# Topological parafermion corner states
in clock-symmetric non-Hermitian second-order topological insulator

Motohiko Ezawa  Department of Applied Physics, University of Tokyo, Hongo
7-3-1, 113-8656, Japan

###### Abstract

Parafermions are a natural generalization of Majorana fermions. We consider a
breathing Kagome lattice with complex hoppings by imposing $\mathbb{Z}_{3}$
clock symmetry in the complex energy plane. It is a non-Hermitian
generalization of the second-order topological insulator characterized by the
emergence of topological corner states. We demonstrate that the topological
corner states are parafermions in the present $\mathbb{Z}_{3}$ clock-symmetric
model. It is also shown that the model is realized in electric circuits
properly designed, where the parafermion corner states are observed by
impedance resonance. We also construct $\mathbb{Z}_{4}$ and $\mathbb{Z}_{6}$
parafermions on breathing square and honeycomb lattices, respectively.

##  I Introduction

Topological quantum computation is a fault-tolerant quantum computationMoore ;
Das ; Kitaev ; TQC ; SternA ; Stern ; NPJ . Majorana fermions provide us with
a most studied platform of topological quantum computationBeen ; Stan ; Elli ;
AA ; Ivanov ; Halperin . The Majorana operator $\gamma$ satisfies
$\gamma^{2}=1$. They are realized as topological boundary states of
topological superconductorsAliceaBraid ; Qi ; Lei ; Alicea ; Sato ; Tanaka
and Kitaev spin liquidsKitaev ; Matsuda . However, it is impossible to perform
universal quantum computation only by the braiding of Majorana fermions since
they can reproduce only a part of Clifford gatesAhl .

Parafermions are straightforward generalization of Majorana fermionsFendJSM ;
Fend ; AliceaPara ; Jerm ; Ebisu , where the parafermion operator $\gamma$
satisfies $\gamma^{d}=1$ for $d\geq 3$. Braiding of parafermions with $d=3$
are known to reproduce all the Clifford gatesHutter  although universal
quantum computation is not yet possible. In this sense, parafermions are more
powerful than Majorana fermions in the context of quantum computation.
Parafermions are realized in clock-spin modelsBaxter ; Fend , fractional
quantum Hall effectsReadR ; Clarke , fractional topological
superconductorsLaub1  and twisted bilayer grapheneLaub2 . Among them, the
$\mathbb{Z}_{d}$ clock-spin modelBaxter  is non-Hermitian and its energy
spectrum is $\mathbb{Z}_{d}$ symmetric in the complex plane. It is an
interesting problem if they also emerge as topological boundary states in
certain lattice structures just as Majorana fermions do.

Higher-order topological insulators and superconductors are generalization of
topological insulators and superconductorsFan ; Science ; APS ; Peng ; Lang ;
Song ; Bena ; Schin ; FuRot ; Kagome ; EzawaPhos ; Gei ; Kha ; EzawaMajo .
They are prominent by the emergence of zero-energy corner states instead of
gapless edge states. These zero-energy corner states are topologically
protected. A typical example is given by the breathing Kagome latticeKagome ,
where three topological corner states emerge. There are some generalization to
non-Hermitian higher-order topological insulatorsLiuSOTI ; EzawaLCR ;
EzawaSkin ; Berg .

In this paper, generalizing the breathing Kagome second-order topological
insulator model by imposing $\mathbb{Z}_{3}$ clock symmetry, we propose a new
type of non-Hermitian higher-order topological insulator, where the
topological corner states are parafermions. This model is non-Hermitian, where
the energy spectrum is $\mathbb{Z}_{3}$ symmetric in the complex energy plane
as in the case of the $\mathbb{Z}_{3}$ clock-spin model. We demonstrate how to
implement the present model of parafermions in an electric circuit. We also
construct $\mathbb{Z}_{4}$ and $\mathbb{Z}_{6}$ parafermions as topological
corner states on breathing square and honeycomb lattices.

##  II Majorana fermion and parafermion

Majorana fermion operators $\gamma_{i}$ satisfy the relations

$\left(\gamma_{j}\right)^{2}=1,\qquad\gamma_{j}\gamma_{k}=-\gamma_{k}\gamma_{j}.$
(1)

Majorana fermions are realized as zero-energy states of a topological
superconductor, where particle-hole symmetry (PHS) preserves. It is understood
as follows. The PHS operator $\Xi$ acts as $\Xi^{-1}H\Xi=-H$ with the eigen
equation $H\left|\psi\right\rangle=E\left|\psi\right\rangle$. If a particle
has an energy $E$, its antiparticle has the energy $-E$ in the presence of
PHS. Namely, the wave functions always appear in a particle-hole pair with a
pair of energies $\left(E,-E\right)$. If the states satisfy the relation
$E=-E$, the particle is identical to its antiparticle, and a pair of Majorana
fermions emerge. Hence, the zero-energy ($E=0$) states respecting PHS are
Majorana states.

Parafermions are natural generalization of Majorana fermions. $\mathbb{Z}_{d}$
parafermions are defined through the relations

$\left(\gamma_{j}\right)^{d}=1,\qquad\gamma_{j}\gamma_{k}=\omega\gamma_{k}\gamma_{j},$
(2)

where $\omega=e^{2\pi i/d}$. The minimal model consists of two elements
$\gamma_{1}$ and $\gamma_{2}$.

##  III $\mathbb{Z}_{3}$ Parafermion

We start with a minimal model by setting $d=3$ in Eq.(2). $\mathbb{Z}_{3}$
parafermions are represented by the shift operatorZohar ; Fend ; AliceaPara

$\gamma_{1}\equiv\tau=\left(\begin{array}[]{ccc}0&0&1\\\ 1&0&0\\\
0&1&0\end{array}\right),$ (3)

and the clock operatorZohar ; Fend ; AliceaPara

$\gamma_{2}\equiv\sigma=\text{diag.}\left(1,\omega,\omega^{2}\right),$ (4)

where $\omega=e^{2\pi i/3}$. Here, $\tau$ and $\sigma$ satisfy the
$\mathbb{Z}_{3}$ parafermion relations,

$\tau^{3}=\sigma^{3}=1,\qquad\sigma\tau=\omega\tau\sigma.$ (5)

In the $\mathbb{Z}_{3}$ clock-symmetric model, the energy spectrum is composed
of tripletsBaxter  $E_{n}^{\left(0,1,2\right)}$, $n=0,1,2,\cdots$,

$E_{n}^{\left(0,1,2\right)}=\varepsilon_{n},\quad\omega\varepsilon_{n},\quad\omega^{2}\varepsilon_{n},$
(6)

satisfying
$\varepsilon_{n}+\omega\varepsilon_{n}+\omega^{2}\varepsilon_{n}=0$. The
system is necessarily non-Hermitian because the eigen energies are complex
except for zero-energy states.

Figure 1: (a) Breathing Kagome lattice, (b) breathing square lattice and (c)
breathing honeycomb lattice. The hopping parameters are $t_{a}=0$ for
(a1)$\sim$(c1), $t_{a}t_{b}\neq 0$ for (a2)$\sim$(c2), and $t_{b}=0$ for
(a3)$\sim$(c3). There emerge three, four and six parafermion corner states in
the topological phase. When $t_{a}=0$, they are isolated as marked by blue
disks in (a1), (b1) and (c1), respectively. The unit cells are given in (a4),
(b4) and (c4) with the hopping parameters indicated.

###  III.1 Zero-energy parafermion states

It follows from Eq.(6) that only the zero-energy states form a set of
degenerate states respecting $\mathbb{Z}_{3}$ clock symmetry. They are
$\mathbb{Z}_{3}$ parafermion states. We denote them as
$\left|\psi_{0}\right\rangle$, $\left|\psi_{1}\right\rangle$ and
$\left|\psi_{2}\right\rangle$. They are characterized by the properties

$\displaystyle\tau\left|\psi_{0}\right\rangle$
$\displaystyle=\left|\psi_{1}\right\rangle,\quad\tau\left|\psi_{1}\right\rangle=\left|\psi_{2}\right\rangle,\quad\tau\left|\psi_{2}\right\rangle=\left|\psi_{0}\right\rangle,$
(7) $\displaystyle\sigma\left|\psi_{0}\right\rangle$
$\displaystyle=\left|\psi_{0}\right\rangle,\quad\sigma\left|\psi_{1}\right\rangle=\omega\left|\psi_{1}\right\rangle,\quad\sigma\left|\psi_{2}\right\rangle=\omega^{2}\left|\psi_{2}\right\rangle,$
(8)

from which the matrix representations (3) and (4) follow. Then, the
$\mathbb{Z}_{3}$ parafermion relations (5) are verifed. Namely, it is
necessary and sufficient to examine Eqs.(7) and (8) for a triplet set of zero-
energy states in order to show that they are $\mathbb{Z}_{3}$ parafermions.

###  III.2 Breathing Kagome lattice

We propose a model possessing parafermions on the breathing Kagome lattice.
The bulk Hamiltonian is given by

$H=\left(\begin{array}[]{ccc}0&h_{12}&\omega h_{13}\\\
h_{12}^{\ast}&0&\omega^{2}h_{23}\\\ \omega
h_{13}^{\ast}&\omega^{2}h_{23}^{\ast}&0\end{array}\right),$ (9)

with

$\displaystyle h_{12}$ $\displaystyle=t_{a}+t_{b}e^{ik_{x}},$ (10)
$\displaystyle h_{23}$
$\displaystyle=t_{a}+t_{b}e^{-i\left(k_{x}/2+\sqrt{3}k_{y}/2\right)},$ (11)
$\displaystyle h_{13}$
$\displaystyle=t_{a}+t_{b}e^{-i\left(k_{x}/2-\sqrt{3}k_{y}/2\right)},$ (12)

where we have introduced two hopping parameters $t_{a}$ and $t_{b}$,
corresponding to the magenta link and the cyan link along the horizontal axis
in Fig.1(a4). The hopping parameters along the other two triangle sides are
given by $\omega t_{a}$ and $\omega^{2}t_{a}$ for a magenta triangle, and
$\omega t_{b}$ and $\omega^{2}t_{b}$ for a cyan triangle. This model is non-
Hermitian due to the presence of $\omega$.

A comment is in order with respect to the breathing Kagome model. It is a
typical model for the conventional second-order topological insulatorKagome ,
where the factor $\omega$ is absent and it is Hermitian. The present
generalization of the breathing Kagome lattice model provides us with a new
type of non-Hermitian second-order topological insulators.

Figure 2:  (a) Topological number (21) as a function of $t_{a}/t_{b}$. It is
quantized in the insulators. (b)$\sim$(c) Energy spectrum as a function of
$t_{a}/t_{b}$ for (b) the bulk, (c) a nanoribbon with width 64, and (d) a
triangle with size 20. In (b), the blue curves represent the bulk energy (20)
at the $K$ and $K^{\prime}$ points, and the cyan lines represent the bulk
energy (19) at the $\Gamma$ point, which are analytically obtained. In the
topological phase, gapless edge states are absent in nanoribbon geometry as in
(c), but zero-energy corner states emerge in triangle geometry as indicated by
the magenta flat-line segment in (d). The segment slightly deviates from the
region $(-1,1/2)$ due to the finite size effect.

###  III.3 Clock symmetry

The Hamiltonian (9) has $\mathbb{Z}_{3}$ clock symmetryFend ,

$\tau H\left(\mathbf{k}\right)\tau^{\dagger}=\omega
H\left(R\mathbf{k}\right),$ (13)

where $R$ rotates the momentum by 120 degrees as

$\displaystyle R\left(k_{x},0\right)$
$\displaystyle=\left(-\frac{k_{x}}{2},\frac{\sqrt{3}k_{y}}{2}\right),$ (14)
$\displaystyle R\left(-\frac{k_{x}}{2},\frac{\sqrt{3}k_{y}}{2}\right)$
$\displaystyle=\left(-\frac{k_{x}}{2},-\frac{\sqrt{3}k_{y}}{2}\right),$ (15)
$\displaystyle R\left(-\frac{k_{x}}{2},-\frac{\sqrt{3}k_{y}}{2}\right)$
$\displaystyle=\left(k_{x},0\right),$ (16)

making the energy spectrum have $\mathbb{Z}_{3}$ symmetry in the complex plane
as in Eq.(6). In addition, there is an anti-unitary symmetry

$KH\left(\mathbf{k}\right)K=H^{\ast}\left(\mathbf{k}\right),$ (17)

where $K$ implies complex conjugate. It leads to reflection symmetry between
$E$ and $E^{\ast}$. As a result, the energy spectrum has $C_{3v}$ symmetry in
the complex plane, which consists of the three-fold rotational symmetry and
three reflection symmetries.

The system also has a generalized chiral symmetry for a three-band modelNi ,

$\displaystyle\sigma H\sigma^{-1}$ $\displaystyle=$ $\displaystyle
H_{1},\qquad\sigma H_{1}\sigma^{-1}=H_{2},$ $\displaystyle H+H_{1}+H_{2}$
$\displaystyle=$ $\displaystyle 0.$ (18)

We show the bulk energy spectrum in Fig.3(a) and Fig.4(a), where $C_{3v}$
symmetry is manifest for all parameters.

###  III.4 Phase diagram

The notion of insulator and metal is generalized to the non-Hermitian
Hamiltonian in two ways. On is a point-gap insulatorGong ; Kawabata , where
$|E|$ has a gap. The other is a line-gap insulatorGong ; Kawabata , where
Re$\left[E\right]$ or Im$\left[E\right]$ has a gap. In our model, we adopt the
definition of the point-gap insulator due to $\mathbb{Z}_{3}$ symmetry. We are
able to determine the energy spectrum analytically at the $\Gamma=(0,0)$ point
as

$E\left(0,0\right)=(t_{a}+t_{b}),\quad\omega(t_{a}+t_{b}),\quad\omega^{2}(t_{a}+t_{b}),$
(19)

and at the $K=(4\pi/3,0)$ and $K^{\prime}=(-4\pi/3,0)$ points as

$E^{3}(\pm
4\pi/3,0)=\left(t_{a}+t_{b}\right)\left(t_{a}-2t_{b}\right)\left(2t_{a}-t_{b}\right).$
(20)

The point gap closes at the $K$ and $K^{\prime}$ points for $t_{a}/t_{b}=1/2$,
$t_{a}/t_{b}=2$, and at the $K$, $K^{\prime}$ and $\Gamma$ points for
$t_{a}/t_{b}=-1$, as in Fig.2(b). This is also confirmed numerically by
calculating the band spectrum as in Fig.2(b).

Figure 3:  Energy spectrum of a triangle, where (a) the vertical axis is
Re[$E$] and the horizontal axis is Im[$E$], while (b) the horizontal axis is
$t_{a}/t_{b}$. Color indicates the value of $t_{a}/t_{b}$, where the color
pallet is the same as in (b). $C_{3v}$ symmetry in the complex energy plane is
manifest in (a). The emergence of zero-energy corner states is clear in (b).
We have used a triangle with size $6$.

Figure 4: Complex energy spectrum (a1)$\sim$ (h1) for the bulk, (a2)$\sim$
(h2) for a nanoribbon and (a3)$\sim$ (h3) for a triangle, where the hopping
parameters are (a1)$\sim$ (a3) $t_{a}=-1.5t_{b}$, (b1)$\sim$ (b3)
$t_{a}=-t_{b}$, (c1)$\sim$ (c3) $t_{a}=-0.25t_{b}$, (d1)$\sim$ (d3)
$t_{a}=0.25t_{b}$, (e1)$\sim$ (e3) $t_{a}=0.5t_{b}$, (f1)$\sim$ (f3)
$t_{a}=1.25t_{b}$, (g1)$\sim$ (g3) $t_{a}=2t_{b}$ and (h1)$\sim$ (h3)
$t_{a}=2.5t_{b}$. The horizontal axis is Im[$E$] and the vertical axis is
Re[$E$]. The numerical value on the horizontal axis in (a1)$\sim$ (h1) is the
energy in unit of $t_{b}$. Color indicates the momentum of along the
nanoribbon direction, where red color indicates $k=\pi$ and blue color
indicates $k=0$ in a nanoribbon. We have used a nanoribbon with width $128$
for (a2)$\sim$ (h2), and a triangle with size $16$ for (a3)$\sim$ (h3). The
magenta dots in (c3) and (d3) represent the topological corner states.

Figure 5: (a1)$\sim$(h1) Bulk band structure, where the horizontal axis is
$k_{x}$. Color in the spectrum indicates $k_{y}$, where red color indicates
$k_{y}=0$ while cyan color indicates $k_{y}=2\pi/\sqrt{3}$. (a2)$\sim$(h2)
Band structure of a nanoribbon, where the horizontal axis is $k$. The vertical
axis is $|E|$. The values of $t_{a}$ and $t_{b}$ are the same as in Fig.4. It
is notable that there are no gapless edge states in the topological phase.

###  III.5 Topological number

The topological number is given by the Berry phase defined by

$Q\equiv\frac{1}{2\pi
i}\int_{0}^{2\pi}\left\langle\psi_{0}\left(k_{x},0\right)\right|\partial_{k_{x}}\left|\psi_{0}\left(k_{x},0\right)\right\rangle
dk_{x},$ (21)

where $\psi_{0}$ is the eigen function of the Hamiltonian (9) for the bulk,
whose eigen energy is real along the $k_{x}$ axis (i.e., $k_{y}=0$). We
calculate it numerically, whose results are shown in Fig.2(a). We find $Q=1$
for $-1<t_{a}/t_{b}<1/2$, and $Q=0$ for $t_{a}/t_{b}<-1$ and $t_{a}/t_{b}>2$,
while it continuously changes from $1$ to $0$ for $1/2<t_{a}/t_{b}<2$. In
fact, $Q$ is quantized in the insulator phases.

###  III.6 Edge states

We calculate the energy spectrum in a nanoribbon numerically. Edge states are
observed in Fig.4(a2)$\sim$(h2), where the complex energy spectrum is shown
for various momentum $k$ specified by color. Three-fold symmetry is slightly
broken in nanoribbon geometry. It is due to the finite size effect of a
nanoribbon.

We present the band structure of a nanoribbon in Fig.5(a2)$\sim$(h2), where we
observe the absence of gapless edge states in the topological phase.

###  III.7 Corner states

We calculate the energy spectrum in triangle geometry numerically. $C_{3v}$
symmetry is manifest as shown in Fig.3(a) and in Fig.4(a3)$\sim$(h3). It is
because the triangle respects $\mathbb{Z}_{3}$ clock symmetry. We find zero-
energy states in the region $-1<t_{a}/t_{b}<1/2$, as indicated by a magenta
line in Fig.2(d). We also show the energy spectrum as a function of
$t_{a}/t_{b}$ in Fig.3(b), where the emergence of the zero-energy states is
manifest in the topological phase.

Consequently, the present model is a second-order topological insulator in the
region $-1<t_{a}/t_{b}<1/2$, being characterized by the emergence of
topological corner states.

It is possible to obtain explicitly the wave functions of the three corner
states by numerical calculation. We have numerically confirmed that they
satisfy the relations (7) and (8). Therefore, they are $\mathbb{Z}_{3}$
parafermions.

###  III.8 Electric-circuit implementation

Electric circuits are governed by the Kirchhoff current law. By making the
Fourier transformation with respect to time, the Kirchhoff current law is
expressed as

$I_{a}\left(\omega\right)=\sum_{b}J_{ab}\left(\omega\right)V_{b}\left(\omega\right),$
(22)

where $I_{a}$ is the current between node $a$ and the ground, while $V_{b}$ is
the voltage at node $b$. The matrix $J_{ab}\left(\omega\right)$ is called the
circuit Laplacian. Once the circuit Laplacian is given, we can uniquely setup
the corresponding electric circuit. By equating it with the Hamiltonian $H$
asTECNature ; ComPhys

$J_{ab}\left(\omega\right)=i\omega H_{ab}\left(\omega\right),$ (23)

it is possible to simulate various topological phases of the Hamiltonian by
electric circuitsTECNature ; ComPhys ; Hel ; Lu ; YLi ; EzawaTEC ; EzawaLCR ;
EzawaSkin ; Garcia ; Hofmann ; EzawaMajo ; Tjunc . The relations between the
parameters in the Hamiltonian and in the electric circuit are determined by
this formula.

The circuit Laplacian is constructed as follows. To simulate the positive and
negative hoppings in the Hamiltonian, we replace them with the capacitance
$i\omega C$ and the inductance $1/i\omega L$, respectively. We note that $\sin
k=(e^{ik}-e^{-ik})/2i$ represents an imaginary hopping in the tight-bind
model. The imaginary hopping is realized by an operational amplifierHofmann .

We thus make the following replacements with respect to hoppings in the
Hamiltonian to derive the circuit Laplacian: (i) $+X\rightarrow i\omega C_{X}$
for $X=t_{a}$ and $t_{b}$, where $C_{X}$ represents the capacitance whose
value is $X$ [pF]. (ii) $-X\rightarrow 1/i\omega L_{X}$ for $X=t_{a}/2$ and
$t_{b}/2$, where $L_{X}$ represents the inductance whose value is $X$
[$\mu$H].

Figure 6: (a) Illustration of the breathing Kagome circuit corresponding to
Fig.1(a). (b) Each node is grounded by an inductor. (c) Complex hoppings
$\propto e^{\pm 2\pi i/3}$ are realized by a parallel connection of an
inductor and an operational amplifier. We show a triangle with size $4$.

We explicitly study the breathing Kagome lattice described by (9), where the
electric circuit is given by Fig.6(a). The Hamiltonian (9) is decomposed into

$H=H_{1}+H_{2},$ (24)

with

$H_{1}=\left(\begin{array}[]{ccc}0&h_{12}&-\frac{1}{2}h_{13}\\\
h_{12}^{\ast}&0&\omega^{2}h_{23}\\\
-\frac{1}{2}h_{13}^{\ast}&\omega^{2}h_{23}^{\ast}&0\end{array}\right)$ (25)

and

$H_{2}=\frac{\sqrt{3}}{2}\left(\begin{array}[]{ccc}0&0&ih_{13}\\\
0&0&-ih_{23}\\\ ih_{13}^{\ast}&-ih_{23}^{\ast}&0\end{array}\right),$ (26)

where $H_{1}$ is Hermitian ($H_{1}^{\dagger}=H_{1}$), and $H_{2}$ is anti-
Hermitian ($H_{2}^{\dagger}=-H_{2}$). It is necessary to construct imaginary
hopping Hamiltonians

$\frac{\sqrt{3}}{2}t_{a}\left(\begin{array}[]{ccc}0&0&i\\\ 0&0&0\\\
i&0&0\end{array}\right),$ (27)

and

$\frac{\sqrt{3}}{2}t_{a}\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&-i\\\
0&-i&0\end{array}\right)$ (28)

for the magenta lines in Fig.6(a). They are constructed by using operational
amplifiers and resistors.

Figure 7: Impedance as a function of $\omega$. The vertical axis is
$\log_{10}Z$ and the horizontal axis is $\omega/{\omega_{0}}$. (a) topological
phase with $t_{a}/t_{b}=0.25$, (b) metallic phase with $t_{a}/t_{b}=1$ and (c)
trivial phase with $t_{a}/t_{b}=2.5$. A prominent peak is found in the
topological phase.

We review a negative impedance converter with current inversion based on an
operational amplifier with resistorsHofmann . The voltage-current relation for
the operational amplifier circuit is given byHofmann

$\left(\begin{array}[]{c}I_{1}\\\
I_{2}\end{array}\right)=\frac{1}{R}\left(\begin{array}[]{cc}-\nu&\nu\\\
-1&1\end{array}\right)\left(\begin{array}[]{c}V_{1}\\\
V_{2}\end{array}\right),$ (29)

with $\nu=R_{b}/R_{a}$, where $R$, $R_{a}$ and $R_{b}$ are the resistances in
an operational amplifier: See Fig.6(d). We note that the resistors in the
operational amplifier circuit are tuned to be $\nu=1$ in the literatureHofmann
so that the system becomes Hermitian, where the corresponding Hamiltonian
represents a spin-orbit interaction.

In this paper, we use two negative impedance converters parallelly connected
with the opposite direction as in Fig.6(c). The circuit Laplacian due to these
two converters is given by

$\frac{1}{R}\left[\left(\begin{array}[]{cc}-\nu&\nu\\\
-1&1\end{array}\right)+\left(\begin{array}[]{cc}1&-1\\\
\nu&-\nu\end{array}\right)\right]=\frac{1}{R}\left(\begin{array}[]{cc}1-\nu&\nu-1\\\
\nu-1&1-\nu\end{array}\right).$ (30)

It corresponds to the Hamiltonian

$H=\frac{1}{i\omega R}\left(\begin{array}[]{cc}1-\nu&\nu-1\\\
\nu-1&1-\nu\end{array}\right).$ (31)

It is embedded in the 3$\times$3 matrix as

$H=\frac{1}{i\omega R}\left(\begin{array}[]{ccc}1-\nu&0&\nu-1\\\ 0&0&0\\\
\nu-1&0&1-\nu\end{array}\right),$ (32)

where we have set

$\frac{\sqrt{3}}{2}t_{a}=\frac{1-\nu}{\omega R}$ (33)

with $\nu<1$, and

$H=\frac{1}{i\omega R}\left(\begin{array}[]{ccc}0&0&0\\\ 0&1-\nu&\nu-1\\\
0&\nu-1&1-\nu\end{array}\right),$ (34)

where we have set

$\frac{\sqrt{3}}{2}t_{a}=\frac{\nu-1}{\omega R}$ (35)

with $\nu>1$. These matrices are different from Eqs.(27) and (28) by the
diagonal terms. They are cancelled by adding a resistor (for $\nu>1$) or an
operational amplifier (for $\nu<1$) with the amount of

$\frac{1-\nu}{i\omega R}$ (36)

between a lattice site and the ground.

###  III.9 Impedance resonance

The zero-energy parafermion corner states are well observed by impedance
resonance, which is definedHel  by

$Z_{ab}=V_{a}/I_{b}=G_{ab}$ (37)

where $G=J^{-1}$ is the Green function. It diverges at the frequency where the
admittance is zero ($J=0$). Taking the nodes $a$ and $b$ at two corners, we
show the impedance in topological, metallic and trivial phases in
Figs.7(a)$\sim$(c), respectively. A strong impedance peak is observed at the
critical frequency $\omega_{0}\equiv 1/\sqrt{LC}$ only in the topological
phase. It signals the emergence of zero-energy parafermion corner states.

Figure 8: Energy spectrum of a square, where (a) the vertical axis is Re[$E$]
and the horizontal axis is Im[$E$], while (b) the horizontal axis is
$t_{a}/t_{b}$. Color indicates the value of $t_{a}/t_{b}$, where the color
pallet is the same as in (b). $C_{4v}$ symmetry in the complex energy plane is
manifest in (a). The emergence of zero-energy corner states is clear in (b).

##  IV $\mathbb{Z}_{4}$ Parafermion

We proceed to a model with $d=4$. $\mathbb{Z}_{4}$ parafermions are
represented by the shift operatorZohar ; Fend ; AliceaPara

$\gamma_{1}\equiv\tau=\left(\begin{array}[]{cccc}0&0&0&1\\\ 1&0&0&0\\\
0&1&0&0\\\ 0&0&1&0\end{array}\right),$ (38)

and the clock operatorZohar ; Fend ; AliceaPara

$\gamma_{2}\equiv\sigma=\text{diag.}\left(1,i,-1,-i\right).$ (39)

Here, $\tau$ and $\sigma$ satisfy the $\mathbb{Z}_{4}$ parafermion relations,

$\tau^{4}=\sigma^{4}=1,\qquad\tau\sigma=\omega\sigma\tau.$ (40)

In the $\mathbb{Z}_{4}$ clock-symmetric model, the energy spectrum is composed
of quartets $E_{n}^{\left(0,1,2,3\right)}$, $n=0,1,2,\cdots$,

$E_{n}^{\left(0,1,2,3\right)}=\varepsilon_{n},\quad
i\varepsilon_{n},\quad-\varepsilon_{n},\quad-i\varepsilon_{n}.$ (41)

The system is necessarily non-Hermitian because the eigen energies are complex
except for zero-energy states.

###  IV.1 Zero-energy parafermion states

It follows from Eq.(41) that only the zero-energy states form a set of
degenerate states respecting $\mathbb{Z}_{4}$ clock symmetry. They are
$\mathbb{Z}_{4}$ parafermion states. We denote them as
$\left|\psi_{0}\right\rangle$, $\left|\psi_{1}\right\rangle$,
$\left|\psi_{2}\right\rangle$ and $\left|\psi_{3}\right\rangle$. They are
characterized by the properties

$\displaystyle\tau\left|\psi_{0}\right\rangle$
$\displaystyle=\left|\psi_{1}\right\rangle,\quad\tau\left|\psi_{1}\right\rangle=\left|\psi_{2}\right\rangle,$
(42) $\displaystyle\tau\left|\psi_{2}\right\rangle$
$\displaystyle=\left|\psi_{3}\right\rangle,\quad\tau\left|\psi_{3}\right\rangle=\left|\psi_{0}\right\rangle,$
(43) $\displaystyle\sigma\left|\psi_{0}\right\rangle$
$\displaystyle=\left|\psi_{0}\right\rangle,\quad\sigma\left|\psi_{1}\right\rangle=i\left|\psi_{1}\right\rangle,$
(44) $\displaystyle\sigma\left|\psi_{2}\right\rangle$
$\displaystyle=-\left|\psi_{2}\right\rangle,\quad\sigma\left|\psi_{3}\right\rangle=-i\left|\psi_{3}\right\rangle,$
(45)

from which the matrix representations (3) and (4) follow. Then, the
$\mathbb{Z}_{4}$ parafermion relations (5) are verified. Namely, it is
necessary and sufficient to examine Eqs.(7) and (8) for a triplet set of zero-
energy states in order to show that they are $\mathbb{Z}_{4}$ parafermions.

###  IV.2 Breathing square lattice

The quadrupole insulator has been proposed on the breathing square
latticeScience . We propose a model possessing $\mathbb{Z}_{4}$ parafermions
on the breathing square lattice. The bulk Hamiltonian is given by

$H=\left(\begin{array}[]{cccc}0&f_{x}^{\ast}&0&if_{y}^{\ast}\\\
-f_{x}&0&if_{y}^{\ast}&0\\\ 0&-if_{y}&0&-f_{x}\\\
-if_{y}&0&f_{x}^{\ast}&0\end{array}\right),$ (46)

with

$f_{x}=t_{a}+t_{b}e^{ik_{x}},\qquad f_{y}=t_{a}+t_{b}e^{ik_{y}},$ (47)

where we have introduced two hopping parameters $t_{a}$ and $t_{b}$, which are
shown in Fig.1(b4).

The Hamiltonian (46) has $\mathbb{Z}_{4}$ clock symmetryFend ,

$\tau H\left(\mathbf{k}\right)\tau^{\dagger}=-iH\left(R\mathbf{k}\right),$
(48)

where $R$ rotates the momentum by $90$ degrees as

$\displaystyle R\left(k_{x},0\right)$ $\displaystyle=\left(0,-k_{y}\right),$
(49) $\displaystyle R\left(0,k_{y}\right)$
$\displaystyle=\left(k_{x},0\right),$ (50)

making the energy spectrum have $\mathbb{Z}_{4}$ symmetry in the complex plane
as in Eq.(41).

###  IV.3 Edge and corner states

The topological number is defined by (21), where $\psi_{0}$ is the eigen
function of the Hamiltonian (46) for the bulk. We find the topological
insulator phase for $\left|t_{a}/t_{b}\right|<1$, where $Q=1$ and the trivial
insulator phase for $\left|t_{a}/t_{b}\right|>1$, where $Q=0$.

We calculate the energy spectrum in square geometry numerically. $C_{4v}$
symmetry is manifest as shown in Fig.8(a). It is because the square lattice
respects $\mathbb{Z}_{4}$ clock symmetry. We also show the energy spectrum as
a function of $t_{a}/t_{b}$ in Fig.8(b), where the emergence of the zero-
energy states is manifest in the topological phase.

Consequently, the present model is a second-order topological insulator in the
region $|t_{a}/t_{b}|<1$, being characterized by the emergence of topological
corner states.

It is possible to obtain numerically the wave functions of the four corner
states. We have confirmed that they satisfy the relations (7) and (8).
Therefore, they are $\mathbb{Z}_{4}$ parafermions.

Figure 9: (a) Electric circuit implementation of the breathing square model
shown in Fig.1(b), where the hopping parameters correspond to those in
Fig.1(b4). (b) Configuration of negative impedance converter with current
inversion realizing imaginary hoppingHofmann , and (c) the one realizing
nonreciprocal hoppingNonR , corresponding to those in (a).

###  IV.4 Electric-circuit implementation

In the Hamiltonian (46), there are nonreciprocal hopping terms along the $x$
axis. Nonreciprocal hopping is constructed by a combination of operational
amplifier and capacitorsNonR ,

$\left(\begin{array}[]{c}I_{ij}\\\ I_{ji}\end{array}\right)=i\omega
C\left(\begin{array}[]{cc}-1&1\\\
-1&1\end{array}\right)\left(\begin{array}[]{c}V_{i}\\\
V_{j}\end{array}\right),$ (51)

It corresponds to the Hamiltonian

$H=C\left(\begin{array}[]{cc}-1&1\\\ -1&1\end{array}\right).$ (52)

We add an inductor or a capacitor in order to cancel the diagonal term.

We construct an electric circuit as shown in Fig.9(a), where the $x$ axis is
constructed by Fig.9(c) and the $y$ axis is constructed by Fig.9(b).

##  V $\mathbb{Z}_{6}$ Parafermion

Finally, we construct a $\mathbb{Z}_{6}$ parafermion model. The parafermion
operators are represented by the shift operatorZohar ; Fend ; AliceaPara

$\gamma_{1}\equiv\tau=\left(\begin{array}[]{cccccc}0&0&0&0&0&1\\\
1&0&0&0&0&0\\\ 0&1&0&0&0&0\\\ 0&0&1&0&0&0\\\ 0&0&0&1&0&0\\\
0&0&0&0&1&0\end{array}\right),$ (53)

and the clock operatorZohar ; Fend ; AliceaPara

$\gamma_{2}\equiv\sigma=\text{diag.}\left(1,\omega,\omega^{2},\omega^{3},\omega^{4},\omega^{5}\right).$
(54)

Here, $\tau$ and $\sigma$ satisfy the $\mathbb{Z}_{6}$ parafermion relations,

$\tau^{6}=\sigma^{6}=1,\qquad\tau\sigma=\omega\sigma\tau.$ (55)

In the $\mathbb{Z}_{6}$ clock-symmetric model, the energy spectrum is composed
of sextets $E_{n}^{\left(0,1,2,3,4,5\right)}$, $n=0,1,2,\cdots$,

$E_{n}^{\left(0,1,2,3,4,5\right)}=\varepsilon_{n},\quad\omega\varepsilon_{n},\quad\omega^{2}\varepsilon_{n},\quad\omega^{3}\varepsilon_{n},\quad\omega^{4}\varepsilon_{n},\quad\omega^{5}\varepsilon_{n}.$
(56)

The system is necessarily non-Hermitian because the eigen energies are complex
except for zero-energy states.

Figure 10: Energy spectrum of a hexagon shown in Fig.1(c2). (a) the vertical
axis is Re[$E$] and the horizontal axis is Im[$E$], while (b) the vertical
axis is $|E|$ and the horizontal axis is $t_{a}/t_{b}$. Color indicates the
value of $t_{a}/t_{b}$ in (a). $C_{6v}$ symmetry in the complex energy plane
is manifest in (a). The emergence of zero-energy corner states is clear in
(b). The green curves are calculated from the bulk Hamiltonian, while red
curves are calculated based on the hexagon. The cyan curve is the energy at
the $\Gamma$ point, while the blue curve is the energy at the $K$ point
calculated based on the bulk Hamiltonian.

A Hermitian second-order topological insulator has been proposed on the
breathing honeycomb latticeMizoguchi . We generalize it to a non-Hermitian
model with $\mathbb{Z}_{6}$ clock symmetry by introducing complex hoppings.
The bulk Hamiltonian is defined on the breathing honeycomb lattice and given
by

$H=\left(\begin{array}[]{cccccc}0&\omega^{2}t_{b}&0&t_{a}e^{ik_{y}}&0&\omega^{4}t_{b}\\\
\omega^{5}t_{b}&0&\omega^{3}t_{b}&0&\omega
t_{a}e^{i\frac{-\sqrt{3}k_{x}+k_{y}}{2}}&0\\\
0&t_{b}&0&\omega^{4}t_{b}&0&\omega^{2}t_{a}e^{-i\frac{\sqrt{3}k_{x}+k_{y}}{2}}\\\
\omega^{3}t_{a}e^{-ik_{y}}&0&\omega t_{b}&0&\omega^{5}t_{b}&0\\\
0&\omega^{4}t_{a}e^{i\frac{\sqrt{3}k_{x}-k_{y}}{2}}&0&\omega^{2}t_{b}&0&t_{b}\\\
\omega
t_{b}&0&\omega^{5}t_{a}e^{i\frac{\sqrt{3}k_{x}+k_{y}}{2}}&0&\omega^{3}t_{b}&0\end{array}\right).$
(57)

We diagonalize this Hamiltonian for a hexagon shown in Fig.1(c2). $C_{6v}$
symmetry is manifest in the complex energy plane as in Fig.10(a). Furthermore,
we find six zero-energy topological corner states for
$\left|t_{a}/t_{b}\right|<1$, as shown in Fig.10(b). We also find the trivial
insulator phase for $t_{a}/t_{b}<-2$ and $t_{a}/t_{b}>1$. Additionally, there
is metallic phase for $-2<t_{a}/t_{b}<-1$.

##  VI Conclusion

We have constructed a $\mathbb{Z}_{3}$ parafermion model on the breathing
Kagome lattice, a $\mathbb{Z}_{4}$ parafermion model on the breathing square
lattice and a $\mathbb{Z}_{6}$ parafermion model on the breathing honeycomb
lattice. These model exhaust all the possible realization of $\mathbb{Z}_{d}$
parafermions since there are only three-fold, four-fold and six-fold
rotational symmetries that are compatible with the periodic lattices. We note
that two-fold symmetry corresponds to Majorana fermions.

The author is very much grateful to Y. Tanaka and N. Nagaosa for helpful
discussions on the subject. This work is supported by the Grants-in-Aid for
Scientific Research from MEXT KAKENHI (Grants No. JP17K05490 and No.
JP18H03676). This work is also supported by CREST, JST (JPMJCR16F1 and
JPMJCR20T2).

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