Multifractality without fine-tuning in a Floquet quasiperiodic chain

An original analysis of a simple model that leads to interesting results.

Some statements in the abstract/intro are not explicitly demonstrated in the main text.

The presentation/layout is not as clear as it could be in places.

This is an interesting paper. It addresses a theoretical question that arises naturally in the context of Floquet systems -- the effect of hybridising localised and extended states through the Floquet drive -- and finds the very interesting result that this leads to eigenstates with multifractal character.

The abstract refers to this as "an entire band of multifractal wavefunctions" and, correctly, emphasises that the multifractality is of interest because it "does not require any fine-tuning of the model parameters".

I do not doubt that these two quoted statements are correct. However, unless I have missed it in the paper, I do not see any direct evidence given for either statement.

To interpret this as "an entire band of multifractal wavefunctions" would surely require the reader to be given some sense of the (Floquet) energy spectrum of the multifractal states, as compared to the localised and delocalised ones. I do not see this is the paper, as the relevant graphs appear to sort the eigenstates by their IPR and not their quasienergy. Are the IPR and the quasienergy closely correlated? In what sense do the multifractal states form a band?

The results showing multifractal behaviour appear to be only for one value of the drive frequency: \Omega = 2.74\pi J. The authors argue why it needs to be close to the bandwidth, and in Appendix A show it should not be larger. But, unless I missed it, there is no statement about how robust the behaviour is to variations in \Omega about this value. Showing results even for one more value of \Omega would be helpful.

In terms of presentation, I did have trouble following some of the ideas, as I felt I needed to flick back and forward to appendices to understand. (The reference to Appendix F is particularly irritating, as one then needs to turn more pages to find the associated Fig 11.)

1) Please provide direct evidence justifying/explaining the statements ("an entire band of multifractal wavefunctions" and "does not require any fine-tuning of the model parameters") which are made in the abstract.

2) Please reduce need to consult appendices where possible (move what is important to the main text), and try to keep figures close to where they are referenced.

3) There are several typographical errors to correct:

page 2: localisation lengths is the unifies these two contexts.

Page 3: generalisations of which is known to host multifractal eigenstates

Page 7: This visible in a plot of σ2(n)/n2,

Page 9: To extract τq shown in Fig. 2(d) of the, we do a linear fit o

Fig 6 caption: The black dashed line denotes the mobility edge, states near which do not participate in any significant way.

Fig 7 caption: where as the columns correspond

The main idea of the paper is very interesting and should be published.

The presentation can be greatly improved, and an important issue needs be clarified.

1) Recommendations to improve the presentation :

1a) The present version of the manuscript contains one main text
and 7 Appendices (from A to G) and 12 figures,
so that the reading is extremely difficult :
to understand a given statement mentioned in the text,
one needs to go to the corresponding Appendix on another page,
and to find the corresponding figures on still another page...
(The three last Figures 10,11,12 are even plotted after the list of references !).

As a consequence, to facilitate the reading, I would recommend
to include most of the material of the Appendices within the main text
in order to obtain a logical presentation that can be followed linearly.

1b) Another difficulty in the reading comes from the mixing between exact and approximate statements
almost everywhere, in particular in the discussion of multifractal exponents, for instance :

1b-i) the statement $D_q \simeq 1/2$ (end of the paragraph after Eq 3)
can only be valid in a certain region of the index $q \geq q_c $ that should be mentioned
(since it cannot be true near $q=0$).
Then in the caption of Fig 2 the authors give $\tau_2 \simeq 0.55$
while after Eq 5, one reads $D_q = 2(1-\gamma_1) \simeq 0.44$.
And in figures 2 and 7 , the authors display the rescaling $\sqrt{L} I_2$
based on the value $1/2$.

1b-ii) the exponent beta \simeq 0.72 first appears in section 4 of the main text
to characterize the sub-diffusive dynamics;
then it appears after Eq 25 to characterize M_0 =L^{-\beta} with the value beta=0.9 \pm 0.1.
Then in Eq 30, the fit value is beta=0.77,
and three lines after Eq 30, the authors give again the value beta=0.9 \pm 1.

To avoid confusion between these different statements, I would recommend
to clearly separate the definitions of the exponents and the exact relations between them,
from the various numerical estimates.

1c) Equations : many equations are written inside the text, while they would be clearer if written as proper equations.
Eq 21 should be written on a single lign.
Eqs (31) and (32) seem to contain too big empty spaces ?

2) Important issue that needs to be clarified :

The multifractal spectrum given in Eq. 5 corresponds to the GAUSSIAN generalized Rosenzweig-Porter model
of Ref [9], as discussed also in the various recent papers that the authors might wish to consider :

[R1] D. Facoetti, P. Vivo and G. Biroli, arxiv:1607.05942.

[R2] C. Monthus, arxiv:1609.01121

[R3] K. Truong and A. Ossipov, arxix:1609.03467.

[R4] B. L. Altshuler, L. B. Ioffe, V. E. Kravtsov, arxiv:1610.00758.

Here, the authors stress that their distribution of M is NOT GAUSSIAN (text around Eq 4 and Fig 4),
and the form given in Eq. 26 actually corresponds to the Levy power-law tail at infinity 1/m^{1+\mu}
of Levy index $\mu=2a-1 \simeq 1.2$ for the fit value $a \simeq 1.1$ given after Eq 26.
As a consequence, the moments of $M$ alone already display the multifractal behaviors given in Eq 29,
in contrast with the Gaussian case, so that the result of Eq 5 concerning the Gaussian case
cannot be valid for the Levy case.
The multifractal properties of some LEVY generalized Rosenzweig-Porter model have been actually discussed
in the ref [R2] mentioned above. So the authors should clarify whether their effective matrix model
with the distribution of Eq 26 has the same multifractal properties as the Levy model analyzed in the ref [R2]
or not , and what are the consequences for their multifractal spectrum.
(In particular, the derivation of Eq 24=Eq 5 in Appendix D
based on the approach of Ref 9 is OK for the Gaussian case, but should be modified for the Levy case;
the other approaches discussed in the refs R1-R2-R3-R4 might be more appropriate for the generalization to the Levy case).

In conclusion, I think that these clarifications on the precise effective matrix model and its multifractal properties
are very important since the Floquet quasi periodic chain proposed by the authors gives a very nice realization
of some generalized Rosenzweig-Porter model, while up to now, these models were only considered as 'toy models
where multifractal spectra can be exactly computed'. It is thus interesting to understand precisely
which variant of these models is realized in this Floquet framework, and what are its multifractal properties.

see the above report