Anomalous mobility edges in one-dimensional quasiperiodic models

Dear Editor,

We thank you for forwarding the Referees' reports. We are grateful to both of them for the positive assessment of our work and for the comments they made. Hereafter we respond the comments from the reports and indicate the corresponding changes whenever applicable. We hope that with the changes, the paper is now suitable for publication in SciPost Physics.

Best regards,
Tong Liu, Xu Xia, Stefano Longhi, and Laurent Sanchez-Palencia

Referee's comment/question 1 : "In the finite-size scaling analysis for the exponent beta (Fig.2a, Fig.3c, Fig.5) the authors observed a linear dependence of beta when plotted as a function of 1/m. Since the system size L=Fm, where Fm is the m-th Fibonacci number, one finds m \propto log(L) for large enough m. The finite-size correction to the exponent beta is therefore of order 1/log(L). Is this behavior general and expected for second order phase transitions or is it specific to the model considered ? "

Answer : The linear finite-scaling with the Fibonacci index is quite standard in quasiperiodic models. It holds for other models as well, including the celebrated Aubry-Andre-Harper (quasi Mathieu) Hamiltonian. However, we are not aware of studies establishing a direct relation between the 1/log(L) scaling of the corrections and the phase transition class.


Referee's comment/question 2 : "A comment to Fig.4. To my understanding, the results shown do not depend on the value of the phase because the irrational parameter alpha used in the calculation is approximated by the ratio of consecutive Fibonacci numbers. This, together with the periodic boundary conditions, makes the choice of the phase irrelevant and therefore disorder averaging can be avoided."

Answer : The argument suggested by the Referee is in principle correct and works in case of pure point or critical spectrum (as in our case). However, as a matter of fact in our simulations the energy spectra shown in Fig.4, for different and some illustrative values of the phase \theta, are obtained assuming a lattice size L=1000 and using open boundary conditions. The lattice size is thus not equal to nor proportional to any Fibonacci number and we directly use the irrational Diophantine number \alpha=(\sqrt{5}-1)/2. Yet, we could not observe any marked dependence of the spectra on \theta. This may be attributed to the irrational Diophantine nature of \alpha: this is why in the numerical computation of the eigenvalues of the matrix Hamiltonian for large L and rational approximant \alpha=p_n/q_n with q_n>L one cannot notice any marked sensitivity of the energy spectrum on \theta. Conversely, if one assume a system size L much larger than q_n, one could observe in the numerical simulations a dependence on \theta since the system becomes closer to periodic rather than aperiodic.


Referee's comment/question 3 : "From the Appendix A it seems that, if the disorder potential v(x) is unbounded and quasi-periodic, the Lyapunov exponent is independent of the specific form of v(x), as the final result coincides with the clean limit case. Is this statement correct?"

Answer : In fact, for a quasi-periodic and unbounded potential, the Lyapunov exponent depends rather generally on the particular form of v(x) and on the energy. For example, for the Maryland model, v(x)=V tan (\pi \alpha n + \theta), which is quasi-periodic and unbounded potential, the Lypaunov exponent L(E) can be computed analytically (see e.g. Ref.[50] in our manuscript) and the result is different than for our model. The main point is that the Lyapunov exponent in the "clean limit model" is the value of L(E) when the complex phase \epsilon added to the argument of the potential goes to infinity, while the Lypaunov exponent L(E) of the physical model, which we are looking for, is the value of L(E) when for $\epsilon=0$. Avila's global theory allows us to connect the two values of L(E), from \epsilon=\infty to \epsilon=0. However in order to use such a theory some strict conditions on the cocycle should be met.

Requested change 1 : "In Fig5: which value of a is used and what is the corresponding value of Em".

Answer : The have added the values of the tuning parameter a and the corresponding mobility edges in the caption of the figure.

2. The authors mentioned a "multifractal theorem". What are they referring to exactly ?

Answer : We thank the referee for raising this point. Indeed, this is an incorrect wording. There is no theorem in the mathematical sense. We have changed it to "multifractal analysis", which refers to the analysis of the scalings of the parameters \beta.