# Shannon Entropy and Diffusion Coefficient in Parity-Time Symmetric Quantum
Walks

Zhiyu Tian 1,†, Yang Liu 1,2,† and Le Luo 1,2,∗ 1 School Of Physics And
Astronomy, Sun Yat-Sen University, Zhuhai, 519082, China
2 Center Of Quantum Information Technology, Shenzhen Research Institute Of Sun
Yat-Sen University, Nanshan Shenzhen 518087, China
† These authors contribute equally to this work
∗ Correspondence<EMAIL_ADDRESS>
###### Abstract

Non-Hermitian topological edge states have many intriguing properties, but
have so far mainly been discussed in terms of bulk-boundary correspondence.
Here we propose to use a bulk property of diffusion coefficients for probing
the topological states and exploring their dynamics. The diffusion coefficient
is found to show unique features with the topological phase transitions driven
by parity-time($PT$)-symmetric non-Hermitian discrete-time quantum walks as
well as by Hermitian ones, despite artificial boundaries are not constructed
by inhomogeneous quantum walk. For a Hermitian system, a turning point and
abrupt change appears in the diffusion coefficient when the system is
approaching the topological phase transition, while it remains stable in the
trivial topological state. For a non-Hermitian system, except for the feature
associated to the topological transition, the diffusion coefficient in the PT-
symmetric-broken phase demonstrates an abrupt change with a peak structure. In
addition, the Shannon entropy of the quantum walk is found to exhibit a direct
correlation with the diffusion coefficient. The numerical results presented
here may open up a new avenue for studying the topological state in Non-
Hermitian quantum walk systems.

##  I Introduction

Quantum walk Childs (2009); Karski  _et al._ (2009); Schmitz  _et al._ (2009);
Lovett  _et al._ (2010); Zähringer  _et al._ (2010); Childs  _et al._ (2013)
is the quantum analog of the classic random walk, which has found wide
applications in many realm of quantum information science. In recent years,
quantum walks have been extensively studied theoretically, and their
experimental realizations have been reported by single neutral atoms in
optical lattices Karski  _et al._ (2009), photons in waveguide lattices Perets
_et al._ (2008), trapped ions Schmitz  _et al._ (2009); Zähringer  _et al._
(2010) and single photons in free space Broome  _et al._ (2010). Since quantum
walk can be constructed with various symmetries by designing appropriate
evolution operators with periodic driving, it enables readily observation of
the topological phenomena with Floquet methods. quantum walk has been proved
to be suitable for simulating topological materials and exploring dynamics in
a wide rangeThouless  _et al._ (1982)

Recently, the use of quantum walks to simulate and explore new topological
phenomenaKitagawa  _et al._ (2010, 2012); Cardano  _et al._ (2017) has
attracted wide interests. Particularly, a new type of quantum walk, open
quantum walksAttal  _et al._ (2012a, b) have been realized and investigated,
where the topological properties can be controlled by dissipation and
decoherence. Open quantum walks show rich dynamical behavior Attal  _et al._
(2012a, b); Sinayskiy and Petruccione (2013); Bauer  _et al._ (2013) and can
be used for quantum algorithms in quantum computation for specific
tasksSinayskiy and Petruccione (2012) and quantum state engineeringAttal  _et
al._ (2012a); Sinayskiy and Petruccione (2012). PT-symmetric quantum walks, as
one type of open quantum walks, have recently been investigated both
theoretically Mochizuki  _et al._ (2016); Mittal  _et al._ (2021) and
experimentally Xiao  _et al._ (2017); Dadras  _et al._ (2018).

However, physical implementation of PT-symmetric quantum walks have been
limited due to a few issues. First is the limited evolution time induced by
dissipation and decoherence effectsRyu and Hatsugai (2002). These effects are
unavoidable in open quantum walksAttal  _et al._ (2012a, b), which is because
of the non-unitary dynamics and is described in the framework of the non-
Hermitian quantum mechanics Bender and Boettcher (1998); Bender  _et al._
(2002). Second, PT-symmetric quantum walks and related topological properties
have only been studied by the bulk-boundary correspondence Yao and Wang
(2018); Kunst  _et al._ (2018); Mochizuki  _et al._ (2020), while the
characterization of bulk property is absent so far, such as diffusion property
and Shannon entropy. Until very recently, the connection between diffusion
property and topological property have been studied for skyrmions Weißenhofer
and Nowak (2020), and the relation between diffusion phenomenon and bulk-edge
correspondence has been discussed Yoshida and Hatsugai (2021).

In this article, we propose to use diffusion coefficient, one of the bulk
properties, for characterizing the topological phases in PT-symmetric quantum
walk. With this new viewpoint, we can address the above-mentioned two issues
and there is no necessity of constructing artificial boundaries. We further
illustrate the direct correlation between Shannon entropy and diffusion
coefficient for the first time. The result suggests a fascinating possibility
for exploration topological properties of non-Hermitian systems Zeuner  _et
al._ (2015); Weimann  _et al._ (2017) using diffusion coefficient. It is noted
that our main tool is the numerical simulation instead of the analytical
method so that the general principle can be further revealed by analytical
calculation.

##  II Discrete-Time Quantum Walk

The quantum walk is governed by the time-evolution operator Xiao  _et al._
(2017)

$U=\operatorname{LTC}\left(\beta\right)L^{\prime}TC\left(\alpha\right)$ (1)

where $T$ is the position operator

$T=\sum_{x}|x+1\rangle\langle
x|\otimes|\uparrow\rangle\langle\uparrow|+|x-1\rangle\langle
x|\otimes|\downarrow\rangle\langle\downarrow|$ (2)

where x is the position of the walker. The position state changes according to
the coin states $|\uparrow\rangle$ and $|\downarrow\rangle$ of the walker. $C$
is the position-dependent coin operator:

$C(\theta)=I\otimes\left(\begin{array}[]{cc}\cos\theta&\sin\theta\\\
\sin\theta&-\cos\theta\end{array}\right)$ (3)

where $I$ is a unit matrix with the same dimension as the number of grid
points. The loss operators are defined as:

$L=I\otimes\left[\begin{array}[]{ll}l_{1}&0\\\
0&l_{2}\end{array}\right],L^{\prime}=I\otimes\left[\begin{array}[]{ll}l_{2}&0\\\
0&l_{1}\end{array}\right]$ (4)

The evolution operator of the discrete-time quantum walk can be written in the
form of $U=e^{-iH_{eff}}$, where $H_{eff}$ represents the equivalent
Hamiltonian of the system. Let $\lambda$ as the eigenvalue of the evolution
operator, then we have

$U|\psi\rangle=\lambda|\psi\rangle,\lambda=e^{-i\epsilon+\epsilon_{0}}$ (5)

where $\epsilon$ is the quasi-energy of the system
$e^{-\epsilon_{0}}=1/l_{1}l_{2}$. When $l_{1}=l_{2}=1$ the evolution operator
$U$ is a unitary matrix with $\epsilon_{0}=0$ and $\absolutevalue{\lambda}=1$.
The system is a Hermitian system. When $l_{1}\neq l_{2}$, it is a non-
Hermitian system and $U$ is a non-unitary operator, which respects PT-symmetry
$(\mathcal{PT})U(\mathcal{PT})^{-1}=U^{-1}$.

##  III Diffusion Coefficient and Shannon Entropy

As demonstrated in Xiao  _et al._ (2017), the topological phase of the
evolution operator can be changed by tuning two variables of the system
evolution operator $\alpha$ and $\beta$. There are boundaries between
topological phases with different topological invariants. When
$\left(l_{1},l_{2}\right)=(1,1)$ which corresponds to no loss, the evolution
operator is the Hermitian operator. In this case, the topological invariant at
the boundary is not defined. When $\left(l_{1},l_{2}\right)=(1,0.8)$, the
evolution operator is a non-Hermitian operator, and there is a boundary area
between different topological phases, where the PT-symmetry of the state is
broken, and the topological invariant is not defined. It has been proved that
whether Hermitian operator or PT-symmetric non-Hermitian operator is used to
construct an artificial boundary in the system, there will always be a
topological edge state at the boundary when evolution operators with different
topological invariants are used on each side of the boundary Xiao  _et al._
(2017), which is related to the bulk-boundary correspondence in the
topological effects Essin and Gurarie (2011).

In order to study the topological effect of the quantum walk, traditionally,
artificial boundaries are constructed by employing different evolution
operators on each side of the boundary, then the probability distribution at
the boundary are probed, thus enabling simulation of the bulk-boundary
correspondence in topological materials Kitagawa  _et al._ (2010). Recently,
time evolution of the position variance Brun  _et al._ (2003); Romanelli  _et
al._ (2005); Xue  _et al._ (2014), statistical moments of the walker position
distribution Cardano  _et al._ (2016); Wang  _et al._ (2018); Xiao  _et al._
(2018), are measured for either investigating the localization effects or
revealing the topological phase transitions in discrete-time photonic quantum
walks.

Here we adopt a different approach, that is using the diffusion coefficient
Romanelli  _et al._ (2005); Xiaoxia and Zhijian (2017), a natural feature of
the quantum walk as the probe of the topological edge states, since the
evolution of the system during the quantum walk would lead to a distinctive
diffusive behavior Romanelli  _et al._ (2005). In addition, we investigate the
properties of Shannon entropy in the quantum walk. In the following sections,
we simulate the quantum walk, both Hermitian and PT-symmetric non-Hermitian,
and observe the behavior of the diffusion coefficient and Shannon entropy. In
all cases we take the initial state as $|\psi\rangle=|0\rangle$.

###  III.1 Hermitian Quantum Walk

Figure 1:  (a) Phase diagram for discrete time quantum walk in unitary
evolution, where winding numbers are function of the spin-rotation angles
$\alpha$ and $\beta$. Topologically distinct gapped phases characterized by
the winding number W are separated by phase-transition lines where the
quasienergy gap closes at either $E=0$ (red) or $E=\pi$ (blue). (b) The
diffusion coefficient of the quantum walk in unitary evolution with different
number of steps for the same $\beta=\pi/4$. Blue, red and yellow lines
correspond to $t=5$, $t=15$ and $t=50$ steps, respectively. (c) The dependence
of diffusion coefficient on the spin-rotation angles $\alpha$ and $\beta$ with
number of steps $t=15$. (d) The dependence of Shannon entropy on the spin-
rotation angles $\alpha$ and $\beta$ with number of steps $t=15$.

First, we study quantum walk with unitary evolution in Hermitian case. The
quasi-energy of the system and the Bloch vector are given by:

$\cos\epsilon(k)=-\sin\alpha\sin\beta+\cos k\cos\alpha\cos\beta,$ (6)

and $\vec{n}(k)=n_{x}(k)\hat{i}+n_{y}(k)\hat{j}+n_{z}(k)\hat{k}$ with

$\begin{array}[]{l}n_{x}(k)=\frac{\sin
k\sin\alpha\cos\beta}{\sin\epsilon(k)}\\\
n_{y}(k)=\frac{\cos\alpha\sin\beta+\cos
k\sin\alpha\cos\beta}{\sin\epsilon(k)}\\\ n_{z}(k)=\frac{-\sin
k\cos\alpha\cos\beta}{\sin\epsilon(k)}\end{array}$ (7)

Thus, we get the winding number with
$\Gamma=\left(\cos\alpha,0,\sin\alpha\right)$ according to:

$W=\frac{1}{2\pi}\oint dk\left(\vec{n}\times\frac{\partial\vec{n}}{\partial
k}\right)\cdot\Gamma$ (8)

The resulted phase diagram for the winding numbers is shown in Fig.1(a). We
observe two topological phases corresponding to $W=0$ and $W=1$ which are
separated by phase-transition lines. Red and blue lines represent closing of
the quasienergy gap at $\epsilon=0$ and $\epsilon=\pi$, respectively. We
simulate the topology of the quantum walk on a one-dimensional lattice. Let
the initial state of the walker as
$|\psi\rangle=|0\rangle\otimes|\uparrow\rangle$, i.e. the initial position of
the walker is $x=0$, initial coil state is $|\uparrow\rangle$.

Then, the calculation of the diffusion coefficient is carried out with
different number of steps $t$. The diffusion coefficient D is calculated
according to Romanelli  _et al._ (2005); Xiaoxia and Zhijian (2017):

$D=\frac{\sigma^{2}(t)}{2t}=\frac{M_{2}(t)-M_{1}^{2}(t)}{2t}$ (9)

where $M_{1}(t)=\sum_{n=-t}^{n=t}nP_{n}(t)$ and
$M_{2}(t)=\sum_{n=-t}^{n=t}n^{2}P_{n}(t)$ are the first and second moment,
$P_{n}(t)$ is the probability that the position of the walker is n at time t.
$\sigma^{2}$ is the variance of the quantum walk. Fig.1(b) shows the results
of $t=5$, $t=15$ and $t=50$ with the same $\beta=\pi/4$. For all three chosen
t, they show a very similar trend with $\alpha$, although the diffusion
coefficient increases with t. In the topological phase of $W=0$, the diffusion
coefficient is quite stable with small fluctuations, while in the topological
phase of $W=1$, it shows a valley structure along with $\alpha$. More
importantly, in the close proximity of the topological phase transition,
turning point appears on the diffusion coefficient with the change of
$\alpha$. In Fig.1(c), the dependence of diffusion coefficient on spin-
rotation angle is given for $\beta=\pi/6$, $\beta=\pi/4$ and $\beta=\pi/3$
with number of steps $t=15$, respectively. On the bottom $\alpha\theta_{2}$
plane, phase-transition lines are shown as Fig.1(a), together with vertical
dashed line in order to clearly distinguish those turning points in the
vicinity to the topological phase transition. It is also noticed that,
diffusion coefficient decreases with the increase of $\beta$ for the same
$\alpha$.

Subsequently, we obtain the corresponding Shannon entropy of the walker
according to:

$S=-\sum_{i=1}^{m}P_{n}\left(x_{i}\right)\log_{2}P_{n}\left(x_{i}\right)$ (10)

where $P_{n}\left(x_{i}\right)$ is the probability that a quantum walker is in
position $x_{i}$ at time $n$. The results are illustrated in Fig.1(d). It is
clear and interesting to find out, the Shannon entropy resembles amazingly
with the diffusion coefficient in terms of the dependence on spin-rotation
angles $\alpha$ and $\beta$, probably due to both of them are related to
$P_{n}(t)$.

###  III.2 PT-Symmetric Non-Hermitian Quantum Walk

Figure 2: (a) The diffusion coefficient of the quantum walk in PT-symmetric
non-unitary evolution with different number of steps for the same
$\beta=\pi/4$. Blue, red and yellow lines correspond to $t=5$, $t=15$ and
$t=30$ steps, respectively. The blue and red shaded regions represent broken
PT-symmetric phases Xiao  _et al._ (2017) with complex eigenenergies whose
real parts are 0 and $\pi$, respectively. (b) The dependence of diffusion
coefficient on the spin-rotation angle $\alpha$ with number of steps $t=15$
for $\beta=\pi/6$, $\beta=\pi/4$ and $\beta=\pi/3$ respectively. (c) The
dependence of Shannon entropy on the spin-rotation angle $\alpha$ with number
of steps $t=15$ for $\beta=\pi/6$, $\beta=\pi/4$ and $\beta=\pi/3$
respectively.

In particular, we are more interested in the behavior of a non-Hermitian
quantum walk, where the quantum coin (or position) carried by the walker
interacts with an environment, resulting in a loss. Such system would allow us
to explore topological properties of non-Hermitian systems and to investigate
Floquet topological phases Mochizuki  _et al._ (2016); Xiao  _et al._ (2017).
Moreover, it is shown that the quantum walk properties are highly sensitive to
decoherent events. As the quantum coil (or position) is coupled to an
environment, decoherence would lead a transition from quantum walk to
classical random walk for large number of steps, and a relation between the
decoherence and the diffusion properties of the system can be established Brun
_et al._ (2003); López and Paz (2003); Kendon and Tregenna (2003); Romanelli
_et al._ (2005). In addition, a weak decoherence could possibly enhance the
properties of the quantum walk and is thus beneficial to the development of
quantum algorithms Kendon and Tregenna (2003).

Here we consider the PT-symmetric non-Hermitian scenario by calculating
$\left(l_{1},l_{2}\right)=(1,0.8)$ with the same calculation and analysis. We
calculate diffusion coefficients for different number of steps for $t=5$,
$t=15$ and $t=30$ three different number of steps with the same $\beta=\pi/4$,
the results are shown in Fig.2(a). Based on the phase diagram of the PT-
symmetric non-Hermitian quantum walk Xiao  _et al._ (2017), the diffusion
coefficient are found to show similar features to Fig.1(b), which is
exhibiting abrupt change at the boundary of the topological phase transition.
As $\alpha$ increases from $-\pi$ to 0, the system starts from PT-symmetric
phase (corresponding to a topologically non-trivial state with topological
number $(\nu_{0},\nu_{\pi})=(1,1)$), and transits broken PT-symmetric phase
which is represented by red region in the vicinity of $\alpha=-3\pi/4$, with
which the real part of the eigenenergy is $\pi$. Then the system enters
another PT-symmetric phase (topologically non-trivial state with topological
number $(\nu_{0},\nu_{\pi})=(1,-1)$), and transits previous broken PTsymmetric
phase again, finally reaches a PT-symmetric phase (topologically non-trivial
state with topological number $(\nu_{0},\nu_{\pi})=(1,1)$). The diffusion
coefficient stays stable until it reaches the broken PT-symmetric phase. When
it first crosses broken PT-symmetric phase, the diffusion coefficient
gradually increases before $\alpha=-3\pi/4$ and then decreases, exhibiting a
peak-valley profile. When it crosses broken PT-symmetric phase again, the
diffusion coefficient increases before $\alpha=-\pi/4$ and then decreases,
exhibiting a valley-peak profile. The varies of the diffusion coefficient from
$\alpha=0$ to $\pi$ followes the same way as from $\alpha=-\pi$ to 0.

Another feature of the diffusion coefficient is that its amplitude is much
smaller than that in the Hermitian case, and it decreases with the increase of
the number of the step t due to the existing of the loss operator. Fig.2(b)
illustrates the dependence of diffusion coefficient on spin-rotation angle
$\alpha$ for $\beta=\pi/6$, $\beta=\pi/4$ and $\beta=\pi/3$ with number of
steps $t=15$, respectively. We choose $t=15$ because of several reasons.
First, we find the effects by varying t is much larger than by varying
$\alpha$. Second, larger $t$ would lead to too small the value of diffusion
coefficient to observe any effect due to excessive loss. Third, the effect of
coherent diffusion would be invisable if the number of steps t is very small.
Therefore, in order to better observe the effect of PT-symmetric quantum walk
with the topological phase in the non-Hermitian scenario, we take $t=15$ for
the following calculations for both diffusion coefficient and Shannon entropy.
From Fig.2(a-b), we notice that, the peak profile of the diffusion coefficient
has direct correspondence to the broken PT-symmetric phase, i.e. the
topological edge state, which suggests a possible intrinsic relation between
them. In Fig.2(c), Shannon entropy as the function of spin-rotation angle
$\alpha$ for $\beta=\pi/6$, $\beta=\pi/4$ and $\beta=\pi/3$ with number of
steps $t=15$ are presented, respectively. It is obvious to find, all of them
exhibit an interesting resemblance to that for diffusion coefficient.

##  IV Conclusion

We have proposed using the diffusion coefficient as a probe for studying the
dynamics of topological edge states. We have conducted simulations of both
Hermitian quantum walk and PTsymmetric non-Hermitian quantum walk, and have
investigated the topological phases driven by such quantum walks. The
properties of diffusion coefficient as a function of two parameters $\alpha$
and $\beta$ in evolution operator of the quantum walk are studied. It has been
found that that diffusion coefficient exhibits qualitatively similar
characteristics as the topological phase transition appears in Hermitian and
PT-symmetric non-Hermitian quantum walks. For a Hermitian system, a turning
point and abrupt change appears in the diffusion coefficient when the system
is at the boundary of the topological phase transition, while it remains
stable in trivial topological state. For a PT-symmetric non-Hermitian system,
however, the diffusion coefficient displays unique peak structure in the
topological phase transition region. In addition, Shannon entropy is found to
illustrate very corresponding behavior as the diffusion coefficient. Such
interesting signatures might give us a unique probe for the topological edge
states, and suggest exciting possibilities for studying the topological
boundary effects and edge states in PT-symmetric non-Hermitian quantum walks
by exploiting diffusion coefficient and Shannon entropy.

## Acknowledgements

This work is supported by the Key-Area Research and Development Program of
Guang Dong Province under Grant No.2019B030330001, the National Natural
Science Foundation of China under Grant No.11774436 and No.11974434. Le Luo
received supports from Guangdong Province Youth Talent Program under Grant
No.2017GC010656, Sun Yat-sen University Core Technology Development Fund. Yang
Liu acknowledges the support from Natural Science Foundation of Guangdong
Province under Grant 2020A1515011159 and Fundamental Research Funds for the
Central Universities of Education of China under Grant No. 191gpy276.

## References

  * Childs (2009) A. M. Childs, Physical review letters 102, 180501 (2009).
  * Karski  _et al._ (2009) M. Karski, L. Förster, J.-M. Choi, A. Steffen, W. Alt, D. Meschede, and A. Widera, Science 325, 174 (2009).
  * Schmitz  _et al._ (2009) H. Schmitz, R. Matjeschk, C. Schneider, J. Glueckert, M. Enderlein, T. Huber, and T. Schaetz, Physical review letters 103, 090504 (2009).
  * Lovett  _et al._ (2010) N. B. Lovett, S. Cooper, M. Everitt, M. Trevers, and V. Kendon, Physical Review A 81, 042330 (2010).
  * Zähringer  _et al._ (2010) F. Zähringer, G. Kirchmair, R. Gerritsma, E. Solano, R. Blatt, and C. Roos, Physical review letters 104, 100503 (2010).
  * Childs  _et al._ (2013) A. M. Childs, D. Gosset, and Z. Webb, Science 339, 791 (2013).
  * Perets  _et al._ (2008) H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, Physical review letters 100, 170506 (2008).
  * Broome  _et al._ (2010) M. A. Broome, A. Fedrizzi, B. P. Lanyon, I. Kassal, A. Aspuru-Guzik, and A. G. White, Physical review letters 104, 153602 (2010).
  * Thouless  _et al._ (1982) D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Physical review letters 49, 405 (1982).
  * Kitagawa  _et al._ (2010) T. Kitagawa, M. S. Rudner, E. Berg, and E. Demler, Physical Review A 82, 033429 (2010).
  * Kitagawa  _et al._ (2012) T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, Nature communications 3, 1 (2012).
  * Cardano  _et al._ (2017) F. Cardano, A. D’Errico, A. Dauphin, M. Maffei, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, L. Marrucci, _et al._ , Nature communications 8, 1 (2017).
  * Attal  _et al._ (2012a) S. Attal, F. Petruccione, and I. Sinayskiy, Physics Letters A 376, 1545 (2012a).
  * Attal  _et al._ (2012b) S. Attal, F. Petruccione, C. Sabot, and I. Sinayskiy, Journal of Statistical Physics 147, 832 (2012b).
  * Sinayskiy and Petruccione (2013) I. Sinayskiy and F. Petruccione, in  _Journal of Physics: Conference Series_ , Vol. 442 (IOP Publishing, 2013) p. 012003.
  * Bauer  _et al._ (2013) M. Bauer, D. Bernard, and A. Tilloy, Physical Review A 88, 062340 (2013).
  * Sinayskiy and Petruccione (2012) I. Sinayskiy and F. Petruccione, Quantum Information Processing 11, 1301 (2012).
  * Mochizuki  _et al._ (2016) K. Mochizuki, D. Kim, and H. Obuse, Physical Review A 93, 062116 (2016).
  * Mittal  _et al._ (2021) V. Mittal, A. Raj, S. Dey, and S. K. Goyal, Scientific Reports 11, 1 (2021).
  * Xiao  _et al._ (2017) L. Xiao, X. Zhan, Z. Bian, K. Wang, X. Zhang, X. Wang, J. Li, K. Mochizuki, D. Kim, N. Kawakami, _et al._ , Nature Physics 13, 1117 (2017).
  * Dadras  _et al._ (2018) S. Dadras, A. Gresch, C. Groiseau, S. Wimberger, and G. S. Summy, Physical review letters 121, 070402 (2018).
  * Ryu and Hatsugai (2002) S. Ryu and Y. Hatsugai, Physical review letters 89, 077002 (2002).
  * Bender and Boettcher (1998) C. M. Bender and S. Boettcher, Physical Review Letters 80, 5243 (1998).
  * Bender  _et al._ (2002) C. M. Bender, D. C. Brody, and H. F. Jones, Physical Review Letters 89, 270401 (2002).
  * Yao and Wang (2018) S. Yao and Z. Wang, Physical review letters 121, 086803 (2018).
  * Kunst  _et al._ (2018) F. K. Kunst, E. Edvardsson, J. C. Budich, and E. J. Bergholtz, Physical review letters 121, 026808 (2018).
  * Mochizuki  _et al._ (2020) K. Mochizuki, D. Kim, N. Kawakami, and H. Obuse, Physical Review A 102, 062202 (2020).
  * Weißenhofer and Nowak (2020) M. Weißenhofer and U. Nowak, New Journal of Physics 22, 103059 (2020).
  * Yoshida and Hatsugai (2021) T. Yoshida and Y. Hatsugai, Scientific reports 11, 1 (2021).
  * Zeuner  _et al._ (2015) J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, Physical review letters 115, 040402 (2015).
  * Weimann  _et al._ (2017) S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, Nature materials 16, 433 (2017).
  * Essin and Gurarie (2011) A. M. Essin and V. Gurarie, Physical Review B 84, 125132 (2011).
  * Brun  _et al._ (2003) T. A. Brun, H. A. Carteret, and A. Ambainis, Physical review letters 91, 130602 (2003).
  * Romanelli  _et al._ (2005) A. Romanelli, R. Siri, G. Abal, A. Auyuanet, and R. Donangelo, Physica A: Statistical Mechanics and its Applications 347, 137 (2005).
  * Xue  _et al._ (2014) P. Xue, H. Qin, and B. Tang, Scientific reports 4, 1 (2014).
  * Cardano  _et al._ (2016) F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. De Lisio, G. De Filippis, V. Cataudella, E. Santamato, and L. Marrucci, Nature communications 7, 1 (2016).
  * Wang  _et al._ (2018) X. Wang, L. Xiao, X. Qiu, K. Wang, W. Yi, and P. Xue, Physical Review A 98, 013835 (2018).
  * Xiao  _et al._ (2018) L. Xiao, X. Qiu, K. Wang, Z. Bian, X. Zhan, H. Obuse, B. C. Sanders, W. Yi, and P. Xue, Physical Review A 98, 063847 (2018).
  * Xiaoxia and Zhijian (2017) Z. Xiaoxia and L. Zhijian, Journal of Shanxi University 40, 100 (2017).
  * López and Paz (2003) C. C. López and J. P. Paz, Physical Review A 68, 052305 (2003).
  * Kendon and Tregenna (2003) V. Kendon and B. Tregenna, Physical Review A 67, 042315 (2003).