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Detecting a manybody mobility edge with quantum quenches
by Piero Naldesi, Elisa Ercolessi, Tommaso Roscilde
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Authors (as registered SciPost users):  Piero Naldesi · Tommaso Roscilde 
Submission information  

Preprint Link:  http://arxiv.org/abs/1607.01285v2 (pdf) 
Date submitted:  20160828 02:00 
Submitted by:  Roscilde, Tommaso 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
The manybody localization (MBL) transition is a quantum phase transition involving highly excited eigenstates of a disordered quantum manybody Hamiltonian, which evolve from "extended/ergodic" (exhibiting extensive entanglement entropies and fluctuations) to "localized" (exhibiting arealaw scaling of entanglement and fluctuations). The MBL transition can be driven by the strength of disorder in a given spectral range, or by the energy density at fixed disorder  if the system possesses a manybody mobility edge. Here we propose to explore the latter mechanism by using "quantumquench spectroscopy", namely via quantum quenches of variable width which prepare the state of the system in a superposition of eigenstates of the Hamiltonian within a controllable spectral region. Studying numerically a chain of interacting spinless fermions in a quasiperiodic potential, we argue that this system has a manybody mobility edge; and we show that its existence translates into a clear dynamical transition in the time evolution immediately following a quench in the strength of the quasiperiodic potential, as well as a transition in the scaling properties of the quasistationary state at long times. Our results suggest a practical scheme for the experimental observation of manybody mobility edges using coldatom setups.
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Reports on this Submission
Anonymous Report 2 on 2016920 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1607.01285v1, delivered 20160920, doi: 10.21468/SciPost.Report.22
Strengths
1Systematic study of energy dependence after a global quench, revealing the manybody mobility edge
2Clearly written and detailed
Weaknesses
1Finite size scaling is not very systematically analyzed.
Report
This is a detailed study of the properties of the interacting quasiperiodic AubryAndre model, which shows a manybody localization (MBL) transition. Starting from a ground state phase diagram (obtained with a combination of Monte Carlo and DMRG) and the observation that reversing the sign of the interaction takes the ground state to the highest excited state, the authors argue that this model necessarily exhibits a manybody mobility edge, since the phase diagram is asymmetric in the interaction strength. That is, for a given set of parameters close to the MBL transition, the manybody spectrum consists of separate regions of extended and localized states. For small system sizes, accessible by exact diagonalization, this observation is corroborated by calculating standard measures of MBLlevel spacing statistics, particle fluctuations and entanglement entropyleading to a energydensity vs. potential strength phase diagram with a clear manybody mobility edge.
At this point, there is perhaps not much new apart from details, but these are useful details. My only criticism here would be that the extraction of critical disorder values seems to be done essentially by eye, and it's not clear that the error bars of this procedure are not much larger than what is given. For example, in Fig. 3, there is never a regime where the level spacing statistics in the localized regime actually gets all the way to the Poisson value. The level spacing statistics in this regime also depends only weakly on the system size, and it's not at all clear that the grey window accurately captures the proper uncertainty in where the transition actually takes place. This is important since the existence of the manybody mobility edge means that the transition depends on energy, and the error bars might be so large that it would be hard ton convincingly conclude from this data that the mobility edge is there. The same can be said about the data for the entanglement entropy. That being said, the data is reasonably good, and it is likely hard to do much better than this for energy densities close to the edges of the spectrum.
The main new element in this work is the systematic study of the energy density dependence of quantum dynamics after a global quench. The particular protocol that is chosen here is to chance the amplitude of the quasiperiodic potential, which is also experimentally possible and relevant, starting from the ground state at \Delta_i and quenching to \Delta_f. Depending on how much this amplitude is changed, the amount of energy density change compared with the ground state of the final Hamiltonian changes, and this is plotted in Fig. 6. One observes that it's possible to tune this energy density from being small (final energy close to the ground state energy) to being large (final energy close to the maximum energy state). Via this quench protocol the authors are able to systematically study the energy dependence of the dynamics and thereby observe the manybody mobility edge in dynamical properties.
The dynamical properties that the authors study are, like the eigenstate properties studies before, standard measure of MBL, namely entanglement entropy and particle number fluctuations. In the ergodic phase the entanglement increases quickly to the expected ergodic value, while in the localized phase it increases logarithmically to a subthermal value. The results that are obtained are what is expected and are consistent with the several prior studies that have studied these quantities. As noted above, the main contribution of this study is the systematic study of the energy dependence in the particular quench they look at.
Again here, as in the eigenstate study, I find the estimation of the critical disorder strength not to be very systematicor at least it is not systematically and clearly explained. The comparison with the grey regions obtained from the level spacing statistic has overlap with peaks in the energy derivative of the entanglement and particle number fluctuations. However, looking at the energy level spacing data it seems equally reasonable that the transition is at somewhat higher energy densities, and then the agreement would seem coincidental.
In any case, a more systematic extraction of critical energy densities and error bar analysis would be nice, but it's unlikely to change things much qualitatively, but I think it will change the certainty with which one can state that there is a manybody mobility edge. The skeptics that don't think that the data obtained in prior work, seemingly demonstrating a manybody mobility edge, are perhaps not likely to be completely swayed be these results. The study is nevertheless solid and detailed and it is a good addition to the literature on manybody localization in the interacting AubryAndre model.
Requested changes
1Explain in more detail how the estimates of the critical energies for the MBL transition is obtained (the grey regions in several plots) and give arguments for why this is a reliable estimate. If possible, present a more systematic finite size scaling analysis to further support the data presented.
Anonymous Report 1 on 201698 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1607.01285v1, delivered 20160908, doi: 10.21468/SciPost.Report.10
Strengths
1 Very thorough analysis of groundstate , fintie energy density
and outequilibrium properties of the model.
2 Broad spectrum of theoretical methods employed in the work
(exact diagonalization, DMRG, QMC)
3 Original proposal for the observation of the manybody mobility
edge
4 Well written manuscript, very good presentation
Weaknesses
1 More concrete predictions for experimentally accessible observalbles
besides particle number fluctuations would be desirable.
2 Some few omissions of related literature
Report
Report
The prediction of the existence of a manybody mobility edge in
interacting systems with quenched disorder is one of the key
results of the theory of manybody localization. So far there,
are no experimental attempts at verifying the existence of the
manybody mobility edge. Moreover, the very existence of a manybody
mobility edge has been questioned by some theoretical papers.
The present manuscript makes a concrete proposal for the observation
of the manybody mobility edge using a quench protocol that involves
quenching the strength of a quasiperiodic potential, which are used
in many optical lattice experiments to study localization physics.
The paper contains a very thorough analysis of the equilibrium properties
of the model, both in the ground state as well as for finite energy densities.
Most notably, this analysis is summarized in groundstate and high energy
phase diagrams, the latter suggesting the existence of a manybody mobility edge in the model.
The authors proceed to study entanglement dynamics to characterize the quantum
quench dynamics as a measure for whether the postquench state is in the
delocalized or localized part of the manybody spectrum. The results suggest that the manybody mobility edge could be resolved from both the longtime limit
and the short timedynamics of entanglement measures.
The paper is very timely, of clear relevance to the field, very well written
and the results and conclusions are convincing. The results will be of interest
to both theoretical and experimental researchers interested in MBL. With regard
to experiments, one must admit that measuring entanglement in a disordered system is not possible yet. Thus the most relevant predictions for an experiment are those for the particle number fluctuations. Perhaps the authors could elaborate more on what an experimentalist should measure in this particular quench setup.
Other than that, I have only very minor suggestions and comments (see below)
and I recommend publications of the manuscript after the authors have given
considerations to the points mentioned here.
Requested changes
1 If possible, then it would be desirable to elaborate more on the
behavior of experimentally accessible observables (other than particle number fluctuations) and signatures of the mobility edge in their dynamics.
2 The observation of an area law in the MBL goes back to a paper by Bauer and
Nayak, which in my opinion should be cited here.
3 TDMRG was developed in parallel efforts by White and Feiguin and Daley et al.
J. Stat. Mech.: Theory Exp. (2004), P04005. The latter reference is missing,
perhaps it would also be fair to cite Vidal Phys. Rev. Lett. 93, 040502 (2004) as well, which was very important in the methodological developments.
4 Figure 2 shows little contrast between the LL and the CDW phase. Isn't that just a coincidence? in the CDW phase, there should be an area law and hence for large L, the halfcut entanglement in the LL phase should exceed that in the CDW phase.
5 By comparing ground state and finite energy density properties, can the authors speculate about the existence of an inverted mobility edgde (i.e., a transition from the LL into the MBL phase as energy density increases)?
6 By inspection of Fig. 8 it seems that the estimate of the position of the manybody mobility edge from the quantum quench disagrees with the result of equilibrium calculations by a factor of 2. Perhaps this could simply be states, the authors on page 8, just before Sec C is perhaps a bit too optimistic.
7 There is no final agreement yet about the existence of a subdiffusive regime in the ergodic phase. Numerical results for sigma_dc consistently report sigma_dc >0, inconsistent with subdiffusive dynamics (see Steinigeweg et al. arXiv:1512.08519 and Barisic et al. Phys. Rev. B 94, 045126 (2016)).
8 I'd like to draw the authors' attention to Phys. Rev. Lett. 115, 230401 as well, another study of MBL in a 1D system with a quasiperiodic potential.
Author: Tommaso Roscilde on 20161021 [id 70]
(in reply to Report 1 on 20160908)
We would like thank the Referee for the appreciation expressed towards our work, and for the useful suggestions. We reply to the requests one by one:
1  This is a delicate point, and, as a matter of fact, it is the subject of our current investigations. As we already stated in the previous version of the manuscript, we believe that quantumgas microscopy can extract salient features of the quenchinduced MBL transition by monitoring the dynamics of correlation spreading, as done in famous previous experiments on nondisordered onedimensional lattice gases. The precise form of the dynamics of correlations is a subject on its own, which in our opinion goes well beyond the scopes of the present manuscript. Nonetheless we have expanded our discussion of this experimental signature in the coda of the manuscript;
2  We agree with the suggestion, and we have added the corresponding citation;
3  Same as at point 2;
4  The little contrast is a coincidence, as discussed already in the text: the slow logarithmic size dependence of entanglement in the LL phase produces a halfchain entropy which, for the system sizes we considered, is nearly equal to that produced by the double degeneracy of the ground state in the CDW phase;
5  We could not observe such an inverted mobility edge in our calculations  in the model under investigation a groundstate LL phase always evolves into an extended phase as the energy density is turned on. It would obviously be very fascinating to observe such a counterintuitive phenomenon, although we are not aware of a microscopic model featuring it;
6  The width of the grayshaded area in Fig. 8 (now Fig. 9) has been corrected, and the agreement between the "transition region" and the onset of ergodicity in our finitesize data has somewhat improved. Nonetheless we already commented diffusely in our original manuscript about this agreement (which we termed as "reasonable"), and in particular on the fact that the entropy density s_Q produced by the quench dynamics might be underestimated, which in turns produces an overestimation of the injected energy density required to reach ergodicity. We do not find our claims to be particularly strong and "optimistic" on this topic  the strongest claim is that two distinct dynamical regimes exist, tuned by the quench amplitude, but we think that such a claim is rather objective in the face of our results;
7  This is an interesting remark, although we are not sure in what sense our observations connect (or clash) with the estimate of a finite conductivity at finite temperature in contact with a heat bath. Here we are exploring the farfromequilibrium unitary dynamics of the system in the absence of any heat bath, whereas the two references cited by the Referee focus on the linear response to an oscillating field for the system in thermal in equilibrium with a bath. The fact that the dc conductivity is always finite at any finite temperature in the latter system is per se not inconsistent with the subdiffusive behavior clearly observed in the unitary dynamics of the system we studied. Perfect insulation does not exist in disordered systems  incoherent hopping assisted by the thermal bath is in principle always possible: see e.g. the famous variablerange hopping mechanism predicted by Mott;
8  We were aware of this reference  which goes along with a number of other ones concerning models with a quasiperiodic modulation of the parameters (either hopping or onsite potential or both). The article in question did not investigate the standard interacting AubryAndrÃ© model, but a variant thereof, which possesses a mobility edge already at the level of the singleparticle spectrum. On the other hand we focused on a situation in which a mobility edge exists uniquely as a manybody effect, whence our choice of the standard AA model. Nonetheless we have added a reference to this work in the context of previous studies of the MBL transition via quenches from random factorized states.
Author: Tommaso Roscilde on 20161021 [id 69]
(in reply to Report 2 on 20160920)We thank the Referee for the positive report, and for the useful remarks. We have revised all our estimates of the transition points and error bars estimated from the scaling of the adjacentgap ratio, now obtained in a systematic way in the data reported on the spectral phase diagram, and correctly reflected by the width of the grayshaded areas in the other figures. Several naive mistakes had been made in the previous version in both aspects, which are now removed. The criteria for estimating the transition point and its error bar are now explicitly stated in the text: they are based on the crossing points of the adjacentgap ratio r among the four largest sizes we considered, namely L=16 with L=14, and L=15 with L=13 (only the crossing between sizes of the same parity is considered, given that evenodd effects are present in our results due to the half filling condition which is differently realized in the two cases). The error bar simply reflects the spreading of these crossing points, and it therefore identifies the region over which we observe an inversion of scaling for the four largest system sizes. The width of all the shaded areas reported in the figures now reflects the error bar estimated in this way  we have modified Figs. 3, 6, 7, 9 and 11 in this sense.
We believe that above criterion leads to a rather honest and conservative estimate of the error bar. Even within this estimate of the error bar, the extendedMBL transition line appears to be clearly dependent on the energy density, revealing the existence of a mobility edge. The fact that the data in Fig. 3 never reach the Poisson value is easily understood in that, unlike in previous studies, here the r ratio is plotted as a function of the energy density at *fixed* strength of the quasidisordered potential \Delta, something which does not allow the localisation length to become arbitrarily small, and, in particular, much smaller than the system size: under this condition, it should not be possible to observe Possonian statistics in the proper way. We have added an explicit remark in the text. On the other hand, as shown in the falsecolor part of the spectral phase diagram (now Fig. 5 of the new version) there are \Delta values (such as \Delta ~ 3) for which the excursion from the levelrepulsion statistics to the Poissonian statistics is better realized. To be even more convincing in this sense, we have added to our manuscript a new figure (Fig. 4) containing the scaling of the r ratio as a function of \Delta for two different values of the energy density. Working at variable \Delta allows to approach arbitrarily well the Poissonstatistics value for r. In particular, this figure shows rather clearly that the crossing regions between the various r curves (estimated as declared above) are nonoverlapping for different energy densities, witnessing again the existence of a mobility edge.
Coming to the data concerning the dynamics, we do not fully understand the remark of the Referee concerning the the overlap between the peaks in the energy derivative of the entanglement and fluctuations, and the estimate of the transition region from the level statistics  the latter is an analysis which pertains uniquely to the study of the spectral properties. We disagree with the idea that the correspondence between the two estimates of the transition could be coincidental, given the clear trend that they both follow in the (\Delta,\epsilon) plane in Fig. 5. The estimate of the mobility edge stemming from the dynamics is shown in Figs. 9 and 11, both for what concerns the longtime dynamics (Fig. 9) as well as the shorttime one (Fig. 11). It seems hardly coincidental to us that full ergodicity at long times is only observed when the injected energy density clearly exceeds the mobility edge estimated from level statistics, and that e.g. the logarithmicint onset of particle number fluctuations appears only once that same threshold has been exceeded.