# Flow Equations for Disordered Floquet Systems

### Submission summary

 As Contributors: Steven Thomson Arxiv Link: https://arxiv.org/abs/2009.03186v3 (pdf) Date submitted: 2021-01-01 13:30 Submitted by: Thomson, Steven Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Quantum Physics Approach: Theoretical

### Abstract

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

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.