Flow Equations for Disordered Floquet Systems

We thank the Referee for their careful reading of our manuscript, and we respond in detail the points raised 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. "

Response:

We thank the referee for this comment and for drawing our attention to several previous works on driven Anderson Insulators, notably Refs. [88-90] of the revised manuscript.

To answer the Referee's main concern, namely that our results are not consistent with existing literature, we highlight the fact that our driving protocol (a step drive, or bang-bang protocol) differs from the traditional monochromatic drive considered in the references above. This has important consequences on the physics of the problem, especially at intermediate and low frequencies. In fact, within our protocol, the effective Floquet Hamiltonian to which the Referee points is a one-dimensional Anderson model only in the high-frequency limit, where indeed our results show localisation.

To further substantiate this point, we have computed (using exact diagonalisation, to demonstrate that this is not a feature of our flow equation method) the level spacing statistics as a function of drive frequency for both monochromatic drive ($F(t)=\cos(\omega t)$) and a step-like drive ($F(t) = \textrm{sign}[\cos(\omega t)]$), for different system sizes (see new Appendix C). We find that the step-like drive exhibits the delocalisation transition discussed in our manuscript, while monochromatic drive (as considered in previous works) does not, in agreement with the literature.

Finally, we emphasize that the main point behind Section 5 of the manuscript was to benchmark the new flow equation approach against exact diagonalisation (ED), rather than provide a detailed discussion of the driven Anderson model in full generality. For these reasons we consider rather small system sizes ($L=12$) and focus on the level statistics, which is particularly challenging to obtain numerically with flow equations since very long flow times are required. A more detailed discussion of the properties of a driven Anderson model is beyond the scope of this work.

1b) "Does one have convergence when the driving frequency lies in the disorder bandwidth? For sufficiently large systems one might expect to find resonances in individual disorder realizations, which lead to degeneracies that cannot be resolved by flow equations. The authors should comment on this point since the analysis and proliferation of such resonances is really the key issue in Refs. [87-89]."

Response:

We believe that this point is already answered by our Fig. 4, which compares the relative error (panel a) in the FE quasienergies measured with respect to the ED results, and the level spacing statistics computed by both ED and FE methods (panel b) for a variety of values of drive frequency at fixed disorder strength W=5. This figure demonstrates that by retaining a sufficiently high number of harmonics, we obtain good convergence to the numerically exact results at all frequencies we have considered, including frequencies smaller than the disorder bandwidth. We have explicitly rewritten all mentions of the drive frequency $\omega$ in dimensionless form, normalised by the disorder bandwidth, as $\omega/W$ in order to make this clearer throughout.

Flow equations methods in disordered systems typically have no significant problems with resonances in the normal sense of the word, e.g. situations where the on-site energy difference is smaller than the hopping $J_{ij} \geq (h_i - h_j)$. In the limit of $l \to \infty$, only an exact degeneracy between the on-site energies of two sites will lead to the failure of an off-diagonal element to decay quickly, however as such degeneracies are never exact to numerical precision, what we instead observe in practice is the asymptotically slow convergence of the system with increasing flow time. As such, this problem can be mitigated by using a sufficiently large flow time. In practice, we do not find this to be an issue. We note that in previous works (e.g. Ref. [60]), the insensitivity of flow equation methods to resonances has already been remarked upon, even in interacting systems.

2a) "The authors use a weak coupling expansion in (39). However, one needs to be very careful because higher order coupling terms contain (higher order) energy denominators, which can invalidate the expansion even for small coupling in case of resonances from the driving. The authors should comment on this admittedly very difficult point (pointing out the possibility of such problems would already be sufficient, everything else can be considered beyond the scope of this manuscript). "

Response:

We believe the referee may have misunderstood certain aspects of our technique. We do not use a weak coupling expansion in Eq. 39, and there are no ‘energy denominators’ at any point in our calculation. The fixed-point Hamiltonian for weak interactions is not the result of perturbation theory, but rather based around a truncation in operator space. The calculation is based on earlier work in Refs. [61] and [65] where this truncated approach was originally developed for many-body localized systems in the localized regime. We have added a new subsection to Appendix B outlining the calculation of the flow equation for the interacting terms (Appendix B3) in order to clearly demonstrate the technique.

There is no significant issue with resonances in the flow equation method, as discussed in Ref. [60] for the case of a static many-body localized system. To quote Ref. [60], Wegner-type flows "only slow down when the problem is nearly diagonal, blithely integrating past would-be resonances that complicate ordinary perturbative treatments". In truncated flow equation methods of the type used in the present work, the main problem is runaway exponential increase of the interaction terms in the delocalized phase due to the structure of the nested commutators. As we discussed in Section 6 of the original manuscript, this is a problem which can be ameliorated in future work by extending the method proposed here to include more advanced techniques such as normal-ordering corrections which couple the flow of the interaction terms to the flow of the quadratic terms and prevent such divergences from occurring. We believe this discussion to be outwith the scope of the current manuscript, but provide several references in the text to other works discussing this in more detail. Here, we restrict to the weakly-interacting regime where such problems are minimized.

2b) "In Fig. 6b it is really not visible that there are significant deviations between ED and flow equations. One needs to present the data differently if one discusses the (not unexpected) deviations for low frequencies."

Response:

We have added an additional figure (Fig. 7 of the revised manuscript) to address this issue and now present the data for the interacting system in a similar manner as the non-interacting system. We have replaced the left column of Fig. 6 with data at an even lower drive frequency where the disagreement between FE and ED methods is clearly visible.

Additionally, we have replaced Figs. 3, 4, 5, 6 and 8 with new data, incorporating a greater number of disorder averages and a wider range of frequencies, as well as other minor cosmetic improvements throughout the manuscript.