Multifractality without fine-tuning in a Floquet quasiperiodic chain

Dear Editor,

We thank your editorial recommendation and also the referees for considering our work suitable for Scipost, and for various useful suggestions. In the resubmitted version, we have taken all the suggestions and questions of the referees. A common suggestion of both the referees was to incorporate much of the appendices in the main text so that the paper is easier to read. In view of this, we have completely restructured the paper so that the main text, now split up into sections, is completely self-contained and appendices are used only to show additional numerical data.

Below, we address the specific questions of the referees in detail.

We hope that the concerns of the referees have been addressed satisfactorily, such that the work can now be published in Scipost.

reply to Report #1

1. Regarding improving the presentation:

   (a) We have completely restructured the main text of the manuscript to make it as self-contained and linear as possible and have used the appendices to present only some additional numerical data.

   (b) We have tried to be very clear and explicit regarding what values are exactly obtained from the numerical results and which of them are approximate values obtained either from the fits and or from the perturbation theory. (i) The quoted values of $D_q$ are valid for $q\gtrsim 1$; we mention this clearly now. (ii) The multifractal exponents show a small spread over all the multifractal states in the spectrum. The value 0.55 is obtained after averaging over all the multifractal states. The scaling with $\sqrt{L}$ based on the value 0.5 is shown to facilitate the demonstration of the non-trivial scaling of the IPRs with system size. The value of 0.44 is obtained from the proposed random matrix model for the Floquet multifractal states which is further analysed perturbatively. We have been explicit in distinguishing these now. (iii) We apologise for reusing the notation $\beta$ in two different contexts. In the previous version of the manuscript, the exponent $\beta$ in the section on wavepacket dynamics is not the same as the one used in the analysis of the distributions of matrix elements. To clear the confusion, we now use the symbol $\nu$ for the latter.

   (c) We have taken care of the formatting issues of equations raised by the referee.

2. Regarding the question about Levy matrices:

   We first thank the referee for bringing to our attention the references on generalised Rosenzweig-Porter and the one on the multifractal states in Levy random matrices. We have cited them appropriately. The Referee is correct in saying that the results and analysis of the Rosenzweig-Porter random matrices do not immediately carry over to our case due to the power-law tails of the distributions of the matrix elements. However, the perturbative analysis we do is robust to that as can be shown by a one-parameter rescaling of the distribution as a function of the system size. We have modified the text in Secs. 4.2 and 4.3 to explicitly state the assumptions (4.2) and their justifications (4.3) and substantiate them with the numerical analysis of the distribution. We would like to mention that the exponents obtained from the rescaling of the distributions are not very accurate, however, there are a couple of points to be noted here. Firstly, the random matrix model proposed for the Floquet multifractality already has a few assumptions built in which could lead to the errors, and secondly, as can be seen in Fig. 8(b) of the current version, it is difficult to get very reliable statistics for the tails of the distributions. On the other hand, the encouraging feature is that the scaling of the mean and typical values of the distributions with system sizes are much more accurate when compared to the exact numerical results. To be consistent, we also cite another realisation of the generalised Rosenzweig-Porter ensemble with a Levy-type distribution of the hopping elements dubbed as “preferred basis Levy matrix ensemble” in V. N. Smelyansky et al. arXiv:1802.09542. In this example, multifractal states also have the same properties as in the generalised Rosenzweig-Porter ensemble. So this validates our analytical perturbative calculations.

reply to Report #2

1. The referee raises a question about the quasienergy structure of the multifractal states and asks for a justification of the statement "an entire band of multifractal wavefunctions". As is the case naturally in disordered/quasiperiodic systems, the energy spectrum varies across disorder realisations and hence sorting the Floquet eigenstates with their IPRs was rather convenient for the purpose of scaling the IPRs and higher moments. However, to address the referee's concern, we have now shown additional data in Appendix A where we have presented the IPRs and their scalings with system sizes as a function of the quasienergy. The results show that the multifractal states indeed appear in a contiguous band of quasienergies thus justifying our statement. To demonstrate this, we bin the Floquet eigenstates over many realisations within windows in quasienergy and average the IPR of the states within the windows.

2. The referee asks about the robustness of the Floquet multifractality to the driving frequency as we had chosen to show data for only one frequency which was rather close to the bandwidth of the undriven system. To address this issue, in Appendix B, we have shown IPRs and their scalings for a much lower value of the driving frequency. We find that not only is the multifractality persistent, but its fraction is enhanced due to more delocalised and localised states taking part in the hybridisations.

3. The referee suggests reorganising the manuscript to minimise the need for appendices and also points out typographical errors. We have now completely restructured the manuscript by incorporating much of the appendices in the main text so that the paper is easier to read, and have also taken care of the typographical errors.

Additional corrections

1) We have slightly modified the derivation in Eqs. (16-19) and (25) to confirm the applicability of our perturbative approach to the case of Levy distribution of hopping elements.

2) We have replaced the notation $\gamma_2$ by $\gamma/2$ to be consistent with the previous papers on the generalised Rosenzweig-Porter ensemble.

3) To clarify the estimation of the exponent $\gamma=2-D$ we have merged Figs. 4 and 10 and modified the panel showing the scaling of the moments with L focusing on the typical and mean instead of higher moments.

4) We have added the references [35-40] and [43] considering the generalised Rosenzweig-Porter ensemble.

1) We have completely restructured the main text of the manuscript to make it as self-contained and linear as possible and have used the appendices to present only some additional numerical data.
2) We have slightly modified the derivation in Eqs. (16-19) and (25) to confirm the applicability of our perturbative approach to the case of Levy distribution of hopping elements.
3) To clarify the estimation of the exponent \gamma=2-D we have merged Figs. 4 and 10 and modified the panel showing the scaling of the moments with L focusing on the typical and mean instead of higher moments.
4) We have added additional data to show the robustness of the Floquet multifractality to variations in the drive frequency.
5) We have shown additional data where the IPRs are shown as a function of quasienergy to justify the term "band of multifractal states".