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Localization control born of intertwined quasiperiodicity and non-Hermiticity
by Junmo Jeon, SungBin Lee
This Submission thread is now published as
|Authors (as registered SciPost users):||Junmo Jeon · SungBin Lee|
|Preprint Link:||scipost_202305_00026v3 (pdf)|
|Date submitted:||2023-09-05 05:11|
|Submitted by:||Lee, SungBin|
|Submitted to:||SciPost Physics|
Quasiperiodic systems are neither randomly disordered nor translationally invariant in the absence of periodic length scales. Based on their incommensurate order, novel physical properties such as critical states and self-similar wavefunctions have been actively discussed. However, in open systems generally described by the non-Hermitian Hamiltonians, it is hardly known how such quasiperiodic order would lead to new phenomena. In this work, we show that the intertwined quasiperiodicity and non-Hermiticity can give rise to striking effects: perfect delocalization of the critical and localized states to the extended states. In particular, we explore the wave function localization character in the Aubry-Andre-Fibonacci (AAF) model where non-reciprocal hopping phases are present. Here, the AAF model continuously interpolates the two different limits between metal to insulator transition and the critical states, and the non-Hermiticity is encoded in the hopping phase factors. Surprisingly, their interplay results in the perfect delocalization of the states, which is never allowed in quasiperiodic systems with Hermiticity. By quantifying the localization via the inverse participation ratio and the fractal dimension, we discuss that the non-Hermitian hopping phase leads to delicate control of localization characteristics of the wave function. Our work offers (1) emergent delocalization transition in quasiperiodic systems via non-Hermitian hopping phase and (2) detailed localization control of the critical states. In addition, we suggest an experimental realization of controllable localized, critical and delocalized states, using photonic crystals.
Published as SciPost Phys. Core 6, 077 (2023)
Author comments upon resubmission
We appreciate the first referee for giving a positive comment on our present manuscript.
*Author response 2
We thank the referee for detailed review. First of all, ``the delocalization of the state is generally observed'' is supported by the decrease of MIPR for both $\beta=0$ and $\beta=2.5$ with $T=2\lambda$ (See the red curves in Fig.3 (a) and Fig.4 (a)). The decrease of MIPR indicates that the delocalization of the states in the spectrum has been enhanced.
Next, "Fig.4(c)'' should be "Fig.4 (d)'' which shows the case of the maximally extended states. Note that if the maximally extended state is the localized state, then there are no extended states. It is because the maximally extended state has minimum value of the IPR. In particular, the sky-colored region in Fig.4 (d) corresponds to the exponentially localized states as we have already specified in the caption of Fig.4. To improve the clarity, we should also emphasize this fact in the main text, and demonstrate the localization of maximally extended state in the sky blue region in Fig.4 (d) in the appendix. The referee suspects that there is no strict disappearance of the extended states, however by investigating the localization properties of the maximally extended state in terms of the minimum value of the IPR in the spectrum, we can numerically specify the conditions where every eigenstate is localized.
The referee also asks the dependence on the system size, $N$. Although we specified $N=987$ for Figs.3 and 4, we should emphasize that the characteristics of the change of IPR as the function of $N$ does not change due to the non-reciprocal hopping phase in the AAF model. In other words, the fractal dimensions of the maximally localized and extended states do not change, except the case of the maximally extended states with $T/\lambda<0.3$ for $\beta=2.5$. Hence, the change of the system size only changes the whole values of the IPR, that is irrelevant to our main interests. Even for the case of the maximally extended states with $T/\lambda<0.3$ for $\beta=2.5$, the localization characteristics change from extended to exponentially localized without exploring the critical state. Since the IPR of the exponentially localized state is independent of $N$, the changing $N$ only decrease IPR of the extended states. This remains the localization and delocalization behavior as the function of $\theta$ shown in Fig.4 (d). Thus, the dependence on the system size, $N$ is irrelvant to our main interests in section 3.1.
Based on the referee's commet, we agree that the further clarifications and explanations are necessary. We modify our manuscript to improve the clarity. For clarification, we emphasize that the sky blue region in Fig. 4 (d) represents the localized state not only in the caption but also in the main text. We also clarify the dependence on the system size $N$, and its irrelevance to our main interests.
*Author response 3
First of all, the different scales between $T=0.2\lambda$ and $2\lambda$ are not significant because depending on the $T/\lambda$, the phases are different. For example, as we have explained in the last second paragraph on page 7, for $T/\lambda<0.5$ and $\beta=0$, every state is localized for the Hermitian case. On the other hand, for $T/\lambda>0.5$, every state is extended. These facts for the Hermitian case are already known, thus, the discussion of the scales of the MIPR for different $T$ is not necessary here.
As the referee has pointed out, one of the notable difference between Figs. 3 and 4 is the different shape of the $T=0.2\lambda$ curve in panel (a). The referee claims that this is a minor effect once the overall scales, however, this is not a minor difference. Although the fraction of the change of MIPR is small, the different types of the change i.e. decreasing and increasing gives totally different results for each state. Specifically, the delocalization of the maximally extended state is observed for $\beta=0$, while the exponential localization tendency of maximally extended states with varying $\theta$ occurs for $\beta=2.5$ case i.e. localization. Note that the number of extended states, which become localized near $\theta=\pi/2$ for $\beta=2.5$ is small for the Hermitian case when $T/\lambda$ is small. Thus, even for the case of $\beta=2.5$, the fraction of change on MIPR is not large. However, this never means that the different types of the changes on the MIPR for $\beta=0$ and $\beta=2.5$ with $T=0.2\lambda$ are not important. We emphasize that the different types of the changes on the MIPR capture the totally different localization tendancies of each state as the functions of $\theta$.
*Author response 4
We appreciate your comment. Indeed, we think it is better to have more explanation for clarification in the main text. We have not used the fractal dimension in section 3.1 because we do not have any critical states and phase transitions exploring unconventional fractal dimensions. Please note that the fractal dimension is important when there are some fractal wave functions such as the critical states. However, in the AAF model discussed in section 3.1 have either localized or extended states only. Furthermore, although the strength of the localization given by the IPR would be changed, the characteristics of the localization, which are termed by extended, localized and critical are not altered by $\theta$ in almost all of the cases. For instance, note that the maximally localized states, which are exponentially localized for $\theta=0$ remain localized as we have mentioned in the main text. The only case where the fractal dimension changes as $\theta$ approaches $\pi/2$ is the maximally extended states for small $T/\lambda$ regime with $\beta=2.5$ shown in Fig. 4 (d). However, again even in this case, we have only two trivial kinds of the localization characteristics, localized and extended whose localization characteristics do not have the fractality. Thus, we believe that it is enough to explicitly show the localized and extended states, respectively in the appendix. The landscape of the fractal dimension does not give any further information to the readers in this case. Instead, for clarity, we specify the localization characteristics of the states either exponentially localized ($D_2=0$) or extended ($D_2=1$) in the captions of Figs 3 and 4.
On the other hand, for the case of the Fibonacci tiling, we have found the important delocalization transition exploring unconventional fractal dimension as shown in Fig. 5. Furthermore, in the case of $\beta\to\infty$, the maximally localized state would be not only localized or extended but also critical states having intermediate fractal dimensions $D_2\approx 0.5$. Thus, the discussion using the fractal dimension is relevant to the case of Fibonacci tiling only. We agree that it is better to explain the importance of different approaches as the referee commented. We briefly specify the significance of the fractal dimension when we discuss the Fibonacci quasicrystal in the main text.
*Author response 5
We thank for pointing it out. Based on the referee's comment, we add the information of $N$ values we have used to compute $D_2$. Surely, the fractal dimension has some numerical errors with finite $N$. However, such error does not change our results shown in Fig. 5. We also add the detailed numerical results relevant to Fig. 5 in the appendix.
*Author response 6
We thank again for your comment. We have asked one of our colleague who is native in English. Based on his comment, we corrected typos and expressions.
List of changes
1. We specify the size of the system in chapter 3.1, and clarify its insignificance.
2. We clarify the transition where the extended states disappear, supported by FIg.4 (d) in the main text.
3. We explain the relevant features in Figs. 3 and 4 better, specifying difference and similarities.
4. We briefly specify the reason why the discussion using the fractal dimension is relevant to the Fibonacci case ($\beta\to\infty$). Also, in the captions of Figs.3 and 4, we specify the localization characteristics of the states discussed in section 3.1, either exponentially localized or extended. In detail, every maximally localized states are exponentially localized regardless of $\beta$ and $T/\lambda$. For $T\ge 0.5\lambda$, every maximally extended states are extended. For $\beta=0$ and $T<0.5\lambda$, every maximally extended states are exponentially localized. On the other hand, for $\beta=2.5$ and $T<0.5\lambda$, the sky blue region indicates the exponentially localized states, while the green and black regions indicate the extended states.
5. We specify the values of $N$ we used to compute the fractal dimensions. In the appendix, we add the figures to clarify the numerical error (which is minor) of the fractal dimensions. In addition, to satisfy the referee's concern, we also add the error bars on Fig. 7 for the random disordered chain. Note that this error bar is originated from the random disorders. We use 20 samples of random local potential distributions.
6. We corrected the typos.
*A more detailed list of changes as a PDF file is attached as the author's response to the second reviewer's comment.
Submission & Refereeing History
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Reports on this Submission
- Cite as: Anonymous, Report on arXiv:scipost_202305_00026v3, delivered 2023-10-15, doi: 10.21468/SciPost.Report.7952
The authors continue to improve their manuscript. Nevertheless, some issues still remain.
1- I believe that it gets increasingly clear that the results of section 3.1 are only qualitative: the data does not allow a strict distinction between localized and extended states, only the identification of qualitative trends. This is aggravated by statements being made that might be supported by the data, but are difficult to extract from the figures that they are attributed to. For example, I believe that it is difficult if not impossible to extract any trend as a function of $\theta$ from Figs. 3(c,d) and Fig. 4(c) scanning across, e.g., $\lambda=0.2$ since the information is simply not contained in the color scale. I admit that I might have made this observation before, but after each round of revision I am slowly understanding better what the authors really mean to say.
2- The discussion in section 3.2 is more quantitative. Still, the new data for the fractal dimension $D_2$ in appendix D shows that it is extracted from sizes $300 \le N \le 987$. This is only a factor just above 3 in system size while it is consensus in critical phenomena that several orders of magnitude are needed to firmly establish power laws. The conclusions may be correct, but the data presented in the manuscript does not suffice to exclude that the estimates for $D_2$ are affected by corrections to scaling. Furthermore, I believe that one should be able to push computations to $N >987$ without too much effort.
We could continue reviewing and revising the manuscript. However, I believe that review should be closed at this point. In my opinion, in view of the above reservations, the manuscript does not meet the standards of SciPost Physics, but it could be published in SciPost Physics Core in its present form.
Minor typographic issues to be corrected on the proofs:
1- Last line of second paragraph on page 2: "quasiperiodic" is misspelled.
2- First line of section 2: no hyphen in "$N$ sites".
3- Fifth line of third paragraph of section 3.2: no comma after "Eq. (8)", i.e., "Eq. (8).".
4- Third line of second paragraph on page 13: I believe that "Recall" would be more appropriate than "Remind".
5- If the production team does not fix punctuation, the authors might pay attention to eliminating full stops inside parentheses if there is one following it (at least two instances of ".).".