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Localization control born of intertwined quasiperiodicity and nonHermiticity
by Junmo Jeon, SungBin Lee
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Authors (as registered SciPost users):  Junmo Jeon · SungBin Lee 
Submission information  

Preprint Link:  scipost_202305_00026v2 (pdf) 
Date submitted:  20230810 07:44 
Submitted by:  Lee, SungBin 
Submitted to:  SciPost Physics 
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Academic field:  Physics 
Specialties: 

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 selfsimilar wavefunctions have been actively discussed. However, in open systems generally described by the nonHermitian 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 nonHermiticity 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 AubryAndreFibonacci (AAF) model where nonreciprocal hopping phases are present. Here, the AAF model continuously interpolates the two different limit between metal to insulator transition and critical states, and the nonHermiticity 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 nonHermitian hopping phase leads to delicate control of localization characteristics of the wave function. Our work offers (1) emergent delocalization transition in quasiperiodic systems via nonHermitian hopping phase, (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.
Current status:
Author comments upon resubmission
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 nonHermitian 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 nonHermitian physics is the interplay of disorder and nonHermiticity. However, the nonHermitian system with nonreciprocal 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 nonreciprocal hopping phases on the quasiperiodic system can give an important contribution in the general research areas related to the nonHermitian 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 second 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 second 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 nonreciprocal 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 highpower applications, for which one desires oneway transmission of light. [Refer to the book, ``Encyclopedia of Physical Science and Technology'', ISBN: 9780122274107]. 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 second 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 second 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 nonreciprocal 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 onsite energies instead of hopping parameters, unlike the SSH model.
Author response 6
The terminology of the “phase rigidity” is widely used in the nonHermitian physics. This measures the rigidity of the phase of eigenfunction. In the nonHermitian 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 nonHermitian 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) jumpswhich 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 7
We appreciate for pointing out the typos and grammar mistakes. We check our manuscript carefully and correct such typos and grammatical errors.
List of changes
1) We remove emphasis of the proposed experiments and add the references of the optical isolators.
(Abstract)
"[$\cdots$](3) 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 nonHermitian optical system with ring resonators."
$\to$ "A proposal for an experimental control of localization characteristics in nonHermitian 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: 9780122274107][$\cdots$] "
(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 nonHermitian 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 nonHermitian 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.
(AddedPage 4, Section 2)
"One of the most important quantity used in the nonHermitian systems is the phase rigidity, which is defined by
$$
r(\psi_k)=\vert\langle\psi_k^{(L)}\psi_k^{(R)}\rangle\vert.
$$
Here, the superscripts $L$ and $R$ stand for left and right eigenstates of the nonHermitian Hamiltonian, and the subscript $k$ is the index of eigenstate. Unlike the Hermitian systems where the phase rigidity is always 1, it could be less than one, and even vanished in the nonHermitian systems. Particularly, when two distinct eigenstates coalesce, the phase rigidity becomes zero\cite{PhysRevX.6.021007,PhysRevA.95.022117}. This unique characteristics of the nonHermitian system is called an exceptional point. Thus, one can use the phase rigidity to quantify the coalesence of the states in the nonHermitian system.
We quantify the localization strength of the state by using the inverse participation ratio (IPR), which is defined for a normalized state $\psi$ as
$$\mbox{IPR}(\psi)=\sum_i\psi(i)^4.$$
Note that the amount of localization for the wave function, $\psi$, can be quantified by the IPR\cite{PhysRevB.83.184206,calixto2015inverse,PhysRevB.100.054301}. In the spectrum, the maximum (minimum) value of the IPR indicates the maximally (minimally) localized state. Let us refer to these states in the spectrum as maximally localized and maximally extended states, respectively. Also, the average localization strength for entire states in the spectrum is given by the mean IPR (MIPR), defined by
$$\mbox{MIPR}=\frac{1}{N}\sum_{k=1}^{N}\mbox{IPR}(\psi_k),$$
where $\psi_k$ is the $k$th eigenstate. The delocalization (localization) can be captured by the reduction (enhancement) of MIPR\cite{PhysRevB.100.054301}."
(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 nonHermitian system, socalled 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 nonHermiticity 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 nonHermitian case, because the right eigenstates could be nonorthogonal 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 nonHermitian case, because the right eigenstates could be nonorthonormal to each other."
(RemovedPage 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
$$
\mbox{IPR}(\psi)=\sum_i\psi(i)^4.
$$
Note that the amount of localization for the wave function, $\psi$, can be quantified by the IPR\cite{PhysRevB.83.184206,calixto2015inverse,PhysRevB.100.054301}. In the spectrum, the maximum (minimum) value of the IPR indicates the maximally (minimally) localized state. Let us refer to these states in the spectrum as maximally localized and maximally extended states, respectively. Also, the average localization strength for entire states in the spectrum is given by the mean IPR (MIPR), defined by
$$\mbox{MIPR}=\frac{1}{N}\sum_{k=1}^{N}\mbox{IPR}(\psi_k),$$
where $\psi_k$ is the $k$th eigenstate. The delocalization (localization) can be captured by the reduction (enhancement) of MIPR\cite{PhysRevB.100.054301}."
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, nonHermiticity 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 powerlaw scaling. Thus, the hybridization of the eigenstates of the Hermitian Hamiltonian due to the nonreciprocal hopping phase occurs mainly between pairs of critical states in a way to compensate the powerlaw decaying probability amplitudes. This gives rise to the general delocalization tendency in terms of the nonreciprocal 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, nonHermiticity 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 nonHermiticity. [$\cdots$]"
(Caption of Fig. 6)
"The mean value of the IPR (MIPR) of the spectrum as the function of the strength of the nonHermiticity given by the phase angle of the hopping parameter, $\theta$. At $\theta=\pi/2$, the nonHermiticity becomes maximum for given $T$. The MIPR which is the amount of the localization in the spectrum decreases as the nonHermiticity becomes stronger."
$\to$ "The mean value of the IPR (MIPR) of the energy spectrum as the function of the strength of the nonHermiticity given by the phase angle of the hopping parameter, $\theta$ in the Fibonacci quasicrystal model. At $\theta=\pi/2$, the nonHermiticity becomes maximum for given $T/V$, where $T$ is the hopping magnitude and $V=(V_AV_B)/2$ is the difference between two kinds of the onsite 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 nonHermiticity 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 onsite 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 nonreciprocal hopping phase ($\theta$). Here, the unit of the localization length is the atomic spacing between neighboring atoms. The nonHermiticity induces the delocalization, so the localization length increases as the nonHermiticity 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 nonHermitian case. The hopping magnitude is $T=3V$. The system size is $N=987$."
$\to$ "[$\cdots$] as a function of the nonreciprocal 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 nonHermiticity induces the delocalization, so the localization length increases as the nonHermiticity 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 nonHermitian 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.
(RemovedPage 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
$$P_i=\left\vert\sum_{m=1}^{\infty}A_m\right\vert^2$$
Due to the nonreciprocal hopping phase, the phase difference between the transition amplitudes arises. Specifically, for $A_m$, the relative phase shift, $m\theta$ can be accumulated from $H_T$. In this case, the return probability of the state becomes smaller compared to the Hermitian case, indicating the delocalization of the state from the $i$th site. On the other hand, the nonreciprocal phase can also increase the return probability due to constructive interference with respect to $\theta$ for some states. In this case, the delocalization is hindered by the interference from the nonreciprocal hopping phase. Thus, the nonreciprocal hopping phase gives rise to the \textit{statedependent} control of the localization properties"
(AddedPage 5, first paragraph)
"[$\cdots$] This interference originated from the nonreciprocal hopping phase essentially gives rise to the delocalization of the state by reducing the return probability. On the other hand, the nonreciprocal 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 nonreciprocal hopping phase. Thus, the nonreciprocal hopping phase gives rise to the \textit{statedependent} control of the localization properties. Moreover, in the nonHermitian systems, such interference effect leads to the unconventional coalesence of the states which have different localization characteristics, socalled 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 $V_AV_B=2V$. "
$\to$ "The potential difference between atoms $A$ and $B$ is given by $V_AV_B=2V$. 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 $T=13V$. $V=(V_AV_B)/2$ is the difference between two kinds of the onsite 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.
Submission & Refereeing History
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Anonymous Report 2 on 2023823 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202305_00026v2, delivered 20230823, doi: 10.21468/SciPost.Report.7702
Report
I would like to thank the authors for improving their presentation. These improvements have allowed me to better appreciate their work. Given that the issue of relevance is still under discussion, I have read the manuscript once again, and I am afraid that I have identified some further issues.
First, I admit that I confused dimerized hopping amplitudes and alternating onsite potentials in my previous Report, but I still believe that both are textbook physics in the Hermitian case.
In section 3.1, I really had some difficulty following the discussion when rereading it. For example, I was not able to verify statements such as "the delocalization of the state is generally observed" at the top of page 9 or "In particular, Fig.4(c) shows that the extended states disappear in the spectrum due to the nonreciprocal hopping phase" further down on the same page. Indeed, the IPR never goes to zero, i.e., the trends are only qualitative, but there is no strict transition and thus no strict "disappearance". Maybe this is due to the finite system size (finite approximant), and I believe that in order to establish a rigorous conclusion, the dependence on this parameter would have to be investigated. However, I have not even seen this parameter specified in section 3.1.
Actually, the biggest difference between the cases $\lambda=0.2$ and $2$ in Figs. 3(a) and 4(a) is the overall scale of the MIPR, but this difference is hidden by using different scales for the two cases. In fact, the biggest difference that I can see between Figs. 3 and 4 is the different shape of the $\lambda = 0.2$ curve in panel (a), but this is in fact a minor effect once the overall scales are considered. I also note that the color scales of panels (c) and (d) might suggest the opposite of what is actually shown (bright colors for small values, lighter ones for large values).
In section 3.2, the authors use a different approach from section 3.1 to characterize the $\beta\to \infty$ limit. In fact, reading statements such as "Although a larger IPR indicates stronger localization, the value of the IPR alone is insufficient to determine the detailed localization characteristics and scaling behavior of the wave function, since the IPR is an averaged quantity over space", I could not help wondering why they used exactly the criticized quantity in section 3.1. I think the relation between the approaches needs to be clarified, including the relation between Figs. 3 and 4 on the one side and Fig. 5 on the other side.
In section 3.2, the authors also include an analysis of the fractal dimension $D_2$. They explain that this quantity is derived from the scaling behavior with $N$, but again they do not specify the values of $N$ that they have used. I also suspect that working with finite $N$ gives rise to numerical errors for $D_2$. If so, these would have to be specified.
A final, more stylistic remark is that the authors write "we check the grammer and typos carefully, and correct them", I still have some issues with these (actually, this very sentence contains a spelling error). I would prefer to avoid providing a complete list of corrections and thus strongly recommend that they ask a colleague. If they cannot find a native English speaker, e.g., someone from Western Europe should already feel more comfortable with articles.
Requested changes
1 Specify size of the system/approximant in chapter 3.1 and discuss the influence of this parameter.
2 Clearly distinguish true transitions and crossovers in the discussion.
3 Explain the relevant features in Figs. 3 and 4 better, including differences and similarities.
4 If the authors do not wish to apply the same careful analysis as for the Fibonacci quasicrystal (section 3.2) also to the AubryAndréFibonacci model (section 3.1), they should at least explain better why they changed their mind when going to the $\beta\to\infty$ limit.
5 The values of $N$ used by the authors need to be specified in section 3.2; the resulting error of $D_2$ also needs to be discussed.
6 More on the level of detail, the "for the first time" on line 6 of the abstract is an unnecessary claim for priority. Consequently, I recommend removing it.
7 The English would still need proofreading. I will not provide a complete list of corrections here, but just some examples:
a) In the Abstract alone, I see at least three corrections:
i) Line 11: "two different limits" (add plural "s").
ii) Line 14: "the inverse participation ratio" (article "the" missing).
iii) Line 18: "and (2)" ("and" instead of comma  now these are just two items).
b) To the best of my knowledge "is happened" is not a correct English term (use "is happening" instead) and "vanishment" is not even an English word (maybe use "vanishing" instead).
I do recommend that the authors get some help with proofreading their article (the misspelled "controlloed" at the beginning of Appendix B should also be detected by a spellchecker).
Report
I would very much appreciate the response and the corresponding revision of the manuscript. The authors have addressed all of my concerns satisfactorily and revised the manuscript accordingly. I would like to recommend publication of this manuscript in SciPost Physics.
Author: SungBin Lee on 20230905 [id 3953]
(in reply to Report 1 on 20230810)We appreciate the referee for giving a positive comment on our present manuscript.
Author: SungBin Lee on 20230905 [id 3952]
(in reply to Report 2 on 20230823)We attach the detailed author responses along with the list of changes as a PDF file.
*Author response 1
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 skycolored 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 nonreciprocal 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 2
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 3
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 4
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 5
We have asked one of our colleague who is native in English. Based on his comment, we corrected typos and expressions.
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Author_response_scipost.pdf