# Exotic Topological Phenomena in Chiral Superconducting States on Doped
Quantum Spin Hall Insulators with Honeycomb Lattices

Junkang Huang1,2 Tao Zhou1,2,3<EMAIL_ADDRESS>Z. D. Wang3<EMAIL_ADDRESS>1Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum
Materials, School of Physics, South China Normal University, Guangzhou 510006,
China
2Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Frontier Research
Institute for Physics, South China Normal University, Guangzhou 510006, China
3Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Department of
Physics, and HK Institute of Quantum Science $\&$ Technology, The University
of Hong Kong, Pokfulam Road, Hong Kong, China

###### Abstract

We have conducted a theoretical investigation of the topological phenomena
associated with chiral superconducting pairing states induced in a doped Kane-
Mele model on a honeycomb lattice. Through numerical analysis, we have
obtained exotic phase diagrams for both the $d+id$ and $p+ip$ superconducting
states. In the case of the $d+id$ pairing state, higher Chern number states
with $\left|C\right|=\pm 4$ emerge. The Chern number decreases as the spin-
orbit coupling is introduced. For the $p+ip$ pairing state, additional phase
transition lines are present in the overdoped region near the Van Hove
singularity point, leading to the emergence of higher Chern number phases with
$\left|C\right|=\pm 6$. These higher Chern number phases are further verified
through the bulk-edge correspondence. To understand the origin of the exotic
topological phase diagrams in the chiral superconducting state, we have
examined the electronic structure at the phase transition lines. This
investigation provides insight into the complex interplay between chiral
superconductivity and topological properties, potentially paving the way for
the discovery of new materials with unique topological properties.

††preprint: APS/123-QED

##  I Introduction

Chiral topological superconductors have been both theoretically and
experimentally investigated, drawing significant research interest due to
their fascinating properties [1, 2]. The chiral pairing term breaks time-
reversal symmetry, leading to chiral Majorana zero modes and edge currents
with promising potential for quantum computation. Various chiral
superconducting pairing states have been proposed in numerous systems [3, 4,
5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19].

Honeycomb lattice systems are excellent candidates for achieving chiral
superconductivity. One well-known material system with honeycomb lattice
symmetry is graphene-based materials. Superconductivity in this family of
materials has previously attracted considerable research interest. Various
theoretical techniques, such as symmetry analysis, exact numerical study,
random phase approximation, renormalization group method, dynamic cluster
approximation, kinetic-energy-driven superconductivity, and quantum Monte
Carlo method, have extensively suggested that the superconducting pairing
symmetry in graphene-based materials could be either $p+ip$ pairing symmetry
or $d+id$ pairing symmetry [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31,
32]. Chiral superconductivity may serve as a useful platform for exploring
nontrivial topology in the honeycomb lattice system [33].

Experimentally, superconductivity signatures have been observed in various
graphene-based materials [34, 35, 36, 37, 38, 39]. Besides graphene-based
materials, superconductivity has also been discovered in $\beta$-MNCl (M = Hf,
Zr) [40, 41], SrPtAs [42, 12], silicene-based materials [43, 44, 45],
hydrogenated germanene [46], and stanene [47, 48]. These materials also
exhibit honeycomb lattice symmetry. On the other hand, it has been reported
that the honeycomb lattice and unconventional superfluidity can be realized in
cold atom systems [49, 50], offering another platform for exploring chiral
pairing states and potential exotic topological behaviors.

Another non-trivial topological state that has garnered significant attention
is the quantum spin Hall insulating (QSHI) state. The QSHI state in a
honeycomb lattice can be characterized by the Kane-Mele (KM) model, which was
originally proposed to describe the topological properties of graphene [51].
It has subsequently been proposed for implementation in materials such as
silicene, germanene, stanene, $\mathrm{Pt}_{2}\mathrm{HgSe}_{3}$, and
$\mathrm{Pd}_{2}\mathrm{HgSe}_{3}$ [52, 53, 54, 55, 56, 57, 58]. In the QSHI
state, energy bands are fully gapped, rendering the material a topological
insulator. The Fermi level of a material can be artificially adjusted through
gate voltage or chemical doping. As a result, the metallic state is realized
when the Fermi level intersects the energy bands. At low temperatures, the
superconducting state should be the ground state, with an additional effective
pairing interaction. It has previously been reported that the Fermi level of
monolayer QSHI material WTe2 can be gate-tuned, subsequently inducing
superconductivity [59, 60]. For graphene, it has also been reported that the
material can be heavily overdoped, with the Fermi level being tuned beyond the
Van Hove Singularity (VHS) point [61]. Consequently, unconventional
superconductivity may be realized due to fermiology near the VHS point [62].
Hence, investigating the topological behavior in the superconducting state of
doped QSHI material on a honeycomb lattice is fundamentally fascinating, as
the interplay between distinct non-trivial topologies could result in a
considerably more complex phase diagram and even more exotic topological
phenomena.

In this paper, motivated by the considerations above, we investigate the
topological properties of a honeycomb lattice system by introducing chiral
superconductivity into the doped KM model. The topological properties can be
characterized by defining the Chern number of the system. We obtain a rich
phase diagram numerically, demonstrating the existence of high Chern number
phases. The presence of higher Chern number states strongly depends on lattice
symmetry and can be investigated by examining the Dirac cones around phase
transition lines. These higher Chern number states can be further confirmed by
exploring the edge states of the system. The existence of multiple Majorana
zero modes at the system edges may have potential applications in topological
quantum computation [63, 64]. Experimentally, the Chern number of a chiral
topological system relates to the edge current or the anomalous hall
conductance [65, 66, 67]. However, its relationship still requires further
investigation. The rich and exotic phase diagram presented here offers a new
platform to explore the relationship between the Chern number and observable
physical quantities in a chiral topological superconducting system.

The organization of this paper is as follows: In Section II, we describe the
model and elaborate on the formalism. In Section III, we present the numerical
results, including phase diagrams, energy spectra, and corresponding boundary
spectral functions for different phases. In Section IV, we discuss the origins
of the numerical results. Finally, we provide a brief summary of our work in
Section V.

##  II Model and Formalism

We begin by defining the Hamiltonian on a honeycomb lattice, which includes
both the normal state component and the chiral superconducting pairing
component. The complete Hamiltonian can be represented in momentum space as
$H=\sum_{\bf k}\Psi^{\dagger}({\bf k})H({\bf k})\Psi({\bf k})$, where $H({\bf
k})$ is expressed in an $8\times 8$ matrix,

$\displaystyle
H\left(\bf{k}\right)=\left(\begin{array}[]{cccc}{H_{0}\left(\bf{k}\right)}&{\Delta_{D/P}\left(\bf{k}\right)}\\\
{\Delta^{\dagger}_{D/P}\left(\bf{k}\right)}&{-H_{0}^{*}\left(\bf{k}\right)}\end{array}\right),$
(3)

The basis vector $\Psi^{\dagger}\left(\bf{k}\right)$ is expressed as
$(C_{\bf{k}A\uparrow}^{\dagger},C_{\bf{k}A\downarrow}^{\dagger},C_{\bf{k}B\uparrow}^{\dagger},C_{\bf{k}B\downarrow}^{\dagger},$
$C_{\bf{k}A\uparrow},C_{\bf{k}A\downarrow},C_{\bf{k}B\uparrow},C_{\bf{k}B\downarrow})^{T}$.
$H_{0}\left(\bf{k}\right)$ is the $4\times 4$ normal states Hamiltonian
matrix, taken as the doped KM model [51], expressed as,

$\displaystyle
H_{0}\left(\bf{k}\right)=\left(\begin{array}[]{cccc}{h_{SO}\left(\bf{k}\right)}&{h_{t}\left(\bf{k}\right)+h_{R}\left(\bf{k}\right)}\\\
{\left[h_{t}\left(\bf{k}\right)+h_{R}\left(\bf{k}\right)\right]^{\dagger}}&{-h_{SO}\left(\bf{k}\right)}\end{array}\right)-\mu,$
(6)

where $h_{t}\left(\bf{k}\right)$, $h_{SO}\left(\bf{k}\right)$ and
$h_{R}\left(\bf{k}\right)$ are the nearest neighbor hopping term, spin-orbit
coupling term and Rashba term, respectively.

$\displaystyle h_{t}\left(\bf{k}\right)$ $\displaystyle=$ $\displaystyle-
ts_{0}\sum_{i}\exp\left(i\bf{k}\cdot\bf{a_{i}}\right),$ (7) $\displaystyle
h_{SO}\left(\bf{k}\right)$ $\displaystyle=$
$\displaystyle-2\lambda_{SO}s_{3}\sum_{i}\sin\left(\bf{k}\cdot\bf{b_{i}}\right),$
(8) $\displaystyle h_{R}\left(\bf{k}\right)$ $\displaystyle=$ $\displaystyle
i\lambda_{R}\sum_{i}\exp\left(i\bf{k}\cdot\bf{a_{i}}\right)\left(\bf{s}\times\frac{\bf{a_{i}}}{\left|\bf{a_{i}}\right|}\right)_{z},$
(9)

where $t$ is the nearest-neighbor hopping constant. $s_{0}$ and $s_{i}$
$(i=1,2,3)$ are the identity matrix and the Pauli matrices in the spin
channel, respectively. Vectors ${\bf a_{i}}$ and ${\bf b_{i}}$ are expressed
as: ${\bf a_{1}}=(\sqrt{3}a/3,0)$, ${\bf a_{2}}=(-\sqrt{3}a/6,a/2)$, ${\bf
a_{3}}=(-\sqrt{3}a/6,-a/2)$, and ${\bf b_{1}}=(0,-a)$, ${\bf
b_{2}}=(\sqrt{3}a/2,a/2)$, ${\bf b_{3}}=(-\sqrt{3}a/2,a/2)$.

$\Delta_{D/P}\left(\bf{k}\right)$ is the $d+id/p+ip$ pairing term, written as,

$\displaystyle\Delta_{D}\left(\bf{k}\right)$ $\displaystyle=$ $\displaystyle
i\Delta_{0}^{D}\sigma_{1}s_{2}(e^{i\sigma\bf{k}\cdot{\bf
a_{1}}}+e^{i4\pi/3}\cdot e^{i\sigma\bf{k}\cdot{\bf a_{2}}}$ (10)
$\displaystyle+e^{i8\pi/3}\cdot e^{i\sigma\bf{k}\cdot{\bf a_{3}}}),$
$\displaystyle\Delta_{P}\left(\bf{k}\right)$ $\displaystyle=$
$\displaystyle-\Delta_{0}^{P}\sigma_{2}s_{2}(e^{i\sigma\bf{k}\cdot{\bf
a_{1}}}+e^{i2\pi/3}\cdot e^{i\sigma\bf{k}\cdot{\bf a_{2}}}$ (11)
$\displaystyle+e^{i4\pi/3}\cdot e^{i\sigma\bf{k}\cdot{\bf a_{3}}}),$

where $\sigma_{i}$ are Pauli matrices in the sublattice channel. $\sigma$
takes $+1$ or $-1$ depending on the pairing orientation: A to B or B to A,
respectively.

$\Delta^{D/P}_{0}$ is the superconducting order parameter magnitude, which can
be calculated self-consistently in the momentum space as,

$\displaystyle\Delta_{0}^{D}$ $\displaystyle=$
$\displaystyle\frac{V}{2}\sum_{{\bf
k}n}u^{*}_{1n\bf{k}}u_{8n\bf{k}}\tanh\left(\frac{\beta E_{n_{\bf
k}}}{2}\right)(e^{i\bf{k}\cdot{\bf a_{1}}}+$ (12) $\displaystyle
e^{i4\pi/3}\cdot e^{i\bf{k}\cdot{\bf a_{2}}}+e^{i8\pi/3}\cdot
e^{i\bf{k}\cdot{\bf a_{3}}}),$ $\displaystyle\Delta_{0}^{P}$ $\displaystyle=$
$\displaystyle\frac{V}{2}\sum_{{\bf
k}n}u^{*}_{1n\bf{k}}u_{8n\bf{k}}\tanh\left(\frac{\beta E_{n_{\bf
k}}}{2}\right)(e^{i\bf{k}\cdot{\bf a_{1}}}+$ (13) $\displaystyle
e^{i2\pi/3}\cdot e^{i\bf{k}\cdot{\bf a_{2}}}+e^{i4\pi/3}\cdot
e^{i\bf{k}\cdot{\bf a_{3}}}),$

with $V$ being the attractive pairing interaction. $u_{in\bf{k}}$ is the
$i$-th component of the vector $u_{n\bf{k}}$. $u_{n\bf{k}}$ and $E_{n_{\bf
k}}$ are the $n$-th eigenvectors and $E_{n_{\bf k}}$ are eigenvectors and
eigenvalues of the Hamiltonian matrix $H({\bf k})$.

When the chiral superconducting pairing term is present, the time reversal
symmetry of the Hamiltonian is broken. As a result, the Chern number can be
utilized to describe the nontrivial topology of the system, express as [68],

$\displaystyle C$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi i}\sum_{\bf
k}\tilde{F}_{xy}\left({\bf{k}}\right).$ (14)

where $\tilde{F}_{xy}$ is the lattice field strength that can be defined by

$\displaystyle\tilde{F}_{xy}\left({\bf{k}}\right)\equiv\ln\frac{U_{x}\left({\bf
k}\right)U_{y}\left({\bf k}+{\hat{x}}\right)}{U_{x}\left({\bf
k}+{\hat{y}}\right)U_{y}\left({\bf k}\right)},$ (15)

with

$\displaystyle U_{\alpha}\left({\bf
k}\right)=\frac{\det\Phi^{\dagger}\left({\bf k}\right)\Phi\left({\bf
k}+{\hat{\alpha}}\right)}{\left|\det\Phi^{\dagger}\left({\bf
k}\right)\Phi\left({\bf k}+{\hat{\alpha}}\right)\right|},$ (16)

where $\Phi({\bf k})$ is an $8\times 4$ matrix, which is constructed by
column-wise packing the four occupied eigenstates with $\Phi({\bf
k})=(u_{1\bf{k}},u_{2\bf{k}},u_{3\bf{k}},u_{4\bf{k}})$.

To investigate the edge states, we apply a periodic boundary condition in the
$x$ direction and a zigzag boundary condition in the $y$ direction $(1\leq
y\leq N_{y})$. The Hamiltonian with a partial open boundary condition can be
expressed as
$H=\sum_{j,{k_{x}}}\Psi^{\dagger}\left(j,{k_{x}}\right)\hat{M}\left({k_{x}}\right)\Psi\left(j,{k_{x}}\right)$,
where $\hat{M}({k_{x}})$ is an $8N_{y}\times 8N_{y}$ matrix and
$\Psi_{j,{k_{x}}}^{\dagger}$ is the basis vector with
$\Psi_{j,{k_{x}}}^{\dagger}=(C_{j,{k_{x}}A\uparrow}^{\dagger},C_{j,{k_{x}}A\downarrow}^{\dagger},C_{j,{k_{x}}B\uparrow}^{\dagger},C_{j,{k_{x}}B\downarrow}^{\dagger},C_{j,{k_{x}}A\uparrow},$
$C_{j,{k_{x}}A\downarrow},C_{j,{k_{x}}B\uparrow},C_{j,{k_{x}}B\downarrow})$
$(j=1,2,\cdots,N_{y})$.

Then the spectral function can be calculated by,

$\displaystyle
A_{j}\left({k_{x}},\omega\right)=-\frac{1}{\pi}\sum_{p=1}^{4}\sum_{n}\mathrm{Im}\frac{\mid
u_{m+p,n}(k_{x})\mid^{2}}{\omega-E_{n}({k_{x}})+i\Gamma},$ (17)

with $m=4(j-1)$. $u_{m+p,n}(k_{x})$ and $E_{n}({k_{x}})$ are eigenvectors and
eigenvalues of the matrix $\hat{M}({k_{x}})$, respectively.

In the following presented results, without loss of generality, we use the
hopping constant $t$ and the lattice constant $a$ as the energy unit $(t=1)$
and the length unit $(a=1)$, respectively. Other parameters are taken as
$\lambda_{R}=0.05$ and $\Gamma=0.01$.

Figure 1:  Topological phase diagrams depicting the dependence of Chern number
$C$ on parameter sets $(\mu,\lambda_{SO})$ in distinct pairing states with the
gap magnitude $\Delta_{0}=0.2$: (a) $d+id$ pairing state and (b) $p+ip$
pairing State.

##  III Results

In the chiral superconducting state, the topological properties are described
by the Chern number [Eq.(8)]. The Chern number should, in principle, depend
strongly on the chemical potential $\mu$ and the spin-orbit coupling strength
$\lambda_{SO}$. We present the $(\lambda_{SO},\mu)$-dependent topological
phase diagram for the $d+id$ and $p+ip$ superconducting states in Figs. 1(a)
and 1(b), respectively.

For the $d+id$ pairing symmetry, as seen in Fig. 1(a), the phase diagram
includes four regions (indicated as ①$-$④) with the Chern number $C$ equal to
$4$, $0$, $-2$, and $-4$, respectively. For the $p+ip$ pairing symmetry, as
seen in Fig. 1(b), the phase diagram comprises nine regions (①$-$⑨). The phase
diagram in the regime $-0.9<\mu<0.9$ is qualitatively consistent with that of
the $d+id$ pairing symmetry. However, additional phase transition lines exist
at the chemical potentials around $\mu=\pm 0.9$. Consequently, the phase
diagram for the heavily overdoped regions with $|\mu|>0.9$ is significantly
different, and higher Chern number phases with $C=6$ and $C=-6$ emerge.

Due to the existence of the Rashba term, the phase diagrams for both pairing
symmetries are electron-hole asymmetric. The phase diagram for the electron-
doped ($\mu>0$) region differs from that for the hole-doped ($\mu<0$) region.
For the $d+id$ pairing symmetry, the $C=-2$ phase in the hole-doped region is
significantly larger than in the electron-doped region.

For the $p+ip$ pairing symmetry, in the heavily over hole-doped region
($\mu<-0.9$), there are three phases with $C=-2$, $-4$, and $-6$, as well as
two phase transition lines. The Chern number decreases by 2 when crossing each
transition line. In the heavily over electron-doped region ($\mu>0.9$), there
are two stable topological phases with $C=2$ and $C=6$. Between these two
phases lies a rather narrow region with $C=8$. This phase is unstable and
exists only at several lattice sites. In fact, as the spin-orbit strength
$\lambda_{SO}$ increases, there are still two phase transition lines. The
Chern number increases by 6 when crossing the first line and decreases by 2
when crossing the second one.

Figure 2:  Energy bands and spectral functions at the system edge $(y=1)$ in
$d+id$ superconducting pairing states with zigzag boundary Condition along the
$y$-Direction, varying with parameter sets $(\lambda_{SO},\mu)$. Panels (a)
and (b) showcase the energy bands and spectral function in region ① $(C=4)$
for $(\lambda_{SO},\mu)=(0.1,1.1)$, respectively. Panels (c) and (d) display
the energy bands and spectral function in region ③ $(C=-2)$ for
$(\lambda_{SO},\mu)=(0.15,-0.6)$, respectively.

According to the bulk-edge correspondence, a topological nontrivial state with
a Chern number $C=N$ should give rise to $2N$ edge states when considering the
aforementioned partial open boundary condition, resulting in $N$ pairs of
Majorana zero modes at the system edges. We now aim to verify the topological
phase with different Chern numbers by investigating the edge states of the
system while employing a zigzag boundary condition along the $y$-direction.
The energy bands and site-dependent spectral functions can then be obtained
through the diagonalization of the Hamiltonian matrix. We present the
numerical results of the energy bands and spectral functions at the system
edge $(y=1)$ in various topological regions for the $d+id$ and $p+ip$ pairing
symmetries in Figs. 2 and 3, respectively.

Figure 3:  Similar to Fig. 2, but for numerical results in $p+ip$
superconducting states. Panels (a) and (b) display results in region ① (C=2)
with $(\lambda_{SO},\mu)=(0.08,1.1)$. Panels (c) and (d) show results in
region ⑧ (C=-4) with $(\lambda_{SO},\mu)=(0.22,-1.1)$. Panels (e) and (f)
depict results in region ② (C=6) with $(\lambda_{SO},\mu)=(0.3,1.1)$.

In the $d+id$ pairing state, there are three topologically nontrivial phases
with Chern numbers of $4$, $-2$, and $-4$. We present the numerical results of
the energy bands for the regions with $C=4$ and $C=-2$ in Figs. 2(a) and 2(c),
respectively. The corresponding spectral functions are shown in Figs. 2(b) and
2(d), respectively. As can be seen, in the $C=4$ region, there are eight
energy bands crossing the Fermi energy, resulting in four pairs of Majorana
zero modes at the momenta $k_{x}=\pm 0.65\pi\pm 0.02\pi$. These energy bands
originate from the edge states, and the edge modes are chiral, as observed
from the spectral function spectra at the system edge [Fig. 2(b)]. The four
energy bands with positive slopes (corresponding to the quasiparticles with
positive velocity) are contributed by the $y=1$ edge. The other four energy
bands with negative slopes are contributed by the $(y=N_{y})$ edge (not
presented here).

In the $C=-2$ region, similarly, there are four energy bands crossing the
Fermi energy, corresponding to two pairs of Majorana zero modes at the momenta
$\bm{k_{x}}=\pm 0.77\pi$. The Chern number is consistent with the number of
edge states at the $y=1$ edge. The quasiparticle energy bands and spectral
functions in the $C=-4$ region are qualitatively similar to those in the $C=4$
region; therefore, the numerical results are not presented here.

We now present the numerical results of the energy bands and spectral
functions in the $p+ip$ superconducting state. The phase diagram in the $p+ip$
state is more complex, with topological phases having higher Chern numbers
$C=\pm 6$ emerging. The numerical results of the energy bands and spectral
functions with Chern numbers $2$, $-4$, and $6$ are displayed in Fig. 3. As
observed, the number of edge states is consistent with the Chern number.

In the $C=2$ region, using the parameter set $(\lambda_{SO},\mu)=(0.08,1.1)$
[Figs. 3(a)], there are four energy bands crossing the Fermi energy at
$k_{x}=0$ momentum, resulting in two pairs of Majorana zero modes at this
momentum. In the $C=-4$ region, using the parameter set
$(\lambda_{SO},\mu)=(0.22,-1.1)$ [Figs. 3(c)], eight energy bands cross the
Fermi energy, among which four energy bands cross at ${k_{x}}=0$ and the other
four cross at the momenta ${k_{x}}=\pm 0.86\pi$. In this case, two pairs of
Majorana zero modes emerge at the momentum ${k_{x}}=0$, and the other two
pairs emerge at ${k_{x}}=\pm 0.86\pi$.

Figure 4:  Electronic structures in the $d+id$ and $p+ip$ superconducting
states. Panels (a) and (b) display intensity plots of the energy gap as
functions of spin-orbit coupling strength $\lambda_{SO}$ and chemical
potential $\mu$ for the $d+id$ pairing state and the $p+ip$ pairing state,
respectively. Panels (c) and (d) illustrate the energy bands along highly
symmetric lines of the first Brillouin zone for both states ($d+id$ and
$p+ip$) without spin-orbital coupling and with a chemical potential of
$\mu=0.9$.  Figure 5:  Electronic structures along various phase transition
points: Panels (a-d) display the intensity plots of zero-energy spectral
functions for parameter sets $(\lambda_{SO},\mu)=(0.11,0.5)$, $(0.05,-0.9)$,
$(0.21,-1.2)$, and $(0.28,-1.2)$, respectively. Panels (e-h) illustrate the
corresponding energy bands along highly symmetric lines within the first
Brillouin zone for these parameter sets.

In the $C=6$ region, using the parameter set $(\lambda_{SO},\mu)=(0.3,1.1)$
[Figs. 3(e)], twelve energy bands cross the Fermi energy, among which four
energy bands cross at ${k_{x}}=0$ and the other eight cross at the momenta
${k_{x}}=\pm 0.9\pi\pm 0.01\pi$. In this case, two pairs of Majorana zero
modes emerge at the momentum ${k_{x}}=0$, and the other four pairs emerge at
${k_{x}}=\pm 0.9\pi\pm 0.01\pi$, resulting in six pairs of Majorana zero
modes. The spectral functions at the system edge, presented in the right
panels of Fig. 3, also indicate that the edge modes are chiral, similar to the
results of the $d+id$ pairing states.

##  IV Discussion

At this stage, we aim to elucidate the origin of the exotic topological phase
diagram in the chiral superconducting state. Generally, a single-band two-
dimensional chiral superconductor is topologically nontrivial with a Chern
number $C=\pm 1$. In the normal state, due to the presence of spin-orbit
coupling and the two sub-lattices, there are four energy bands in the normal
state. As a result, it is understandable that the $C=\pm 4$ topological phase
initially emerges in the phase diagram.

Topological phase transitions may occur when the parameters $\mu$ or
$\lambda_{SO}$ change. Generally, along the topological transition lines, the
energy gap tends to close. The intensity plots of the energy gap in the two
pairing states, as functions of $\mu$ and $\lambda_{SO}$, are displayed in
Figs. 4(a) and 4(b), where the blue lines indicate the gap closing lines. In
comparison to the phase diagrams shown in Fig. 1, the energy gaps of the
system indeed close along the phase transition lines. The differences between
the phase diagrams of the $d+id$ state and the $p+ip$ state can be effectively
explained by examining the band structures of these two pairing states.
Previous studies have extensively investigated the electronic structures in
the superconducting state on the honeycomb lattice. The normal state energy
band exhibits the VHS at the chemical potential around $\mu=1$. Near the VHS
point, the energy gap in the $p+ip$ superconducting state is generally smaller
than that in the $d+id$ superconducting state [69]. Furthermore, the nodal or
fully gapped states near the VHS point for graphene material have been
previously investigated [70]. It was indicated that for the $p+ip$ pairing
symmetry, additional nodal points emerge near the $M$ point when the Fermi
level is around the VHS point. In this work, we present numerical results of
the energy bands along the highly symmetric lines near the VHS point
$(\mu=0.9)$ and without spin-orbital coupling $(\lambda_{R}=\lambda_{SO}=0)$
for these two pairing states in Figs. 4(c) and 4(d), respectively. As
observed, in the $p+ip$ superconducting state, the energy gap closes and nodal
points indeed emerge near the $M$ point. In contrast, in the $d+id$
superconducting state, the system remains fully gapped along the entire highly
symmetrical lines. These results are consistent with the previous study [70].
As a result, additional phase transition lines near the VHS point appear for
the $p+ip$ pairing states, as illustrated in Fig. 1.

We now attempt to elucidate the origin of the exotic topological phase diagram
in the chiral superconducting state by examining the Dirac cones along phase
transition lines in the phase diagrams. Within the topologically nontrivial
phase, the bulk energy bands of the system are generally fully gapped,
allowing the Chern number of the system to be defined. At the phase transition
points, the energy gap typically closes, often accompanied by Dirac cones. The
change in the Chern number should be related to the number of Dirac cones in
the energy bands [71].

Numerically, the position and number of Dirac cones can be explored by
calculating the zero-energy spectral function. The phase diagram of the $d+id$
pairing state is relatively simple, and the main characteristics are involved
in that of the $p+ip$ state. We focus on the phase transitions in the $p+ip$
pairing state. The intensity plots of the zero-energy spectral function at
different phase transition points are presented in Figs. 5(a)-5(d),
respectively. The corresponding energy bands along the highly symmetric lines
of the Brillouin zone are presented in Figs. 5(e)-5(h).

As seen in Fig. 1(b), the point $(\lambda_{SO},\mu)=(0.11,0.5)$ connects two
topological phases of $C=4$ and $C=-2$. The numerical results of the
electronic structure at this point are displayed in Figs. 5(a) and 5(e). The
gap is closed, and a Dirac cone emerges along the $\Gamma-K$ line. Due to the
six-fold symmetry of the honeycomb lattice, there are actually six different
Dirac cones in the first Brillouin zone, as seen in Fig. 5(e). This explains
why the Chern number decreases from 4 to -2 when crossing this point.
Similarly, the point $(0.05,-0.9)$ connects the two phases with $C=4$ and
$C=-2$. The corresponding numerical results of the electronic structure at
this point are displayed in Figs. 5(b) and 5(f). A Dirac cone emerges along
the $\Gamma-M$ line. Also, in the first Brillouin zone, there are six
different Dirac points due to the six-fold symmetry of the energy bands,
consistent with the decrease in the Chern number when crossing this point.

Figure 6:  Numerical results from self-consistent calculations: Panels (a) and
(b) display the self-consistently obtained phase diagram and pairing
magnitudes for the $d+id$ pairing state. Panels (c) and (d) show the
corresponding results for the $p+ip$ pairing state.

We now discuss the energy bands at the points $(0.21,-1.2)$ and $(0.28,-1.2)$.
Crossing these two points, the Chern number changes from $-2$ to $-4$ and from
$-4$ to $-6$, respectively. As seen in Figs. 5(g) and 5(h), the Dirac cones
emerge at the $K$ points, namely, the vertices of the Brillouin zone. For the
hexagonal Brillouin zone, among the six vertices, only two neighboring
vertices belong to the first Brillouin zone. The other four vertices can be
obtained by translating these two points through reciprocal lattice vectors.
Therefore, at these two points, there are merely two different Dirac cones in
the first Brillouin zone, resulting in a change of 2 in the Chern number when
crossing these points.

It is necessary to discuss the robustness of the phase diagram against the
superconducting pairing magnitude. In Fig. 1, we consider a constant
superconducting gap magnitude with $\Delta_{0}=0.2$. We have checked
numerically that when the gap magnitude varies, the main features of the phase
diagram remain qualitatively the same. Moreover, for an intrinsic
superconductor, the superconducting magnitudes may strongly depend on the
parameters. We also calculate the superconducting gap magnitudes self-
consistently based on Eqs.(8) and (9) by considering an effective attractive
interaction $V=2$. The self-consistently obtained phase diagram and the
superconducting order parameters are plotted in Fig. 6. As seen, the order
parameters indeed depend strongly on the parameters, while the main features
of the phase diagram remain qualitatively similar. For the $d+id$ pairing
state, the original $C=\pm 4$ topological phases emerge. The Chern number
reduces as the spin-orbit coupling strength increases, and a large
topologically trivial region appears in the phase diagram. For the $p+ip$
pairing states, additional phase transition lines appear near the VHS. The
higher Chern number regions with $C=\pm 6$ emerge in the phase diagram. Our
numerical results indicate that the exotic topological properties of the phase
diagram are indeed quite robust, at least qualitatively.

##  V Summary

In summary, we have explored the topological properties of the chiral
superconducting state in the doped Kane-Mele model, considering two different
pairing symmetries: the $d+id$ state and the $p+ip$ state. The phase diagram
with different chemical potentials and spin-orbit coupling strengths is
displayed, using the Chern number to describe various topological phases.

In the $d+id$ pairing state, higher Chern number states with $C=\pm 4$ emerge
in the phase diagram. As the spin-orbit coupling strength increases, the Chern
number decreases, and ultimately the system becomes topologically trivial. In
the $p+ip$ state, the phase diagram at small chemical potentials is
qualitatively similar to that in the $d+id$ state, while additional phase
transition lines emerge in the heavily overdoped region with chemical
potentials around $\mu=\pm 0.9$. Higher Chern number phases with $C=\pm 6$
emerge. We also examine the number of edge states to verify the Chern number
of different phases. All the exotic topological properties can be understood
by studying the electronic structures at the phase transition points.

###### Acknowledgements.

We thank Wei Chen and Wen Huang for useful discussions. This work was
supported by the NSFC (Grant No.12074130), the Natural Science Foundation of
Guangdong Province (Grant No. 2021A1515012340), the Key-Area Research and
Development Program of Guangdong Province (Grant No.2019B030330001), and the
CRF of Hong Kong (C6009-20G).

## References

  * Ghosh  _et al._ [2020a] S. K. Ghosh, M. Smidman, T. Shang, J. F. Annett, A. D. Hillier, J. Quintanilla, and H. Yuan, Recent progress on superconductors with time-reversal symmetry breaking, J. Phys.: Condens.Matter 33, 033001 (2020a).
  * Kallin and Berlinsky [2016] C. Kallin and J. Berlinsky, Chiral superconductors, Rep. Prog. Phys. 79, 054502 (2016).
  * Schemm  _et al._ [2014] E. R. Schemm, W. J. Gannon, C. M. Wishne, W. P. Halperin, and A. Kapitulnik, Observation of broken time-reversal symmetry in the heavy-fermion superconductor ${UPt}_{3}$, Science 345, 190 (2014).
  * Avers  _et al._ [2020] K. E. Avers, W. J. Gannon, S. J. Kuhn, W. P. Halperin, J. A. Sauls, L. DeBeer Schmitt, C. D. Dewhurst, J. Gavilano, G. Nagy, U. Gasser, and M. R. Eskildsen, Broken time-reversal symmetry in the topological superconductor ${UPt}_{3}$, Nat. Phys. 16, 531 (2020).
  * Hayes  _et al._ [2021] I. M. Hayes, D. S. Wei, T. Metz, J. Zhang, Y. S. Eo, S. Ran, S. R. Saha, J. Collini, N. P. Butch, D. F. Agterberg, A. Kapitulnik, and J. Paglione, Multicomponent superconducting order parameter in ${UTe}_{2}$, Science 373, 797 (2021).
  * Jiao  _et al._ [2020] L. Jiao, S. Howard, S. Ran, Z. Wang, J. O. Rodriguez, M. Sigrist, Z. Wang, N. P. Butch, and V. Madhavan, Chiral superconductivity in heavy-fermion metal ${UTe}_{2}$, Nature 579, 523 (2020).
  * Ishihara  _et al._ [2023] K. Ishihara, M. Roppongi, M. Kobayashi, Y. Mizukami, H. Sakai, Y. Haga, K. Hashimoto, and T. Shibauchi, Chiral superconductivity in ${UTe}_{2}$ probed by anisotropic low-energy excitations, Nat. Commun. 14, 2966 (2023).
  * Hillier  _et al._ [2009] A. D. Hillier, J. Quintanilla, and R. Cywinski, Evidence for time-reversal symmetry breaking in the noncentrosymmetric superconductor ${LaNiC}_{2}$, Phys. Rev. Lett. 102, 117007 (2009).
  * Quintanilla  _et al._ [2010] J. Quintanilla, A. D. Hillier, J. F. Annett, and R. Cywinski, Relativistic analysis of the pairing symmetry of the noncentrosymmetric superconductor ${LaNiC}_{2}$, Phys. Rev. B 82, 174511 (2010).
  * Ghosh  _et al._ [2020b] S. K. Ghosh, G. Csire, P. Whittlesea, J. F. Annett, M. Gradhand, B. Újfalussy, and J. Quintanilla, Quantitative theory of triplet pairing in the unconventional superconductor ${LaNiGa}_{2}$, Phys. Rev. B 101, 100506 (2020b).
  * Pustogow  _et al._ [2019] A. Pustogow, Y. Luo, A. Chronister, Y. S. Su, D. A. Sokolov, F. Jerzembeck, A. P. Mackenzie, C. W. Hicks, N. Kikugawa, S. Raghu, E. D. Bauer, and S. E. Brown, Constraints on the superconducting order parameter in sr2ruo4 from oxygen-17 nuclear magnetic resonance, Nature 574, 72 (2019).
  * Biswas  _et al._ [2013] P. K. Biswas, H. Luetkens, T. Neupert, T. Stürzer, C. Baines, G. Pascua, A. P. Schnyder, M. H. Fischer, J. Goryo, M. R. Lees, H. Maeter, F. Brückner, H.-H. Klauss, M. Nicklas, P. J. Baker, A. D. Hillier, M. Sigrist, A. Amato, and D. Johrendt, Evidence for superconductivity with broken time-reversal symmetry in locally noncentrosymmetric srptas, Phys. Rev. B 87, 180503 (2013).
  * Fischer  _et al._ [2014] M. H. Fischer, T. Neupert, C. Platt, A. P. Schnyder, W. Hanke, J. Goryo, R. Thomale, and M. Sigrist, Chiral $d$-wave superconductivity in ${SrPtAs}$, Phys. Rev. B 89, 020509 (2014).
  * K. P.  _et al._ [2019] S. K. P., D. Singh, P. K. Biswas, G. B. G. Stenning, A. D. Hillier, and R. P. Singh, Investigations of the superconducting ground state of ${Zr}_{3}{Ir}$: Introducing a new noncentrosymmetric superconductor, Phys. Rev. Mater. 3, 104802 (2019).
  * Shang  _et al._ [2020] T. Shang, S. K. Ghosh, J. Z. Zhao, L.-J. Chang, C. Baines, M. K. Lee, D. J. Gawryluk, M. Shi, M. Medarde, J. Quintanilla, and T. Shiroka, Time-reversal symmetry breaking in the noncentrosymmetric ${Zr}_{3}{Ir}$ superconductor, Phys. Rev. B 102, 020503 (2020).
  * Schemm  _et al._ [2015] E. R. Schemm, R. E. Baumbach, P. H. Tobash, F. Ronning, E. D. Bauer, and A. Kapitulnik, Evidence for broken time-reversal symmetry in the superconducting phase of ${\mathrm{uru}}_{2}{\mathrm{si}}_{2}$, Phys. Rev. B 91, 140506 (2015).
  * Kittaka  _et al._ [2016] S. Kittaka, Y. Shimizu, T. Sakakibara, Y. Haga, E. Yamamoto, Y. Onuki, Y. Tsutsumi, T. Nomoto, H. Ikeda, and K. Machida, Evidence for chiral d-wave superconductivity in uru2si2 from the field-angle variation of its specific heat, J. Phys. Soc. Jpn. 85, 033704 (2016).
  * Singh  _et al._ [2020] D. Singh, M. S. Scheurer, A. D. Hillier, D. T. Adroja, and R. P. Singh, Time-reversal-symmetry breaking and unconventional pairing in the noncentrosymmetric superconductor ${\mathrm{la}}_{7}{\mathrm{rh}}_{3}$, Phys. Rev. B 102, 134511 (2020).
  * Barker  _et al._ [2015] J. A. T. Barker, D. Singh, A. Thamizhavel, A. D. Hillier, M. R. Lees, G. Balakrishnan, D. M. Paul, and R. P. Singh, Unconventional superconductivity in ${\mathrm{la}}_{7}{\mathrm{ir}}_{3}$ revealed by muon spin relaxation: Introducing a new family of noncentrosymmetric superconductor that breaks time-reversal symmetry, Phys. Rev. Lett. 115, 267001 (2015).
  * Ma  _et al._ [2011] T. Ma, Z. Huang, F. Hu, and H.-Q. Lin, Pairing in graphene: A quantum monte carlo study, Phys. Rev. B 84, 121410 (2011).
  * Kiesel  _et al._ [2012] M. L. Kiesel, C. Platt, W. Hanke, D. A. Abanin, and R. Thomale, Competing many-body instabilities and unconventional superconductivity in graphene, Phys. Rev. B 86, 020507(R) (2012).
  * Nandkishore  _et al._ [2014] R. Nandkishore, R. Thomale, and A. V. Chubukov, Superconductivity from weak repulsion in hexagonal lattice systems, Phys. Rev. B 89, 144501 (2014).
  * Ma  _et al._ [2014] T. Ma, F. Yang, H. Yao, and H.-Q. Lin, Possible triplet $p+ip$ superconductivity in graphene at low filling, Phys. Rev. B 90, 245114 (2014).
  * Faye  _et al._ [2015] J. P. L. Faye, P. Sahebsara, and D. Sénéchal, Chiral triplet superconductivity on the graphene lattice, Phys. Rev. B 92, 085121 (2015).
  * Nandkishore  _et al._ [2012] R. Nandkishore, L. S. Levitov, and A. V. Chubukov, Chiral superconductivity from repulsive interactions in doped graphene, Nat. Phys. 8, 158 (2012).
  * Xiao  _et al._ [2016] L.-Y. Xiao, S.-L. Yu, W. Wang, Z.-J. Yao, and J.-X. Li, Possible singlet and triplet superconductivity on honeycomb lattice, Europhys. Lett. 115, 27008 (2016).
  * Huang  _et al._ [2019] T. Huang, L. Zhang, and T. Ma, Antiferromagnetically ordered mott insulator and d+id superconductivity in twisted bilayer graphene: a quantum monte carlo study, Sci. Bull. 64, 310 (2019).
  * Chen  _et al._ [2020] W. Chen, Y. Chu, T. Huang, and T. Ma, Metal-insulator transition and dominant $d+id$ pairing symmetry in twisted bilayer graphene, Phys. Rev. B 101, 155413 (2020).
  * Xu  _et al._ [2016] X. Y. Xu, S. Wessel, and Z. Y. Meng, Competing pairing channels in the doped honeycomb lattice hubbard model, Phys. Rev. B 94, 115105 (2016).
  * Ying and Yang [2020] T. Ying and S. Yang, Superconducting pairing symmetries of the hubbard model on the honeycomb lattice with inhomogeneous hopping strength, Phys. Rev. B 102, 125125 (2020).
  * Dai  _et al._ [2023] H. Dai, R. Ma, X. Zhang, T. Guo, and T. Ma, Quantum monte carlo study of superconductivity in rhombohedral trilayer graphene under an electric field, Phys. Rev. B 107, 245106 (2023).
  * Lan  _et al._ [2023] Y. Lan, X.-F. Yu, and L.-T. Zhang, Doping dependence of superconductivity on a honeycomb lattice within the framework of kinetic-energy-driven superconductivity (2023), arXiv:2312.01972 [cond-mat.supr-con] .
  * Crépieux  _et al._ [2023] A. Crépieux, E. Pangburn, L. Haurie, O. A. Awoga, A. M. Black-Schaffer, N. Sedlmayr, C. Pépin, and C. Bena, Superconductivity in monolayer and few-layer graphene. ii. topological edge states and chern numbers, Phys. Rev. B 108, 134515 (2023).
  * Xue  _et al._ [2012] M. Xue, G. Chen, H. Yang, Y. Zhu, D. Wang, J. He, and T. Cao, Superconductivity in potassium-doped few-layer graphene, J. Am. Chem. Soc. 134, 6536 (2012).
  * Ichinokura  _et al._ [2016] S. Ichinokura, K. Sugawara, A. Takayama, T. Takahashi, and S. Hasegawa, Superconducting calcium-intercalated bilayer graphene, ACS Nano 10, 2761 (2016).
  * Ludbrook  _et al._ [2015] B. M. Ludbrook, G. Levy, P. Nigge, M. Zonno, M. Schneider, D. J. Dvorak, C. N. Veenstra, S. Zhdanovich, D. Wong, P. Dosanjh, C. Stra$\beta$er, A. Stohr, S. Forti, C. R. Ast, U. Starke, and A. Damascelli, Evidence for superconductivity in li-decorated monolayer graphene, P. Natl. Acad. Sci. U.S.A. 112, 11795 (2015).
  * Chapman  _et al._ [2016] J. Chapman, Y. Su, C. A. Howard, D. Kundys, A. N. Grigorenko, F. Guinea, A. K. Geim, I. V. Grigorieva, and R. R. Nair, Superconductivity in ca-doped graphene laminates, Sci. Rep. 6, 23254 (2016).
  * Cao  _et al._ [2018] Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero, Unconventional superconductivity in magic-angle graphene superlattices, Nature 556, 43 (2018).
  * Zhou  _et al._ [2022] H. Zhou, L. Holleis, Y. Saito, L. Cohen, W. Huynh, C. L. Patterson, F. Yang, T. Taniguchi, K. Watanabe, and A. F. Young, Isospin magnetism and spin-polarized superconductivity in bernal bilayer graphene, Science 375, 774 (2022).
  * Yamanaka  _et al._ [1996] S. Yamanaka, H. Kawaji, K.-i. Hotehama, and M. Ohashi, A new layer-structured nitride superconductor. lithium-intercalated $\beta$-zirconium nitride chloride, lixzrncl, Adv. Mater. 8, 771 (1996).
  * Yamanaka  _et al._ [1998] S. Yamanaka, K.-i. Hotehama, and H. Kawaji, Superconductivity at 25.5k in electron-doped layered hafnium nitride, Nature 392, 580 (1998).
  * Nishikubo  _et al._ [2011] Y. Nishikubo, K. Kudo, and M. Nohara, Superconductivity in the honeycomb-lattice pnictide srptas, J. Phys. Soc. Jpn. 80, 055002 (2011).
  * Chen  _et al._ [2013] L. Chen, B. Feng, and K. Wu, Observation of a possible superconducting gap in silicene on Ag(111) surface, Appl. Phys. Lett. 102, 081602 (2013).
  * Zhang  _et al._ [2015] L.-D. Zhang, F. Yang, and Y. Yao, Possible electric-field-induced superconducting states in doped silicene, Sci. Rep. 5, 8203 (2015).
  * Liu  _et al._ [2013] F. Liu, C.-C. Liu, K. Wu, F. Yang, and Y. Yao, $d+i{d}^{{}^{\prime}}$ chiral superconductivity in bilayer silicene, Phys. Rev. Lett. 111, 066804 (2013).
  * Xi  _et al._ [2022] Y. Xi, X. Jing, Z. Xu, N. Liu, Y. Liu, M.-L. Lin, M. Yang, Y. Sun, J. Zhuang, X. Xu, W. Hao, Y. Li, X. Li, X. Wei, P.-H. Tan, Q. Li, B. Liu, S. X. Dou, and Y. Du, Superconductivity in layered van der waals hydrogenated germanene at high pressure, J. Am. Chem. Soc. 144, 18887 (2022).
  * Liao  _et al._ [2018] M. Liao, Y. Zang, Z. Guan, H. Li, Y. Gong, K. Zhu, X.-P. Hu, D. Zhang, Y. Xu, Y.-Y. Wang, K. He, X.-C. Ma, S.-C. Zhang, and Q.-K. Xue, Superconductivity in few-layer stanene, Nat. Phys. 14, 344 (2018).
  * Zhao  _et al._ [2022] C. Zhao, L. Li, L. Zhang, J. Qin, H. Chen, B. Xia, B. Yang, H. Zheng, S. Wang, C. Liu, Y. Li, D. Guan, P. Cui, Z. Zhang, and J. Jia, Coexistence of robust edge states and superconductivity in few-layer stanene, Phys. Rev. Lett. 128, 206802 (2022).
  * Soltan-Panahi  _et al._ [2012] P. Soltan-Panahi, D.-S. Luhmann, J. Struck, P. Windpassinger, and K. Sengstock, Quantum phase transition to unconventional multi-orbital superfluidity in optical lattices, Nat. Phys. 8, 71 (2012).
  * Tarruell  _et al._ [2012] L. Tarruell, D. Greif, T. Uehlinger, G. Jotzu, and T. Esslinger, Creating, moving and merging dirac points with a fermi gas in a tunable honeycomb lattice, Nature 483, 302 (2012).
  * Kane and Mele [2005] C. L. Kane and E. J. Mele, Quantum spin hall effect in graphene, Phys. Rev. Lett. 95, 226801 (2005).
  * Ezawa [2015] M. Ezawa, Monolayer topological insulators: Silicene, germanene, and stanene, J. Phys. Soc. Jpn. 84, 121003 (2015).
  * Marrazzo  _et al._ [2019] A. Marrazzo, M. Gibertini, D. Campi, N. Mounet, and N. Marzari, Relative abundance of z2 topological order in exfoliable two-dimensional insulators, Nano Lett. 19, 8431 (2019), pMID: 31658415\.
  * Molle  _et al._ [2017] A. Molle, J. Goldberger, M. Houssa, Y. Xu, S.-C. Zhang, and D. Akinwande, Buckled two-dimensional xene sheets, Nat. Mater. 16, 163 (2017).
  * Liu  _et al._ [2011] C.-C. Liu, H. Jiang, and Y. Yao, Low-energy effective hamiltonian involving spin-orbit coupling in silicene and two-dimensional germanium and tin, Phys. Rev. B 84, 195430 (2011).
  * Ezawa [2012] M. Ezawa, Valley-polarized metals and quantum anomalous hall effect in silicene, Phys. Rev. Lett. 109, 055502 (2012).
  * Marrazzo  _et al._ [2018] A. Marrazzo, M. Gibertini, D. Campi, N. Mounet, and N. Marzari, Prediction of a large-gap and switchable kane-mele quantum spin hall insulator, Phys. Rev. Lett. 120, 117701 (2018).
  * Facio  _et al._ [2019] J. I. Facio, S. K. Das, Y. Zhang, K. Koepernik, J. van den Brink, and I. C. Fulga, Dual topology in jacutingaite ${\mathrm{pt}}_{2}{\mathrm{hgse}}_{3}$, Phys. Rev. Mater. 3, 074202 (2019).
  * Sajadi  _et al._ [2018] E. Sajadi, T. Palomaki, Z. Fei, W. Zhao, P. Bement, C. Olsen, S. Luescher, X. Xu, J. A. Folk, and D. H. Cobden, Gate-induced superconductivity in a monolayer topological insulator, Science 362, 922 (2018).
  * Fatemi  _et al._ [2018] V. Fatemi, S. Wu, Y. Cao, L. Bretheau, Q. D. Gibson, K. Watanabe, T. Taniguchi, R. J. Cava, and P. Jarillo-Herrero, Electrically tunable low-density superconductivity in a monolayer topological insulator, Science 362, 926 (2018).
  * Rosenzweig  _et al._ [2020] P. Rosenzweig, H. Karakachian, D. Marchenko, K. Küster, and U. Starke, Overdoping graphene beyond the van hove singularity, Phys. Rev. Lett. 125, 176403 (2020).
  * Lin  _et al._ [2021] W.-J. Lin, W. LiMing, and T. Zhou, Spin excitations in the heavily overdoped monolayer graphene superconductor: An analog to the cuprates, Phys. Rev. B 103, 174513 (2021).
  * Qin  _et al._ [2019] W. Qin, L. Li, and Z. Zhang, Chiral topological superconductivity arising from the interplay of geometric phase and electron correlation, Nat. Phys. 15, 796 (2019).
  * Lian  _et al._ [2018] B. Lian, X.-Q. Sun, A. Vaezi, X.-L. Qi, and S.-C. Zhang, Topological quantum computation based on chiral majorana fermions, P. Natl. Acad. Sci. U.S.A 115, 10938 (2018).
  * Huang  _et al._ [2015] W. Huang, S. Lederer, E. Taylor, and C. Kallin, Nontopological nature of the edge current in a chiral $p$-wave superconductor, Phys. Rev. B 91, 094507 (2015).
  * Huang  _et al._ [2014] W. Huang, E. Taylor, and C. Kallin, Vanishing edge currents in non-$p$-wave topological chiral superconductors, Phys. Rev. B 90, 224519 (2014).
  * Liu  _et al._ [2023] H.-T. Liu, W. Chen, and W. Huang, Impact of random impurities on the anomalous hall effect in chiral superconductors, Phys. Rev. B 107, 224517 (2023).
  * Fukui  _et al._ [2005] T. Fukui, Y. Hatsugai, and H. Suzuki, Chern numbers in discretized brillouin zone: Efficient method of computing (spin) hall conductances, J. Phys. Soc. Jpn. 74, 1674 (2005).
  * Li and Zhou [2021] Y.-Q. Li and T. Zhou, Impurity effect as a probe for the pairing symmetry of graphene-based superconductors, Front. Phys. 16, 43502 (2021).
  * Pangburn  _et al._ [2023] E. Pangburn, L. Haurie, A. Crépieux, O. A. Awoga, A. M. Black-Schaffer, C. Pépin, and C. Bena, Superconductivity in monolayer and few-layer graphene. i. review of possible pairing symmetries and basic electronic properties, Phys. Rev. B 108, 134514 (2023).
  * Liu  _et al._ [2016] X.-P. Liu, Y. Zhou, Y.-F. Wang, and C.-D. Gong, Multifarious topological quantum phase transitions in two-dimensional topological superconductors, Sci. Rep. 6, 28471 (2016).