Exponentially long lifetime of universal quasi-steady states in topological Floquet pumps

Dear editor and referees,

We thank all of you for the meticulous and detailed analysis of our work. The referees raised a few valid questions which we address in this reply and clarify at the relevant points in the revised version of the manuscript.


Reply to the comments of referee 2:

(1) The hybridization of the instantaneous bands to form Floquet bands is actually not exponentially small - the mixing of the instantaneous bands is of the order of \omega divided by the instantaneous gap \Delta. The existence of quantized pumping relies on this fact: the induced current in a quantized pump must be proportional to \omega to yield a pumped charge over a full period that is independent of \omega. The hybridization gap that is exponentially small in \Delta/\omega is the avoided crossing gap between the two Floquet bands (the crossing of the red and blue lines in Fig. 2; the gaps are too small to be visible in the figure). This is discussed in depth in Ref. [57] and supported by Ref. [68]. To define the right- and left- moving Floquet bands, these small gaps must be ignored. As the referee states, these gaps are only relevant at exponentially long time scales.

We added a clarification of this point to the discussion of the Floquet spectrum in Fig. 2.

(2) We thank the referee for asking for clarification.The perturbation theory we are using throughout the paper to estimate the inter-band scattering rate is just the standard Fermi's golden rule obtained from time-dependent perturbation theory (Eq. 4). The introduction of M time derivatives and factors of \Delta^{-1} [for example in Eq. (32)] is just a mathematical tool to bound the matrix element appearing in Fermi’s golden rule, with the value of M chosen such that the resulting bound is as tight as possible.

We have added a clarification of this point in the text, both before Eq. (32) and after Eq. (51).

(3) As explained in our answer to point 1, in order to define the right and left moving band it is necessary to neglect the exponentially small avoided-crossing gaps between the Floquet bands. This is shown in Figure 2, where the red color indicates the right moving band, and the blue color indicates the left moving band. If a small electric field is applied, there is an exponentially small probability to make a transition between right and left moving Floquet bands, which by Landau-Zener theory scales as (\Delta_ac)^2/E, where \Delta_ac is the exponentially small avoided-crossing gap, and E is the strength of the electric field. Correspondingly, with a probability which is exponentially close to 1, the particle would keep its identity as a right or a left mover as it sweeps through the crossing.

We have added a clarification in which we explain the avoided crossing between the Floquet bands in the first paragraph of sub-section 2.2.


Reply to the comments of referee 1:

We thank the referee for the attentive reading of our manuscript. We corrected the noted typos in the final version. Regarding \theta_k(t): its periodicity condition can be easily derived in the same way as the conditions for \alpha and \beta, $\theta_k(t+T) = \theta_k(t) + (\epsilon_{L,k}+\epsilon_{R,k})T mod 2\pi$. This condition is not relevant for the rest of the paper, therefore we opted not to include it.


We hope this reply and associated changes in the text have addressed all comments and questions raised by the two referees. We therefore believe that the updated manuscript should now be suitable for publication in SciPost, and look forward to its further processing.

With kind regards,

Tobias Gulden, Erez Berg, Mark Rudner, Netanel Lindner