Flow Equations for Disordered Floquet Systems

Dear Editor,

Please find attached the revised version of our manuscript "Flow Equations for Disordered Floquet Systems".

We are happy to see that both referees found our manuscript clear and well written, and we appreciate their constructive suggestions. The main feedback on the previous version from both referees was that this work would be significantly stronger with an improved error estimate that did not rely on comparison with exact numerics, such that when we consider larger systems than accessible with exact numerics we can still assess the accuracy of the method in a self-consistent way. We have conducted a detailed and thorough investigation of this point, and the major change in the revised version of our manuscript is the addition of several new sections (Sections 4.3.1 and 6.4) which introduce the concept of flow invariants and demonstrate how they may be used to serve this purpose. We thank the referees for motivating us to address this.

We have additionally made many other smaller changes to the manuscript, detailed in our full response to the Referees below.

Yours sincerely, S. J. Thomson, D. Magano & M. Schiró

--- Response to Second Report of First Referee ---

We thank the Referee for their continued detailed reading of our manuscript and their constructive suggestions.

"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."

The Referee is quite correct to say that divergences may arise due to long flow times, as also seen in Ref. 61 for a time-independent system of disordered, interacting fermions. In our previous response, we focused on issues due to resonances in the sense of energy denominators of the form $1/(E_i-E_j)$ becoming ill-defined when the two energies coincided, and leading to divergent terms. As no such energy denominators occur in this method, this precise problem is not encountered. However, as the Referee notes, at low drive frequencies (or small disorder strengths), the long flow times required can lead to eventual divergences of the interacting terms: the sharp-eyed reader may note that this is visible in Fig. 9 for small distances at the lowest drive frequencies considered, and briefly discussed in the context of normal-ordering corrections. We thank the Referee for pointing out that this is indeed a consequence of (near) resonances, and in our revised manuscript we have given a more precise discussion of this point.

Such problems were encountered in previous work by two of us on localization in time-independent systems (Refs 61 and 65). We found that the generation of new terms tended to remain well-controlled in localized phases, and could be quantified by looking at so-called `invariants of the flow', i.e. quantities which would be conserved by an exact unitary transform, but which are not conserved exactly by our approximate ansatz (Eq. 44). We have now extended the discussion of flow invariants to the Wegner Floquet flow (see the new Section 4.3.1) and have used this as a diagnostic to assess the validity of our ansatz (see also next point). We previously found that normal-ordering is a useful technical tool in taming these divergences as it typically leads to more stable numerical behaviour, as discussed in Refs 61 and 65, though its implementation in the driven case is beyond the scope of the present manuscript.

We have amended the term `variational', as also noted by the second referee.

"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."

We have acted on the Referee's suggestion and have included a new Section 6.4 where an error analysis is presented based on the flow invariant, which would be exactly conserved in absence of any truncation of newly generated terms and therefore provides us with an estimate of the internal consistency of the method, along the lines of previous methods used in Refs. 61 and 65. The summary of this analysis is given in the new Figure 8 where the violation from exact invariance, $\delta I$, is plotted for the weakly interacting case as a function of the drive frequency, showing the frequency range in which the method remains controlled. Interestingly this range extends well within the disorder bandwidth and it is not therefore limited to the high-frequency regime.

"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."

We have removed the word transition' when referring to the delocalized system; we thank the Referee for spotting this. We now use the wordtransition' only when discussing the interacting system, where we speculate that the breakdown of our ansatz Hamiltonian may signal the onset of a phase transition, as in the time-independent case (Refs 61 and 65).

--- Response to First Report of Second Referee ---

We thank the Referee for their careful reading of the manuscript, and for their detailed feedback.

"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."

We thank the Referee for pointing this out, and we have corrected this notation in the revised version.

"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."

The Referee is correct that this is not, strictly speaking, the static limit and we have amended this sentence accordingly. In the limit of $\omega \to 0$, all terms with an explicit frequency dependence vanish and the flow equations take on the form of the time-independent system, but indeed with an additional sum over the harmonic index n. In this manner, the equations more closely resemble the flow equations for a time-independent two-dimensional system.

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

This is a very interesting suggestion: we thank the Referee for drawing our attention to this possibility, and plan to return to it in a future work.

"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."

Fig. 3 is intended to give the reader an overview of the behaviour of different classes of terms under the action of the continuous unitary transform, e.g. panels (c) and (c) show the flow of the zero-frequency on-site terms (solid) and hopping terms (dashed), while panels (e) and (f) show the flow of the higher frequency terms. The aim of these figures is to demonstrate that the flow is smooth and well-controlled, and also that the higher harmonics decay quickly to zero at high drive frequencies and more slowly in the case of low drive frequencies.

We do, however, agree with the Referee that Fig. 3 (and the similar Fig. 6) is difficult to read, and so we have streamlined this figure in the revised version, now showing the flow of only a small representative subset of the running couplings. We believe that the revised figure is clearer, while still conveying the same qualitative insight into the behaviour of the flow in different regimes.

"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."

In fact, the central claim of Fig. 4b is that the flow equations and exact diagonalization results match extremely well, as this is a highly non-trivial quantity to compute using continuous unitary transforms. Additional results are shown in Appendix C from exact diagonalization demonstrate the behaviour over a greater range of frequencies and system sizes to illustrate the behaviour across a wider parameter. We believe that the data shown in Fig. 4b is sufficient to demonstrate our main point that flow equations give the same results as numerically exact methods in this regime.

We have nonetheless added an additional data point at lower frequencies to show that the level statistics reach a plateau. For this data point, the flow equations do not reproduce the exact diagonalization result, as a larger number of harmonics would be required to accurately capture the behaviour in this low frequency region. We have also updated the colour scheme of Fig. 4a.

"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."

We thank the Referee for pointing this out, and have rectified this error in the revised manuscript.

"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."

The Floquet eigenstates are time-independent, and therefore should not have contributions from higher harmonics. In this section, we are approximating the eigenstates of the block-diagonal time-independent matrices sketched in Fig. 1c, not the full time-dependent eigenstates of Fig. 1a. Strictly speaking, to do so exactly would require two unitary transforms: one `reverse' transform to give us the full time-dependent eigenstates of the Floquet operator shown in Fig. 1a, followed by a second transform into the basis of Fig. 1c to give us the time-independent Floquet eigenstates of the effective Floquet Hamiltonian. We included this section to show that at high frequencies, we can obtain the Floquet eigenstates to high accuracy without requiring this two-step procedure.

"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."

This point was also mentioned in the second report of the first referee, and we have removed the word `variational'. This ansatz has been shown to be valid in the localized phase of the corresponding time-independent model, only breaking down close to the transition (see Ref. 61), however has also been used to study the delocalization transition in a time-independent model with long-range couplings (Ref. 65), where it was able to capture the delocalized phase in the weakly-interacting regime.

"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."

We thank the Referee for motivating us to further develop our error analysis. To this extent we have extended the concept of flow invariants, conserved quantities of the exact unitary flow used in static (time-independent) applications of the flow equation method, to the Wegner Floquet generator used in this work (see new Sect.4.3.1). This allows us to assess the stability of our method in a case where truncation of the newly generated terms in the flow is necessary, such as in presence of interactions, which would result in a violation of the exact conservation law. In particular, as we now discuss in the new Sec.6.4, by monitoring the flow invariant as a function of the external parameters we can assess the regimes in which our truncated ansatz remains under control, as shown in Figure 8.

New Section 4.3.1 on flow invariants and their uses.
Added brief discussion of the conservation of the flow invariant in Section 5.3 for the non-interacting system.
New Section 6.4 where we show the flow invariant for the interacting system, and added new Fig. 8.
Replaced Figs. 3 and 6 with cleaner figures showing fewer lines.
Corrected the notation in Eq. 12.
Minor changes to notation in Section 4.3 to be consistent with the new Sec. 4.3.1.
Replaced Fig. 4a with new colour scheme, and Fig. 4b now includes an extra point at low frequency.
Added sentence regarding the generation of higher-order terms by the flow, referencing work where this is discussed.
Added an additional citation in the introduction (Ref 73).