††thanks: Corresponding author<EMAIL_ADDRESS>
# Non-Abelian operation through scattering between chiral Dirac edge modes

Zhi-Xing Lin  International Center for Quantum Materials, School of Physics,
Peking University, Beijing 100871, China  Yijia Wu  International Center for
Quantum Materials, School of Physics, Peking University, Beijing 100871, China
X. C. Xie  International Center for Quantum Materials, School of Physics,
Peking University, Beijing 100871, China  CAS Center for Excellence in
Topological Quantum Computation, University of Chinese Academy of Sciences,
Beijing 100190, China

###### Abstract

We theoretically demonstrate that non-Abelian braiding operation can be
realized through the scattering between chiral Dirac edge modes (CDEMs) in
quantum anomalous Hall insulators by analytically deriving its $S$-matrix.
Based on the analytical model, we propose a viable device for the experimental
realization and detection of the non-Abelian braiding operations. Through
investigating the tunneling conductance in a discretized lattice model, the
non-Abelian properties of CDEMs could also be verified in a numerical way. Our
proposal for the CDEM-based braiding provides a new avenue for realizing
topologically protected quantum gates.

##  I Introduction

Majorana zero modes (MZMs) are self-conjugate excitations which appear as
vortex-bound statesIvanov (2001) or topologically protected end statesKitaev
(2001); Alicea et al. (2011) in topological superconductors. MZMs are regarded
to possess non-Abelian statisticsAlicea et al. (2011) due to its topology-
related non-Abelian geometric phase accumulated during the braiding
operations. Owing to these charming properties, MZMs have been widely
investigatedWang et al. (2018); Zhu et al. (2020); Kong and Ding (2021);
Mourik et al. (2012); Rokhinson et al. (2012); Deng et al. (2012); Churchill
et al. (2013); Das et al. (2012); Finck et al. (2013) and regarded as the most
promising candidate for topological quantum computationSarma et al. (2015).

The experimental signals for MZMs have been experimentally reported in various
platforms such as the iron-based superconductors with vorticesWang et al.
(2018); Zhu et al. (2020); Kong and Ding (2021) and the semiconductor
nanowires proximate to $s$-wave superconductorMourik et al. (2012); Rokhinson
et al. (2012); Deng et al. (2012); Churchill et al. (2013); Das et al. (2012);
Finck et al. (2013). Since MZMs are “zero-dimensional” localized states bound
to the vortex or the end of the nanowire, the braiding operation of MZMs are
expected to be conducted through spatially moving the vortices (in the
assistance of the probing technology such as scanning tunneling microscope
tips) or modulating the gate voltages in superconductor-semiconductor nanowire
junctionsAlicea et al. (2011); Sau et al. (2011); Van Heck et al. (2012).
However, the adiabatic conditionAmorim et al. (2015) will impose restriction
on the braiding operation that the typical time cost for each braiding
operation should be in the scale of nanosecondAmorim et al. (2015). Moreover,
extra instability could also be introduced since the braiding operations are
conducted through external manipulations, in which some undesirable coupling
or dephasing may also appear.

The one-dimensional counterpart of the MZM is the chiral Majorana edge mode
(CMEM), which emerges as the topological edge state in two-dimensional
$p$-wave superconductorsRead and Green (2000). Such one-dimensional CMEMs
usually possess linear dispersion near the Fermi energy and hence are more
favorable in electrical transport. It has been theoretically proved that the
propagation of the CMEMs could also lead to the similar non-Abelian
transformations as those in the braiding process of the “zero-dimensional”
MZMsLian et al. (2018); Zhou et al. (2019). Instead of the delicate
manipulations required for braiding MZM, the braiding operation can be
naturally accomplished through the swift propagation of CMEMs, which
efficiently enhance the speed and stability of the quantum gatesLian et al.
(2018). Moreover, the possible signals of the CMEMs have been reported in the
hybrid system of quantum anomalous Hall insulator (QAHI) and superconductorHe
et al. (2017). Based on the above reasons, the CMEMs now has also served as a
possible platform for non-Abelian braiding.

Nonetheless, the experimental signalsHe et al. (2017) for CMEMs are still
controversial since some non-Majorana mechanismKayyalha et al. (2020) cannot
be excluded yet. Such controversy in Majorana-based braiding inspire ones to
explore the possible non-Abelian statistics in non-Majorana systems where the
superconductivity is absent. Remarkably, topological insulators supporting
Dirac edge states share a similar Hamiltonian with the topological
superconductors supporting Majorana edge states, although they are described
in two different Hamiltonian basis. Such an analogy implies the similarities
in both their topological and transport propertiesLee et al. (2007); Wu et al.
(2020a). Actually, the “zero-dimensional” Dirac fermionic mode as the
topological end states of one-dimensional topological insulators, which is
regarded as the “Dirac counterpart” of the MZM, also obeys the non-Abelian
braiding statisticsYasui et al. (2012); Wu et al. (2020a, b) as one might have
expected.

In contrast to the Majorana case in which more experimental evidences for both
the MZMLiu et al. (2012); Ruby et al. (2015); Moore et al. (2018); Liu et al.
(2018); Vuik et al. (2019) and the CMEMKayyalha et al. (2020) are still highly
requested, the topological chiral Dirac edge modes (CDEMs) have been
repeatedly verified in a number of experimental platforms including integer
quantum Hall (IQH)Klitzing et al. (1980); Novoselov et al. (2007); Tsukazaki
et al. (2007) insulators and QAHIsChang et al. (2013); Kou et al. (2014);
Bestwick et al. (2015); Checkelsky et al. (2014) in a solid way. Since Dirac
fermionic modes have been demonstrated to obey non-Abelian braiding
statisticsYasui et al. (2012); Wu et al. (2020a, b), in analogy to the idea
“promoting” MZMs into CMEMs, we can naturally wonder whether we can extend
topological Dirac fermionic modes from “zero-dimension” to one-dimension. Such
a one-dimensional extension is exactly the CDEM discussed above. By noticing
again the similarity between the topological insulators and the topological
superconductors, one can also expect that the CDEMs will exhibit non-Abelian
properties as CMEMs do during their scattering and propagation process.

In this paper, we investigate the CDEM-based braiding both analytically and
numerically. We also propose a viable experimental scheme for its realization.
In Sec. II, we analytically calculate the scattering matrix between CDEMs and
discuss the resonant condition for realizing the non-Abelian braiding
operation; in Sec. III, based on our experimental proposal, we discover that
the resonant condition is accompanied with the conductance peak, which could
be satisfied by tuning the back gate voltages; in Sec. IV, we conduct a
numerical simulation of our experimental device and investigate the effects of
different parameters, which corroborates our analytical conclusion. in Sec. V,
we make a brief discussion comparing the braiding properties of the CDEMs and
the MZMs; and finally in Sec. VI, we present a short conclusion.

##  II Analytical study for the scattering between chiral Dirac edge modes

Following a similar path of the CMEM-based quantum gates, we intend to
implement the braiding operation through the scattering and propagation of
CDEMs. Therefore, we begin with the scattering process of CDEMs coupled in the
manner shown in Fig. 1, which bears a resemblance to the Andreev reflection
induced by the CMEMLaw et al. (2009). A QAHI island with some magnetic
vortices penetrated serves as the scattering center here, where the edge state
of this QAHI island is a CDEM. Another two CDEMs as the edge states of another
two QAHI bulks are coupled to the scattering center at points $a$ and $b$ with
tunneling amplitudes $t_{1}$ and $t_{2}$, respectively. The total Hamiltonian
of the system is

Figure 1: A QAHI island with magnetic vortices penetrated is placed in the
center, which possesses a branch of CDEM $\varphi$ at its outer boundary.
Another two CDEMs are coupled to the QAHI islands through weak tunneling at
points $a$ and $b$ with tunneling amplitudes $t_{1}$ and $t_{2}$,
respectively.

$\displaystyle H_{\mathrm{tot}}=-i\hbar
v_{f}\int_{-\infty}^{\infty}[\psi^{\dagger}_{1}(x)\partial_{x}\psi_{1}(x)+\psi^{\dagger}_{2}(x)\partial_{x}\psi_{2}(x)]\mathrm{d}x\quad$
$\displaystyle+i\hbar
v\int_{0}^{L}\varphi^{\dagger}(x)\partial_{x}\varphi(x)\mathrm{d}x-it_{1}[\varphi^{\dagger}(a)\psi_{1}(0)+\varphi(a)\psi_{1}^{\dagger}(0)]$
$\displaystyle-
it_{2}[\varphi^{\dagger}(b)\psi_{2}(0)+\varphi(b)\psi_{2}^{\dagger}(0)]\quad\quad$
(1)

where $\varphi$ stands for the CDEM in the QAHI island and $\psi_{1},\psi_{2}$
represents the CDEMs of QAHI coupled to the center.

Denoting the incoming and outgoing scattering states of the electrons by
$\psi_{1/2\omega}(0-),\psi_{1/2\omega}(0+)$, respectively ($\omega$ stands for
the electron’s energy), the scattering matrix can be written as

${\left(\begin{array}[]{c}\psi_{1\omega}(0+)\\\
\psi_{2\omega}(0+)\end{array}\right)}=S{\left(\begin{array}[]{c}\psi_{1\omega}(0-)\\\
\psi_{2\omega}(0-)\end{array}\right)}$ (2)

Using the anticommutation relation and the equation of motion for the field
operators, the explicit form of the $S$-matrixDatta (1997) can be derived as:

$\displaystyle
S=\frac{1}{Z}{\left(\begin{array}[]{ccc}i\sin(\frac{\alpha+\beta}{2})(1-\tilde{t}_{1}^{2}\tilde{t}_{2}^{2})-(\tilde{t}_{1}^{2}-\tilde{t}_{2}^{2})\cos(\frac{\alpha+\beta}{2})&-2e^{i(\beta-\alpha)/2}\tilde{t}_{1}\tilde{t}_{2}\\\
-2e^{i(\alpha-\beta)/2}\tilde{t}_{1}\tilde{t}_{2}&i\sin(\frac{\alpha+\beta}{2})(1-\tilde{t}_{1}^{2}\tilde{t}_{2}^{2})-(\tilde{t}_{2}^{2}-\tilde{t}_{1}^{2})\cos(\frac{\alpha+\beta}{2})\end{array}\right)}$
(5)

where
$Z=i\sin(\frac{\alpha+\beta}{2})(1+\tilde{t}_{1}^{2}\tilde{t}_{2}^{2})+(\tilde{t}_{1}^{2}+\tilde{t}_{2}^{2})\cos(\frac{\alpha+\beta}{2})$.
$e^{i\alpha}(e^{i\beta})$ is the phase factor acquired by a CDEM propagating
from point $a(b)$ to point $b(a)$. $\tilde{t}_{1}=t_{1}/2\hbar\sqrt{vv_{f}}$,
and $\tilde{t}_{2}=t_{2}/2\hbar\sqrt{vv_{f}}$ are dimensionless coupling
amplitudes.

Then we denote $\alpha+\beta$ by $\theta$, which stands for the overall phase
acquired by the CDEMs when they make a complete circle around the QAHI island.
For a CDEM with energy $\omega$, we have $\theta=\omega L/v+\Phi+\pi$, where
$v$ is the Fermi velocity of the QAHI island, $L$ stands for the perimeter of
the QAHI island, $\Phi$ is the phase induced by the magnetic flux penetrating
the QAHI island, and the last term comes from the Berry phase contribution of
the spinLaw et al. (2009).

In some special cases, the $S$-matrix [Eq. (5)] can reduce to a simple form
which is significant for the realization of non-Abelian braiding. We first
consider a symmetric case in which $\tilde{t}_{1}=\tilde{t}_{2}$, with the
overall phase $\theta=2m\pi$ ($m\in\mathbb{Z}$), then the $S$-matrix becomes
$S=\left(\begin{matrix}0&(-1)^{m+1}e^{i(\beta-\alpha)/2}\\\
(-1)^{m+1}e^{i(\alpha-\beta)/2}&0\\\ \end{matrix}\right)$, which implies a
resonant exchange with $\psi_{1}\to(-1)^{m+1}e^{i(\alpha-\beta)/2}\psi_{2}$
and $\psi_{2}\to(-1)^{m+1}e^{i(\beta-\alpha)/2}\psi_{1}$. Furthermore, for a
single-coupled case where $\tilde{t}_{1}=0$, and the overall phase
$\theta=2m\pi(m\in\mathbb{Z})$ is the same as the previous case, it is evident
that $S=\left(\begin{matrix}1&0\\\ 0&-1\\\ \end{matrix}\right)$, indicating
that the phase shift induced by the scattering is $\pi$. Remarkably, if we
conduct the above two manipulations successively, the CDEMs $\psi_{1}$ and
$\psi_{2}$ will be scattered into $(-1)^{m+1}e^{i(\alpha-\beta)/2}\psi_{2}$
and $(-1)^{m}e^{i(\beta-\alpha)/2}\psi_{1}$, respectively. After applying a
gauge transformation that $\tilde{\psi}_{1}=e^{-i\alpha/2}\psi_{1}$,
$\tilde{\psi}_{2}=e^{-i\beta/2}\psi_{2}$, finally we realize a non-Abelian
braiding between two branches of CDEMs as
$\tilde{\psi}_{1}\to(-1)^{m+1}\tilde{\psi}_{2}$ and
$\tilde{\psi}_{2}\to(-1)^{m}\tilde{\psi}_{1}$.

According to the calculation above, in order to realize the non-Abelian
braiding operation, it is essential for our system to satisfy the resonant
condition $\theta=\alpha+\beta=2m\pi$ ($m\in\mathbb{Z}$). Since $\theta=\omega
L/v+\Phi+\pi$ as mentioned above, such a resonant condition can be achieved by
tuning the chemical potential of the QAHI island (the scattering center) to
choose the specific electron energy that satisfies the requirement, or to
change the number of the magnetic flux vortices penetrating through the QAHI
island.

So the crucial point is how to distinguish whether the resonant condition is
satisfied or not experimentally, that is, how to manipulate the experimental
parameters to conduct the braiding operation in a correct manner.

##  III Proposal for experimental scheme

Based on the discussion above, we propose a four-lead device to observe the
non-Abelian braiding of CDEMs experimentally. As shown in Fig. 2, the device
consists of four blocks of QAHI with Chern number $\mathcal{C}=1$, which
correspond to one CDEM branch in each QAHI block. Two QAHI islands (denoted by
I, II) with magnetic vortices penetrated serve as two scattering centers, and
another two QAHI regions coupled to them provide two branches of CDEMs
$\tilde{\psi}_{1},\tilde{\psi}_{2}$. A bias $\Delta V$ is applied between lead
1 and lead 4, while lead 2 and lead 3 are shorted. These two CDEMs are
scattered and manipulated during its propagation along the QAHI edges, and
their transport behaviors can be detected by measuring the conductance. Two
pairs of split gates are deposited at the coupling sites to form the gate-
defined confinementFu et al. (2019); Bid et al. (2009). In such a way, the
tunneling between the scattering centers and
$\tilde{\psi}_{1},\tilde{\psi}_{2}$ could be manipulated by tuning the
corresponding split gate voltages.

Noticeably, we only take QAHIs here as an example, all the QAHI regions in
Fig. 2 can be replaced by other materials supporting CDEMs, e.g., gallium
arsenide heterostructure supporting IQH effect. In addition, the device in
Fig. 2 could be experimentally realized by depositing a top gate with specific
shape above a whole QAHI sample. By depleting the electrons underlying such a
top gate, the whole QAH sample can be separated into four blocks as we
desired. Lin et al. (2012)

Figure 2: (a) An experimental device based on QAHI to realize braiding
operations on two CDEM branches $\tilde{\psi}_{1},\tilde{\psi}_{2}$. Two QAHI
islands are denoted by I and II, the corresponding phases accumulated by
circulating these two QAHI islands are denoted by $\theta_{1}$ and
$\theta_{2}$, respectively. Back gate $V_{bg1}$ ($V_{bg2}$) is attached to
QAHI island I (II) for tuning $\theta_{1}$ ($\theta_{2}$). Two pairs of split
gates (denoted as “Gate 1” and “Gate 2”) are deposited to manipulate the
coupling between $\tilde{\psi}_{1}$ and QAHI islands II, and the coupling
between $\tilde{\psi}_{2}$ and QAHI islands I, respectively. We define the
state when gate voltage is applied and the coupling is cut off as “Gate off”,
otherwise as “Gate on”. (b) The tunneling conductance $G=I_{2}/\Delta V$ as a
function of $\theta_{1}$.

Figure 3: The numerical results of the tunneling conductance $G$ as a function
of the electron’s energy $\omega$, with respect to different connection width
$w$, coupling strength $t_{\mathrm{cp}}$, the phase $\Phi_{1}$ induced by the
magnetic flux in the QAHI island I, and the energy-band parameter $A$. (a)
Tunneling conductance $G$ versus $\omega$ with respect to different $w$, with
$t_{\mathrm{cp}}=1.0t_{0},\Phi_{1}=\pi,A=0.5$.
($t_{0}=-\frac{i}{2}A\sigma_{y}+B\sigma_{z}$ with $B$=0.5, and the same
below.) (b) Tunneling conductance $G$ versus $\omega$ with respect to
different $t_{\mathrm{cp}}$, with $w=2,\Phi_{1}=\pi,A=0.5$. (c) Tunneling
conductance $G$ versus $\omega$ with respect to different $\Phi_{1}$, with
$w=1,t_{\mathrm{cp}}=1.0t_{0},A=0.5$. (d) Tunneling conductance $G$ versus
$\omega$ with respect to different $A$, with
$w=2,t_{\mathrm{cp}}=1.0t_{0},\Phi_{1}=\pi$.

To observe the effect of the phase $\theta_{1}$ (the phase acquired by the
CDEMs when circulating the QAHI island I), we first consider the case with
Gate 2 off and Gate 1 on so that the tunneling amplitude between
$\tilde{\psi}_{1}$ and QAHI islands II (controlled by Gate 2) is pinched off,
while the tunneling between $\tilde{\psi}_{2}$ and QAHI islands I (manipulated
by Gate 1) is allowed [see Fig. 2]. Under this condition, the scattering of
island II just applies an additional phase to $\tilde{\psi}_{2}$, thus the
tunneling conductance is independent of $\theta_{2}$. The measured currents of
lead $n$ can be calculated using the Landauer-Buttiker formulaButtiker (1986):
$I_{n}=\frac{e^{2}}{h}\sum_{m}T_{nm}(V_{n}-V_{m})$, where tunneling
coefficients $T_{nm}$ can be deduced from the scattering amplitude calculated
above.

Therefore the tunneling conductance $G$, which is defined as $I_{2}/\Delta V$,
can be expressed as a function of $\theta_{1}$ as:

$\displaystyle G$ $\displaystyle=\frac{I_{2}}{\Delta
V}=\frac{e^{2}}{2h}(T_{32}-T_{31})$ (6)
$\displaystyle=\frac{e^{2}}{2h}\frac{4\tilde{t}_{1}^{2}\tilde{t}_{2}^{2}-(1-\tilde{t}_{1}^{2}\tilde{t}_{2}^{2})^{2}\sin^{2}\frac{\theta_{1}}{2}-(\tilde{t}_{1}^{2}-\tilde{t}_{2}^{2})^{2}\cos^{2}\frac{\theta_{1}}{2}}{(1+\tilde{t}_{1}^{2}\tilde{t}_{2}^{2})^{2}\sin^{2}\frac{\theta_{1}}{2}+(\tilde{t}_{1}^{2}+\tilde{t}_{2}^{2})^{2}\cos^{2}\frac{\theta_{1}}{2}}$

Fig. 2 represents the oscillation behavior of the tunneling conductance $G$ as
a function of $\theta_{1}$, in the condition that
$\tilde{t}_{1}^{4}=\tilde{t}_{2}^{4}=\tilde{t}^{4}=0.1$.

From Fig. 2, one can find that when $\theta_{1}$ approaches $2m\pi$ ($m\in
Z$), the tunneling conductance $G$ shows a peak of one half of
$\frac{e^{2}}{h}$, that is, the resonant condition $\theta_{1}=2m\pi$ is
satisfied when the conductance reaches its maximum. Such a resonant condition
can be achieved by tuning the electron’s energy $\omega$ through the back gate
$V_{\mathrm{bg1}}$.

The tuning of $\theta_{2}$ can be accomplished in the same fashion as
$\theta_{1}$. We can first switch on Gate 2 and switch off Gate 1, thus the
dependence of $G$ on $\theta_{2}$ is the same as that on $\theta_{1}$ in the
previous case; then the resonant condition of $\theta_{2}$ can be achieved by
tuning $V_{\mathrm{bg2}}$, and finally close Gate 2 and open Gate 1.
Consequently, in our experimental scheme, the resonant condition can be
reflected by the conductance peaks which are experimental observables. By
switching on and off the gate-defined confinement and then modulating the back
gate voltages, we can tune the phases $\theta_{1}$ and $\theta_{2}$ for both
two QAHI islands to satisfy the resonant condition. Therefore we have proposed
a set of viable procedures for the realization of the CDEM-based non-Abelian
braiding operation.

##  IV Numerical Evidence

To verify the validity of our braiding proposal, we utilize a discretized
lattice version of the Bernevig-Hughes-Zhang (BHZ) modelBernevig et al. (2006)
to perform numerical simulation. The effective lattice Hamiltonian in each
QAHI block is

$\displaystyle H_{\mathrm{eff}}$ $\displaystyle=\sum_{\vec{k}}[A\sigma_{x}\sin
k_{x}+A\sigma_{y}\sin k_{y}$ (7)
$\displaystyle+(\Delta-4B\sin^{2}\frac{k_{x}}{2}-4B\sin^{2}\frac{k_{y}}{2})\sigma_{z}]|\vec{k}\rangle\langle\vec{k}|$

In addition to the four QAHI blocks, the coupling of the CDEMs
$\tilde{\psi}_{1},\tilde{\psi}_{2}$ and the two QAHI islands can be included
by adding corresponding hopping terms connecting the corresponding QAHI block
boundaries with connection width $w$ and hopping strength $t_{\mathrm{cp}}$.
To obviate the effects of bulk states, the energy $\omega$ of the lattice
Green function in our numerical investigation lies within the bulk gap.

Our numerical results are shown in Fig. 3, the tunneling conductance $G$
exhibits a periodic oscillation as expected, and its maximum coincides with
our theoretical prediction. The amplitude and the full width at half maximum
(FWHM) of each peak varies with the connection width $w$ [Fig. 3] as well as
the hopping strength $t_{\mathrm{cp}}$ [Fig. 3], which implies that the
effective coupling strength $t_{1},t_{2}$ in the analytic model [see Eq. (II)]
depends on both $w$ and $t_{\mathrm{cp}}$. With the increase of the connection
width $w$, the CDEM will tunnel into the QAHI island from different lattice
sites with different phases and then a self-interference is presented. Hence
the FWHM decreases with the increase of $w$, as shown in Fig. 3.

However, an unexpected damping of the oscillation amplitude accompanies with
the oscillating behavior, where the damping rate differs with respect to
different parameter choices. A reasonable explanation is that the Fourier
transform of the coupling strength from the form in the tight-binding model
$t_{\mathrm{cp}}$ to the form in the analytic model $t_{1},t_{2}$ depends on
the energy $\omega$. Hence the effective coupling strength in the lattice
model varies with the energy $\omega$.

Nonetheless, the phase $\theta_{1}$ remains unchanged with the variance of the
coupling strength transformation, so that both the oscillation period and the
phase shift are independent of the energy $\omega$, the coupling width $w$ and
the coupling strength $t_{\mathrm{cp}}$, which is consistent with the
simulation results shown in Fig. 3, 3.

As shown in Fig. 3, the phase shift of the conductance curve can be induced by
the magnetic flux $\Phi_{1}$ penetrating through the QAHI island I. The phase
shift is defined as
$\Delta\theta_{1}=\theta_{1}(\Phi_{1})-\theta_{1}(\Phi_{1}=\pi)$. If the CDEMs
are well localized at the edge so that all the magnetic flux penetrated is
encircled, then the phase shift will follow such a compact relation:
$\Delta\theta_{1}=\Phi_{1}-\pi$. We calculate the phase shifts with respect to
different magnetic fluxes and different coupling parameters [Fig. 4], and all
of them are in good agreement with the formula
$\Delta\theta_{1}=\Phi_{1}-\pi$, corroborating that the tunneling conductance
obtained in the numerical simulation is contributed from the CDEMs which
circulate all the magnetic flux penetrated through the QAHI island.

Furthermore, in the low-energy limit, the oscillation frequency $f_{\omega}$
with respect to energy $\omega$ equals to $\frac{L}{2A}$. Thus by tuning the
energy-band parameter $A$, we can modulate the oscillation period of the
tunneling conductance $G$ as the numerical results shown in Fig. 3. As shown
in Fig. 4, our simulation results exhibit a strong linear relation in log-log
plot with a slope that is approximately equal to $1$. Such results corroborate
our analytical formula. Furthermore, based on the fitting results shown in
Fig. 4, we can obtain an effective perimeter $L_{\mathrm{eff}}$ of the QAHI
island, which roughly equals to the perimeter of our simulation model $L$ as
$L_{\mathrm{eff}}\gtrsim L$, confirming the contribution from edge physics.
The slight deviation between $L_{\mathrm{eff}}$ and $L$ may contributes to the
tunneling points near which the CDEMs may make a detour so that the effective
perimeter becomes longer.

Now we conclude that the numerical results based on the BHZ model are in good
agreement with the analytical ones, providing a more solid demonstration for
our experimental scheme.

Figure 4: (a) The phase shift with respect to the phase induced by the flux
inside the QAHI island I $\Phi_{1}$ under different $w$ and $t_{\mathrm{cp}}$.
The consistency between the numerical simulation and the theoretical formula
confirms that the numerical results come from the physics of edge states. (b)
The oscillation frequency of the current with respect to energy-band parameter
$A$ under different coupling conditions.

##  V Discussions

In the previous sections, we have investigated the CDEM-based non-Abelian
braiding both analytically and numerically, and proposed viable device for
further experiments. Compared with other braiding schemes, the CDEM-based
braiding inherit the favorable transport properties of the CMEMs that the
braiding operation can be accomplished via its natural edge propagation other
than the external manipulation. Furthermore, the experimental platforms
supporting CDEMs possess desirable robustness for their larger bulk gapWu et
al. (2014); Qi et al. (2016); Li et al. (2020) compared with the Majorana
system. The absence of the superconductivity in the CDEM-based braiding not
only significantly reduce the complexity of the experimental setup, but also
exclude other possible states Moore et al. (2018); Liu et al. (2018); Vuik et
al. (2019) that may obstruct the non-Abelian braiding.

In addition, we have demonstrated the non-Abelian properties of CDEMs, which
bears resemblance to the braiding properties of the renowned MZMs. Whereas, it
is noteworthy that the corresponding basis of non-Abelian braiding operation
for CDEMs Yasui et al. (2012); Wu et al. (2020b) differ from those for MZMs.
The basis for CDEM-based braiding are all single-particle operators, thus the
braiding operation preserves the particle number. In comparison, the MZM-based
braiding preserves the electron parity, which allows particles created or
annihilated in pairsIvanov (2001).

Furthermore, in our experimental proposal, the braiding effect is detected by
observing the conductance peak due to resonant tunneling. Although resonant
tunneling could also be exhibited by a localized state bound in a quantum dot
(QD)Zhou et al. (2019), the QD will only induce a single peak rather than a
series of equally-spaced conductance peaks with the same height shown in our
model. Hence the conductance signal characteristics possessed by the CDEMs in
our proposal is distinguishable from those come from the other trivial
mechanisms.

##  VI Conclusion

In conclusion, we have demonstrated the non-Abelian braiding operation can be
conducted utilizing CDEMs. Through two successive scattering process between
two CDEMs, the overall operation recovers the standard non-Abelian form under
a gauge transformation. Additionally, the effect of the braiding operation can
be experimentally detected by observing the conductance peaks in QAHI
junctions, which has been demonstrated by simulations. Compared with the MZM-
based braiding proposal, our scheme overcomes two main obstacles in
topological quantum computing: the speed of braiding operation and the
robustness of edge modes. By utilizing the high velocity of the CDEMs and the
robust experimental platforms supporting CDEMs, our braiding proposal enables
us to manipulate CDEMs with desirable speed and stability, thus may sheds some
light on the topological quantum computation.

##  VII Acknowledgements

We thank for the fruitful discussion with Hua Jiang. This work is financially
supported by the Strategic Priority Research Program of Chinese Academy of
Sciences (Grant No. XDB28000000), the National Basic Research Program of China
(Grants No. 2015CB921102), and the China Postdoctoral Science Foundation
(Grant No. 2021M690233)

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