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|As Contributors:||Piero Naldesi · Tommaso Roscilde|
|Submitted by:||Roscilde, Tommaso|
|Submitted to:||SciPost Physics|
|Domain(s):||Theor. & Comp.|
|Subject area:||Quantum Physics|
The many-body localization (MBL) transition is a quantum phase transition involving highly excited eigenstates of a disordered quantum many-body Hamiltonian, which evolve from "extended/ergodic" (exhibiting extensive entanglement entropies and fluctuations) to "localized" (exhibiting area-law 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 many-body mobility edge. Here we propose to explore the latter mechanism by using "quantum-quench 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 quasi-periodic potential, we argue that this system has a many-body 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 quasi-periodic potential, as well as a transition in the scaling properties of the quasi-stationary state at long times. Our results suggest a practical scheme for the experimental observation of many-body mobility edges using cold-atom setups.
1-Systematic study of energy dependence after a global quench, revealing the many-body mobility edge
2-Clearly written and detailed
1-Finite size scaling is not very systematically analyzed.
This is a detailed study of the properties of the interacting quasi-periodic Aubry-Andre model, which shows a many-body 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 many-body 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 many-body 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 MBL---level spacing statistics, particle fluctuations and entanglement entropy---leading to a energy-density vs. potential strength phase diagram with a clear many-body 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 many-body 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 quasi-periodic 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 many-body 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 sub-thermal 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 systematic---or 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 many-body mobility edge. The skeptics that don't think that the data obtained in prior work, seemingly demonstrating a many-body 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 many-body localization in the interacting Aubry-Andre model.
1-Explain 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.
1- Very thorough analysis of ground-state , fintie energy density
and out-equilibrium 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 many-body mobility
4- Well written manuscript, very good presentation
1- More concrete predictions for experimentally accessible observalbles
besides particle number fluctuations would be desirable.
2- Some few omissions of related literature
The prediction of the existence of a many-body mobility edge in
interacting systems with quenched disorder is one of the key
results of the theory of many-body localization. So far there,
are no experimental attempts at verifying the existence of the
many-body mobility edge. Moreover, the very existence of a many-body
mobility edge has been questioned by some theoretical papers.
The present manuscript makes a concrete proposal for the observation
of the many-body mobility edge using a quench protocol that involves
quenching the strength of a quasi-periodic 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 ground-state and high energy
phase diagrams, the latter suggesting the existence of a many-body mobility edge in the model.
The authors proceed to study entanglement dynamics to characterize the quantum
quench dynamics as a measure for whether the post-quench state is in the
delocalized or localized part of the many-body spectrum. The results suggest that the many-body mobility edge could be resolved from both the long-time limit
and the short time-dynamics 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 set-up.
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.
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 half-cut 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 many-body 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 sub-diffusive 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 quasi-periodic potential.