# Second Order Topological Insulator State in Hexagonal Lattices and its
Abundant Material Candidates

Shifeng Qian  Centre for Quantum Physics, Key Laboratory of Advanced
Optoelectronic Quantum Architecture and Measurement (MOE), School of Physics,
Beijing Institute of Technology, Beijing, 100081, China  Beijing Key Lab of
Nanophotonics & Ultrafine Optoelectronic Systems, School of Physics, Beijing
Institute of Technology, Beijing, 100081, China  Cheng-Cheng Liu
<EMAIL_ADDRESS>Centre for Quantum Physics, Key Laboratory of Advanced
Optoelectronic Quantum Architecture and Measurement (MOE), School of Physics,
Beijing Institute of Technology, Beijing, 100081, China  Beijing Key Lab of
Nanophotonics & Ultrafine Optoelectronic Systems, School of Physics, Beijing
Institute of Technology, Beijing, 100081, China  Yugui Yao<EMAIL_ADDRESS>Centre for Quantum Physics, Key Laboratory of Advanced Optoelectronic Quantum
Architecture and Measurement (MOE), School of Physics, Beijing Institute of
Technology, Beijing, 100081, China  Beijing Key Lab of Nanophotonics &
Ultrafine Optoelectronic Systems, School of Physics, Beijing Institute of
Technology, Beijing, 100081, China

###### Abstract

We propose two mechanisms to realize the second order topological insulator
(SOTI) state in spinless hexagonal lattices, viz., chemical modification and
anti-Kekulé/Kekulé distortion of hexagonal lattice. Correspondingly, we
construct two models and demonstrate the nontrivial band topology of the SOTI
state characterized by the second Stiefel-Whitney class $w_{2}$ in the
presence of inversion symmetry (P) and time-reversal symmetry (T). Based on
the two mechanisms and using first-principles calculations and symmetry
analysis, we predict three categories of real light element material
candidates, i.e., hydrogenated and halogenated 2D hexagonal group IV materials
XY (X=C, Si, Ge, Sn, Y=H, F, Cl), 2D hexagonal group V materials (blue
phosphorene, blue arsenene, and black phosphorene, black arsenene), and the
recent experimentally synthesized anti-Kekulé/Kekulé order graphenes and the
counterparts of silicene/germanene/stanene. We explicitly demonstrate the
nontrivial topological invariants and existence of the protected corner states
with fractional charge for these candidates with giant bulk band gap (up to
3.5 eV), which could facilitate the experimental verification by STM. Our
approaches and proposed abundant real material candidates will greatly enrich
2D SOTIs and promote their intriguing physics research.

###  Introduction.—

Topological classification of states and search for the material candidates
are among the most activate research fields in condensed matter physics [1, 2,
3, 4, 5]. Recently, high-order topological insulators (HOTIs) have been
proposed and attracted great interest [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16,
17]. The high-order of topological insulator (TI) is reflected in special
bulk-boundary correspondence. A TI in _d_ dimensions has protected gapless
states on its (_d_ -1)-dimensional boundary, while an _n_ th (_n_ = 2, 3
…)-order (high-order) TI in _d_ dimensions has protected gapless states on its
(_d_ -_n_)-dimensional boundary. For instance, a three dimensional (3D)
second-order topological insulator (SOTI) displays 1D gapless modes along its
hinges, and a 2D SOTI hosts zero-energy states localized at its corners. Until
now, SOTIs have only been realized in 3D bismuth single crystal [18] and some
artificial systems [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. In
addition, in the literature there are a few theoretical material proposals for
3D and 2D SOTIs [31, 32, 33, 34, 35, 36, 37, 38, 39, 38, 40, 41]. However, a
simple and feasible program to achieve SOTI with abundant real material
candidates is still absent, which greatly impeded the experimental and further
theoretical studies on SOTIs, especially for 2D.

2D hexagonal lattice systems provide us a platform to realize intriguing
phenomena and have always received widespread attention [42, 43, 44, 45, 46,
47, 48, 49, 50, 51, 52, 53]. On one hand, graphene, as a superstar material,
takes a hexagonal lattice structure, opening the curtain of the 2D material
research boom [54]. On the other hand, the important typical topological
phases originate in hexagonal lattice, such as quantum anomalous Hall effect
[55], valley Hall effect [56] and quantum spin Hall effect [57].

In this Letter, we put forward three ways to realize SOTI states in hexagonal
lattice based on two different mechanisms, and propose three categories of
real materials. First, we construct two kinds of spinless tight-binding (TB)
models and demonstrate that the two mechanisms are feasible to realize the
SOTI state. The nontrivial band topology is characterized by the second
Stiefel-Whitney class in the presence of inversion symmetry (P) and time-
reversal symmetry (T). Based on combined density function theory (DFT)
simulation and theoretical analysis, we predict three major types of real
light elememt material families in which spin-orbit coupling (SOC) is
irrelevent, i.e., hydrogenated and halogenated 2D hexagonal group IV materials
XY (X=C, Si, Ge, Sn, Y=H, F, Cl) [58, 59, 60, 61], 2D hexagonal group V
materials (blue phosphorene, blue arsenene, and black phosphorene, black
arsenene) [62, 63, 64, 65, 66, 67, 68], and the recent experimentally
synthesized anti-Kekulé/Kekulé order graphenes [69, 70, 71, 72] and Kekulé
order silicene/germanene/stanene. These materials have giant bulk band gap (up
to 3.5 eV), which will facilitate the experimental detection and exploration
of the SOTI phase.

###  Symmetry and topological classifaction.—

Let us first review the symmetry of hexagonal lattices. A planar hexagonal
lattice is made of two equivalent sublattices and has six fold rotation
symmetry, P and T. For a buckled hexagonal lattice, P and T are preserved and
the six fold rotation symmetry is degraded to three fold rotation symmetry, as
shown in Fig. S1(a). Here, we focus on the light elememt systems with P and T
symmetry.

In the spinless system with PT symmetry, the corresponding Bloch wave
functions is real-valued. PT is an antiunitary symmetry operator that is local
in momentum space and satisfies the relation of
$(\textit{P}\textit{T})^{2}=1$. The PT can be represented by P$\textit{T}=K$,
where $K$ is the complex conjugation. The invariance of the Hamiltonian
$H(\boldsymbol{k})$ under PT imposes the reality condition to $H(\textbf{k})$
and real eigenstates $|u(\boldsymbol{k})\rangle$ [73], such that
$(\textit{PT})H(\boldsymbol{k})(\textit{PT})^{-1}=H^{*}(\boldsymbol{k})=H(\boldsymbol{k})$,
$(\textit{PT})|u(\boldsymbol{k})\rangle=|u(\boldsymbol{k})\rangle^{*}=|u(\boldsymbol{k})$.
Since the real occupied states are orientable on a sphere, transition
functions can be restricted to special orthogonal group
$\mathrm{SO}\left(N_{\mathrm{occ}}\right)$ with the number of occupied bands
$N_{\mathrm{occ}}$. From the mathematical results of homotopy groups,
$\pi_{1}[\mathrm{SO}(2)]=\mathbb{Z}$,
$\pi_{1}\left[\mathrm{SO}\left(N_{\mathrm{occ}}>2\right)\right]=\mathbb{Z}_{2}$.
There exist PT-invariant topological insulators in 2D, characterized by
topological invariant corresponding to the above homotopy class. In the case
of $N_{\mathrm{occ}}=2$, the topological invariant is thus
$\mathbb{Z}$-valued, which corresponds to the Euler characteristic class. The
Euler class is not stable in $K$-theory but adding a trivial valence band can
collapse the $\mathbb{Z}$ into $\mathbb{Z}_{2}$ topological invariants, i.e.,
the so-called second Stiefel-Whitney number $w_{2}$. The $w_{2}$ can measure
the higher-order band topology of P-symmetric spinless fermion systems [74,
75, 76].

One can use three methods to calculate the second Stiefel-Whitney number
$w_{2}$, namely, the parity criterion, Wilson loop method, and nested Wilson
loop method [77, 76, 78]. The value of $w_{2}$ can be calculated by parity
eigenvalues in the presence of P symmetry,

$(-1)^{w_{2}}=\prod_{i=1}^{4}(-1)^{\left\lfloor N_{\text{occ
}}^{-}\left(\Gamma_{i}\right)/2\right\rfloor},$ (1)

where $N_{\text{occ }}^{-}\left(\Gamma_{i}\right)$ is the number of occupied
bands with odd parity at time-reversal invariant momentum (TRIM) $\Gamma_{i}$
and $\left\lfloor\right\rfloor$ is the floor function. We also use Wilson loop
method and nested Wilson loop method to check the second Stiefel-Whitney
number $w_{2}$ [79].

###  Two mechanisms, three ways to achieve the second order topological
insulator state in hexagonal lattices.—

The parity formula provides us an intuitive knob to engineer the second
Stiefel-Whitney number $w_{2}$. For pristine hexagonal lattice (e.g.
graphene), $w_{2}=1$ for the $\sigma$ bands ($sp^{2}$) alone, while $w_{2}=0$
for the $\pi$ bands ($p_{z}$) alone. Fortunately, the $w_{2}$ of the $\pi$
bands can be engineered to nontrivial value through a simple
$\sqrt{3}\times\sqrt{3}$ distortion. As a result, we propose two physical
mechanisms to approach the SOTI state, i.e., chemical modification to remove
the $p_{z}$ orbitals with a gap left and anti-Kekulé/Kekulé distortion[Fig.
S1]. According to the two mechanisms, we give three ways, i.e., group IV
hexagonal lattices with chemical modification, group V hexagonal lattices with
appropriate electron filling, and group IV hexagonal lattices anti-
Kekulé/Kekulé distortion, to realize the SOTI state in planar, buckled or
puckered hexagonal lattices.

FIG. 1: Schematic illustration of two mechanisms and three ways to achieve the
second order topological insulator state. (a) Hexagonal lattices with solid
and hollow circles labeling two sublattices. (b) Band structure of the eight-
band model. For a planar hexagonal structure, the $p_{z}$ orbitals (blue) is
decoupled with the $sp^{2}$ ($s$, $p_{x}$, $p_{y}$) orbitals (green). (c) Same
as (b) but with $p_{z}$ orbitals (blue) shifted under the Fermi level. (d) The
energy spectrum of the finite-size hexagonal flake plotted in the inset. The
charge spatial distribution of the six zero-energy modes, which are localized
at corners, is shown in the inset. Blue, red and grey dots represent corner,
edge and bulk states, respectively. (e) (f) Same as (c) (d) but for anti-
Kekulé distortion hexagonal lattices. The parameters for (b)-(f) are given in
Suppl. Mater. [79].

For Mechanism I, we build an eight-band TB model (Model I) for both planar or
buckled hexagonal lattices [79]. Figure S1(b) shows the band structure and
irreducible representations (IR) at high symmetric points for Model I, similar
to graphene. The $p_{z}$ orbitals (blue lines) are decoupled with $sp^{2}$
orbitals (green lines). There is a Dirac cone at the Fermi level located at
the $K$ point. One can not open a gap at $K$ point between two $p_{z}$
orbitals with the PT preserved in the spinless case. However, one can shift
the two $p_{z}$ orbitals entirely up or down the Fermi level to obtain a gap
by chemical modification. The corresponding band structure and parity
eigenvalues at TRIMs are exhibited in Fig. S1(c), where the onsite energy of
$p_{z}$ orbitals is set to $-5t_{1}$ to force the $p_{z}$ orbitals shift down
the Fermi level. No matter the two $p_{z}$ orbitals is shifted up or down,
which depends on the relative electronegativity, the second Stiefel-Whitney
number $w_{2}$ of the bands under the Fermi level is nontrivial according to
the parity eigenvalues or Wilson loop spectrum, indicating a SOTI state. The
nontrivial band topology is also preserved in the buckled structure as long as
the buckling height does not close the gap. To explore the hallmark corner
states of 2D SOTIs, we calculate the discrete energy spectrum of a hexagonal
finite-size flake, as shown in Fig. S1(d). There are six states degenerate at
zero energy in the spectrum, whose charge spatial distribution shown in the
inset is well located at the corners of the flake, corresponding to the corner
states. The red (gray) dots represent the surface (bulk) states and the blue
dots represent the corner states which are situated up, middle and down the
surface states [79].

For Mechanism II, we develop anti-Kekulé and Kekulé distortion hexagonal
lattice model (Model II) [79]. The $\pi$ bands at original $K$ point are
folded onto the new $\Gamma$ point with a gap opened. Figure S1(e) shows band
structure of the anti-Kekulé distortion hexagonal lattice with a gap opened at
$\Gamma$. From the parity criterion or two Wilson loop methods, the second
Stiefel-Whitney number $w_{2}$ is also nontrivial, which means a SOTI state.
We build a hexagonal flake and calculate its discrete energy spectrum (Fig.
S1(f)). The charge spatial distribution of the six zero-energy states shown in
the inset are well located at the flake’s corners, i.e. the corner states.

###  Material Realization with abundant real candidates.—

Based on the above two mechanisms and three ways to achieve the SOIT state, we
propose the following three major types of real material candidate systems.

FIG. 2: (a) Top view and side view of geometry structures of hydrogenated or
halogenated 2D hexagonal group IV materials XY (X = C, Si, Ge, Sn, Y = H, F,
Cl). (b) (c) Bulk bands and edge states of graphane from DFT calculation. (d)
Wilson loop of graphane. The number of crossing on $\theta=\pi$ is 1,
therefore the $w_{2}$ is 1. (e) Energy spectrum of hexagonal finite-size flake
shown in (f). Blue, red and grey dots represent the corner, edge, and bulk
states, respectively. (f) Charge spatial distribution of blue states in (e).

### Hydrogenated and halogenated 2D hexagonal group IV materials XY(X=C, Si,
Ge, Sn, Y=H, F, Cl).—

These group IV hydrides and halides take a buckled hexagonal geometry [Fig.
2(a)]. We take graphane, i.e., hydride-graphene, as an example. Figure 2(b)
shows the band structure of graphane with a direct band gap (3.5 eV) at
$\Gamma$. There are two gapped armchair edge states at the middle of the gap,
as shown in Fig. 2(c). Next, we explore the SOTI state in graphane. The
nontrivial second Stiefel-Whitney number is demonstrated by the Wilson loop
spectrum as shown in Fig. 2(d), where the spectrum exhibits a crossing point
at $k_{y}=0$, $\theta=\pi$, indicating $w_{2}=1$. The nonzero $w_{2}$ is
checked by the nested Wilson loop method [Table 1] and parity criterion
[Table. S1]. The $w_{2}$ of all SOTI candidate materials are verified in the
three methods. To explicitly reveal the corner states, we calculate the
discrete spectrum [Fig. 2(e)] of the hexagonal finite-size flake based on DFT
calculation and Wannier function. Up and below the edge states exit six
degenerate states (blue dots in Fig. 2(d)), whose charge spatial distribution
is plotted in Fig. 2(e). Such six degenerate states are well located at the
six corners of the flake [Fig. 2(f)], i.e., the corner states. The bulk bands,
edge states and corner states of other 2D group IV hydrides/halides are given
in Suppl. Mater. [79].

FIG. 3: Top view and side view of geometry structures of blue phosphene or
blue arsenene (a) and black phosphene or arsenene (e). Bulk band structures of
blue phosphene (orange), blue arsenene (black) (b), and black phosphene
(orange) and black arsenene (black) (f). Energy discrete spectra of the
finite-size hexagonal/rectangle flakes of blue/black phosphene (c)/(g) and
blue/black arsenene (d)/(h) calculated from DFT. The corresponding charge
spatial distribution of blue states in (c), (d), (g) and (h) plotted in their
insets are well localized at the corners, i.e., the corner states, and the
blue, red and grey dots represent the corner, edge, and bulk states,
respectively.

### 2D hexagonal group V materials (blue phosphorene, blue arsenene, and black
phosphorene, black arsenene).—

The blue phosphorene and arsenene [Fig. 3(a)] and black phosphorene and
arsenene [Fig. 3(e)] are also good SOTI candidates. Although black phosphorene
and arsenene do not take a normal hexagonal lattice, they can be seen as
puckered honeycomb structures with P and T symmetry. The band structures of
blue phosphene, blue arsenene, black phosphene and black arsenene are given in
Figs. 3(b, f). The second Stiefel-Whitney number $w_{2}$ of these four 2D
hexagonal group V materials are all nontrivial. We calculate the energy
spectra of the finite-size flakes for blue phosphene, blue arsenene, black
phosphene and black arsenene, and their hallmark corner states are explicitly
displayed from DFT calculation, as shown in Figs. 3(c, d) and Figs. 3(g, h).
We notice that black phosphorene was also predicted as a SOTI based on TB
model recently [91, 92].

FIG. 4: Structures of anti-Kekulé (a) and Kekulé (e) distortion lattices of
graphene and silicene/germanene/stanene. Black dots label the atoms, and thick
and thin lines represent the strong and weak bonds. Bulk bands of anti-Kekulé
graphene/silicene (orange/black) (b), and the Kekulé ones (orange/black) (f).
Energy discrete spectra of the hexagonal nanoflakes of anti-Kekulé/Kekulé
graphenes (c)/(g) and silicenes (d)/(h), calculated by DFT and Wanner
function. The corresponding charge spatial distribution of blue states in (c),
(d), (g) and (h) plotted in their insets are well localized at the corners,
i.e., the corner states, and the blue, red and grey dots represent the corner,
edge and bulk states, respectively.

Bulk band structures of blue phosphene (orange), blue arsenene (black) (b),
and black phosphene (orange) and black arsenene (black) (f). Energy discrete
spectra of the finite-size hexagonal/rectangle flakes of blue/black phosphene
(c)/(g) and blue/black arsenene (d)/(h) calculated from DFT. The corresponding
charge spatial distribution of blue states in (c), (d), (g) and (h) plotted in
their insets are well localized at the corners, i.e., the corner states, and
the blue, red and grey dots represent the corner, edge, and bulk states,
respectively.

### The experimentally synthesized anti-Kekulé/Kekulé order graphenes and the
counterparts of silicene/germanene/stanene.—

According to Mechanism II, the recent experimentally synthesized anti-Kekulé
order graphene with a 0.38 eV gap at $\Gamma$ [69] provides perfect material
candidates. Here, we use first-principle calculation to demonstrate that anti-
Kekulé/Kekulé order graphenes and the counterparts of
silicene/germanene/stanene are 2D SOTI candidates (See details in Suppl.
Mater. [79]). We take anti-Kekulé/Kekulé graphenes (silicenes) as examples,
which have P, T and six-fold (three-fold) rotation symmetry [Figs. 4(a, e)].
The thin and thick lines represent the weak and strong bonds and black dots
label the C/Si atoms. The second Stiefel-Whitney number $w_{2}$ of anti-Kekulé
and Kekulé are 0 and 1 for the occupied bands, indicating that the anti-Kekulé
(Kekulé) order graphene is trivial (nontrivial). However, for anti-Kekulé
order graphene, the $p_{z}$ orbitals and $sp^{2}$ orbitals ($s$, $p_{x}$,
$p_{y}$) are decoupled and either of the two parts is nontrivial and both
parts have corner states [79]. Therefore, the anti-Kekulé graphene is
topological nontrivial. To further demonstrate the SOTI state in anti-
Kekulé/Kekulé graphenes and silicenes, we calculate the discrete spectra
[Figs. 4(c, d, g, h)] of their finite-size hexagonal flakes from DFT
calculation and Wannier function. The conner states are clearly shown in the
insets.

###  The corner states with fractional charge.—

The topological origin of these corner states can arises from filling anomaly,
which keeps track of mismatch between the number of electrons required to
simultaneously satisfy charge netrality and the crystal symmetry [93]. From
the rotation symmetry eigenvalues and elementary band representations, these
systems can be taken as obstructed atomic insulators [94]. The Wannier centers
and the atoms do not overlap as shown in Figs. 5(a-c). For the candidate
materials based on Mechanism I, the SOTI state is derived from $sp^{2}$
orbitals. The Wannier centers of the $sp^{2}$ orbitals of every unit cell
(shadow area) are located at the bond centers [Fig. 5(a)]. In the finite-size
hexagonal flake composed of the unit cells, as shown in Fig. 5(d), each corner
has a charge of $e$/2. The Wanner centers of Model II and anti-Kekulé graphene
with $p_{z}$ orbitals are shown in Fig. 5(b) and Fig. 5(d), where each corner
also has $e$/2. We can use the similar procedure to demonstrate the $e$/2
corner charge for the candidate materials with $p_{z}$ orbitals based on
Mechanism II [Figs. 5(b, d)], and black phosphorene and arsenene [Figs. 5(c,
e)]. In addition, the Wannier centers are also distributed at the flake’s
edges, which is the origin of the edge states in the two Models and all the
candidate materials.

FIG. 5: Atom structure and Wannier centers at maximal Wyckoff position for the
SOTI candidates materials based on (a) Mechanism I, (b) Mechanism II, and for
(c) black phosphorene and arsenene. (d) Fractional Corner charge for system
(a) and (b). (e) Fractional corner charge for (c). Corner charges are in units
of the electron charge $e$ and mod 1. The black solid and hollow circles
represent the atoms and green stars denote the Wannier centers.

The fractional corner charge is related with the $C_{n}$-symmetry
representations of the occupied energy bands [93]. The eigenvalues of $n$-fold
rotation symmetry at high symmetry points $\mathbf{\Pi}^{(n)}$ is
$\Pi_{p}^{(n)}=e^{2\pi i(p-1)/n},\quad$ for $p=1,2,\ldots n$. The integer
topological invariants are defined by using the rotation eigenvalues at
$\mathbf{\Pi}^{(n)}$ compared to a reference point
$\boldsymbol{\Gamma}=(0,0)$, which is
$\left[\Pi_{p}^{(n)}\right]\equiv\\#\Pi_{p}^{(n)}-\\#\Gamma_{p}^{(n)}$, where
$\\#\Pi_{p}^{(n)}$ is the number of energy bands below the Fermi level with
eigenvalue $\Pi_{p}^{(n)}$. Our proposed three kinds of SOTI materials have
$C_{6}$, $C_{3}+I$ and $C_{2}$ symmetry. Therefore, the fractional corner
charge can be given by $Q_{\text{corner
}}^{(6)}=\frac{e}{4}\left[M_{1}^{(2)}\right]+\frac{e}{6}\left[K_{1}^{(3)}\right]\bmod
e$, $Q_{\text{corner
}}^{(3+I)}=\frac{e}{4}\left[M_{1}^{(I)}\right]+\frac{e}{6}\left[K_{1}^{(3)}\right]\bmod
e$, and $Q_{\text{corner
}}^{(2)}=\frac{e}{4}\left(-\left[X_{1}^{(2)}\right]-\left[Y_{1}^{(2)}\right]+\left[M_{1}^{(2)}\right]\right)\bmod
e$, where the superscript $n$ of $Q_{\text{corner }}^{(n)}$ labels the $C_{n}$
symmetry. We calculated the fractional corner charge of all candidate
materials and the corner charge of these materials are all 1/2 [Table. 1],
which agrees with the above filling anomaly analysis and the direct numerical
calculation of corner charges [79].

TABLE. 1: Corner charge, second Stiefel-Whitney number $w_{2}$, nest Wilson loop (det($W_{2}$)) and bulk gaps of graphane (CH), graphene fluoride (CF), graphene chloride (CCl), silicene hydride (SiH), silicene chloride (SiCl), germanene hydride(GeH), stanene hydride (SnH), blue phosphorene/arsenene (blue P/As), black phosphorene/arsenene (black P/As), anti-Kekulé/Kekulé graphenes, silicenes, germanenes and stanenes(anti-KekuléGr/KekuléGr, anti-KekuléSi/KekuléSi, anti-KekuléGe/KekuléGe, anti-KekuléSn/KekuléSn). The gap of anti-Kekulé distortion graphene is taken as the experimental value of 0.38 eV. |  | $Q_{c}$ | $w_{2}$ | det($W_{2}$) | Gap (eV)
---|---|---|---|---|---
$C_{6}$ | Anti-KekuleGr | $sp^{2}$ | 1/2 | 1 | $\pi$ | 0.38
$p_{z}$ | 1/2 | 1 | $\pi$
KekuleGr | 1/2 | 1 | $\pi$ |
$C_{3}+I$ | CH | 1/2 | 1 | $\pi$ | 3.5
CF | 1/2 | 1 | $\pi$ | 3.1
CCl | 1/2 | 1 | $\pi$ | 1.6
SiH | 1/2 | 1 | $\pi$ | 2.2
SiCl | 1/2 | 1 | $\pi$ | 1.3
GeH | 1/2 | 1 | $\pi$ | 1.0
SnH | 1/2 | 1 | $\pi$ | 0.5
Blue P | 1/2 | 1 | $\pi$ | 1.9
Blue As | 1/2 | 1 | $\pi$ | 1.6
| Anti-KekuléSi/Ge/Sn [79, 95] | 0 | 0 | 0 |
| KekuléSi/Ge/Sn [79, 96] | 1/2 | 1 | $\pi$ |
$C_{2}$ | Black P | 1/2 | 1 | $\pi$ | 0.9
Black As | 1/2 | 1 | $\pi$ | 0.7

We will develop a $k\cdot p$ effective model and edge theory to capture the
low-energy physics and corner states. The symmetry group at $\Gamma$ of our
systems are $D_{6h}$ or $D_{3d}$ besides $T$ symmetry. The generators for
$D_{6h}$ or $D_{3d}$ can be chosen as $C_{3z}$, $P$ and $M_{y}$. We consider
the four low-energy bands with two doublets $(E_{1g},E_{2u})^{T}$ or
$(E_{2g},E_{1u})^{T}$ [Figs. S1 (c, e)], and the symmetry operators can be
represented by
$C_{3z}=\tau_{0}e^{-i(\pi/3)\sigma_{y}},\quad\mathcal{P}(C_{2z})=\tau_{z}\sigma_{0},\quad\mathcal{M}_{y}=\tau_{0}\sigma_{z}$
with Pauli matrices $\tau$ acting on the two doublets and $\sigma$ acting on
the two degenerate states within each doublet. Under these symmetry
operations, momentum and pseudospin are transformed as
$C_{3}:k_{\pm}\rightarrow e^{\pm i2\pi/3}k_{\pm},\sigma_{\pm}\rightarrow
e^{\pm
i2\pi/3}\sigma_{\pm},\sigma_{y}\rightarrow\sigma_{y};P(C_{2}):k_{\pm}\rightarrow-
k_{\pm},\sigma_{\pm}\rightarrow\sigma_{\pm},\sigma_{y}\rightarrow\sigma_{y};M_{y}:k_{\pm}\rightarrow
k_{\mp},\sigma_{\pm}\rightarrow\sigma_{\mp},\sigma_{y}\rightarrow-\sigma_{y}$.
$T=K$ in spinless systems, with $K$ as the complex conjugation. Constrained by
these symmetries, the bulk model expanded to $k$-quadratic order is

$\displaystyle\mathcal{H}_{eff}(\boldsymbol{k})=$ $\displaystyle
E_{0}+\left(m_{0}-m_{1}k^{2}\right)\tau_{z}+v\left(k_{x}\sigma_{z}+k_{y}\sigma_{x}\right)\tau_{y}$
(2)
$\displaystyle+\left[\left(k_{x}^{2}-k_{y}^{2}\right)\sigma_{z}-2k_{x}k_{y}\sigma_{x}\right]\left(\alpha_{1}\tau_{0}+\alpha_{2}\tau_{z}\right),$

where $E_{0}=c_{0}+c_{1}k^{2},k=|\boldsymbol{k}|$; $c_{i},m_{i},\alpha_{i}$
and $v$ are real parameters. The chiral symmetry is an approximate symmetry
and represented as $C=\tau_{x}$, when the first term and the $\alpha_{1}$ term
can be ignored.

In the basis of the two zero-energy edge modes, the edge Hamiltonian is given
by [79] $\mathcal{H}_{\text{edge }}(k)=vks_{y}$, where $k$ is the wave vector
along the edge, the Pauli matrices $s$ act on the space of
$\left(\psi_{+},\psi_{-}\right)^{T}$ and $v$ is Fermi velocity. In the basis,
the symmetry operators can be represented by
$\mathcal{M}_{y}=\mathcal{C}=s_{z}$. In the presence of $M_{y}$ and $C$
symmetry, the mass term is forced to be zero at edge Hamiltonian. However, for
an edge that does not preserve $M_{y}$, the edge Hamiltonian will generally be
gaped with the mass term $\Delta_{M}=m_{M}s_{x}$. Two edges related by $M_{y}$
must have opposite Dirac mass. Therefore, the protected 0D corner mode must
exit at the intersection between the two edges, as shown in the finite-size
flakes of the two mechanisms and candidate materials.

###  Conclusion and discussion.—

We put forward guidelines for designing the SOTI state in 2D light element
materials with honeycomb lattice structure. Based on the guidelines, we have
proposed a series of candidate materials with excellent performance, namely,
hexagonal group IV materials hydrides and halides (graphane, halogenated
graphenes, silicene hydride, silicene chloride, germanene hydride, stannene
hydride), hexagonal group V materials (blue phosphorene, blue arsenene, black
phosphorene, black arsenene), and hexagonal anti-Kekulé/Kekulé distortion
group IV materials (anti-Kekulé/Kekulé distortion graphenes, silicenes,
germanenes and stanenes). The second Stiefel-Whitney number $w_{2}$,
characterizing the bulk topology of these candidate materials, is calculated
by parity criterion and two Wilson loop methods. The hallmark corner states
have been explicitly calculated from DFT and analyzed by effective model and
edge theory. The fractional corner charge is obtained by the $C_{n}$-symmetry
representations of the occupied bands and explicit numerical calculation.

Experimentally, graphane and halogenated graphenes [58, 59, 60] were prepared
shortly after the discovery of graphene. Anti-Kekulé or Kekulé ordered
graphenes can also be induced experimentally through an external super-lattice
potential [70, 71, 72] . Blue phosphorene/arsenene and black
phosphorene/arsenene were synthesized with high quality [62, 63, 64, 65, 66,
67, 68]. In addition, silicene hydride has also been successfully prepared
experimentally [61]. These materials with giant gaps (up to 3.5 eV), which are
larger than that of the 2D SOTI materials predicted previously [32, 33, 34,
35, 97], will facilitate the experimental detection of the corner states as
sharp peaks in the scanning tunneling spectroscopy measurement when the
scanning tip moves close to the corners. As the majority of these materials
are easy to grow and have large gaps, they are ideal candidates to explore the
HOTI state and related novel properties.

###### Acknowledgements.

S. Qian and C.-C. Liu are supported by the NSF of China (Grants No. 11922401,
No. 11774028). Y. Yao is supported by the National Key R&D Program of China
(Grant No. 2020YFA0308800), the NSF of China (Grants No. 11734003, No.
12061131002), the Strategic Priority Research Program of Chinese Academy of
Sciences (Grant No. XDB30000000).

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  * not [a] The Anti-KekuléSi/Ge/Sn have trivial second Stiefel-Whitney number but still have corner states [79] .
  * not [b] The KekuléGe/Sn have nontrivial second Stiefel-Whitney number. However, since the bulk band gaps are too small and the dispersion of edge states are too large, the corner states are mixed with bulk states or edge states [79] .
  * not [c] Recently proposed SOTI materials graphdiyne [32, 33] and graphyne [34, 35], whose gaps are induced by the band folding and alternate equivalent C-C bond lengths, can be considered as belonging to Mechanism II of anti-Kekulé and Kekulé distortion .

Supplemental Material — Second Order Topological Insulator State in Hexagonal
Lattices and its Material Realization

###### Contents

  1. I Calculation methods
  2. II Projection band structures of graphane (CH) and graphene fluoride (CF)
  3. III Model I: Eight-band TB model for planar or buckled hexagonal lattice
  4. IV Model II: Anti-Kekulé and Kekulé distortion hexagonal lattice model
  5. V Wilson loop and nested Wilson loop
  6. VI The geometry structures, bulk bands, edge states, Wilson loop spectra, energy spectra of hexagonal finite-size flakes, and charge spatial distribution of corner states of the SOTI material candidates (2D hexagonal group IV hydrides/halides, group V materials)
  7. VII The $p_{z}$ orbitals and $sp^{2}$ orbitals decomposition of SOTI anti-Kekulé and Kekulé distortion graphenes
  8. VIII The geometry structures, bulk bands, edge states, Wilson loop spectra, energy spectra of hexagonal finite-size flakes, and charge spatial distribution of corner states of the SOTI material candidates (anti-Kekulé and Kekulé distortion group IV materials)
  9. IX $k\cdot p$ Hamiltonian and edge theory
  10. X Corner charge, second Stiefel-Whitney number, nested Wilson loop and bulk gaps of all the SOTI material candidates
  11. XI Charge spatial distribution of corner states
  12. XII Numerical calculation of corner charge

##  I Calculation methods

The calculations of the band structures were performed using DFT in the
Perdew-Becke-Ernzerhof (PBE) generalized gradient approximation (GGA) [80]
implemented in the Vienna $ab$ $initio$ simulation package (VASP) [81]. The
plane-wave energy cutoff is set to 600 eV and the Brillouin zone is sampled by
a 12 $\times$ 12 $\times$ 1 mesh. The vacuum layer is up to 20 Å to avoid the
interactions between the layers. The symmetry adapted Wannier functions are
constructed using the WANNIER90 code [82, 83, 84] and quantum espresso [85].
Based on the constructed Wannier functions, we obtain the edge states, corner
states and Wilson loop. The parity eigenvalues and irreducible representations
of electronic states from DFT results were calculated using irvsp code [86],
which relies on the space-group character tables published on the Bilbao
Crystallographic server.

##  II Projection band structures of graphane (CH) and graphene fluoride (CF)

Due to the different relative electronegativity, the $p_{z}$ orbitals of
hexagonal group IV hydrides are pushed down the Fermi level, while those of
hexagonal group IV halides are pushed up. Figure S1 shows the projection band
structures of graphane and graphene fluoride. The hydrogen has weaker
electronegativity than that of carbon which induce the $p_{z}$ orbitals of
carbon shift down the Fermi level, while the halide has stronger
electronegativity which induce the $p_{z}$ orbitals shift up the Fermi level.

FIG. S1: Projection band structures of graphane (CH) and graphene fluoride
(CF).

##  III Model I: Eight-band TB model for planar or buckled hexagonal lattice

For Mechanism I, we build an eight-band tight binding (TB) model with one $s$
orbital and three $p$ orbitals per site [87, 88]. In the basis of
$\left\\{\left|p_{z}^{A}\right\rangle,\left|p_{z}^{B}\right\rangle,\left|p_{y}^{A}\right\rangle,\left|p_{x}^{A}\right\rangle,\left|s^{A}\right\rangle,\left|p_{y}^{B}\right\rangle,\left|p_{x}^{B}\right\rangle,\left|s^{B}\right\rangle\right\\}$
with A and B labeling two sublattices, the Hamiltonian has the form

$H_{I}=\left(\begin{array}[]{cc}\mathbf{H}_{\pi}&\mathbf{H}_{{n}}\\\
\mathbf{H}_{{n}}{}^{\dagger}&\mathbf{H}_{\sigma}\end{array}\right),$ (1)

where

$\mathbf{H}_{\sigma}=\left(\begin{array}[]{cc}\mathbf{E}&\mathbf{T}\\\
\mathbf{T}^{\dagger}&\mathbf{E}\end{array}\right).$ (2)

The 2 $\times$ 2 diagonal block $\mathbf{H}_{\pi}$ and 6 $\times$ 6
$\mathbf{H}_{\sigma}$ describe the Hamiltonian for $p_{z}$ and $sp^{2}$ ($s$,
$p_{x}$, $p_{y}$), respectively. The nondiagonal block $\mathbf{H}_{n}$ is a 2
$\times$ 6 matrix which couples the $\mathbf{H}_{\pi}$ and
$\mathbf{H}_{\sigma}$. The matrix elements of $\mathbf{H}_{n}$ are nonzero in
the buckled structures. The matrix $\mathbf{E}$ describes the on-site energy
of $\left\\{p_{y},p_{x},s\right\\}$ orbitals and can be written as

$\mathbf{E}=\left(\begin{array}[]{lll}\epsilon_{p}&0&0\\\ 0&\epsilon_{p}&0\\\
0&0&\epsilon_{s}\end{array}\right).$ (3)

$\mathbf{T}$ matrix describes the nearest neighbour hopping between A and B
sublattices. Therefore, the matrix $\mathbf{T}$ and $\mathbf{H}_{n}$ can be
written as

$\mathbf{T}=\begin{bmatrix}t_{p_{y}^{A},p_{y}^{B}}&t_{p_{y}^{A},p_{x}^{B}}&t_{p_{y}^{A},s^{B}}\\\
t_{p_{x}^{A},p_{y}^{B}}&t_{p_{x}^{A},p_{x}^{B}}&t_{p_{x}^{A},s^{B}}\\\
t_{s^{A},p_{y}^{B}}&t_{s^{A},p_{x}^{B}}&t_{s^{A},s^{B}}\end{bmatrix},$ (4)
$\mathbf{H}_{n}=\begin{bmatrix}0&0&0&t_{p_{z}^{A},p_{y}^{B}}&t_{p_{z}^{A},p_{x}^{B}}&t_{p_{z}^{A},s^{B}}\\\
t_{p_{z}^{B},p_{y}^{A}}&t_{p_{z}^{B},p_{x}^{A}}&t_{p_{z}^{B},s^{A}}&0&0&0\end{bmatrix},$
(5)

where the $t_{\alpha,\beta}$ represents the $\alpha$ orbital hopping to
$\beta$ and can be written as

$t_{\alpha,\beta}=\sum_{i=1}^{3}t_{\alpha,\beta}\left(\vec{d}_{i}\right)e^{i\vec{k}\cdot\vec{d}_{i}},$
(6)

where $\alpha$ and $\beta$ are the orbitals in different sublattices and
$\vec{d}_{i}$ is the vector between the two sublattices. We use Slater-Koster
parameters for the TB model and represent the hopping integrals
$t_{\alpha,\beta}\left(\vec{d}_{i}\right)$ using direction cosine ($l,m,n$)
and bond integral ($V$), which are given in Table. S1. The diagonal block
$\mathbf{H}_{\pi}$ can be written as

$H_{\pi}=\begin{bmatrix}\epsilon_{p}&t_{p_{z}^{A},p_{z}^{B}}\\\
t_{p_{z}^{B},p_{z}^{A}}&\epsilon_{p}\end{bmatrix},$ (7)

where the diagonal elements $\epsilon_{p}$ are the on-site energy of $p_{z}$
orbitals and the non-diagonal elements represent hopping between the $p_{z}$
orbitals given in the form of Eq. (6). For Figs. 1 (b-d) in the text, the
parameters $V_{ss\sigma}$, $V_{sp\sigma}$, $V_{pp\sigma}$ and $V_{pp\pi}$ are
-2$t_{1}$, 2$t_{1}$, 2$t_{1}$ and -$t_{1}$, respectively. $t_{1}$ is about 3
eV for graphene [89]. The onsite energy $\epsilon_{s}$ and $\epsilon_{p}$ of
$s$ and $p$ orbitals are -3$t_{1}$ and 0, respectively. We can change the on-
site energy of $p_{z}$ orbitals to push the $p_{z}$ orbitals up or down the
Fermi level. In Figs. 1(c, d) in the text, the onsite energy energy of $p_{z}$
orbitals is -5$t_{1}$.

TABLE. S1: The hopping integrals between $s$ and $p$ orbitals are described as functions of the direction cosine (${l,m,n}$). $t_{s,s}$ | $V_{ss\sigma}$
---|---
$t_{x,x}$ | $l^{2}V_{pp\sigma}+\left(1-l^{2}\right)V_{pp\pi}$
$t_{s,x}$ | $lV_{sp\sigma}$
$t_{x,y}$ | $\operatorname{lm}\left(V_{pp\sigma^{-}}V_{pp\pi}\right)$
$t_{x,s}$ | $-lV_{sp\sigma}$
$t_{y,z}$ | $mn\left(V_{pp\sigma^{-}}V_{pp\pi}\right)$

##  IV Model II: Anti-Kekulé and Kekulé distortion hexagonal lattice model

For Mechanism II, we consider anti-Kekulé and Kekulé distortion hexagonal
lattice model. The unit cell of anti-Kekulé or Kekulé distortion hexagonal
lattice is $\sqrt{3}\times\sqrt{3}$ supercell of hexagonal unit cell, as shown
in Fig. 4(a) in the text. The TB Hamiltonian with one $p_{z}$ orbital per site
reads

$H_{II}(\mathbf{k})=\left(\begin{array}[]{cccccc}0&t_{3}&0&t_{2}e^{i\mathbf{k}\cdot\mathbf{a}_{2}}&0&t_{3}\\\
&0&t_{3}&0&t_{2}e^{-i\mathbf{k}\cdot\mathbf{a}_{3}}&0\\\
&&0&t_{3}&0&t_{2}e^{-i\mathbf{k}\cdot\mathbf{a}_{1}}\\\ &\dagger&&0&t_{3}&0\\\
&&&&0&t_{3}\\\ &&&&&0\end{array}\right),$ (8)

where $\mathbf{a}_{1}$ and $\mathbf{a}_{2}$ are the unit lattice vectors, and
$\mathbf{a}_{3}=\mathbf{a}_{1}-\mathbf{a}_{2}$. $t_{2}$ and $t_{3}$ represent
the inter-cell hopping and intra-cell hopping, respectively. For anti-Kekulé
distortion, $t_{3}/t_{2}<1$, while $t_{3}/t_{2}>1$ for Kekulé distortion.
Figure 1(e) shows band structures of the anti-Kekulé distortion hexagonal
lattice with $t_{3}/t_{2}=0.3$.

##  V Wilson loop and nested Wilson loop

One can use Wilson loop and nested Wilson loop to calculate the second
Stiefel-Whitney number $w_{2}$. The Wilson loop operator of $k_{x}$ is defined
by

$W_{1\left(k_{x},k_{y}\right)\rightarrow\left(k_{x}+2\pi,k_{y}\right)}=\lim_{N\rightarrow\infty}F_{0}F_{1}\cdots
F_{N-2}F_{N-1},$ (9)

where $F_{j}$ is the overlap matrix whose matrix elements are given by

$\left[F_{i}\right]_{mn}=\left\langle u_{m}\left(2\pi(i)/N,k_{y}\right)\mid
u_{n}\left(2\pi(i+1)/N,k_{y}\right)\right\rangle,$ (10)

and its spectrum is gauge-invariant. The topological information is encoded in
the phase factors $\theta_{m}\left(k_{y}\right)\in(-\pi,\pi]$ of the
eigenvalues $\lambda_{m}\left(k_{y}\right)$ of Wilson operator:

$\theta_{m}\left(k_{y}\right)=\operatorname{Im}\left[\log\lambda_{m}\left(k_{y}\right)\right].$
(11)

The spectrum in the $\theta-k_{y}$ diagram represents evolution of the Wanner
centers. The topological classification is indicated by the number of linear
crossing points on $\theta=\pi$. A spectrum corresponds to $w_{2}=0$
($w_{2}=1$) when it has even (odd) linear crossing points on $\theta=\pi$.

The calculation of nested Wilson loop is similar to Wilson loop, by projecting
onto the eigenstates $\left|\nu_{i}\left(k_{y}\right)\right\rangle$ of
$W_{1k_{y}}$. We use the set of eigenvectors corresponding to the sub-band
centered at $\theta=\pi$, the nested Wilson loop operator $W_{2}$ along the
$k_{y}$ is given by

$W_{2\left(k_{y}\right)\rightarrow\left(k_{y}+2\pi\right)}=\lim_{N\rightarrow\infty}\tilde{F}_{0}\tilde{F}_{2}\cdots\tilde{F}_{N-2}\tilde{F}_{N-1},$
(12)

where

$\left[\tilde{F}_{i}\right]_{mn}=\left\langle\nu_{m}\left(2\pi(i)/N\right)\mid\nu_{n}\left(2\pi(i+1)/N\right)\right\rangle.$
(13)

The determinant of the nested Wilson loop operator det($W_{2}$) is identical
to $(-1)^{w_{2}}$.

##  VI The geometry structures, bulk bands, edge states, Wilson loop spectra,
energy spectra of hexagonal finite-size flakes, and charge spatial
distribution of corner states of the SOTI material candidates (2D hexagonal
group IV hydrides/halides, group V materials)

Here, we demonstrate that other 2D hexagonal group IV hydrides/halides, group
V materials are also SOTI material candidates with global gaps at Fermi level.
Figures S2-S11 show the geometry structures, bulk bands, edge states, Wilson
loop spectra, discrete energy spectra and charge distribution of corner states
of these SOTI material candidates, which include graphene fluoride (CF),
graphene chloride (CCl), silicene chloride (SiCl), silicene hydride (SiH),
germanene hydride (GeH), stanene hydride (SnH), and blue phosphorene (BlueP),
black phosphene (BlackP), blue arsenene (BlueAs), black arsenene (BlackAs).
These materials all have nontrivial second Stiefel-Whitney number and hallmark
corner states.

FIG. S2: (a) Top view and side view of geometry structures of graphene
fluoride (CF). (b) (c) Bulk bands and edge states of CF from DFT calculation.
(d) Wilson loop of CF. The number of crossing on $\theta=\pi$ is 1 and
therefore the second Stiefel-Whitney number $w_{2}$ is 1. (e) Energy spectrum
of hexagonal finite-size flake shown in (f). Blue, red and grey dots represent
the corner, edge and bulk states, respectively. (f) Charge spatial
distribution of blue states in (e). FIG. S3: Top view and side view of
geometry structures of graphene chloride (CCl). (b) (c) Bulk bands and edge
states of CCl from DFT calculation. (d) Wilson loop of CCl. The number of
crossing on $\theta=\pi$ is 1 and therefore the second Stiefel-Whitney number
$w_{2}$ is 1. (e) Energy spectrum of hexagonal finite-size flake shown in (f).
Blue, red and grey dots represent the corner, edge and bulk states,
respectively. (f) Charge spatial distribution of blue states in (e). FIG. S4:
Top view and side view of geometry structures of silicene chloride (SiCl). (b)
(c) Bulk bands and edge states of SiCl from DFT calculation. (d) Wilson loop
of SiCl. The number of crossing on $\theta=\pi$ is 1 and therefore the second
Stiefel-Whitney number $w_{2}$ is 1. (e) Energy spectrum of hexagonal finite-
size flake shown in (f). Blue, red and grey dots represent the corner, edge
and bulk states, respectively. (f) Charge spatial distribution of blue states
in (e). FIG. S5: Top view and side view of geometry structures of silicene
hydride (SiH). (b) (c) Bulk bands and edge states of SiH from DFT calculation.
(d) Wilson loop of SiH. The number of crossing on $\theta=\pi$ is 1 and
therefore the second Stiefel-Whitney number $w_{2}$ is 1. (e) Energy spectrum
of hexagonal finite-size flake shown in (f). Blue, red and grey dots represent
the corner, edge and bulk states, respectively. (f) Charge spatial
distribution of blue states in (e). FIG. S6: Top view and side view of
geometry structures of germanene hydride (GeH). (b) (c) Bulk bands and edge
states of GeH from DFT calculation. (d) Wilson loop of GeH. The number of
crossing on $\theta=\pi$ is 1 and therefore the second Stiefel-Whitney number
$w_{2}$ is 1. (e) Energy spectrum of hexagonal finite-size flake shown in (f).
Blue, red and grey dots represent the corner, edge and bulk states,
respectively. (f) Charge spatial distribution of blue states in (e). FIG. S7:
Top view and side view of geometry structures of stanene hydride (SnH). (b)
(c) Bulk bands and edge states of SnH from DFT calculation. (d) Wilson loop of
SnH. The number of crossing on $\theta=\pi$ is 1 and therefore the second
Stiefel-Whitney number $w_{2}$ is 1. (e) Energy spectrum of hexagonal finite-
size flake shown in (f). Blue, red and grey dots represent the corner states,
edge states and bulk states, respectively. (f) Charge spatial distribution of
blue states in (e). FIG. S8: Top view and side view of geometry structures of
blue phosphorene (BlueP). (b) (c) Bulk bands and edge states of BlueP from DFT
calculation. (d) Wilson loop of BlueP. The number of crossing on $\theta=\pi$
is 3 and therefore the second Stiefel-Whitney number $w_{2}$ is 1. (e) Energy
spectrum of hexagonal finite-size flake shown in (f). Blue, red and grey dots
represent the corner, edge and bulk states, respectively. (f) Charge spatial
distribution of blue states in (e). FIG. S9: Top view and side view of
geometry structures of black phosphene (BlackP). (b) (c) Bulk bands and edge
states of BlackP from DFT calculation. (d) Wilson loop of BlackP. The number
of crossing on $\theta=\pi$ is 3 and therefore the second Stiefel-Whitney
number $w_{2}$ is 1. (e) Energy spectrum of rectangle finite-size flake shown
in (f). Blue, red and grey dots represent the corner, edge and bulk states,
respectively. (f) Charge spatial distribution of blue states in (e). FIG. S10:
Top view and side view of geometry structures of blue arsenene (BlueAs). (b)
(c) Bulk bands and edge states of BlueAs from DFT calculation. (d) Wilson loop
of BlueAs. The number of crossing on $\theta=\pi$ is 3 and therefore the
second Stiefel-Whitney number $w_{2}$ is 1. (e) Energy spectrum of hexagonal
finite-size flake shown in (f). Blue, red and grey dots represent the corner,
edge and bulk states, respectively. (f) Charge spatial distribution of blue
states in (e). FIG. S11: Top view and side view of geometry structures of
black arsenene (BlackAs). (b) (c) Bulk bands and edge states of BlackAs from
DFT calculation. (d) Wilson loop of BlackAs. The number of crossing on
$\theta=\pi$ is 3 and thus the second Stiefel-Whitney number $w_{2}$ is 1. (e)
Energy spectrum of rectangle finite-size flake shown in (f). Blue, red and
grey dots represent the corner, edge and bulk states, respectively. (f) Charge
spatial distribution of blue states in (e).

##  VII The $p_{z}$ orbitals and $sp^{2}$ orbitals decomposition of SOTI anti-
Kekulé and Kekulé distortion graphenes

The $p_{z}$ orbitals and $sp^{2}$ orbitals ($s$, $p_{x}$ and $p_{y}$) of anti-
Kekulé distortion graphene are decoupled, because the anti-Kekulé distortion
graphene is fully planar. The $w_{2}$ of Wilson loop is 0 if we consider all
occupied states [Fig. S12 (j)], which indicates the anti-Kekulé distortion
graphene is topological trivial. However, since the $p_{z}$ orbitals and
$sp^{2}$ orbitals are independent, one can consider them separately. Both of
them are topological nontrivial with $w_{2}=1$ [Figs. S12 (k, l)]. We also
calculated the edge states and corner states of anti-Kekulé distortion
graphene with only $p_{z}$ orbitals and $sp^{2}$ orbitals [Figs. S12(e, h, f,
i)]. The anti-Kekulé distortion graphene with all occupied states can be seen
as the direct summation of two independent topological phases. Therefore,
anti-Kekulé order graphene is topological nontrivial.

For Kekulé distortion graphene, the $p_{z}$ orbitals and $sp^{2}$ orbitals are
also decoupled. The $p_{z}$ orbitals are trivial and the $sp^{2}$ orbitals are
nontrivial according to the Wilson loop [Figs. S13 (k, l)], edge states [Figs.
S13 (h, i)] and corner states [Figs. S13 (h, i)]. As a result, Kekulé order
graphene is also topological nontrivial with corner states.

FIG. S12: Bulk bands, edge statets, corner states and Wilson loop of anti-
Kekulé distortion graphene based on DFT calculations and Wannier functions.
Band structures of anti-Kekulé graphene with all $s$ and $p$ orbitals (a),
only $s$, $p_{x}$ and $p_{y}$ ($sp^{2}$) orbitals (b) and only $p_{z}$
orbitals (c). Green lines represent the band structures contributed by
$sp^{2}$ orbitals while blue lines represent the band structures contributed
by $p_{z}$ orbitals. Edge states of anti-Kekulé graphene with all $s$ and $p$
orbitals (d), only $sp^{2}$ orbitals (e) and only $p_{z}$ orbitals (f). Energy
discrete spectra of hexagonal finite-size flake with all $s$ and $p$ orbitals
(g), only $sp^{2}$ orbitals (h) and only $p_{z}$ orbitals (i). Green dots
represent the corner states with only $sp^{2}$ orbitals, blue dots represent
the corner states with only $p_{z}$, red dots represent the edge states and
grey dots represent the bulk states. The corresponding charge spatial
distribution of blue states and green states in (g), (h) and (i) plotted in
their insets are well localized at the corners, i.e., the corner states.
Wilson loop of anti-Kekulé graphene with all $s$ and $p$ orbitals (j), only
$sp^{2}$ orbitals (k) and only $p_{z}$ orbitals (l). The number of crossing on
$\theta=\pi$ of anti-Kekulé graphene with only $sp^{2}$ orbitals (k) and only
$p_{z}$ orbitals (l) are both 1. FIG. S13: Bulk bands, edge statets, corner
states and Wilson loop of Kekulé distortion graphene based on DFT calculations
and Wannier functions. Band structures of Kekulé graphene with all $s$ and $p$
orbitals (a), only $s$, $p_{x}$ and $p_{y}$ ($sp^{2}$) orbitals (b) and only
$p_{z}$ orbitals (c). Green lines represent the band structures contributed by
$sp^{2}$ orbitals while blue lines represent the band structures contributed
by $p_{z}$ orbitals. Edge states of Kekulé graphene with all $s$ and $p$
orbitals (d), only $sp^{2}$ orbitals (e) and only $p_{z}$ orbitals (f). Energy
discrete spectra of hexagonal finite-size flake with all $s$ and $p$ orbitals
(g), only $sp^{2}$ orbitals (h) and only $p_{z}$ orbitals (i). Green dots
represent the corner states with $sp^{2}$ orbitals, red dots represent the
edge states and grey dots represent the bulk states. The corresponding charge
spatial distribution of green states in (g), (h) and (i) plotted in their
insets are well localized at the corners, i.e., the corner states. Wilson loop
of Kekulé graphene with all $s$ and $p$ orbitals (j), only $sp^{2}$ orbitals
(k) and only $p_{z}$ orbitals (l). The number of crossing on $\theta=\pi$ of
Kekulé graphene with only $sp^{2}$ orbitals (k) and only $p_{z}$ orbitals are
1 and 0, respectively.

##  VIII The geometry structures, bulk bands, edge states, Wilson loop
spectra, energy spectra of hexagonal finite-size flakes, and charge spatial
distribution of corner states of the SOTI material candidates (anti-Kekulé and
Kekulé distortion group IV materials)

The anti-Kekulé/Kekulé distortion group IV materials are also SOTI material
candidates with global gaps at Fermi level. Figures S14-S19 show the geometry
structures, bulk bands, edge states, Wilson loop spectra, discrete energy
spectra and charge distribution of corner states of these SOTI material
candidates, which include anti-Kekulé/Kekulé distortion graphenes (anti-
KekuléGr/KekuléGr) and silicenes (anti-KekuléSi/KekuléSi), anti-Kekulé
distortion germanene (anti-KekuléGe), anti-Kekulé distortion stanene (anti-
KekuléSn). It should be noticed that anti-KekuléSi/Ge/Sn have the trivial
second Stiefel-Whitney number $w_{2}=0$, different from the anti-KekuléGr due
to their buckled geometry. The buckling of the anti-KekuléSi/Ge/Sn structures
makes the $p_{z}$ orbitals couple with $sp^{2}$ orbitals and therefore one
should consider both $p_{z}$ and $sp^{2}$ orbitals together, which results in
a trivial second Stiefel-Whitney number. However, the buckling do not close
the gaps and one can still observe the robust corner states in the anti-
KekuléSi/Ge/Sn, as shown in Fig. S16, Fig. S18, and Fig. S19. Figures S20-S21
show the geometry structures, bulk bands, edge states, and Wilson loops of
Kekulé distortion germanene (KekuléGe) and Kekulé distortion stanene
(KekuléSn). The KekuléGe/Sn have nontrivial second Stiefel-Whitney number
$w_{2}=1$. However, since the bulk band gaps are too small and the dispersion
of edge states are too large, the corner states are mixed bulk states or edge
states.

FIG. S14: Top view of geometry structures of anti-Kekulé distortion graphene
(anti-KekuléGr). In order to reduce the computational burden, we set the weak
(strong) bond length 1.6 (1.4) Å and thus get a larger band gap at $\Gamma$
than the experimental value but with the same $w_{2}$. (b) (c) Bulk bands and
edge states of anti-KekuléGr from DFT calculation. (d) Wilson loop of $sp^{2}$
orbitals (green) and $p_{z}$ orbitals (blue) of anti-KekuléGr. The number of
crossing on $\theta=\pi$ is 2 and therefore the second Stiefel-Whitney number
$w_{2}$ is 0. (e) Energy spectrum of hexagonal finite-size flake shown in (f).
Blue, red and grey dots represent the corner states, edge states and bulk
states, respectively. (f) Charge spatial distribution of blue states in (e).
FIG. S15: Top view of geometry structures of Kekulé distortion graphene
(KekuléGr). In order to reduce the computational burden, we set the weak
(strong) bond length 1.6 (1.4) Å and thus get a larger band gap at $\Gamma$
than the experimental value but with the same $w_{2}$. (b) (c) Bulk bands and
edge states of KekuléGr from DFT calculation. (d) Wilson loop of $sp^{2}$
orbitals (green) and $p_{z}$ orbitals (blue) of KekuléGr. The number of
crossing on $\theta=\pi$ is 1 and therefore the second Stiefel-Whitney number
$w_{2}$ is 1. (e) Energy spectrum of hexagonal finite-size flake shown in (f).
Blue, red and grey dots represent the corner, edge and bulk states,
respectively. (f) Charge spatial distribution of blue states in (e). FIG. S16:
Top view and side view of geometry structures of anti-Kekulé distortion
silicene (anti-KekuléSi). The weak (strong) bond length is set as 2.44 (2.05)
Å. (b) (c) Bulk bands and edge states of anti-KekuléSi from DFT calculation.
(d) Wilson loop of anti-KekuléSi. The number of crossing on $\theta=\pi$ is 2
and therefore the second Stiefel-Whitney number $w_{2}$ is 0. (e) Energy
spectrum of hexagonal finite-size flake shown in (f). Blue, red and grey dots
represent the corner states, edge states and bulk states, respectively. (f)
Charge spatial distribution of blue states in (e). FIG. S17: Top view and side
view of geometry structures of Kekulé distortion silicene (KekuléSi). The weak
(strong) bond length is 2.44 (2.05) Å. (b) (c) Bulk bands and edge states of
KekuléSi from DFT calculation. (d) Wilson loop of KekuléSi. The number of
crossing on $\theta=\pi$ is 1 and therefore the second Stiefel-Whitney number
$w_{2}$ is 1. (e) Energy spectrum of hexagonal finite-size flake shown in (f).
Blue, red and grey dots represent the corner states, edge states and bulk
states, respectively. (f) Charge spatial distribution of blue states in (e).
FIG. S18: Top view and side view of geometry structures of anti-Kekulé
distortion germanene (anti-KekuléGe). The weak (strong) bond length is 2.54
(2.30) Å. (b) (c) Bulk bands and edge states of anti-KekuléGe from DFT
calculation. (d) Wilson loop of anti-KekuléGe. The number of crossing on
$\theta=\pi$ is 0 and therefore the second Stiefel-Whitney number $w_{2}$ is
0. (e) Energy spectrum of hexagonal finite-size flake shown in (f). Blue, red
and grey dots represent the corner states, edge states and bulk states,
respectively. (f) Charge spatial distribution of blue states in (e). FIG. S19:
Top view and side view of geometry structures of anti-Kekulé distortion
stanene (anti-KekuléSn). The weak (strong) bond length is 2.93 (2.70) Å. (b)
(c) Bulk bands and edge states of anti-KekuléSn from DFT calculation. (d)
Wilson loop of anti-KekuléSn. The number of crossing on $\theta=\pi$ is 2 and
therefore the second Stiefel-Whitney number $w_{2}$ is 0. (e) Energy spectrum
of hexagonal finite-size flake shown in (f). Blue, red and grey dots represent
the corner states, edge states and bulk states, respectively. (f) Charge
spatial distribution of blue states in (e). FIG. S20: Top view and side view
of geometry structures of Kekulé distortion germanene (KekuléGe). The weak
(strong) bond length is set as 2.54 (2.30) Å(̇b) (c) Bulk bands and edge
states of KekuléGe from DFT calculation. (d) Wilson loop of KekuléGe. The
number of crossing on $\theta=\pi$ is 3 and therefore the second Stiefel-
Whitney number $w_{2}$ is 1.  FIG. S21: Top view and side view of geometry
structures of Kekulé distortion stanene (KekuléSn). The weak (strong) bond
length is 3.11 (2.70) Å. (b) (c) Bulk bands and edge states of KekuléSn from
DFT calculation. (d) Wilson loop of KekuléSn. The number of crossing on
$\theta=\pi$ is 3 and therefore the second Stiefel-Whitney number $w_{2}$ is
1.

##  IX $k\cdot p$ Hamiltonian and edge theory

We constructed a $k\cdot p$ model to capture the bulk low-energy band
structure, which is around $\Gamma$ point and features a band inversion. The
$\Gamma$ point of TB model I and model II both have $D_{6h}$ and $T$ symmetry.
The generators for $D_{6h}$ are chosen as the three fold rotation $C_{3z}$,
inversion $P$ and the vertical mirror $M_{y}$ perpendicular to $y$ direction.
For the buckled hexagonal lattice, the symmetry of $\Gamma$ point changes from
$D_{6h}$ to $D_{3d}$, but the generators for $D_{3d}$ can also be chosen as
$C_{3z}$, $P$ and $M_{y}$. We only consider the four lowest energy bands and
in the basis of the two doublets $(E_{1g},E_{2u})^{T}$ or
$(E_{2g},E_{1u})^{T}$ [Figs. 1(c, e) in the text], the symmetry operators can
be represented by

$C_{3z}=\tau_{0}e^{-i(\pi/3)\sigma_{y}},\quad\mathcal{P}(C_{2z})=\tau_{z}\sigma_{0},\quad\mathcal{M}_{y}=\tau_{0}\sigma_{z},$
(14)

where Pauli matrices $\tau$ act on the two doublets and $\sigma$ act on the
two degenerate states within each doublet. In addition, $T=K$ in spinless
system, where $K$ is the complex conjugation. The form of Hamiltonian is
constrained by crystal symmetry and time-reversal symmetry. Under the
operation of these symmetry operators, momentum and pseudospin are transformed
as follows:

$\displaystyle C_{3}:k_{\pm}\rightarrow e^{\pm
i2\pi/3}k_{\pm},\sigma_{\pm}\rightarrow e^{\pm
i2\pi/3}\sigma_{\pm},\sigma_{y}\rightarrow\sigma_{y},$ (15) $\displaystyle
P(C_{2}):k_{\pm}\rightarrow-
k_{\pm},\sigma_{\pm}\rightarrow\sigma_{\pm},\sigma_{y}\rightarrow\sigma_{y},$
$\displaystyle M_{y}:k_{\pm}\rightarrow
k_{\mp},\sigma_{\pm}\rightarrow\sigma_{\mp},\sigma_{y}\rightarrow-\sigma_{y},$

where $k_{\pm}=k_{x}\pm ik_{y}$ and $\sigma_{\pm}=\sigma_{z}\pm i\sigma_{x}$.
The Hamiltonian must be invariant under Eq. 15. In addition, the Hamiltonian
is is also constrained under time-reversal symmetry

$TH_{eff}(\boldsymbol{k})T^{-1}=H_{eff}(-\boldsymbol{k}).$ (16)

Considered by these symmetries, the bulk effective model expanded to
$k$-quadratic order is

$\displaystyle\mathcal{H}_{eff}(\boldsymbol{k})=$ $\displaystyle
E_{0}+\left(m_{0}-m_{1}k^{2}\right)\tau_{z}+\frac{1}{2}v\left(k_{+}\sigma_{-}+k_{-}\sigma_{+}\right)\tau_{y}$
(17)
$\displaystyle+\frac{1}{2}\left(k_{+}^{2}\sigma_{+}+k_{-}^{2}\sigma_{-}\right)\left(\alpha_{1}\tau_{0}+\alpha_{2}\tau_{z}\right),$

where $E_{0}=c_{0}+c_{1}k^{2},k=|\boldsymbol{k}|$; $c_{i},m_{i},\alpha_{i}$
and $v$ are real parameters. The $k\cdot p$ Hamiltonian also has the form

$\displaystyle\mathcal{H}_{eff}(\boldsymbol{k})=$ $\displaystyle
E_{0}+\left(m_{0}-m_{1}k^{2}\right)\tau_{z}+v\left(k_{x}\sigma_{z}+k_{y}\sigma_{x}\right)\tau_{y}$
(18)
$\displaystyle+\left[\left(k_{x}^{2}-k_{y}^{2}\right)\sigma_{z}-2k_{x}k_{y}\sigma_{x}\right]\left(\alpha_{1}\tau_{0}+\alpha_{2}\tau_{z}\right).$

The $m_{0,1}<0$ for Mechanism I, while $m_{0,1}>0$ for Mechanism II, since
their low-energy bands host opposite parity. The two mechanisms have an
approximate chiral symmetry. The chiral symmetry is represented as
$C=\tau_{x}$, when the first term and the $\alpha_{1}$ term can be ignored.

To obtain the intuitive understanding of the corner states, we study the edge
theory based on the $k\cdot p$ effective model. To simplify the picture, we
only consider the first order of the $k\cdot p$ Hamiltonian. We replace
$k_{x}\rightarrow-i\partial_{x}$ and the edge states are solved from

$\tilde{\mathcal{H}}\psi=E\psi,$ (19)

where

$\tilde{\mathcal{H}}_{eff}=m(x)\tau_{z}+v\left(-i\partial_{x}\sigma_{z}+k_{y}\sigma_{x}\right)\tau_{y}.$
(20)

with $m(x<0)$ for the sample, $m(x>0)$ for the vacuum side, and
$m(x<0)m(x>0)<0$. At $k_{y}=0$, the states with $\sigma_{z}=\pm 1$ are
decoupled with opposite $M_{y}$ eigenvalues. First, considering the
$\sigma_{z}=+1$ and the equation reduced to a Jackiw-Rebbi [90] problem with
the zero-energy edge mode

$\psi_{+}=\frac{1}{A}\exp\left(-\frac{m(x)x}{v}\right)\left(\begin{array}[]{l}1\\\
0\end{array}\right)_{\sigma}\otimes\left(\begin{array}[]{c}1\\\
-1\end{array}\right)_{\tau},$ (21)

where A is the normalization factor. The other zero mode with $\sigma_{z}=-1$
is

$\psi_{-}=\frac{1}{A}\exp\left(-\frac{m(x)x}{v}\right)\left(\begin{array}[]{l}0\\\
1\end{array}\right)_{\sigma}\otimes\left(\begin{array}[]{l}1\\\
1\end{array}\right)_{\tau}.$ (22)

Then, in the basis of above two states, the edge Hamiltonian is given by

$\mathcal{H}_{\text{edge }}(k)=vks_{y},$ (23)

where $k$ is the wave vector along the edge, the Pauli matrices $s$ act on the
space of $\left(\psi_{+},\psi_{-}\right)^{T}$ and $v$ is Fermi velocity. In
the basis, the symmetry operators can be represented by
$\mathcal{M}_{y}=\mathcal{C}=s_{z}$. In the presence of $M_{y}$ and
$\mathcal{C}$ symmetry, the mass term is forced to be zero at edge
Hamiltonian. However, for an edge that does not preserve $M_{y}$, the edge
Hamiltonian will generally be gaped with the mass term
$\Delta_{M}=m_{M}s_{x}$. Two edges related by $M_{y}$ must have opposite Dirac
mass. Therefore, the protected 0D corner modes must exit at the intersection
between two edges.

##  X Corner charge, second Stiefel-Whitney number, nested Wilson loop and
bulk gaps of all the SOTI material candidates

The nontrivial topology of these SOTI material candidates is verified by the
second Stiefel-Whitney number $w_{2}$ and fractional corner charge. The number
of odd parity of at time-reversal invariant momentum and nested Wilson loop
demonstrate the nontrivial value of $w_{2}$, and the topological invariants of
$C_{n}$ symmetry at high symmetric points determine the fractional corner
charge [Table. S2]. Table S2 also presents the gaps of these SOTI material
candidates.

TABLE. S2: Topological invariants of $C_{n}$ symmetry at high symmetric points, the number of occupied bands with odd parity at time-reversal invariant momenta, corner charge $Q_{c}$, the second Stiefel-Whitney number $w_{2}$, nested Wilson loop, and gaps of the predicted SOTI material candidates, which include graphane (CH), graphene fluoride (CF), graphene chloride (CCl), silicene hydride (SiH), silicene chloride (SiCl), germanene hydride(GeH), stanene hydride (SnH), blue phosphorene (blue P), blue arsenene (blue As), black phosphorene (black P), black arsenene (black As), anti-Kekulé/Kekulé distortion graphenes, silicenes, germanenes and stanenes (anti-KekuléG/KekuléG, anti-KekuléSi/KekuléSi, anti-KekuléGe/KekuléGe, anti-KekuléSn/KekuléSn). The $C_{n}$ symmetry eigenvalues of XY X=(C, Si, Ge, Sn, Y=H, F, Cl) materials are calculated using a $\sqrt{3}\times\sqrt{3}$ supercell. The gap of anti-Kekulé distortion graphene is taken as the experimental value of 0.38 eV. |  | Invariants | Number of Odd Parity | $Q_{c}$ | $w_{2}$ | det($W_{2}$) | Gap (eV)
---|---|---|---|---|---|---|---
|  |  | $[K_{1}^{(3)}]$ | $[M_{1}^{(2)}]$ |  | $\\#M_{2}^{(I)}$ | $\\#\Gamma_{2}^{(I)}$ |  |  |  |  |  |
$C_{6}$ | Anti-Kekulé | $sp^{2}$ | 0 | 2 |  | 5 | 3 |  |  | 1/2 | 1 | $\pi$ | 0.38
$p_{z}$ | 0 | -2 |  | 1 | 3 |  |  | 1/2 | 1 | $\pi$
Kekulé | 0 | -2 |  | 6 | 4 |  |  | 1/2 | 1 | $\pi$ |
|  |  | $[K_{1}^{(3)}]$ | $[M_{1}^{(I)}]$ |  | $\\#M_{2}^{(I)}$ | $\\#\Gamma_{2}^{(I)}$ |  |  |  |  |  |
$C_{3}$ | CH | 0 | -2 |  | 3 | 1 |  |  | 1/2 | 1 | $\pi$ | 3.5
CF | 0 | -2 |  | 6 | 4 |  |  | 1/2 | 1 | $\pi$ | 3.1
CCl | 0 | -2 |  | 6 | 4 |  |  | 1/2 | 1 | $\pi$ | 1.6
SiH | 0 | -2 |  | 3 | 1 |  |  | 1/2 | 1 | $\pi$ | 2.2
SiCl | 0 | -2 |  | 6 | 4 |  |  | 1/2 | 1 | $\pi$ | 1.3
GeH | 0 | -2 |  | 3 | 1 |  |  | 1/2 | 1 | $\pi$ | 1.0
SnH | 0 | -2 |  | 3 | 1 |  |  | 1/2 | 1 | $\pi$ | 0.5
BlueP | 0 | -2 |  | 3 | 1 |  |  | 1/2 | 1 | $\pi$ | 1.9
BlueAs | 0 | -2 |  | 3 | 1 |  |  | 1/2 | 1 | $\pi$ | 1.6
| Anti-KekuleSi | 0 | 0 |  | 6 | 6 |  |  | 0 | 0 | 0 |
| Anti-KekuleGe | 0 | 0 |  | 6 | 6 |  |  | 0 | 0 | 0 |
| Anti-KekuleSn | 0 | 0 |  | 6 | 6 |  |  | 0 | 0 | 0 |
| KekuleSi | 0 | -2 |  | 6 | 4 |  |  | 1/2 | 1 | $\pi$ |
| KekuleGe | 0 | -2 |  | 6 | 4 |  |  | 1/2 | 1 | $\pi$ |
| KekuleSn | 0 | -2 |  | 6 | 4 |  |  | 1/2 | 1 | $\pi$ |
|  |  | $[X_{1}^{(2)}]$ | $[Y_{1}^{(2)}]$ | $[M_{1}^{(2)}]$ | $\\#M_{2}^{(I)}$ | $\\#\Gamma_{2}^{(I)}$ | $\\#X_{2}^{(I)}$ | $\\#Y_{2}^{(I)}$ |  |  |  |
$C_{2}$ | BlackP | -2 | -2 | -2 | 4 | 4 | 5 | 5 | 1/2 | 1 | $\pi$ | 0.9
BlackAs | -2 | -2 | -2 | 4 | 4 | 5 | 5 | 1/2 | 1 | $\pi$ | 0.7

##  XI Charge spatial distribution of corner states

There are usually more than one corner state at each corner in the two models
and SOTI candidate materials. For example, three corner states located at
different energy in the eight-band TB model. The three corner states are
labeled as 1, 2 and 3 at the Fig. S22 and the charge distribution pattern of
the three corner states are slight different but all are well localized at the
corners of the finite-size flake.

FIG. S22: (a) Energy Spectrum of the finite-size flake of the eight-band TB
model. Corner states are located up, middle and below of the edge states,
which are labeled as 1, 2, 3 in (a). The charge spatial distribution of the
three corner states (b, c, d).

##  XII Numerical calculation of corner charge

FIG. S23: Charge distribution of model I (a) and model II (b). Light blue and
dark blue represent the edge and corner regions. We consider $sp^{2}$ orbitals
in model I.

The charge accumulated in a corner region of finite-size flake is

$Q_{c}=\sum_{i\in\text{ occupied
}}\int_{Corner}d\boldsymbol{r}\left|\psi_{i}(\boldsymbol{r})\right|^{2}\bmod
1,$ (24)

where $\psi_{i}(\boldsymbol{r})$ is the wavefunction of the occupied
eigenstates and the integration is performed within the corner region. We
calculated the charge distribution of every unit cell of finite-size flake of
model I and model II (Fig. S23). The corner charge of model I and model II are
0.504 and 0.512 respectively, which are close to the theoretical value 1/2.