From Dynamical Localization to Bunching in interacting Floquet Systems

Dear Editor and Referees,

We are grateful for your efforts in reviewing our paper for submission. We especially thank the referees for their constructive comments and assesment of our manuscript. In the following, we believe we answer all comments made and concerns raised by the referees, and list all the changes we made in the revised version of our paper.

Sincerely yours, Yuval Baum, Evert van Nieuwenburg and Gil Refael

Response to Referee 1

The referee raised two concerns.

1. The referee wrote, ``The authors use several times the phrase “non-interacting hard core bosons”...hard core bosons in higher dimension actually are strongly interacting''.

   The referee is certainly correct. Indeed, hard-core bosons in $d > 1$ cannot be mapped to spinless fermions and cannot be considered non-interacting. In all dimensions, the addition of a uniform force (linear field) leads to single-doublon localization. However, while we expect a non-diffusive behavior in the dilute limit, for $d>1$ and at finite densities a diffuse behavior may arise due to the presence of interactions.

   We make this point clear in the main text. Moreover, we omitted the term 'non-interacting hard core bosons' whenever $d>1$ is considered.

2. The other concern the referee raised is regarding the doublon localization, ``The authors claim that for a static linear doublon potential there is no diffusion “to all orders in perturbation theory”. This statement is not explained and perhaps misleading.''.

   Indeed the statement above can not hold for all times and is valid as long as doublons are stable. If the effective doublon-Hamiltonian was the true Hamiltonian of the system, then the presence of a linear field leads to localization (due to the Stark-effect) of the doublons. This localization is exact and true to all orders in perturbation theory. Of course, the effective Hamiltonian is not exact and the stability time of the doublons is finite. Therefore, the absence of diffusion is not complete even in the presence of a uniform field. Yet, for times $t$ much smaller than $\tau$, where the effective doublon-Hamiltonian is a valid description of the system, we expect the deviations from the non-diffusive states to be small.

   This is now also elaborated upon in the main text.

Response to Referee 2

First and foremost, we would like to emphasize that the comparison to Ref. (22) (now Ref (24)) was made not in order to challenge it, but rather to show the agreement between our analysis and their work. We think of our work as complementing the results in Ref. (22).

We wish to start with the main critique the Referee raised regarding the comparison with Ref. (22) (in the old version). After private communication with the authors of Ref. (22), we are certain that the transient diffusive behavior they observed is a true effect and it is not due to finite size effects. This was the only question we raised on Ref. (22), and was not meant as a challenge.

We revised the main text accordingly and wrote: ``\textbf{Unlike in the spinful case, neighborons are interacting particles. While in dilute systems we expect the ballistic dynamics to last for long times, in dense systems a diffusive behavior is expected to arise at times much shorter than the stability time $\tau$. In a half-filled system of particles with nearest neighbor interactions, the average mean-free-path is expected to be of the order of a single lattice site, and hence we expect a good quantitative match between $D$ in Ref. (22) and the effective hopping calculated in our model}''.

We would like to stress that the main point in our paper is that, surprisingly, dynamical localization breaks already at the extremely dilute case of a single doublon (spinful) or neighborons (spinless). Indeed, as the referee wrote, we support these statements by numerics of dilute systems. We rely on the impressive numerical effort of Ref. (22) to claim that our model can hold even beyond the dilute limit by showing that the diffusion constant calculated in Ref. (22) perfectly matches the hopping amplitude of the neighborons. While the mean-free-path is expected to be of the order of a single lattice site, it seems that it is identically equal to one. Since this is not the focus of our paper, we choose not to discuss the exact match which requires an exact calculation of the mean-free-path for a specific model.

The nature of our work is not numerical and we do not see the fact that we simulate only dilute systems as a weakness as the referee suggested. We believe that Ref. (22) and our work compliment and support each other. While the stability time in the spinless non-dilute case may be short, the comparison to Ref. (22) indicates that the validity of our simplified model may extend beyond the dilute limit.

Finally, by suggestion of the referee, whenever we refer to the spinless case we emphasize the differences with respect to the spinful case.

1. The referee is concerned with our Floquet derivation. In particular, he wrote: ``The authors use a transformation of Floquet systems into a static problem, which to the best of my knowledge was first introduced by Jon H. Shirley''.

   The idea that a time-periodic function or Hamiltonian may be mapped into a static (infinite matrix) Hamiltonian can be traced back to seminal works of Floquet, Bloch and others (including Shirley). The growing field of Floquet physics has produced hundreds of papers, and common practice in the field is not to cite all these seminal works, most of which are considered textbook materials nowadays. Upon closer investigation however, it seems common in the cold-atom and ion-trap community to include this particular reference, and we have done so now too. We thank the referee for this suggestion.

2. The referee also wrote: ``...the non standard form is e.g. Eq. (8), where it remains obscure where the photons come from and what the 'Floquet form' is''.

   Here we would beg to differ. Equation (8) in our paper is simply the Floquet theorem, stating that a solution to the time-periodic Schrodinger equation is a phase factor times a periodic function. The latter is written as a Fourier series. In fact, this particular form can be found in Eq. (3) of the above-mentioned reference to Shirley. We refer to this solution form as the 'Floquet form' and to the integers in the Fourier expansion as the 'photon number', according to common terminology. Hence Eqs. (8) and (9) were meant as a short recap of a heavily used formalism.

3. Finally, the referee wrote ``Eq. (8) corresponds only to one solution of Eq. (3)''.

   This might be the origin of some confusion. All the solutions to Eq. (3) have the form of Eq (8). While, indeed, each eigenstate of the Floquet Hamiltonian corresponds to one solution of Eq. (3), all eigenstates are accessible by this procedure.

4. The referee wrote ``The authors write e.g. two lines after the heading B. Doublon Localization “and the system becomes diffusive due to the free motion of doublons”, which is clearly incompatible with Eq. (15).''

   We thank the referee for pointing this out. The word diffusive is wrong in this context, and we've replaced it with the word 'delocalized'. In practice, for $t$ much smaller than $\tau$ we expect to observe a ballistic behavior (see the answer to Referee 1, and the list of changes).

We omitted the term 'non-interacting hard core bosons' whenever $d>1$ is considered.
We added relevant references based on referee reports.
Page 6, after Eq. 18: we comment about the first point the referee made (see above).
Page 7, second column, second paragraph: we comment about the second point referee 1 made (see above).
Page 6, first column, second paragraph: we comment about the main point of referee 2 regarding the interacting nature of the composite particles in the spinless case and the expectation to find a diffusive behavior.
Below Eq. (12) ``The last term...'' was replaced by ``The first term...''.