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Flow Equations for Disordered Floquet Systems

by S. J. Thomson, D. Magano, M. Schiró

Submission summary

As Contributors: Steven Thomson
Arxiv Link: (pdf)
Date submitted: 2021-01-01 13:30
Submitted by: Thomson, Steven
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Quantum Physics
Approach: Theoretical


In this work, we present a new approach to disordered, periodically driven (Floquet) quantum many-body systems based on flow equations. Specifically, we introduce a continuous unitary flow of Floquet operators in an extended Hilbert space, whose fixed point is both diagonal and time-independent, allowing us to directly obtain the Floquet modes. We first apply this method to a periodically driven Anderson insulator, for which it is exact, and then extend it to driven many-body localized systems within a truncated flow equation ansatz. In particular we compute the emergent Floquet local integrals of motion that characterise a periodically driven many-body localized phase. We demonstrate that the method remains well-controlled in the weakly-interacting regime, and allows us to access larger system sizes than accessible by numerically exact methods, paving the way for studies of two-dimensional driven many-body systems.

Current status:
Editor-in-charge assigned

Author comments upon resubmission

We would like to thank the Editor for clarifying a point raised by the Referee that we have misunderstood in our first version of the reply. In our previous response, we focused on demonstrating that the emergence of CE level statistics at low drive frequencies was a genuine feature of the model and a direct consequence of the drive protocol, but we did not address the question of whether this is a true transition in the thermodynamic limit.

We would like to kindly ask the Referee to take into account the following reply in addition to our previous one, specifically concerning the Referee's first comment, which we reproduce below.

1a) "The authors discuss the 1d model and state that they find a transition as a function of driving frequency to a delocalized phase. To my understanding this is inconsistent with the literature. The Floquet-Hamiltonian is quasi one-dimensional and even weak disorder will lead to localization. As discussed in Ref. [88] the dynamics appears diffusive over a single drive cycle, but the system remains localized. How does this fit the numerical results in this paper? Also, one really needs to look at different system sizes L before drawing any conclusions. The authors only vary $N_h$ which defines the truncation in Sambe space but not L. Why? Larger systems are discussed for the interacting model, but not here for the non-interacting model. These points need to be clarified/improved/corrected."

The Referee is correct in noticing that what we had previously identified as a delocalization transition as a function of the drive frequency is in fact a crossover.

In the new Appendix C, following the Referee's suggestion, we present results for the level statistics which show how increasing the system size changes the position of the crossover such that the system in the thermodynamic limit remains localised at all frequencies. We have modified the manuscript accordingly (main text and new Appendix C).

We notice nevertheless that the properties of the finite-size driven Anderson model depend strongly on the nature of the driving protocol. In particular we show in Appendix C that this crossover is absent for a monochromatic drive, where the frequency and size dependence of the level statistics is very weak and the system always remains localised.

List of changes

Added clarifying remarks about the nature of the delocalization crossover seen in the driven non-interacting system as the system size is changed.

Submission & Refereeing History

Resubmission 2009.03186v2 on 23 December 2020

Reports on this Submission

Anonymous Report 2 on 2021-2-8 (Invited Report)


This is a well written manuscript that serves as a very good introduction to the flow equation formalism applied to Floquet systems. However, for the same reason it is also necessary to be very careful in describing strengths and weaknesses of the method.

1) My main criticism centers around the sentences:
- Page 13: “There is, however, no divergence encountered due to resonant terms, as in typical perturbative treatments of disordered systems: the sole effect of closely separated on-site energies is the need for a large flow time lmax”
- Page 20: “Note that this truncated form of K(l) is not the result of a perturbative analysis, but instead arises from a truncation in operator space which is valid in the localized phase. Flow equation methods based around Wegner-type generators therefore avoid problems with resonances [60] which can otherwise lead to divergent terms in perturbative treatments of interacting systems” in combination with “variational manifold” below Eq. (39).

Indeed, for a quadratic Hamiltonian like the driven Anderson model (page 13) there are no problems with divergences because the unitary flow is exact in the sense that no higher order terms are being generated. So the only complication due to resonances are long flow times, exactly like the authors say. However, for interacting systems (page 20) the situation is different. The very slow decay of couplings in any given order of the flow equation calculation because of (near) resonances generically leads to large new interactions being generated in the next order. So for the interacting system the situation is worse than just needing long flow times, and the problem with resonances is not avoided (unfortunately). The way to expicitly see this is by going beyond the terms taken into account in the ansatz (39), which is of course a bit of work.

However, on the plus side this does provide a check for the internal consistency of the method if the newly generated and neglected terms indeed remain small. It is not even necessary to take the whole flow of the newly generated terms into account, looking at the source term in their flow equations will be sufficient. One can speculate that normal ordering might be helpful, but there is no obvious reason why this should be the case for a localization-delocalization scenario. For the above reasons the term “variational” is also somehow misleading.

At a minimum the authors should change the sentences on page 13 and 20 to correctly describe the limitation of the method. Additionally, I would encourage them to do an error analysis by looking at the size of the newly generated terms in order to show the internal consistency of the approach. This would also address point 9) raised by the second referee. Once this is done I strongly support publication of this well written manuscript.

2) I appreciate the authors’ clarifying comments regarding the localization-delocalization crossover vs. transition. On page 21 (Sect. 6.3) the authors still use the term transition, this should be changed.

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Anonymous Report 1 on 2021-2-1 (Invited Report)


Very nice and detailed overview of a flow equation method for Floquet systems.


1. No self consistent way to estimate an error within the method in larger systems (beyond ED) is presented.

2. No examples of non trivial reliable results beyond exact diagonalization


I really enjoyed reading this paper. It is written in a very pedagogical way. I learned a lot. I have several questions/comments. Some of them minor, some are more conceptual. I will list them together in the order they appear in the text.

1) Equation (12). I think the notation is not the best. LHS is a state, RHS is an operator. I do not think this equality is mathematically correct.

2) I am not sure I agree with the discussion after Eqs. (26) and (27). The authors say that $\omega\to 0$ corresponds to the static limit. Is that so? I would say $\omega\to \infty$ does as the authors later explain. I would think the zero frequency limit is ill defined without cutoff in $N_h$, there is an extra infinite sum compared to the static case.

3) I wonder if in these equations there is any advantage in going to the co-moving frame $h_i^{(n)}\to h_i^{(n)} \exp[-n^2 \omega^2 l]$ and similarly for $J_i^{(n)}$. Then there is an explicit exponential decay in all the non-zero terms in the sum and various approximations are more transparent.

4) It is very difficult to interpret different lines in Fig. 3 without legends/explanations. In the printed version in black and white the figure is simply unreadable.

5) It looks to me that the bottom plot in Fig. 4 does not support very well the author's claim. If I only look at the figure then I would guess that the line does not stop at the asymptotes and keep going up (at small $\omega$) and keep decreasing (at large $\omega$). More points are needed to justify the conclusion. Also the choice of colors for the top panel is far from optimal with largest and smallest frequency being identical.

6) There is a mistake in the inline expression for the Wigner-Dyson distribution on page (17). First of all there is a wrong sign and second this is not the Wigner-Dyson distribution, this is its approximation, i.e. it is the Wigner surmise.

7) I am completely lost in the motivation behind Sec. 5.4, i.e. truncating the flow to a single harmonic and the top panel in Fig. 5. To me this approximation strongly devaluates the method. Why not use the same number of harmonics as everywhere. Also as I mention below keeping all the harmonics should allow, I believe, to have an independent way of estimating the error without need of any ED.

8) What is variational in the ansatz (39)? Is it a "better" word for "truncated" or there is some variational optimization involved? What is minimized then? Also later the authors say that this ansatz is valid in the localized phase. Probably they mean "deeply localized" phase.

9) Let me finish with what I think is the biggest issue with the paper. It does not provide any internal way of estimating the error within the method. Sometimes this is impossible and we have to do what the authors do, compare with ED in small systems and hope the error does not change with the system size. But this does not seem to be the case here. E.g. one can properly evolve the eigenstates backwards, as the authors explain, and then compute the variance of the photon number. This should be available for system sizes well beyond ED. One can probably do something different with the same idea. Right now the authors advocate the method as an efficient way to overcome difficulties of ED but never show any advantages as all the examples except the last one are confined to very small systems and the last example does not have any independent error estimate. To me the step of finding a clear and controlled example, where the method goes beyond other methods is a crucial part for advocating that it offers real advantages.

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  • significance: high
  • originality: high
  • clarity: top
  • formatting: excellent
  • grammar: perfect

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