, which will be added to a future submission. In addition, we will improve the presentation of our work and the clarity of our plots, as well as providing more details about our numerical method and the benefits thereof compared to older methods like time-evolving block decimation (TEBD). On the other hand, we disagree with the referee's assessment of our work as having low significance and validity. We address all critical remarks of the referee below.

REFEREE:
"The manuscript addresses many-body localization (MBL) in quasi-periodic on-site disorder in isotropic Heisenberg chain model - a paradigmatic model used for MBL studies. While the manuscript lists some of earlier works on this very topic [20,23,26] it misses, however, at least one early study PRB 87, 134202 (2013)."

We thank the referee for pointing out this reference that is indeed relevant - a future submission will include a citation.

REFEREE:
"Those earlier studies used typically exact diagonalization combined with the finite size scaling to estimate the transition to MBL. The present work considers experimentally accessible observables such as imbalance and distribution of spin densities whose time dynamics is evaluated using time-dependent variational principle with matrix product states. The authors report as their main finding (abstract and conclusion) the strong dependence of the system properties on the periodicity of the quasi-periodic potential which they trace to properties of the non-interacting problem. They stress that the decay of the anti-ferromagnetic order is faster than a power law (I believe on the delocalized side of the MBL transition)."

There are in fact three main conclusions that are listed clearly in the summary and are mentioned also in the abstract: the dependence of the MBL transition on $\Phi$, the faster-than-power law decay on the delocalized side and the relatively weak dependence of the transition on system size. We will improve the wording in the abstract to make this clearer.

REFEREE:
"The field of MBL is quite well established by now so making a significant contribution in the field is not as straightforward as say 5 years ago. In my opinion the work presented while providing some incremental value to the field, certainly does not meet the Scipost Physics criterion: "Articles provide in-depth, detailed reports of groundbreaking research within one or more subject areas." "

We disagree with this assessment. In our view our study provides an important addition to the existing literature by studying larger systems up to late times, which is of obvious relevance to current experiments. The differences between the MBL transition in quasiperiodic and purely random systems is currently under active discussion and our work provides much needed numerical analysis to help settle this debate.

REFEREE:
"The main claim, a strong dependence of the system dynamics on the periodicity of the quasi-periodic potential is a bit trivial and not new. It is explained more than 10 years ago in detail in V. Guearrera et al. New. J. Phys. 9, 107 (2007). In fact the on-site energy difference distribution (plotted in Fig.2 right panel of the manuscript) may be easily found analytically as proportional to $1/\sqrt{1−x^2}$ distribution with the amplitude of the effective disorder $W\sin(\pi \Phi)$ as it follows from that paper. Similar strong dependence on the quasi-periodicity was found in an interaction free problem by Major et al. Phys. Rev. A 98 053633 (2018). "

We thank the referee for pointing out the important reference New. J. Phys. 9, 107 (2007), of which we were not aware. The effective disorder mentioned in that work matches with our Fig. 2b after correcting with an additional factor of 2 due to their potential using a squared sine. Our "main claim" is not about the single-particle problem; indeed, that the periodicity strongly affects dynamics was pointed out, as mentioned in our manuscript, also in Ref. [36], Phys. Rev. A 80, 021603 (2009) for the extended AA model. Rather, one of the key results is that this single-particle feature carries over to the many-problem directly and leads to an apparent dependence of the critical disorder on the periodicity $\Phi$ - a non-trivial and to our knowledge new result contrary to the non-interacting case where the critical disorder is always $W = 2$ (in our units) for almost any irrational $\Phi$. That there are still interesting results from recent publications such as in [Phys. Rev. A 98 053633] even for the non-interacting problem underscores that there is still much to learn in the interacting problem.

REFEREE:
"While the application of the variational principle MPS approach to time dynamics seems to be an original contribution, this has been already reported in [22] for a purely random case. So one can hardly give credit for that. "

In Ref. [22] we demonstrated that our method works well for the MBL problem with purely random disorder, allowing to reliably study large systems and long times on the ergodic side of the transition. Hence, it is natural and important to use it also for the quasi-periodic case and compare the results.

REFEREE:
"Especially, as time dependent dynamics using MPS (with e.g. TEBD) for MBL has been studied in a routine way, see [8], also
J. H. Bardarson, F. Pollmann, and J. E. Moore, Phys. Rev. Lett. 109, 017202 (2012)
P. Sierant, D. Delande, and J. Zakrzewski,Phys. Rev. A 95, 021601(R) (2017)
for different implementations.

The variational approach might have some advantage for long times due to the inherent energy conservation, the problem deserves a separate careful study in the MBL regime and close to it - comparisons as those in 1901.05824 do not suggest a clear advantage of the variational approach."

We thank the referee for pointing out these references, which will be cited appropriately. Our method provides considerable advantages over TEBD specifically for the MBL problem (see also our response to point 3 below). Indeed, the main advantage of the method is the inherent energy conservation, as opposed to the non-conservation of energy in e.g. TEBD. As a side note, in the language of 1901.05824 we use a 1TDVP algorithm, not a 2TDVP algorithm. The latter does not conserve energy for otherwise energy-conserving Hamiltonian dynamics due to the truncation step induced by the two-site approach.

REFEREE:
"There is a strong believe that the system studied shows the mobility edge - the transition to MBL depends on energy (see Luitz et al Phys. Rev. B 91, 081103 (2015)). The present study considers a single initial anti-ferromagnetic state whose energy depends strongly on the disorder realization. This may affect the value of the critical disorder obtained. The question of the mobility edge is entirely ignored in the manuscript."

A dependence of the critical disorder on energy in this system is indeed interesting, but we do not expect the essential properties of the MBL transition to depend on energy (as long as we are not too close to the ground state). A large majority of studies of MBL only target the middle of the spectrum and likewise ignore the question of a mobility edge. We thus leave the investigation of the energy-dependence of the critical point and verification of the universal character of the transition to future work.

REFEREE:
"Apart from these general remarks I believe that the authors should (in some future submission) consider the following remarks
1. Results for $\chi=32$ and $\chi=64$ seem unconverged (Fig. 1a and Fig.1b). Why not show convergent results for sufficiently large $\chi$?"

We might have shown only $\chi = 64$ and terminate the lines where convergence with bond dimension is lost, but opted to show results for both $\chi = 32$ and $\chi = 64$ for transparency. We believe that this additional information is useful for a reader. Concerning the value of $\chi$: we find that our TDVP implementation is slower than TEBD using the same $\chi$, so we cannot compute results for these long times with $\chi$ much higher than $64$ with realistic computational resources. This is due to the implicit time integration scheme used in the split-step 1TDVP algorithm, meaning we have to solve a set of equations at each time step. For this purpose evoMPS uses an iterative scheme, which does not always converge quickly. This is a "price" that one pays for the energy conservation in this scheme. The energy conservation, is, however, very important: the energy-non-conserving TEBD scheme becomes unreliable at shorter times (despite large values of $\chi$) - at least for this system - see our response to question 3 below.

REFEREE:
"2. The present Fig.1 is unreadable - it requires huge enlarging to distinguish different curves. "

Indeed, in each of the panels of Fig. 1 there are two curves that lay essentially on top of each other. In the left ($W=4$) and middle ($W=5$) panel this happens because the convergence with bond dimension has been reached, and we want to demonstrate this. In the right ($W=8$) panel these are two curves with lengths $L=16$ and $L=50$; here this demonstrates very weak length dependence. We will update our figures in a resubmission to enhance clarity (by adding some insets) and will also make clarifying comments in the figure caption.

REFEREE:
"3. Let me mention that standard TEBD results with $\chi=250$ do not agree with those reported in Fig.1."

Indeed, the results from TEBD and TDVP on the delocalized side of the transition begin to deviate for long times. This is related to the non-conservation of energy in the TEBD method.
Motivated by the referee's comment, we have computed dynamics for $\Phi = [\sqrt{5}-1]/2$ and $W=4$ (delocalized side of the transition) using TEBD with $\chi = 128$ and $\chi = 256$. The plots are added as attachments, with TDVP results ($\chi = 64$) shown in blue and TEBD results in green and red. In addition to the imbalance, we show there also the energy as well the bond dimension used by TEBD (which saturates at the maximum value, which is 128 for one plot and 256 for the other).
Here the TEBD results are computed using the OSMPS library using default convergence parameters using around 150 realizations. As is visible from the plots, TEBD results start to deviate significantly from TDVP exactly where large errors in the energy become apparent in the TEBD results. The corresponding time is $t \approx 70$ for TEBD bond dimension 128, and somewhat later for TEBD bond dimension 256, so that the TEBD results are reliable for longer times when TEBD bond dimension is increased, as expected.

REFEREE:
"4. Why the authors do not provide information that would allow a reader to reproduce their results? What was the time step in time-dependent integration? What was the accuracy criterion (none - just $\chi$ was defined is one of the options)?"

As was briefly mentioned in the manuscript, our implementation of the method follows Ref. [22] and the bond dimension is indeed the only parameter used to establish convergence. We agree, however, with the Referee that a more detailed exposition of the method will be beneficial for the reader. We will provide some more details and references about the numerical method and how it compares to "traditional" MPS methods in a resubmission. The time-step used is 0.1 in lattice units. We use open source MPS libraries and are happy to share our implementations thereof upon reasonable request.

REFEREE:
"5. The faster than power-law decay of the imbalance on the delocalized side, the main conclusion of the manuscript, is deduced from not converged results."

The faster-than-power law decay is apparent in the converged results. In Fig. 3 we show the "flowing power law" $\beta$, which is an effective time-dependent power law. As is seen from the results, for $\Phi = (\sqrt{5}-1)/2$ convergence with $\chi$ is achieved up to $t \approx 200$, and a clear increase in $\beta(t)$ (roughly by a factor 2.5) is observed already by this time. Note that we did not observe this trend for purely random disorder in Ref. [22], so that we see a clear difference between purely random and quasiperiodic systems within the same approach. The non-converged results in the left panel of Fig. 1 also show a strong curvature, i.e. a faster-than-power law decay. While it is expected that the curvature remains when the converged is reached, we do not draw any conclusions from those results. We will further clarify this in the text in the new version.

REFEREE:
"6. The claim for the critical disorder to be $W_c=5$ is a rough estimate only. The text suggest error bar of the order of 0.5. Can the authors do better with the converged results?"

We still find a very weak decay of imbalance for $W = 4.5$, $L = 50$, and no visible decay for $W=5$, which yields the estimate for the critical disorder and the error bar of order 0.5. It is very difficult to improve the error bar substantially and, at the same time, reliably. In fact, 10% is a pretty good accuracy for determination of the MBL transition. We remind the referee that, for the purely random case, the best value obtained by Luitz et al. from exact diagonalization was $\approx 3.7$, whereas the true transition (based on our previous results in Ref. [22]) is in fact near 5.5, i.e. at 50% higher disorder. In many cases, small error bars quoted in the literature do not correspond to reality.

REFEREE:
"7. Authors claim a weak dependence on the system size for quasi-periodic potential in the introduction but they do not substantiate this claim further - showing just a single curve for $L=16$. How $W_c$ depends on the system size?"

In the case of purely random disorder, the apparent position of the transition changed from $W = 3.5-4$ for $L = 16$ to $W \approx 5.5$ using $L=50$ and $L = 100$, which is a 50% increase. For the quasi-periodic case, we do not observe such a strong increase: the value remains the same within 10% accuracy. We will add a plot to a future resubmission showing more results for different $L$ and $W$, to show this more clearly.