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Localization control born of intertwined quasiperiodicity and non-Hermiticity
by Junmo Jeon, SungBin Lee
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Submission summary
| Authors (as registered SciPost users): | Junmo Jeon · SungBin Lee |
| Submission information | |
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| Preprint Link: | scipost_202305_00026v1 (pdf) |
| Date submitted: | 2023-05-17 06:22 |
| Submitted by: | Lee, SungBin |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
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 for the first time 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 limit between metal to insulator transition and 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 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, (2) detailed localization control of the critical states, (3) experimental realization of controllable localized, critical and delocalized states, using photonic crystals.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2023-8-6 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202305_00026v1, delivered 2023-08-06, doi: 10.21468/SciPost.Report.7613
Report
The authors explore the wave-function localization in the Aubry-André-Fibonacci model where non-reciprocal hopping phases are present, i.e., the non-Hermiticity is encoded in the phases of the hopping terms. The authors find that the interplay between quasi-periodicity and the complex hopping amplitudes can lead to the perfect delocalization of the states.
I have to confess that I am not an expert in the field and thus have some difficulty assessing the relevance of the manuscript, i.e., I am not sure if it is really suitable for SciPost Physics proper or should rather go into SciPost Physics Core.
A possible experimental realization is emphasized both in the abstract and in Fig. 1. However, I could not help the impression that this is only an afterthought that the authors use to sell their work. It is not clear to me if this is experimentally feasible (are experimental colleagues really able to propagate light only in one direction?) and in any case, no experiments are presented in the present manuscript. I thus think that emphasis should be removed from possible experimental realizations.
The work itself appears to be solid. The presentation is sometimes nice, but at other places also difficult to follow. One issue is that the structure of the manuscript is not always optimal. For example, section 2 and the appendices A-C are very short. I recommend specifically:
a) Expand section 2 by the basic definitions. For example, the rigidity $r$ is introduced twice, once on page 6 and once on page 13, but both times only in the text. Collecting such definitions once at the beginning would remove redundancy and could improve clarity.
b) Move the appendices into the main text. For example, appendices A and C have only 7 lines of text such that moving them into the main text might improve readability. Of course, in particular in the case of the material presented in appendix A, relevant quantities would have to be defined first, but see suggestion a).
The "return probability" is introduced at the beginning of section 3, but not used in later analysis. I thus recommend to remove this definition and the related discussion, and rather use the notions that appear later for the main results also at the beginning of section 3.
The model Eq. (3) with $\theta=0$ would be the textbook example of a dimerized chain. It might helpful to add this term to the discussion.
I have a few further more specific comments that I list under "Requested changes".
Requested changes
1- Remove emphasis of the proposed experiments.
2- Collect all relevant definitions (rigidity, inverse participation ratio, ...) in section 2.
3- Move appendices into the main text.
4- Remove the "return probability" Eq. (2) and related text.
5- Mention that Eq. (3) is also known as a "dimerized chain" (?).
6- The abbreviation "NHSE" is only used in the second paragraph of the Introduction. I recommend avoiding the use of abbreviations that are only needed once.
7- Fig. 5: The meaning of the parameter $V$ is not clear; in particular no $V$ appears in the model Eq. (6). I admit that a $V$ is introduced in the text of the third paragraph of section 3.2, but the relation to Eq. (6) is still unclear. Please clarify.
8- Again Fig. 5: Better insert later such that it does not appear before it is discussed.
9- Caption of Fig. 5: The authors probably mean "panel (a)" and "panel (e)" rather than "Fig. (a)" and "Fig. (e)", respectively.
10- Second paragraph of page 13: I believe that the authors mean "non-orthonormal" rather than "non-orthogonal".
11- The abbreviation "EP" appears on page 13, but might actually not have been introduced.
12- Fig. 7: It is not clear from the caption what this figure shows. Please specify the model and relevant parameters in the caption.
13- Fig. 8: It is again not clear from the caption what this figure shows. Please specify the model and relevant parameters in the caption.
14- References:
a) Always provide a DOI when possible, in particular for Refs. [6-11,13,35,38,41,42,53,57,60]
b) Avoid spurious lowercasing of names in titles. In particular, "Aubry", "André", "Fibonacci", and "Anderson" should not be lowercased.
c) It seems that Ref. [20] is published in Ann. Israel Phys. Soc. 3, 18 (1980). Given that this reference is central to the present work, the authors should make an effort to provide full details.
d) It is not necessary to provide a link to the PDF if a DOI is specified (Refs. [33,55,56,76]).
e) It seems that Ref. [79] is vol. 758 of the "Memoirs" of the American Mathematical Society. Actually, at first sight, it was not clear to me if this is a journal article with incomplete data or a book. I think that it would really help if the authors could present this more clearly.
15- The English is generally good, but sometimes there are minor grammatical errors or strange constructions. Let me just mention the "study" that could be "studied" in the last paragraph of the Discussion and Conclusion and the "drastically controlled" on line 2 of appendix B (I do not think that one can say this in English).
Report #1 by Anonymous (Referee 1) on 2023-7-23 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202305_00026v1, delivered 2023-07-23, doi: 10.21468/SciPost.Report.7554
Strengths
1. The manuscript studies a new model of non-Hermitian quasicrystal (non-Hermitian extension of the Aubry-André-Fibonacci model).
2. The manuscript is written in an accessible manner.
Weaknesses
1. More thorough analytical and numerical investigations should be needed to understand the phase transitions and critical behavior in the authors’ model, although they may go beyond the aim and scope of the present manuscript.
Report
The authors introduce and study a new model of non-Hermitian quasicrystal. Specifically, they introduce a non-Hermitian extension of the Aubry-André-Fibonacci model and study its inverse participation ratio and fractal dimension. The authors find unique behavior that has no counterparts in the corresponding Hermitian quasicrystal, such as the unconventional delocalization of the originally critical and localized states. Through these analyses, the authors claim that the localization properties of quasiperiodic systems can be nontrivially controlled by engineering non-Hermiticity, which may be of experimental relevance.
Recently, the physics of non-Hermitian Hamiltonians has attracted growing interest. One of the major current focuses is the interplay of disorder and non-Hermiticity. In this context, I believe that this manuscript, which faithfully studies a new model of non-Hermitian quasicrystal, can make a significant contribution in the research fields of non-Hermitian physics, and I would like to recommend publication of this manuscript in SciPost Physics.
Requested changes
I only have a couple of relatively minor comments, as explained below.
1. In Sec. 3, the authors call “$r \left( k \right) = \left| \langle v_{+} \left( k \right)_L | v_{+} \left( k \right) _R \rangle \right|$” the phase rigidity. While I understand that this quantity can detect the chaotic or localized behavior of eigenstates [e.g., N. Hatano and D. R. Nelson, Phys. Rev. B 58, 8384 (1998); J. T. Chalker and B. Mehlig, Phys. Rev. Lett. 81, 3367 (1998)], I fail to clearly understand why the authors call it the phase rigidity. I would appreciate clarification and justification.
2. The manuscript contains some typographic or grammatical errors, which the authors should correct carefully. For example, above Eq. (5), “Remarkably, the zero values of $\left| \langle \sigma_z \rangle \right|$ appears only for the complex valued energies” should be “Remarkably, the zero values of $\left| \langle \sigma_z \rangle \right|$ appear only for the complex valued energies”. In the second last paragraph of Sec. 3.2, “… due to the non-Hermitianity” should be “… due to the non-Hermiticity”.
Author: SungBin Lee on 2023-08-10 [id 3891]
(in reply to Report 1 on 2023-07-23)
We appreciate the referee for giving a positive comment on our present manuscript.
Author response 1.
The terminology of the “phase rigidity” is widely used in the non-Hermitian physics. This measures the rigidity of the phase of eigenfunction. In the non-Hermitian system, the eigenfunctions are biorthonormal i.e.
$$\langle\psi_n^L|\psi_m^R\rangle=\delta_{nm}, $$ where $$\vert\psi^R\rangle=\frac{\vert\phi^R\rangle}{\sqrt{\langle\phi^L|\phi^R\rangle}} \ \mbox{and} \ \vert\psi^L\rangle=\frac{\vert\phi^L\rangle}{\sqrt{\langle\phi^L|\phi^R\rangle}} $$
Here, $R$ and $L$ stand for the right and left eigenstates, $H\vert\phi^R\rangle=E\vert\phi^R\rangle$ and $\langle\phi^L\vert H=E\langle\phi^L\vert$, where $H$ is the non-Hermitian Hamiltonian. $n,m$ are indices of the eigenstates.
However, at the exceptional point (zero phase rigidity), where two states coalescence, the states become linearly dependent, and the biorthonormal condition is failed. At this point, their phase (in a sense of the complex number) jumps---which is known as the characteristics of the branch point. Due to this jump of the phase, the phase of the eigenfunction is not rigid at vicinity of the exceptional points. Thus, we call it as the phase rigidity. Further details are explained in the review paper, [Ingrid Rotter 2009 J. Phys. A: Math. Theor. 42 153001].
Author response 2
We appreciate for pointing out the typos and grammar mistakes. We check our manuscript carefully and correct such typos and grammatical errors.
Author: SungBin Lee on 2023-08-10 [id 3892]
(in reply to Report 2 on 2023-08-06)Author response 1.
We thanks for the detailed review of our manuscript. Let us briefly emphasize the novelty and importance of our work. Recently, the non-Hermitian physics is widely studied in physics of the open systems such as optics and condensed matter physics. One of the major stream of the research in the non-Hermitian physics is the interplay of disorder and non-Hermiticity. However, the non-Hermitian system with non-reciprocal hopping phases has never been explored, and its interplay with the disorder or quasiperiodic order has been elusive. In this context, our findings of the phase transition from localized phase to delocalized phase due to the non-reciprocal hopping phases on the quasiperiodic system can give an important contribution in the general research areas related to the non-Hermitian physics and open systems. Hence, we believe that our work is suitable for SciPost Physics.
Author response 2. We agree with the comment on our Fig. 1 and abstract raised by the referee. Thus, we edited our manuscript in the section where we discuss the possible experimental applications and noted that Fig.1 is the potential experimental application.
The referee asked if it is really possible to propagate light only in one direction. This is possible by using the optical isolator (or sometimes called optical diode), which is generally used to study the non-reciprocal optics. An optical isolator is a device that allows light to propagate through it only in one direction, but not in the opposite direction. Isolators are useful as valves that allow propagation in only one direction. They are used in high-power applications, for which one desires one-way transmission of light. [Refer to the book, ``Encyclopedia of Physical Science and Technology'', ISBN: 978-0-12-227410-7]. We have added this reference, in the section we introduce the optical isolator. Hence, we believe that our suggestion of the possible experiment realization shown in Fig. 1 would be reasonable, and we are currently in communication with experimentalists in this field.
Author response 3 Based on the referee's comments, we have edited our manuscript for better understanding. Specifically, we expand the section 2 with including the basic definitions of the phase rigidity and inverse participation ratio (IPR). We also removed the redundant expressions in our manuscript. We also move the appendix C to the main text for better understanding. The relevant quantities such as mean inverse participation ratio (MIPR) are also introduced in section 2. We leave the appendix A as the appendix because it would interrupt the flow of the main text. We specify the model (random disordered chain or the Fibonacci chain), and added detailed numerical values in the captions of the figures.
Author response 4 As the referee has pointed out, the definition of the ``return probability'' in Eq.(2) has not been directly used except specifying the meaning of the interference due to the non-reciprocal hopping phases. Thus, based on the referee's comment, we remove the definition (Eq. (2)) and the related detailed discussion. Instead, we briefly explain how the coalesence of the states would change the localization characteristics of the states, which is one of the important notions that appear to explain our results.
Author response 5 The model Eq. (3) with $\theta=0$ is an alternating periodic chain model whose unit cell contains two sublattices called $A$ and $B$ sites. However, this is different from a dimerized chain because the hopping parameter is uniform throughout the chain. Please note that we have two different on-site energies instead of hopping parameters, unlike the SSH model.
Author response 6 We really appreciate the referee for giving detailed review of our manuscript. We follow the "Requested changes" and improve our manuscript. We summarize our changes following the "Requested changes'' in the section of ``List of important changes'' along with possible short comments.
List of important changes
1) We remove emphasis of the proposed experiments and add the references of the optical isolators. (Abstract) "$\cdots$ experimental realization of controllable localized, critical and delocalized states, using photonic crystals." $\to$ "[$\cdots$]. In addition, we suggest an experimental realization of controllable localized, critical and delocalized states, using photonic crystals.
(Caption of Fig.1) "Experimental control of localization characteristics in non-Hermitian optical system with ring resonators." $\to$ "A proposal for an experimental control of localization characteristics in non-Hermitian optical system with ring resonators."
"[$\cdots$] critical states are emphasized by [$\cdots$]" $\to$ "[$\cdots$] critical states are drawn by [$\cdots$]"
(Page 3, second paragraph) "Our main result is illustrated in Fig.1, with potential experimental implications." $\to$ "Fig. 1 illustrates our main results with sketch of potential experimental implications. "
"[$\cdots$] optical isolators [$\cdots$]'' $\to$ "[$\cdots$] optical isolators[``Encyclopedia of Physical Science and Technology'', ISBN: 978-0-12-227410-7][$\cdots$] "
(Page 15, second paragraph) "Importantly, our theoretical study could be studied by the photonic crystal\cite{PhysRevB.104.125416} or electrical circuits similar to other open system models governed by the Lindblad master equation or the effective non-Hermitian Hamiltonian. In particular, we propose an experimental setup to demonstrate the control of the localization of the wave function in the quasiperiodic system [$\cdots$]" $\to$ "Our theoretical work could be studied by the photonic crystal\cite{PhysRevB.104.125416} or electrical circuits similar to other open system models governed by the Lindblad master equation or the effective non-Hermitian Hamiltonian. In particular, we suggest an experimental setup to demonstrate the control of the localization of the wave function in the quasiperiodic system [$\cdots$]"
2) We collect all relevant definitions of phase rigidity, inverse participation ratios in section 2. We remove their detailed definitions in the section 3.
(Added---Page 4, Section 2) "One of the most important quantity used in the non-Hermitian systems is the phase rigidity, which is defined by
We quantify the localization strength of the state by using the inverse participation ratio (IPR), which is defined for a normalized state $\psi$ as
(Page 6, third paragraph) "Note that when $T=\frac{\Delta V}{4\cos(Ka/2)}$, $v_+(K)$ coalesces into $v_-(K)$. This coalescence is a unique feature of the non-Hermitian system, so-called an exceptional point (EP). When the states coalesce, the right and left eigenstates become orthogonal to each other, and hence the phase rigidity, $r(k)=\vert\langle v_+(k,L)|v_+(k,R)\rangle\vert$ becomes zero [PhysRevX.6.021007,PhysRevA.95.022117]. Here the subscripts $L$ and $R$ stand for the left and right eigenstates. Thus, the phase rigidity can be used to indicate the delocalization phase transition where the probability distribution becomes perfectly uniform." $\to$ "Note that when $T=\frac{\Delta V}{4\cos(Ka/2)}$, $v_+(K)$ coalesces into $v_-(K)$, and hence the phase rigidity defined in Eq. (2) for $v_\pm(K)$ becomes zero. Thus, the unconventional coalesence of the states due to the non-Hermiticity is happened when the state is uniformly delocalized. It turns out that the vanishment of the phase rigidity indicates the delocalization transition."
(Page 13, third paragraph) "[$\cdots$] To capture this, we compute the phase rigidity of the maximally localized state, given by $r(\Psi) =\left\vert\langle\Psi_L\vert\Psi_R\rangle\right\vert$, where $\Psi$ is the maximally localized state\cite{PhysRevX.6.021007,PhysRevA.95.022117}. The subscripts $L$ and $R$ denote the left and right eigenstates, respectively. Note that $r(\Psi)=1$ for the Hermitian case, while $r(\Psi)\le 1$ for the non-Hermitian case, because the right eigenstates could be non-orthogonal to each other. At the EPs where the multiple right eigenstates coalesce, the phase rigidity vanishes\cite{PhysRevX.6.021007,doi:10.1126/science.aar7709}." $\to$"[$\cdots$] To capture this, we compute the phase rigidity, $r(\Psi)$ of the maximally localized state. Remind that $r(\Psi)=1$ for the Hermitian case, while $r(\Psi)\le 1$ for the non-Hermitian case, because the right eigenstates could be non-orthonormal to each other."
(Removed---Page 7 in the previous version) "Specifically, we quantify the localization strength using the inverse participation ratio (IPR), which is defined for a normalized state $\psi$ as
3) We move part of appendices into the main text. In addition, we clarify the captions of the figures.
(Page 13, last second paragraph) "[$\cdots$] For a given finite $T\ge 0.2V$, we generally see that the MIPR decreases as $\theta$ gets closer to $\pi/2$ in the Fibonacci quasicrystal. Thus, non-Hermiticity leads to the delocalization of states (see Appendix C.). [$\cdots$]"%Note that in the Fibonacci quasicrystal, even for the small $T$ regime, most of the states are critical states, which are delocalized as a power-law scaling. Thus, the hybridization of the eigenstates of the Hermitian Hamiltonian due to the non-reciprocal hopping phase occurs mainly between pairs of critical states in a way to compensate the power-law decaying probability amplitudes. This gives rise to the general delocalization tendency in terms of the non-reciprocal hopping phase." $\to$"[$\cdots$] For a given finite $T\ge 0.2V$, we generally see that the MIPR decreases as $\theta$ gets closer to $\pi/2$ in the Fibonacci quasicrystal (See Fig. 6). Thus, non-Hermiticity leads to the delocalization of states. In detail, Fig. 6 shows the MIPR as a function of the phase of the hopping parameter, $\theta$, for different hopping magnitudes $T$ in the Fibonacci quasicrystal. For the general $T$, the MIPR decreases as $\theta$ gets closer to $\pi/2$. Thus, the localization strength in the spectrum is suppressed in the Fibonacci quasicrystal due to the non-Hermiticity. [$\cdots$]"
(Caption of Fig. 6) "The mean value of the IPR (MIPR) of the spectrum as the function of the strength of the non-Hermiticity given by the phase angle of the hopping parameter, $\theta$. At $\theta=\pi/2$, the non-Hermiticity becomes maximum for given $T$. The MIPR which is the amount of the localization in the spectrum decreases as the non-Hermiticity becomes stronger." $\to$ "The mean value of the IPR (MIPR) of the energy spectrum as the function of the strength of the non-Hermiticity given by the phase angle of the hopping parameter, $\theta$ in the Fibonacci quasicrystal model. At $\theta=\pi/2$, the non-Hermiticity becomes maximum for given $T/V$, where $T$ is the hopping magnitude and $V=(V_A-V_B)/2$ is the difference between two kinds of the on-site energies, $V_A$ and $V_B$ in the Fibonacci quasicrystal model. Here, $V_A=1$ and $V_B=-1$. The MIPR (Eq. (4)) which is the amount of the localization in the spectrum decreases as the non-Hermiticity becomes stronger."
(Caption of Fig.7) "[$\cdots$] The degree of disorder is $50\%$. The system size, $N=233$, and the hopping parameter value, $T=4V$." $\to$ "[$\cdots$] The degree of disorder of the on-site potential energy is $50\%$. The system size, $N=233$, and the hopping parameter value, $T=4V=4$."
(Caption of Fig. 8) "[$\cdots$] as a function of the non-reciprocal hopping phase ($\theta$). Here, the unit of the localization length is the atomic spacing between neighboring atoms. The non-Hermiticity induces the delocalization, so the localization length increases as the non-Hermiticity becomes stronger. (b) Comparison of the probability distribution of the maximally localized states for (blue) $\theta=\pi/2$ and (red) $\theta=0$ in the logarithmic scale. The linear scaling in the figure indicates the exponential decay. The smaller slope indicates the larger localization length for the non-Hermitian case. The hopping magnitude is $T=3V$. The system size is $N=987$." $\to$ "[$\cdots$] as a function of the non-reciprocal hopping phase ($\theta$) in the Fibonacci quasicrystal model. Here, the unit of the localization length is the atomic spacing between neighboring atoms, which is set to be 1. The non-Hermiticity induces the delocalization, so the localization length increases as the non-Hermiticity becomes stronger. (b) Comparison of the probability distribution of the maximally localized states for (blue) $\theta=\pi/2$ and (red) $\theta=0$ in the logarithmic scale. The linear scaling in the figure indicates the exponential decay. The smaller slope indicates the larger localization length for the non-Hermitian case. The hopping magnitude is $T=3V=3$. The system size is $N=987$."
4) We remove the "return probability" Eq. (2) and related text. Instead, we explain the significance of the exceptional point at the beginning of section 3.
(Removed---Page 4, second paragraph of section 3 in the previous version) "More specifically, we denote $A_m$ as the transition amplitudes of the path traveling $m$ steps. In the discretized system, one can group the possible paths depending on the number of steps. Then the return probability for the $i$-th site, $P_i$, is given by
(Added---Page 5, first paragraph) "[$\cdots$] This interference originated from the non-reciprocal hopping phase essentially gives rise to the delocalization of the state by reducing the return probability. On the other hand, the non-reciprocal phase can also lead to the constructive interference with respect to $\theta$ for some states. In this case, the delocalization is hindered by the interference from the non-reciprocal hopping phase. Thus, the non-reciprocal hopping phase gives rise to the \textit{state-dependent} control of the localization properties. Moreover, in the non-Hermitian systems, such interference effect leads to the unconventional coalesence of the states which have different localization characteristics, so-called exceptional points. Hence, before and after this exceptional point, the localization characteristics would be changed drastically as we will demonstrate."
5) We do not use the unnecessary abbreviations such as "NHSE" and "EP".
6) We place Fig. 5 on the right place. We clarify the definition of $V$ and the relation to the AAF potential function in both caption and the main text.
(Page 10, last paragraph) "[$\cdots$] In detail, the Fibonacci quasicrystal consists of two different atoms, $A$ and $B$, which have onsite potentials $V_A$ and $V_B$, respectively. " $\to$"[$\cdots$] In detail, the Fibonacci quasicrystal consists of two different atoms, $A$ and $B$, which have onsite potentials $V_A$ and $V_B$, respectively. From Eq. (8), $V_A=\lambda$ and $V_B=-\lambda$. "
(Page 12, third paragraph) "The potential difference between atoms $A$ and $B$ is given by $V_A-V_B=2V$. " $\to$ "The potential difference between atoms $A$ and $B$ is given by $V_A-V_B=2V$. From Eq. (8), $V=\lambda$."
(Caption of Fig. 5) "[$\cdots$] with $T=13V$. The fractal dimensions are (b) $D_2=0$, (c) $D_2=0.411$ and (d) $D_2=1$. [$\cdots$] Along the boundary where the fractal dimension changes rapidly in Fig. (a), the phase rigidity also drops steeply in Fig. (e). [$\cdots$]" $\to$ "[$\cdots$] with $T=13V$. $V=(V_A-V_B)/2$ is the difference between two kinds of the on-site energies, $V_A$ and $V_B$ in the Fibonacci quasicrystal model. Here, $V_A=1$ and $V_B=-1$. The fractal dimensions are (b) $D_2=0$, (c) $D_2=0.411$ and (d) $D_2=1$. [$\cdots$] Along the boundary where the fractal dimension changes rapidly in panel (a), the phase rigidity also drops steeply in panel (e). [$\cdots$]"
7) We correct the list of references. We add missing DOIs and add ISBN for the books to clarify the types of the references.
8) We check the grammer and typos carefully, and correct them.