# General corner charge formula in two-dimensional $C_{n}$-symmetric higher-
order topological insulators

Ryo Takahashi  Department of Physics, Tokyo Institute of Technology, 2-12-1
Ookayama, Meguro-ku, Tokyo 152-8551, Japan
Tiantian Zhang  Department of Physics, Tokyo Institute of Technology, 2-12-1
Ookayama, Meguro-ku, Tokyo 152-8551, Japan
TIES, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo
152-8551, Japan
Shuichi Murakami  Department of Physics, Tokyo Institute of Technology, 2-12-1
Ookayama, Meguro-ku, Tokyo 152-8551, Japan
TIES, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo
152-8551, Japan

###### Abstract

In this paper, we derive a general formula for the quantized fractional corner
charge in two-dimensional $C_{n}$-symmetric higher-order topological
insulators. We assume that the electronic states can be described by the
Wannier functions and that the edges are charge neutral, but we do not assume
vanishing bulk electric polarization. We expand the scope of the corner charge
formula obtained in previous works by considering more general surface
conditions, such as surfaces with higher Miller index and surfaces with
surface reconstruction. Our theory is applicable even when the electronic
states are largely modulated near system boundaries. It also applies to
insulators with non-vanishing bulk polarization, and we find that in such
cases the value of the corner charge depends on the surface termination even
for the same bulk crystal with $C_{3}$ or $C_{4}$ symmetry, via a difference
in the Wyckoff position of the center of the $C_{n}$-symmetric crystal.

##  I Introduction

Recent studies have revealed that some topological crystalline insulators
exhibit higher-order bulk boundary correspondence, which are known as higher-
order topological insulators PhysRevLett.111.047006 ; PhysRevB.89.224503 ;
PhysRevB.95.165443 ; fang2019new ; PhysRevLett.119.246401 ; PhysRevB.97.155305
; PhysRevB.97.241402 ; PhysRevB.97.205135 ; PhysRevB.97.205136 ;
schindler2018higherTI ; schindler2018higher ; PhysRevB.98.081110 ;
PhysRevX.8.031070 ; PhysRevB.98.205129 ; wieder2018axion ; PhysRevB.98.245102
; PhysRevX.9.011012 ; PhysRevB.99.235125 ; PhysRevB.100.205126 ;
PhysRevB.100.235302 ; PhysRevB.101.115120 ; PhysRevResearch.2.013300 ;
PhysRevResearch.2.043274 ; Benalcazar61 ; PhysRevLett.119.246402 ;
PhysRevB.96.245115 ; serra2018observation ; PhysRevLett.120.026801 ;
peterson2018quantized ; PhysRevB.98.045125 ; imhof2018topolectrical ;
PhysRevB.99.245151 ; PhysRevLett.122.086804 ; PhysRevResearch.2.012009 ;
doi:10.7566/JPSJ.88.104703 ; PhysRevResearch.1.033074 ; PhysRevB.101.241109 ;
PhysRevB.101.115140 ; peterson2020fractional ; PhysRevResearch.2.043131 ;
PhysRevB.102.165120 . For example, three-dimensional second-order topological
insulators with inversion symmetry (and time-reversal symmetry) have protected
anomalous gapless mode along the hinges PhysRevB.97.205136 ;
PhysRevB.98.081110 ; PhysRevX.8.031070 ; PhysRevB.98.205129 ;
PhysRevB.98.245102 ; PhysRevB.101.115120 ; PhysRevResearch.2.013300 ;
PhysRevResearch.2.043274 , and two-dimensional second-order topological
insulators with rotation symmetry have protected quantized corner charges
Benalcazar61 ; PhysRevLett.119.246402 ; PhysRevB.96.245115 ;
serra2018observation ; PhysRevLett.120.026801 ; peterson2018quantized ;
PhysRevB.98.045125 ; imhof2018topolectrical ; PhysRevB.99.245151 ;
PhysRevLett.122.086804 ; PhysRevResearch.2.012009 ; doi:10.7566/JPSJ.88.104703
; PhysRevResearch.1.033074 ; PhysRevB.101.115140 ; PhysRevB.101.241109 ;
peterson2020fractional ; kooi2020bulk ; PhysRevResearch.2.043131 ;
PhysRevB.102.165120 . Such fractionally quantized corner charges are
generalizations of the quantized surface charge caused by the quantized bulk
electric polarizationPhysRevB.48.4442 .

As discussed in the previous works, the quantized fractional corner charge in
$C_{n}$-symmetric crystalline insulators are associated with rotational
eigenvalues of the bulk wavefunctionsPhysRevB.99.245151 ;
PhysRevResearch.1.033074 ; kooi2020bulk ; PhysRevResearch.2.043131 ;
nonetheless, they are limited to special cases. For example, in
$C_{6}$-symmetric systems, Ref. [PhysRevB.99.245151, ] considers the case with
hexagonal unit cells that cover the whole crystal, while in Ref.
[PhysRevResearch.2.043131, ] systems covered by triangular building blocks are
considered, and so the formulas of the corner charge for these two cases are
different. In Ref. [PhysRevB.102.165120, ], although a general formula of the
quantized corner charge is discussed, the authors assume that the surface is
flat and the electronic charge distribution is the same between the finite
$C_{6}$-symmetric system and an infinite system. Most importantly, the formula
in Ref. [PhysRevB.102.165120, ] treats the contributions of electrons and ions
to corner charges on an equal footing.

In this paper, we derive a general formula for the quantized fractional corner
charge in two-dimensional $C_{n}$-symmetric systems. We assume that the
electronic states can be described by the Wannier functions and that the edges
are charge neutral, but unlike previous worksPhysRevB.99.245151 ;
PhysRevResearch.1.033074 ; kooi2020bulk ; PhysRevResearch.2.043131 , we do not
assume vanishing bulk electric polarization, i.e. our theory applies to a
wider range of systems. We expand the scope of the corner charge formula by
considering more general surface conditions, such as surfaces with higher
Miller index and surfaces with surface reconstruction. Our study also shows
that even when the positions of the nuclei and the electronic states are
largely modulated near the boundaries, the corner charge is quantized to
fractional values as long as the system is $C_{n}$-symmetric and the edge
charge density is zero. Thus, unlike previous theories, our theory also
includes some insulators with non-vanishing bulk polarization, and in such
cases with $C_{3}$ or $C_{4}$ symmetries, we find that the value of the corner
charge depends on the crystal termination for the same bulk crystal.

This paper is organized as follows. In Sec. II, we derive the general corner
charge formula and clarified the assumptions required for the quantization of
the corner charge. We show that as long as the edge is gapped and charge
neutral in $C_{n}$-symmetric finite system, the corner charge is fractionally
quantized modulo $|e|/n$. We also discuss the relation of the general formula
and the bulk rotational eigenvalues. In Sec. III, we consider systems with
nonzero electric polarization, which is beyond the scope of the previous
corner charge formula. In Sec. IV, we consider more general surface
conditions, such as general edge orientations and surface reconstructions.
Conclusion and discussion are given in Sec. V.

##  II Corner charge formula

Here, we show a general theory to calculate the corner charges in an insulator
with a $C_{n}$-rotation symmetry. In the previous worksPhysRevB.99.245151 ;
PhysRevResearch.1.033074 ; kooi2020bulk ; PhysRevResearch.2.043131 ;
PhysRevB.102.165120 , the proof of the quantized corner charge is based on
idealized assumptions that the electronic charge distribution is perfectly
periodic even near the system boundaries, and the quantization will survive
continuous deformation of the system while preserving the $C_{n}$ symmetry and
a nonzero gap. However, it is not clear whether any $C_{n}$-symmetric crystal
can be deformed to a system satisfying the above idealized assumptions. In
real materials, nuclei can be treated as point particles, but electronic
states are extended in general, and it is not clear how and whether one can
deform such electronic wavefunctions to those satisfying the above assumption.
Moreover, the electronic states are determined as solutions of the Schrödinger
equation, which leads to deformation of electronic wavefunctions near the
boundaries, and it is very far from the above idealized assumption. Thus, a
general proof for $C_{n}$-symmetric insulators is desired. In the following,
we give a general proof for this problem.

###  II.1 Derivation of the general corner charge formula

We consider a crystal of finite size which preserves the $C_{n}$ symmetry. We
assume that the system is insulating, in the sense that the Fermi level is
within the energy gap for both the bulk and the boundaries. In some cases,
degenerate corner states exist at the Fermi energy, which violates the above
assumption, when charge neutrality is assumed. However, we can add or remove
several electrons so that no states are at the Fermi energy. This deviation
from charge neutrality is called filling anomalyPhysRevB.99.245151 .

Figure 1: (Color online) A conceptual picture of the region division for (a)
$C_{3}$, (b) $C_{4}$ and (c) $C_{6}$-symmetric systems. In each system, the
width of the region (I), $d_{\text{edge}}$, is taken to be large enough so
that both the electronic states and the ionic positions in the region (II) are
the same as those at the bulk.

In order to calculate the filling anomaly, we divide the system conceptually
into two $C_{n}$-symmetric regions (I) and (II), as shown in Fig. 1. The
regions (I) and (II) can be regarded as surface and bulk regions,
respectively, and we explain this division in the following. The width of the
surface region (I) is taken to be large enough so that both the electronic
states and the ionic positions in the region (II) are the same as those at the
bulk.

To calculate the filling anomaly, we proceed with the following steps. First,
we classify the ions into two groups according to which region they belong to,
so that no ions are at the junctures of two regions.

Second, we classify the electronic states. We assume that the occupied states
of this system can be described by a $C_{n}$-symmetric basis set
$\\{\phi_{i}\\}$ composed of exponentially localized basis
orbitalsPhysRevB.48.4442 . Well within the crystal, the electronic states can
be well described by bulk Wannier functions $W_{nl}$, and we take these
Wannier functions $W_{nl}$ to be among $\\{\phi_{i}\\}$. The remaining
$\phi_{i}$’s are additional localized basis orbitals
$\varphi_{i}^{(\text{add})}$ near the boundary. We classify them into regions
(I) and (II) so that the set of orbitals in region (I) and that in region (II)
are $C_{n}$ symmetric. By construction the numbers of ions and electronic
localized basis orbitals in the region (I) are integer multiples of $n$. Here,
we impose a condition that all the additional localized basis orbitals
$\varphi_{i}^{(\text{add})}$ are classified into the region (I), and we choose
the region (I) large enough so that this condition is satisfied. Thus, even if
the electronic states near the boundary are largely deformed, our theory is
still applicable, by incorporating these deformed wavefunctions into region
(I). We note that this construction of localized basis functions is similar to
that used in the argument in Ref. [PhysRevB.48.4442, ] to show that the bulk
polarization in terms of the Zak phase gives surface charge density. We note
that some degrees of freedom remain in this division into two regions, but
they do not affect the following argument.

Third, we calculate the total charge of the finite crystal in terms of mod
$n|e|$. First, the total charge contribution from the region (I) should be $0$
(mod $n|e|$) because the number of the nuclei and the electronic basis
orbitals within region (I) is an integer multiple of $n$. Next, we calculate
total charges of electrons and ions in the region (II). First, in the region
(II), all the orbitals are Wannier orbitals $W_{nl}$, and therefore positions
of the ions and electrons are classified in terms of Wyckoff positions.
Because the whole crystal is $C_{n}$ symmetric, the center of the crystal
should be located at the Wyckoff position with $C_{n}$ local symmetry. Let
$1a$ be the Wyckoff position with $C_{n}$ symmetry located at the center of
the crystal. Then, the ions not located at $1a$ contribute $nM|e|$
($M\in\mathbb{Z}$) to the net charge in the region (II), while those at 1a
contribute $nM^{\prime}|e|+n_{1a}^{(\text{ion})}|e|$
($M^{\prime}\in\mathbb{Z}$), where $n_{1a}^{(\text{ion})}|e|$ is the charge of
the ion at the $1a$ position. Therefore, the total ionic charge in the region
(II) is equal to $n_{1a}^{(\text{ion})}|e|$ mod $n|e|$ and so is that in the
whole crystal. Similarly, the total electron charge is equal to
$-n_{1a}^{\text{(e)}}|e|$ (mod $n|e|$) where $n_{1a}^{\text{(e)}}$ is the
number of electron Wannier orbitals at the $1a$ position. From the above
discussion, the total charge of the system in terms of modulo $n|e|$ is
calculated as follows,

$\displaystyle
Q_{\text{tot}}\equiv(n_{1a}^{\text{(ion)}}-n_{1a}^{\text{(e)}})|e|\quad(\text{mod}\
n|e|),$ (1)

where $n_{1a}^{\text{(ion)}}|e|$ is an ionic charge at $1a$, and
$n_{1a}^{\text{(e)}}$ is the number of electronic Wannier functions at $1a$.
This formula represents a filling anomaly in the crystal.

Figure 2: (Color online) A conceptual picture of the region division for
calculating the corner charge for a $C_{4}$-symmetric system as an example.
The total charge in each region is calculated by integrating the moving
average of the total charge density $\overline{\rho}(\bm{x})$ in each region.
In the bulk region (C), $\overline{\rho}(\bm{x})=0$, and the total charge in
the bulk region $Q_{\text{bulk}}$ is zero. The edge region (B) includes an
integer number of edge unit cells. The charge of each edge unit cell is
calculated by the product of the edge charge density
$\sigma=\bm{P}\cdot\bm{n}$ and the edge period $a_{\text{edge}}$. The total
charge is represented as summation of the edge charges and the corner charges:
$Q_{\text{tot}}=n(Q_{\text{edge}}+Q_{\text{corner}})$. Therefore, when
$Q_{\text{edge}}$ vanishes, $Q_{\text{corner}}\equiv Q_{\text{tot}}/n$ (mod
$e$).

Fourth, we discuss how $Q_{\text{tot}}$ is distributed in the crystal. As a
first step, we define the moving average of the total charge density
$\overline{\rho}(\bm{x})$ as follows:

$\displaystyle\overline{\rho}(\bm{x})=\frac{1}{|\Omega|}\int_{\Omega}\rho(\bm{x}+\bm{x}^{\prime})\
d\bm{x}^{\prime}.$ (2)

Here, $\rho(\bm{x})$ is the total charge density including ions and electrons,
$\Omega$ is a $C_{n}$-symmetric unit cell whose center is at the origin and
$|\Omega|$ is the area of $\Omega$. From the definition,
$\overline{\rho}(\bm{x})$ have the following properties:

$\displaystyle\overline{\rho}(\bm{x})$
$\displaystyle=0\quad(\bm{x}\in\text{bulk}),$ (3)
$\displaystyle\int_{\text{crystal}}d\bm{x}^{\prime}\
\overline{\rho}(\bm{x}^{\prime})\ $ $\displaystyle=Q_{\text{tot}}.$ (4)
$\displaystyle\overline{\rho}(C_{n}\cdot\bm{x})$
$\displaystyle=\overline{\rho}(\bm{x})$ (5)

We note that as long as $\overline{\rho}(\bm{x})$ satisfies Eqs. (3), (4) and
(5), we can also use other definitions for moving averages. In a sufficiently
large system, we expect that electronic states in the middle part of the edges
far away from the corners are almost the same as those on an edge of a semi-
infinite system with an open boundary condition in one direction. To calculate
the corner charge based on this expectation, we divide the system into three
types of regions: the (A) corner, (B) edge, and (C) bulk regions in a
$C_{n}$-symmetric way as schematically shown in Fig. 2. The edge region (B)
should be chosen to satisfy the following conditions. First, it should be
sufficiently far away from the corners, such that the electronic states in the
edge region (B) are the same as those on an edge of the corresponding semi-
infinite system. Second, it should have a width equal to the period along the
edge and a sufficiently long depth into the bulk, such that in the remaining
bulk region (C), the electronic states are the same as those for a bulk
infinite system. Since $\overline{\rho}(\bm{x})$ takes nonzero values only in
the (A) corner and (B) edge regions, its integral can be calculated as the sum
of the total charges in these regions:

$\displaystyle\int_{\text{crystal}}d\bm{x}^{\prime}\
\overline{\rho}(\bm{x}^{\prime})\ $
$\displaystyle=n\Bigl{(}Q_{\text{edge}}+Q_{\text{corner}}\Bigr{)},$ (6)

where $Q_{\text{edge}}$ and $Q_{\text{corner}}$ are the total charges at one
of the (B) edge regions and one of the (C) corner regions, respectively. Here,
we assume that one edge region includes an integer number of edge unit cells.
Then $Q_{\text{edge}}$ can be calculated as:

$\displaystyle Q_{\text{edge}}$
$\displaystyle=N_{\text{edge}}\sigma_{\text{edge}}a_{\text{edge}}.$ (7)

Here, $N_{\text{edge}}$ is the number of edge unit cells within one edge
region, $\sigma_{\text{edge}}$ is the edge charge density, and
$a_{\text{edge}}$ is the period of the edge unit cell.

Finally, we calculate the corner charge. We assume that the edge is charge
neutral, $\sigma_{\text{edge}}=0$. Then the extra charge $Q_{\text{tot}}$ is
distributed into the $n$ corners. Since the crystal is assumed to be
$C_{n}$-symmetric, the distributed charges are equal among the corners, and
this gives a corner charge formula:

$\displaystyle
Q_{\text{corner}}=Q_{c\\_1a}^{(n)}\equiv\frac{(n_{1a}^{\text{(ion)}}-n_{1a}^{\text{(e)}})|e|}{n}\quad(\text{mod}\
|e|).$ (8)

If the edge charge density $\sigma_{\text{edge}}\neq 0$, the extra charge
$Q_{\text{tot}}$ is distributed not only to the corners, but also to the
edges. Since the separation of charges between the corners and the edges
depends on the choice of the corner regions, the corner charge is not well
defined.

###  II.2 Scope of the general corner charge formula

While the same formula is given in Ref. PhysRevB.102.165120, , our theory
shows that the corner charge formula Eq. (8) can be applied to a wider range
of systems which are not discussed in the previous works. Namely, unlike the
previous worksPhysRevB.99.245151 ; PhysRevResearch.1.033074 ;
PhysRevResearch.2.043131 ; PhysRevB.102.165120 , our theory applies to (i)
systems with electronic states and/or ion positions largely modulated near the
boundaries, (ii) systems with boundaries with high Miller indices and with
surface reconstructions, and (iii) systems with nonzero $\bm{P}$ but with
insulating edges having $\sigma=0$. In the following, we explain these three
cases.

First, our proof applies to cases (i) with modulation of electronic states and
ion positions near the boundaries. For example, in the previous
worksPhysRevB.99.245151 ; PhysRevResearch.1.033074 ; PhysRevResearch.2.043131
; PhysRevB.102.165120 , only the systems with perfect periodicity for the
nuclei and the electrons are considered, including edges and corners. In
reality, however, the electronic states are modified close to the system
boundary, and the positions of the nuclei may deviate from those with perfect
periodicity. In such cases, one can argue that the quantization of the corner
charges persists under a continuous change of the system. Nonetheless, it is
not clear which system can be continuously connected to a perfectly periodic
system including its boundaries. In this respect, our proof above justifies
the formula (8) for the corner charge, as long as the system is
$C_{n}$-symmetric and is insulating over the whole system.

Second is the case (ii) with the boundaries with higher Miller indices. In the
previous works, the arguments are limited to boundaries with low Miller
indices such as (10) and (01) edges, but our theory applies to general Miller
indices. For example, a $C_{4}$-symmetric square system with edges along (34)
and ($4\bar{3}$) directions is covered by our theory.

Third, our theory covers also the cases (iii) with $\bm{P}\neq 0$. In the
previous worksPhysRevB.99.245151 ; PhysRevResearch.1.033074 ;
PhysRevResearch.2.043131 ; PhysRevB.102.165120 , the bulk polarization
$\bm{P}$ is assumed to be 0, to ensure that the edge charge density
$\sigma_{\text{edge}}$ is 0. However, as we discussed in the above, our
assumptions for the edges are gapped and charge neutral, i.e.
$\sigma_{\text{edge}}=0$, which does not necessarily mean $\bm{P}=0$. To be
more precise, the bulk polarization $\bm{P}$ is related to the edge charge
densityPhysRevB.48.4442  via

$\displaystyle\sigma_{\text{edge}}=\bm{P}\cdot\bm{n}\ (\text{mod}\
|e|/a_{\text{edge}}),$ (9)

where $\bm{n}$ is the normal vector of the edge and $a_{\text{edge}}$ is the
lattice periodicity along the edge. Thus, $\sigma_{\text{edge}}=0$ means
$\bm{P}\cdot\bm{n}\equiv 0$ (mod $|e|/a_{\text{edge}}$). In $C_{n}$-symmetric
systems, the bulk polarization is quantized to be fractional, meaning that
$\bm{P}\cdot\bm{n}$ is also quantized to be $|e|/a_{\text{edge}}$ times a
simple fraction for any orientation of the edgesPhysRevB.99.245151 ;
PhysRevB.102.165120 , as we discuss in Sec. IV B. When
$\bm{P}\cdot\bm{n}\not\equiv 0$ (mod $|e|/a_{\text{edge}}$),
$\sigma_{\text{edge}}$ is nonzero and our theory cannot be applied.
Nonetheless, in such cases the edge can undergo Peierls transition or surface
reconstruction to multiply edge periodicity and open a gap, because
$\bm{P}\cdot\bm{n}/(|e|/a_{\text{edge}})$ is a simple fraction. Our theory can
be applied to such cases, which are discussed in detail in Sec. IV. As we
discuss later, the formula (8) gives the same result with the previous works
which deal with special casesPhysRevB.99.245151 ; PhysRevResearch.1.033074 ;
PhysRevResearch.2.043131 .

###  II.3 Corner-charge formula in terms of rotation eigenvalues

In this section, we rewrite our formula for the corner charge in terms of
rotation eigenvalues of the Bloch eigenstates at high-symmetry wavevectors
following Refs. [PhysRevB.99.245151, ,PhysRevResearch.1.033074,
,PhysRevResearch.2.043131, ], in three different systems with and without
time-reversal symmetry, and with and without spin-orbit coupling, separately.
We need to rewrite it, because eigenstates in a crystal are Bloch
wavefunctions, extended over the crystal, and one cannot directly calculate
$n_{1a}^{(e)}$ from the Bloch wavefunctions. Then, we discuss a relation
between our formula and the formulas in the previous worksPhysRevB.99.245151 ;
PhysRevResearch.1.033074 ; PhysRevResearch.2.043131 . The notable difference
from the previous works is that the Wyckoff positions of the ions are not
always 1a, and the corner charge generally depends on the filling $\nu$ as in
Ref. [PhysRevResearch.2.043131, ]. The detail of the calculation is shown in
the Appendix.

By following Refs. [PhysRevB.99.245151, ,PhysRevResearch.1.033074,
,PhysRevResearch.2.043131, ], we introduce topological invariants which
distinguish the Wannier functions with different Wyckoff positions. With a
sufficient number of topological invariants, there can be a one-to-one
correspondence between the values of the topological invariants and Wyckoff
positions of the occupied Wannier orbitals. We assume that the occupied bands
can be written in terms of Wannier functions. Then, the number of Wannier
orbitals at each Wyckoff position is determined from the topological invariant
for the occupied bands.

Figure 3 shows the Wyckoff positions under $C_{n}$ rotational symmetry
($n=3,4,6$). In each system with $C_{n}$ symmetry, there are four kinds of
Wyckoff positions. We define the number of occupied Wannier functions at
Wyckoff position $mX$ as $n_{mX}^{(e)}$. Here, $X=a,b,c,d$ represent the types
of the Wyckoff positions, and $m$ represents the multiplicity of a Wyckoff
position. The weighted sum of the $n_{mX}^{(e)}$ is equal to the electron
filling, i.e. the number of occupied bands $\nu$:

$\displaystyle\sum_{X}mn_{mX}^{(e)}=\nu.$ (10)

For the calculation of the corner charge, the value of $n_{1a}^{(e)}$ is
needed. As we show in the appendix, $n_{mX}^{(e)}$ modulo $n$ ($X\neq a$) is
directly calculated from the rotational topological invariants defined from
rotation eigenvalues of the Bloch wavefunctions, and $n_{1a}^{(e)}$ is
determined from $\nu-\sum_{X\neq a}mn_{mX}^{(e)}$. In the appendix, we show
the detailed calculation of $n_{mX}^{(e)}$.

Figure 3: (Color online) Wyckoff positions in (a) $C_{3}$, (b) $C_{4}$, (c)
$C_{6}$-symmetric systems. The points with the same colors belong to the same
Wyckoff positions. Here, $\bm{a}_{1}$ and $\bm{a}_{2}$ are primitive
translation vectors, and (a) $\bm{a}_{1}=(\frac{a}{2},\frac{\sqrt{3}}{2}a)$,
$\bm{a}_{2}=(-\frac{a}{2},\frac{\sqrt{3}}{2}a)$, (b) $\bm{a}_{1}=(a,0)$,
$\bm{a}_{2}=(0,a)$, (c) $\bm{a}_{1}=(a,0)$,
$\bm{a}_{2}=(-\frac{a}{2},\frac{\sqrt{3}}{2}a)$.

####  II.3.1 class A

First, we consider a system called class A. Unlike the class AI
systemsPhysRevB.99.245151 ; PhysRevResearch.2.043131  or class AII
systemsPhysRevResearch.1.033074 ; kooi2020bulk  considered in previous
studies, the class A systems do not have time-reversal symmetry. From Eq. (8)
and Appendix A, the corner charge is calculated as follows:

$\displaystyle Q_{c\\_1a}^{(3)}$
$\displaystyle\equiv\frac{|e|}{3}\Big{(}n_{1a}^{(\text{ion})}-\nu-[K_{1}^{(3)}]-[K_{2}^{(3)}]-[K_{1}^{\prime(3)}]-[K_{2}^{\prime(3)}]\Big{)}$
$\displaystyle\
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\
(\text{mod}\ e),$ (11) $\displaystyle Q_{c\\_1a}^{(4)}$
$\displaystyle\equiv\frac{|e|}{4}\Big{(}n_{1a}^{(\text{ion})}-\nu+[X_{1}^{(2)}]-\frac{1}{2}[M_{1}^{(4)}]+\frac{3}{2}[M_{3}^{(4)}]\Big{)}$
$\displaystyle\
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\
(\text{mod}\ e),$ (12) $\displaystyle Q_{c\\_1a}^{(6)}$
$\displaystyle\equiv\frac{|e|}{6}\Big{(}n_{1a}^{(\text{ion})}-\nu+2[K_{1}^{(3)}]+\frac{3}{2}[M_{1}^{(2)}]\Big{)}\
(\text{mod}\ e).$ (13)

Here, the rotation topological invariant $[\Pi_{p}^{(n)}]$ represents the
difference of the number of states with $C_{n}$ eigenvalue
$e^{\frac{2\pi(p-1)i}{n}}$ for spinless systems, and
$e^{\frac{2\pi(p-1)i}{n}}e^{\frac{\pi i}{n}}$ for spinful systems, between the
rotation-invariant $\bm{k}$-points $\Pi$ and $\Gamma$.

In addition to Eqs. (11)-(13), where the center of the $C_{n}$-symmetric
crystal is at $1a$, in appendix A, we also discuss the formula when the center
of the crystal is at other Wyckoff positions, i.e., $1b$ and $1c$ for $C_{3}$
and $1b$ for $C_{4}$. The formulae are summarized as follows:

$\displaystyle Q_{c\\_1b}^{(3)}$
$\displaystyle\equiv\frac{|e|}{3}\Big{(}n_{1b}^{(\text{ion})}+[K_{1}^{(3)}]+[K_{2}^{\prime(3)}]\Big{)}\
(\text{mod}\ e),$ (14) $\displaystyle Q_{c\\_1c}^{(3)}$
$\displaystyle\equiv\frac{|e|}{3}\Big{(}n_{1c}^{(\text{ion})}+[K_{2}^{(3)}]+[K_{1}^{\prime(3)}]\Big{)}\
(\text{mod}\ e),$ (15) $\displaystyle Q_{c\\_1b}^{(4)}$
$\displaystyle\equiv\frac{|e|}{4}\Big{(}n_{1b}^{(\text{ion})}-[X_{1}^{(2)}]+\frac{3}{2}[M_{1}^{(4)}]-\frac{1}{2}[M_{3}^{(4)}]\Big{)}$
$\displaystyle\
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\
(\text{mod}\ e).$ (16)

####  II.3.2 class AI

Here, we consider a spinless system with time-reversal symmetry called class
AI. In this case, there are some constraints on the topological invariants
$[\Pi_{p}^{(n)}]$, and the corner charge formula is somewhat simplified.

$\displaystyle Q_{c\\_1a}^{(3)}$
$\displaystyle\equiv\frac{|e|}{3}(n_{1a}^{(\text{ion})}-\nu)-\frac{|e|}{3}[K_{1}^{(3)}]\
(\text{mod}\ e),$ (17) $\displaystyle Q_{c\\_1a}^{(4)}$
$\displaystyle\equiv\frac{|e|}{4}\Big{(}n_{1a}^{(\text{ion})}-\nu+[X_{1}^{(2)}]-2[M_{1}^{(4)}]-3[M_{2}^{(4)}]\Big{)}$
$\displaystyle\
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\
(\text{mod}\ e),$ (18) $\displaystyle Q_{c\\_1a}^{(6)}$
$\displaystyle\equiv\frac{|e|}{6}\Big{(}n_{1a}^{(\text{ion})}-\nu+2[K_{1}^{(3)}]+\frac{3}{2}[M_{1}^{(2)}]\Big{)}\
(\text{mod}\ e).$ (19)

In particular, these formulae are reproduced to the results in Ref.
[PhysRevB.99.245151, ] only if all the ions are located at the center of the
$C_{n}$-symmetric unit cell (i.e., Wyckoff position $1a$) leading to
$n_{1a}^{(\text{ion})}=\nu$ and if the electric polarization is zero. The
formulae are reduced to Eq. (11) of Ref. [PhysRevB.99.245151, ], i.e.,

$\displaystyle Q_{c\\_1a}^{(3)}$
$\displaystyle\equiv-\frac{|e|}{3}[K_{2}^{(3)}]\ (\text{mod}\ e),$ (20)
$\displaystyle Q_{c\\_1a}^{(4)}$
$\displaystyle\equiv-\frac{|e|}{4}\Big{(}[X_{1}^{(2)}]+2[M_{1}^{(4)}]+3[M_{2}^{(4)}]\Big{)}\
(\text{mod}\ e),$ (21) $\displaystyle Q_{c\\_1a}^{(6)}$
$\displaystyle\equiv-\frac{|e|}{4}[M_{1}^{(2)}]-\frac{|e|}{6}[K_{1}^{(3)}]\
(\text{mod}\ e).$ (22)

As shown in appendix B, we also discuss the formula when the center of the
$C_{n}$-symmetric crystal is not at the Wyckoff position $1a$. The formulae
are summarized as follows:

$\displaystyle Q_{c\\_1b}^{(3)}$
$\displaystyle\equiv\frac{|e|}{3}\Big{(}n_{1b}^{(\text{ion})}-[K^{\prime(3)}_{2}]\Big{)}\
(\text{mod}\ e),$ (23) $\displaystyle Q_{c\\_1c}^{(3)}$
$\displaystyle\equiv\frac{|e|}{3}\Big{(}n_{1c}^{(\text{ion})}-[K^{(3)}_{2}]\Big{)}\
(\text{mod}\ e),$ (24) $\displaystyle Q_{c\\_1b}^{(4)}$
$\displaystyle\equiv\frac{|e|}{4}\Big{(}n_{1b}^{(\text{ion})}-[X^{(2)}_{1}]+2[M^{(4)}_{1}]+[M^{(4)}_{2}]\Big{)}$
$\displaystyle\
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\
(\text{mod}\ e).$ (25)

####  II.3.3 class AII

Here, we consider a spinful system with time-reversal symmetry called class
AII. In Refs. [PhysRevResearch.1.033074, , kooi2020bulk, ], in the class AII
systems, the corner charge is defined modulo $2e$ due to the Kramers theorem.
However, as pointed out in Ref. [PhysRevB.102.165120, ], we argue that it
should be defined modulo $e$ in general cases. For example, the nuclei can
have an odd number of charges in the unit of electronic charge $|e|$, which
leads to a change of the corner charge by $e$ without changing the bulk. It
occurs when some surface adatoms are allowed, as discussed in Ref.
[PhysRevB.102.165120, ]. It is also the case in compounds containing elements
with odd atomic numbers; a change in crystal terminations results in a change
of the corner charge by $e$. For the above reasons, we deal with corner
charges defined by modulo $e$, even in the class AII.

As with class AI systems, the topological invariants $[\Pi_{p}^{(n)}]$ have
some constraints due to time-reversal symmetry, and the corner charge formula
is somewhat simplified.

$\displaystyle Q_{c\\_1a}^{(3)}$
$\displaystyle=\frac{|e|}{3}(n_{1a}^{(\text{ion})}-\nu)-\frac{|e|}{3}[K_{2}^{(3)}]\
(\text{mod}\ e),$ (26) $\displaystyle Q_{c\\_1a}^{(4)}$
$\displaystyle\equiv\frac{|e|}{4}(n_{1a}^{(\text{ion})}-\nu)+\frac{|e|}{2}[M_{1}^{(4)}]\
(\text{mod}\ e),$ (27) $\displaystyle Q_{c\\_1a}^{(6)}$
$\displaystyle=\frac{|e|}{6}(n_{1a}^{(\text{ion})}-\nu)+\frac{|e|}{3}[K_{1}^{(3)}]\
(\text{mod}\ e).$ (28)

In appendix C, we also discuss the formula when the center of the
$C_{n}$-symmetric crystal is not at the Wyckoff position $1a$. The formulae
are summarized as follows:

$\displaystyle Q_{c\\_1b}^{(3)}$
$\displaystyle\equiv\frac{|e|}{3}\Big{(}n_{1b}^{(\text{ion})}-[K^{(3)}_{1}]\Big{)}\
(\text{mod}\ e),$ (29) $\displaystyle Q_{c\\_1c}^{(3)}$
$\displaystyle\equiv\frac{|e|}{3}\Big{(}n_{1c}^{(\text{ion})}-[K^{\prime(3)}_{1}]\Big{)}\
(\text{mod}\ e),$ (30) $\displaystyle Q_{c\\_1b}^{(4)}$
$\displaystyle\equiv\frac{|e|}{4}\Big{(}n_{1b}^{(\text{ion})}-2[M^{(4)}_{1}]\Big{)}\
(\text{mod}\ e).$ (31)

##  III Corner charge with nonzero bulk polarization

In this section, we consider the case where the bulk polarization is nonzero.
In the previous worksPhysRevB.99.245151 ; PhysRevResearch.1.033074 ;
PhysRevResearch.2.043131 ; PhysRevB.102.165120 , the bulk polarization is
assumed to be 0, to ensure that the edge charge density per edge unit cell is
0 modulo $|e|$. However, as we explain in the following, even if the bulk
polarization is nonzero, as long as the edge charge density is 0, our general
formula is applicable. For example, we consider a $C_{4}$-symmetric tight-
binding model with nonzero bulk polarization. We show that the fractional
corner charge appears even though the bulk polarization is nonzero, as long as
the edge charge density is 0. Moreover, we also show that two finite
$C_{4}$-symmetric flakes with different terminations can have different corner
charges due to difference in the $C_{4}$ centers. This property is specific to
cases with nonzero polarization.

We consider a simple spinless tight-binding model. It has background positive
ions with $2|e|$ charge at $1a$, being the center of the square unit cell (see
Fig. 4(a),(b)). The Hamiltonian is given as $\hat{H}=\sum_{\langle
ij\rangle}t_{ij}c_{i}^{\dagger}c_{j}$, where $t_{ij}=t_{s}\ (t_{w})$ for
inter-unit-cell (intra-unit-cell) bonds colored in red (blue) in Fig. 4(a). We
set $t_{s}=2$ and $t_{w}=0.8$ in the numerical calculation. The bulk bands are
gapped, and we set the Fermi energy $E_{F}=0$ within the gap. The parameters
are chosen so that the occupied Wannier orbitals are located at $2c$ Wyckoff
positions, in the middle of the red bonds. In this example, the bulk
polarization is $\bm{P}=(e/2a)(1,1)$ (mod $e/a$), where $a$ is the lattice
constant. Therefore, if we cut the system along the $x$ and $y$ axis as shown
in Fig. 4(c), nonzero fractional edge charge appears, $\sigma
a_{\text{edge}}=\bm{P}\cdot(1,0)a=\frac{e}{2}$, where the size of the edge
unit cell $a_{\text{edge}}$ is taken to be equal to $a$. This model has half-
filled edge states within the bulk gap, and they are metallic (see Fig. 4(d)).
Thus, this is out of the scope of the corner charge formula. On the other
hand, if the system is cut along the $(11)$ direction as shown in Fig. 4(e),
the edge charge is 0 modulo $e$: $\sigma
a_{\text{edge}}=\bm{P}\cdot\bm{n}\sqrt{2}a\equiv 0$ (mod $e$) per edge unit
cell, where $\bm{n}=(1/\sqrt{2},1/\sqrt{2})$ and $a_{\text{edge}}=\sqrt{2}a$.
Then the edge is insulating as shown in Fig. 4(f), and we can apply our
general formula.

In $C_{4}$-symmetric systems, there are two $C_{4}$-symmetric Wyckoff
positions, $1a$ and $1b$. As shown in Fig. 3(b), their positions are
$\bm{r}_{1a}=\bm{0}$ and
$\bm{r}_{1b}=\frac{1}{2}\bm{a}_{1}+\frac{1}{2}\bm{a}_{2}$, respectively.
Therefore, which formula to be used depends on whether the $C_{4}$ center of
the finite crystal is $1a$ or $1b$. If the center of the finite crystal is at
$1b$, we need to use a formula with $1a$ replaced by $1b$ in Eq. (8):

$\displaystyle
Q_{c\\_1b}^{(4)}\equiv\frac{(n_{1b}^{\text{(ion)}}-n_{1b}^{\text{(e)}})|e|}{4}\quad(\text{mod}\
|e|).$ (32)

The difference between Eqs. (8) and (32) is expressed as

$\displaystyle Q_{c\\_1a}^{(4)}-Q_{c\\_1b}^{(4)}$
$\displaystyle=\frac{q_{a}-q_{b}}{4}=\frac{-2q_{b}-2q_{c}-4q_{d}}{4}$
$\displaystyle\equiv\frac{q_{b}+q_{c}}{2}\quad(\text{mod}\ e).$ (33)

Here, $q_{X}=(n_{mX}^{\text{(ion)}}-n_{mX}^{\text{(e)}})|e|$
($mX=1a,1b,2c,4d$). The r.h.s of Eq. (33) is equal to $\bm{P}\cdot(a,0)$ and
$\bm{P}\cdot(0,a)$. Therefore, when the quantized polarization along (10) or
(01) direction is nonzero, the corner charge is different between a
$C_{4}$-symmetric crystal with its center at $1a$ and one at $1b$.

We note that in Ref. [PhysRevB.102.165120, ], systems without polarization are
studied, and

$\displaystyle\frac{1}{4}q_{a}$
$\displaystyle=\frac{1}{4}q_{b}\quad(\text{mod}\ e)\quad(n=4),$ (34)

is shown, meaning that the corner charge is independent of whether the crystal
center is at $1a$ or $1b$. It is a special case of our result.

This difference between $Q_{c\\_1a}^{(4)}$ and $Q_{c\\_1b}^{(4)}$ can be
attributed to a difference in the shape of the corners depending on the
$C_{4}$ center. For example, when the shape of the crystal is as shown in Fig.
5(a), the $C_{4}$ center is at $1a$, and the corner charge is calculated as
$Q_{c\\_1a}^{(4)}=(n_{1a}^{\text{(ion)}}-n_{1a}^{(e)})|e|/4=|e|/2$. On the
other hand, when the crystal shape is as shown in Fig. 5(c), the $C_{4}$
center is at $1b$, and the corner charge is calculated as
$Q_{c\\_1b}^{(4)}=(n_{1b}^{\text{(ion)}}-n_{1b}^{(e)})|e|/4=0$.

Correspondingly, in the system shown in Fig. 5(a), where the $C_{4}$ center is
at $1a$, four in-gap corner states appear (see Fig. 5(b)). Here, we briefly
explain the relation of the in-gap corner states and the corner charge. In the
case of a charge-neutral filling indicated by the line (i) in Fig. 5(b), the
corner states are at the Fermi energy. In this case, we cannot define its
corner charge. By adding or extracting two electrons as shown by lines (ii)
and (iii), the whole system becomes gapped and the total charge deviates from
charge neutrality by $\pm 2|e|$. As discussed in Sec. II, the nonzero total
charge is equally distributed to the four corners, resulting in a corner
charge $|e|/2$ (mod $|e|$). In contrast, in the system in Fig. 5(c), where the
$C_{4}$ center is at $1b$, there are no corner states (see Fig. 5(d)). This
agrees with our result in Eq. (33).

In $C_{3}$-symmetric systems, there are three $C_{3}$-symmetric Wyckoff
positions, $1a$, $1b$ and $1c$. As shown in Fig. 3(a), their positions are
$\bm{r}_{1a}=\bm{0}$,
$\bm{r}_{1b}=\frac{1}{3}\bm{a}_{1}+\frac{1}{3}\bm{a}_{2}$ and
$\bm{r}_{1c}=\frac{2}{3}\bm{a}_{1}+\frac{2}{3}\bm{a}_{2}$, respectively.
Therefore, similar to the case of $C_{4}$-symmetric systems, the formula for
the corner charge depends on whether the $C_{3}$ center of the finite crystal
is $1a$, $1b$ or $1c$, and let $Q^{(3)}_{c\\_1a}$, $Q^{(3)}_{c\\_1b}$,
$Q^{(3)}_{c\\_1c}$ represent the corresponding corner charges respectively.
Then, we obtain

$\displaystyle Q^{(3)}_{c\\_1a}-Q^{(3)}_{c\\_1b}$
$\displaystyle=\frac{q_{a}-q_{b}}{3}=\frac{-2q_{b}-q_{c}-3q_{d}}{3}$
$\displaystyle\equiv\frac{q_{b}+2q_{c}}{3}\ (\text{mod}\ e),$ (35)
$\displaystyle Q^{(3)}_{c\\_1a}-Q^{(3)}_{c\\_1c}$
$\displaystyle=\frac{q_{a}-q_{c}}{3}=\frac{-q_{b}-2q_{c}-3q_{d}}{3}$
$\displaystyle\equiv-\frac{q_{b}+2q_{c}}{3}\ (\text{mod}\ e).$ (36)

As discussed in Ref. [PhysRevB.102.165120, ], the r.h.s of Eq. (35) is equal
to both the electric polarization $|\Omega|\bm{P}\cdot\frac{\bm{b}_{1}}{2\pi}$
and $|\Omega|\bm{P}\cdot\frac{\bm{b}_{2}}{2\pi}$, where $\bm{b}_{i}$ ($i=1,2$)
is the reciprocal lattice vector defined as
$\bm{a}_{i}\cdot\bm{b}_{j}=2\pi\delta_{ij}$. Therefore, when the quantized
polarization along $\bm{b}_{1}$ or $\bm{b}_{2}$ is nonzero, the corner charge
depends on the $C_{3}$-center of the finite crystal. In particular, when the
polarization is zero, $\frac{q_{a}}{3}=\frac{q_{b}}{3}=\frac{q_{c}}{3}$ (mod
$e$) holds as shown in Ref. [PhysRevB.102.165120, ], and the corner charge is
independent of the $C_{3}$-center.

In a $C_{6}$-symmetric system, there is only one $C_{6}$-symmetric Wyckoff
position, $1a$. In this case, the corner charge formula is unique.

We discuss the role of a choice of the unit cell in our theory. In our theory,
the unit cell should always be taken to be the primitive one in the bulk. Our
theory is general and we do not need to take larger unit cells, in contrast to
Refs. [PhysRevResearch.1.033074, ,PhysRevB.102.165120, ]. The edge charge
density $\sigma_{\text{edge}}$ and the corner charge $Q_{c}$ are observables,
and therefore they cannot depend on a choice of the unit cell. Thus, a choice
of the unit cell is a gauge degree of freedom, and by taking a larger unit
cell, the formula of the corner charge (8) should be invariant, as long as the
Wyckoff position $1a$ is unchanged. Furthermore, we note that taking a larger
unit cell than the primitive one is misleading in some cases. For example, in
the $C_{4}$-symmetric model in Fig. 4(a), the polarization $\bm{P}$ is equal
to $\bm{P}=\frac{e}{2a}(1,1)$ (mod $e/a$). Meanwhile, if we take a larger unit
cell of $2a\times 2a$ instead of the primitive one, $a\times a$, then $\bm{P}$
becomes trivial because $\bm{P}$ is evaluated modulo $e/(2a)$, and one cannot
distinguish between the cases with $\bm{P}=\frac{e}{2a}(1,1)$ (mod $e/a$) and
$\bm{P}=0$ (mod $e/a$). In fact, it is related to the possibility of surface
reconstruction, which we will discuss in Sec. IV. In the present model, the
edge charge density is $\sigma a_{\text{edge}}=\frac{e}{2}$ (mod $e$) for
$a_{\text{edge}}=a$, whereas it becomes trivial, $\sigma
a_{\text{edge}}^{\prime}=e\equiv 0$ (mod $e$) for the larger unit cell
$a_{\text{edge}}^{\prime}=2a$.

Figure 4: (Color online) Numerical calculation for the $C_{4}$-symmetric
tight-binding model in the ribbon geometry. (a) Hopping terms of the model.
The hopping amplitude is $t_{s}$ ($t_{w}$) for bonds colored in red (blue). We
set $t_{s}=2$ and $t_{w}=0.8$ in the numerical calculation. (b) Wyckoff
positions of the $C_{4}$ symmetric system. (c,d) The ribbon system cut along
(01) direction and its energy spectrum. Due to the nonzero electric
polarization, fractional edge charge appears, and the edge energy spectrum is
metallic. (e,f) The ribbon system cut along (11) direction and its energy
spectrum. Unlike that along (01) direction, the edge energy spectrum is
gapped.

Figure 5: (Color online) Numerical calculation for the $C_{4}$-symmetric
tight-binding model in the flake geometry. (a,b) The finite system with the
$C_{4}$ center at $1a$ and its energy spectrum. In the numerical calculation,
the system size is $N=8$, and it contains $113$ unit cells in total. The
charge-neutral filling shown by line (i) in the figure makes the system
metallic. In an insulating filling as shown by lines (ii) and (iii), the total
charge deviates from charge neutrality by $\pm 2|e|$. As discussed in the main
text, the nonzero total charge is equally distributed to the four corners,
resulting in a corner charge $|e|/2$ (mod $|e|$). (c,d) The finite system with
the $C_{4}$ center at $1b$ and its energy spectrum. In the numerical
calculation, the system size is $N^{\prime}=8$, and it contains $112$ unit
cells in total.

##  IV General edge orientations and surface reconstructions

In this section, we consider edge charge densities for edges with general
orientations. When the edge charge density is nonzero, the corner charge is
nonzero. Nonetheless, even in these cases, surface reconstructions may make
the edge density to vanish, and the corner charge becomes quantized. In this
section, we discuss this in general systems.

###  IV.1 surface reconstruction

Figure 6: (Color online) Conceptual picture of the surface reconstruction and
corresponding edge band structures. (a) Before the surface reconstruction,
each edge unit cell contains $\sigma_{\text{edge}}a_{\text{edge}}=e/2$
charges. We consider the case where the edge is metallic and is half filled in
accordance with the edge charge. (b) After the surface reconstruction, the
edge periodicity is doubled ($a^{\prime}_{\text{edge}}=2a_{\text{edge}})$.
Then, each edge unit cell contains integer charges, and the edge can be gapped
and insulating.

Here we consider systems with surface reconstruction. Conceptual pictures of
the surface reconstruction is shown in Fig. 6. We assume that the system has
nonzero quantized electric polarization in the bulk. Then the edge have
nonzero fractional edge charge $\sigma_{\text{edge}}$. In Fig. 6(a) we show
its band structure when the edge states are metallic. However, it can under go
Peierls transition or surface reconstruction to multiply edge periodicity and
open a gap. In Fig. 6(b), due to the surface reconstruction, the surface
periodicity is doubled, and the edge charge becomes an integer in the enlarged
surface unit cells. Then, the edge energy spectrum can be gapped and
insulating. In this case, one can apply our general formula.

###  IV.2 General edge orientations

Figure 7: (Color online) Conceptual picture of the surface with high Miller
index ($\alpha,\beta$). The gray squares are unit cells.

Here, we consider the surface with a high Miller index. First, we consider the
$C_{4}$-symmetric system with basis vectors of $\bm{a}_{1}=(a,0)$ and
$\bm{a}_{2}=(0,a)$, as shown in Fig. 3(b). In the $C_{4}$-symmetric system,
the bulk electric polarization is quantized to be $\bm{P}\equiv(0,0)$ or
$(\frac{e}{2a},\frac{e}{2a})$ (mod $\frac{e}{a}(m_{1},m_{2})$),
($m_{1},m_{2}\in\mathbb{Z}$)PhysRevB.99.245151 ; PhysRevB.102.165120 ;
PhysRevB.48.4442 . In the following, we consider the case with
$\bm{P}\equiv(\frac{e}{2a},\frac{e}{2a})$. Next, for the surface with a Miller
index ($\alpha,\beta$) ($\alpha,\beta\in\mathbb{Z}$, and $\alpha$ and $\beta$
are coprime), the surface normal vector $\bm{n}$ is defined as follows:

$\displaystyle\bm{n}=\Bigg{(}\frac{\alpha}{\sqrt{\alpha^{2}+\beta^{2}}},\
\frac{\beta}{\sqrt{\alpha^{2}+\beta^{2}}}\Bigg{)}.$ (37)

Therefore, the edge charge in one edge unit cell is calculated as follows:

$\displaystyle\sigma
a_{\text{edge}}\equiv\bm{P}\cdot\bm{n}a_{\text{edge}}\equiv\frac{e}{2}(\alpha+\beta)\
(\text{mod}\ e).$ (38)

Here, we set the size of the edge unit cell to be the minimal one,
$a_{\text{edge}}=|\beta\bm{a}_{1}-\alpha\bm{a}_{2}|=\sqrt{\alpha^{2}+\beta^{2}}a$,
as shown in Fig. 7. If both $\alpha$ and $\beta$ are odd integers, $\sigma
a_{\text{edge}}\equiv 0$ (mod $e$), and otherwise $\sigma
a_{\text{edge}}\equiv e/2$ (mod $e$). Obviously, the edge can be charge
neutral when $\sigma a_{\text{edge}}\equiv 0$ (mod $e$). Even when $\sigma
a_{\text{edge}}\equiv e/2$ (mod $e$), after surface reconstruction with
doubling the surface period $a_{\text{edge}}^{\prime}=2a_{\text{edge}}$, we
get $\sigma a_{\text{edge}}^{\prime}\equiv 0$ (mod $e$), and the edge can be
charge neutral. In particular, when the system has edge states in the gap, and
the fractional edge charge is accommodated in the edge states, the edge states
are half-filled and it always has Peierls instability, meaning that the edge
becomes insulating via spontaneous lattice deformation with doubling the edge
unit cell at sufficiently low temperature.

Next, we consider the $C_{3}$-symmetric system with basis vectors of
$\bm{a}_{1}=(\frac{a}{2},\frac{\sqrt{3}}{2}a)$ and
$\bm{a}_{2}=(-\frac{a}{2},\frac{\sqrt{3}}{2}a)$, as shown in Fig. 3(a). In the
$C_{3}$-symmetric system, the bulk electric polarization is $\bm{P}=\bm{0}$,
$\frac{e}{3\Omega}(\bm{a}_{1}+\bm{a}_{2})$ or
$\frac{-e}{3\Omega}(\bm{a}_{1}+\bm{a}_{2})$ (mod
$\frac{e}{\Omega}\bm{R}$)PhysRevB.99.245151 ; PhysRevB.102.165120 ;
PhysRevB.48.4442 , where $\Omega=\frac{\sqrt{3}}{2}a^{2}$ is the area of the
bulk unit cell and $\bm{R}=m_{1}\bm{a}_{1}+m_{2}\bm{a}_{2}$
($m_{1},m_{2}\in\mathbb{Z}$) is a lattice vector. In the following, we
consider the case with $\bm{P}=\frac{\pm
e}{3\Omega}(\bm{a}_{1}+\bm{a}_{2})=(0,\frac{\pm 2e}{3a})$. The surface normal
vector $\bm{n}$ for the surface with Miller index ($\alpha,\beta$)
($\alpha,\beta\in\mathbb{Z}$, and $\alpha$ and $\beta$ are coprime) is defined
as follows:

$\displaystyle\bm{n}$
$\displaystyle=\frac{1}{\sqrt{\alpha^{2}-\alpha\beta+\beta^{2}}}\Bigg{(}\frac{\sqrt{3}}{2}(\alpha-\beta),\
\frac{1}{2}(\alpha+\beta)\Bigg{)}.$ (39)

Therefore, the edge charge in one edge unit cell is calculated as follows:

$\displaystyle\sigma
a_{\text{edge}}\equiv\bm{P}\cdot\bm{n}a_{\text{edge}}\equiv\pm\frac{e}{3}(\alpha+\beta)\
(\text{mod}\ e).$ (40)

Here, we used
$a_{\text{edge}}=|\beta\bm{a}_{1}-\alpha\bm{a}_{2}|=\sqrt{\alpha^{2}-\alpha\beta+\beta^{2}}a$.
If $\alpha+\beta\in 3\mathbb{Z}$, $\sigma a_{\text{edge}}\equiv 0$ (mod $e$),
and otherwise $\sigma a_{\text{edge}}\equiv e/3$ or $(2e)/3$ (mod $e$).
Obviously, edge can be charge neutral when $\sigma a_{\text{edge}}\equiv 0$
(mod $e$). Even when $\sigma a_{\text{edge}}\equiv e/3$ or $(2e)/3$ (mod $e$),
after surface reconstruction with tripling the surface period
$a_{\text{edge}}^{\prime\prime}=3a_{\text{edge}}$, we get $\sigma
a_{\text{edge}}^{\prime\prime}\equiv 0$ (mod $e$), and the edge can be charge
neutral.

Finally, we consider the $C_{6}$-symmetric system with basis vectors of
$\bm{a}_{1}=(a,0)$ and $\bm{a}_{2}=(-\frac{a}{2},\frac{\sqrt{3}}{2}a)$, as
shown in Fig. 3(c). In the $C_{6}$-symmetric system, the bulk electric
polarization is always $\bm{P}=\bm{0}$ (mod
$\frac{e}{\Omega}\bm{R}$)PhysRevB.99.245151 ; PhysRevB.102.165120 ;
PhysRevB.48.4442 , where $\Omega=\frac{\sqrt{3}}{2}a^{2}$ is the area of the
bulk unit cell and $\bm{R}=(m_{1}a-\frac{m_{2}}{2}a,\frac{\sqrt{3}}{2}m_{2}a)$
($m_{1},m_{2}\in\mathbb{Z}$) is a lattice vector. Therefore,
$\sigma_{\text{edge}}=\bm{P}\cdot\bm{n}$ is 0 for surfaces in any direction
(mod $e/a_{\text{edge}}$).

To summarize, in either case, even when $\bm{P}\neq 0$ in the
$C_{n}$-symmetric systems, $\sigma a_{\text{edge}}\equiv 0$ (mod $e$) can be
achieved with proper surface reconstruction thanks to the fractional
quantization of $\bm{P}$.

##  V Conclusion and Discussion

In the present paper, we derived the general formula for the quantized
fractional corner charge under the assumptions that the electronic states can
be described by Wannier orbitals and the edges are charge neutral. These
assumptions are more general than the previous studies, which assumed
vanishing bulk electric polarization, and these formulas are more general to
be used in different systems with/without time-reversal symmetry/spin-orbit
coupling. We expanded the scope of the corner charge formula by considering
more general surface conditions, such as surfaces with higher Miller index and
surfaces with surface reconstruction.

Our proof also shows that even when the positions of the nuclei and the
electronic states are largely modulated near the boundaries, the corner charge
is still quantized to fractional values as long as the system is
$C_{n}$-symmetric and the edge charge density is zero. Thus, unlike previous
theories, our theory also includes some insulators with non-vanishing bulk
polarization, and in such cases with $C_{3}$ or $C_{4}$ symmetries, we find
that the value of the corner charge depends on the Wyckoff position of the
center of the $C_{n}$-symmetric crystal.

We note that the corner charge formula has a simple form like Eq. (8) only
when the system has $C_{n}$ rotational symmetry. The corner charge in systems
without rotational symmetry is discussed in Refs. [PhysRevResearch.2.043012, ,
ren2020quadrupole, ].

###### Acknowledgements.

This work was supported by JSPS KAKENHI Grant Numbers JP18J23289, JP18H03678,
and JP20H04633.

##  Appendix A Relationship between the topological invariants
$[\Pi_{p}^{(n)}]$ and $n^{(e)}_{mX}$ in class A systems

Here, we discuss the relationship between the topological invariants
$[\Pi_{p}^{(n)}]$ and the occupation number for Wyckoff positions
$n^{(e)}_{mX}$, in the similar manner as in Refs. [PhysRevB.99.245151, ,
PhysRevResearch.1.033074, , PhysRevResearch.2.043131, ]. The topological
invariant $[\Pi_{p}^{(n)}]$ represents the difference of the number of states
with $C_{n}$ eigenvalue $e^{\frac{2(p-1)\pi i}{n}}$ for spinless systems, and
$e^{\frac{(2p-1)\pi i}{n}}$ for spinful systems, between the rotation-
invariant $\bm{k}$-points $\Pi$ and $\Gamma$. First, we assume that the
eigenstates can be described by well-localized Wannier orbitals. Thanks to the
discussion based on lattice homotopy, we can limit ourselves to Wannier
orbitals localized at one of the high-symmetry Wyckoff
positionsPhysRevLett.119.127202 . Thus, we calculate the topological
invariants for such Wannier configurations $W_{j}$PhysRevB.99.245151 ;
PhysRevResearch.1.033074 , and Tables 1, 2 and 3 summarize the values of the
topological number $[\Pi_{p}^{(n)}]$ for Wannier configurations for $C_{3}$-,
$C_{4}$\- and $C_{6}$-symmetric systems, respectively. Different from the
Refs. [PhysRevB.99.245151, ,PhysRevResearch.1.033074, ], we calculate the
general corner charge formula including the case where the ion position is not
limited to $1a$ Wyckoff positions. Since the eigenstates are expressed in
terms of Wannier orbitals, the topological invariant $\bm{\chi}$ for the
occupied bands is represented as the summation of the topological invariants
of the Wannier configurations $\bm{\chi}^{(j)}$:

$\displaystyle\bm{\chi}$
$\displaystyle=\sum_{j=1}^{M}\alpha_{j}\bm{\chi}^{(j)}\quad(\alpha_{j}\in\mathbb{Z}_{\geqslant}),$
(41) $\displaystyle=C_{\bm{\chi}}\bm{\alpha}.$ (42)

Here, $M$ is the number of Wannier configurations with nonzero topological
invariants,
$C_{\bm{\chi}}=(\bm{\chi}^{(1)},\bm{\chi}^{(2)},\cdots,\bm{\chi}^{(M)})$ and
$\bm{\alpha}=(\alpha_{1},\alpha_{2},\cdots,\alpha_{M})^{\text{T}}$, which
represents the numbers of occupied bands corresponding to the Wannier
configurations $W_{j}$. By definition, $n^{(e)}_{mX}$ is represented by the
sum of $\alpha_{j}$ belonging to the Wyckoff position $mX$. Since
$C_{\bm{\chi}}$ is generally not a square matrix and is not always invertible,
$\alpha_{j}$ itself cannot be uniquely determined from $\bm{\chi}$. However,
in a $C_{N}$ symmetric system, $n^{(e)}_{mX}$, expressed in terms of
$\alpha_{j}$, can be determined modulo $N$ ($N=3,4,6$) through direct
calculations, which is sufficient for the calculation of the corner charge.

Figure 8: (Color online) Highly-symmetric $\bm{k}$-points in (a) $C_{3}$, (b)
$C_{4}$, (c) $C_{6}$-symmetric systems. The points with same colors belong to
same $\bm{k}$-vector star, and related by $C_{n}$-operation. Here,
$\bm{b}_{1}$ and $\bm{b}_{2}$ are reciprocal lattice vector, and (a)
$\bm{b}_{1}=(\frac{\sqrt{3}}{2},\frac{1}{2})\frac{2}{\sqrt{3}a}$,
$\bm{b}_{2}=(-\frac{\sqrt{3}}{2},\frac{1}{2})\frac{2}{\sqrt{3}a}$, (b)
$\bm{b}_{1}=(1,0)\frac{1}{a}$, $\bm{b}_{2}=(0,1)\frac{1}{a}$, (c)
$\bm{b}_{1}=(\frac{\sqrt{3}}{2},\frac{1}{2})\frac{2}{\sqrt{3}a}$,
$\bm{b}_{2}=(0,1)\frac{2}{\sqrt{3}a}$.

###  A.1 $C_{3}$-symmetry

Label | WC | invariants
---|---|---
$j$ |  | $[K^{(3)}_{1}]$ | $[K^{(3)}_{2}]$ | $[K^{\prime(3)}_{1}]$ | $[K^{\prime(3)}_{2}]$
1 | $W_{1b|_{0}}^{(3)}$ | $-1$ | 1 | $-1$ | 0
2 | $W_{1b|_{2\pi/3}}^{(3)}$ | 0 | $-1$ | 1 | $-1$
3 | $W_{1b|_{4\pi/3}}^{(3)}$ | 1 | 0 | 0 | 1
4 | $W_{1c|_{0}}^{(3)}$ | $-1$ | 0 | $-1$ | 1
5 | $W_{1c|_{2\pi/3}}^{(3)}$ | 1 | $-1$ | 0 | $-1$
6 | $W_{1c|_{4\pi/3}}^{(3)}$ | 0 | 1 | 1 | 0
Table 1:  Wannier configurations (WC) with labels and their invariants with
$C_{3}$ symmetry in class A systems. The first column is the label for the
WCs. The Wannier configuration belonging to Wyckoff position $1a$ is omitted
in the table because all topological invariants are 0.

In $C_{3}$-symmetric systems, we need to consider six Wannier configurations
$W_{j}$ ($j=1,2,\cdots,6$) in Table 1. For example, $W_{1}=W_{1b|_{0}}^{(3)}$
represents a Wannier configuration localized at Wyckoff position $1b$ with a
$C_{3}$-eigenvalue of $e^{i\times 0}$. Likewise, $W_{mX|_{\theta}}^{(n)}$
denotes a Wannier configuration localized at Wyckoff position $mX$ with a
$C_{n}$-eigenvalue of $e^{i\theta}$. Table 1 is summarized as follows:

$\displaystyle\begin{pmatrix}[K^{(3)}_{1}]\\\ [K^{(3)}_{2}]\\\
[K^{\prime(3)}_{1}]\\\
[K^{\prime(3)}_{2}]\end{pmatrix}=\begin{pmatrix}-1&0&1&-1&1&0\\\
1&-1&0&0&-1&1\\\ -1&1&0&-1&0&1\\\
0&-1&1&1&-1&0\end{pmatrix}\begin{pmatrix}\alpha_{1}\\\ \alpha_{2}\\\ \vdots\\\
\alpha_{6}\end{pmatrix},$ (43)

Here, $K=\frac{2}{3}\bm{b}_{1}+\frac{1}{3}\bm{b}_{2}$ and
$K^{\prime}=\frac{1}{3}\bm{b}_{1}+\frac{2}{3}\bm{b}_{2}$ (see Fig. 8(a)). From
the definition, $n^{(e)}_{1b}=\alpha_{1}\\!+\\!\alpha_{2}\\!+\\!\alpha_{3}$
and $n^{(e)}_{1c}=\alpha_{4}\\!+\\!\alpha_{5}\\!+\\!\alpha_{6}$. From Eq.
(43), the following relation holds:

$\displaystyle[K^{(3)}_{1}]+[K^{\prime(3)}_{2}]$
$\displaystyle=-\alpha_{1}-\alpha_{2}+2\alpha_{3}$
$\displaystyle\equiv-(\alpha_{1}+\alpha_{2}+\alpha_{3})\ (\text{mod}\ 3),$
(44) $\displaystyle[K^{(3)}_{2}]+[K^{\prime(3)}_{1}]$
$\displaystyle=-\alpha_{4}-\alpha_{5}+2\alpha_{6}$
$\displaystyle\equiv-(\alpha_{4}+\alpha_{5}+\alpha_{6})\ (\text{mod}\ 3).$
(45)

Therefore, $n^{(e)}_{1b}$ and $n^{(e)}_{1c}$ (mod $3$) can be determined from
$[\Pi_{p}^{(n)}]$ as follows:

$\displaystyle n^{(e)}_{1b}=-[K_{1}^{(3)}]-[K_{2}^{\prime(3)}]\ (\text{mod}\
3),$ (46) $\displaystyle n^{(e)}_{1c}=-[K_{2}^{(3)}]-[K_{1}^{\prime(3)}]\
(\text{mod}\ 3).$ (47)

Then, $n^{(e)}_{1a}=\nu-n^{(e)}_{1b}-2n^{(e)}_{2c}$ is also determined from
$\nu$ and $[\Pi_{p}^{(n)}]$. Finally, the corner charge formulas are
represented as follows:

$\displaystyle Q_{c\\_1a}^{(3)}$
$\displaystyle\equiv\frac{|e|}{3}\Big{(}n_{1a}^{(\text{ion})}-\nu-[K_{1}^{(3)}]-[K_{2}^{(3)}]-[K_{1}^{\prime(3)}]-[K_{2}^{\prime(3)}]\Big{)}$
$\displaystyle\
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\
(\text{mod}\ e),$ (48) $\displaystyle Q_{c\\_1b}^{(3)}$
$\displaystyle\equiv\frac{|e|}{3}\Big{(}n_{1b}^{(\text{ion})}+[K_{1}^{(3)}]+[K_{2}^{\prime(3)}]\Big{)}\
(\text{mod}\ e),$ (49) $\displaystyle Q_{c\\_1c}^{(3)}$
$\displaystyle\equiv\frac{|e|}{3}\Big{(}n_{1c}^{(\text{ion})}+[K_{2}^{(3)}]+[K_{1}^{\prime(3)}]\Big{)}\
(\text{mod}\ e).$ (50)

###  A.2 $C_{4}$-symmetry

Label | WC | invariants
---|---|---
$j$ |  | $[X^{(2)}_{1}]$ | $[M^{(4)}_{1}]$ | $[M^{(4)}_{2}]$ | $[M^{(4)}_{3}]$
1 | $W_{1b|_{0}}^{(4)}$ | $-1$ | $-1$ | 0 | 1
2 | $W_{1b|_{\pi/2}}^{(4)}$ | 1 | 0 | $-1$ | 0
3 | $W_{1b|_{\pi}}^{(4)}$ | $-1$ | 1 | 0 | $-1$
4 | $W_{1b|_{3\pi/2}}^{(4)}$ | 1 | 0 | 1 | 0
5 | $W_{2c|_{0}}^{(4)}$ | $-1$ | $-1$ | 1 | $-1$
6 | $W_{2c|_{\pi}}^{(4)}$ | 1 | 1 | $-1$ | 1
Table 2: Wannier configurations (WC) with labels and their invariants with
$C_{4}$ symmetry in class A systems. The first column is the label for the
WCs. The Wannier configuration belonging to Wyckoff position $1a$ is omitted
in the table because all topological invariants are 0.

In $C_{4}$-symmetric systems, we need to consider six Wannier configurations
in Table 2. Table 2 is summarized as follows:

$\displaystyle\begin{pmatrix}[X^{(2)}_{1}]\\\ [M^{(4)}_{1}]\\\
[M^{(4)}_{2}]\\\ [M^{(4)}_{3}]\end{pmatrix}=\begin{pmatrix}-1&1&-1&1&-1&1\\\
-1&0&1&0&-1&1\\\ 0&-1&0&1&1&-1\\\
1&0&-1&0&-1&1\end{pmatrix}\begin{pmatrix}\alpha_{1}\\\ \alpha_{2}\\\ \vdots\\\
\alpha_{6}\end{pmatrix}.$ (51)

Here, $X=\frac{1}{2}\bm{b}_{1}$ and
$M=\frac{1}{2}\bm{b}_{1}+\frac{1}{2}\bm{b}_{2}$ (see Fig. 8(b)). From the
definition,
$n^{(e)}_{1b}=\alpha_{1}\\!+\\!\alpha_{2}\\!+\\!\alpha_{3}\\!+\\!\alpha_{4}$
and $n^{(e)}_{2c}=\alpha_{5}\\!+\\!\alpha_{6}$. From Eq. (51), the following
relation holds:

$\displaystyle[X^{(2)}_{1}]\\!-\\!\frac{3}{2}[M^{(4)}_{1}]\\!+\\!\frac{1}{2}[M^{(4)}_{3}]$
$\displaystyle=\alpha_{1}+\alpha_{2}-3\alpha_{3}+\alpha_{4}$
$\displaystyle\equiv\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4}\ (\text{mod}\
4),$ (52) $\displaystyle[M^{(4)}_{1}]+[M^{(4)}_{3}]$
$\displaystyle=-2\alpha_{5}+2\alpha_{6}$ $\displaystyle\equiv
2(\alpha_{5}+\alpha_{6})\ (\text{mod}\ 4),$ (53)

Therefore, $n^{(e)}_{1b}$ (mod $4$) and $n^{(e)}_{1c}$ (mod $2$) can be
determined from $[\Pi_{p}^{(n)}]$ as follows:

$\displaystyle
n^{(e)}_{1b}=[X_{1}^{(2)}]-\frac{3}{2}[M_{1}^{(4)}]+\frac{1}{2}[M_{3}^{(4)}]\
(\text{mod}\ 4),$ (54) $\displaystyle
n^{(e)}_{2c}=\frac{1}{2}[M_{1}^{(4)}]+\frac{1}{2}[M_{3}^{(4)}]\ (\text{mod}\
2).$ (55)

Then, $n^{(e)}_{1a}=\nu-n^{(e)}_{1b}-2n^{(e)}_{2c}$ is also determined from
$\nu$ and $[\Pi_{p}^{(n)}]$. Finally, the corner charge formulas are
summarized as follows:

$\displaystyle Q_{c\\_1a}^{(4)}$
$\displaystyle\equiv\frac{|e|}{4}\Big{(}n_{1a}^{(\text{ion})}-\nu+[X_{1}^{(2)}]-\frac{1}{2}[M_{1}^{(4)}]+\frac{3}{2}[M_{3}^{(4)}]\Big{)}$
$\displaystyle\
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\
(\text{mod}\ e),$ (56) $\displaystyle Q_{c\\_1b}^{(4)}$
$\displaystyle\equiv\frac{|e|}{4}\Big{(}n_{1b}^{(\text{ion})}-[X_{1}^{(2)}]+\frac{3}{2}[M_{1}^{(4)}]-\frac{1}{2}[M_{3}^{(4)}]\Big{)}$
$\displaystyle\
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\
(\text{mod}\ e).$ (57)

###  A.3 $C_{6}$-symmetry

Label | WC | invariants
---|---|---
$j$ |  | $[M^{(2)}_{1}]$ | $[K^{(3)}_{1}]$ | $[K^{(3)}_{2}]$
1 | $W_{2b|_{0}}^{(6)}$ | 0 | $-2$ | 1
2 | $W_{2b|_{2\pi/3}}^{(6)}$ | 0 | 1 | 1
3 | $W_{2b|_{-2\pi/3}}^{(6)}$ | 0 | 1 | $-2$
4 | $W_{3c|_{0}}^{(6)}$ | $-2$ | 0 | 0
5 | $W_{3c|_{\pi}}^{(6)}$ | 2 | 0 | 0
Table 3:  Wannier configurations (WC) with labels and their invariants with
$C_{6}$ symmetry in class A systems. The first column is the label for the
WCs. The Wannier configuration belonging to Wyckoff position $1a$ is omitted
in the table because all topological invariants are 0.

In $C_{6}$-symmetric systems, we need to consider five Wannier configurations
in Table 3. Table 3 is summarized as follows:

$\displaystyle\begin{pmatrix}[M^{(2)}_{1}]\\\ [K^{(3)}_{1}]\\\
[K^{(3)}_{2}]\end{pmatrix}=\begin{pmatrix}0&0&0&-2&2\\\ -2&1&1&0&0\\\
1&1&-2&0&0\end{pmatrix}\begin{pmatrix}\alpha_{1}\\\ \alpha_{2}\\\ \vdots\\\
\alpha_{5}\end{pmatrix}.$ (58)

Here, $M=\frac{1}{2}\bm{b}_{1}$ and
$K=\frac{1}{3}\bm{b}_{1}+\frac{1}{3}\bm{b}_{2}$ (see Fig. 8(c)). From the
definition of $\alpha_{j}$, $n^{(e)}_{2b}=\alpha_{1}+\alpha_{2}+\alpha_{3}$
and $n^{(e)}_{3c}=\alpha_{4}+\alpha_{5}$. From Eq. (58), the following
relation holds:

$\displaystyle[K_{1}^{(3)}]$
$\displaystyle=-2\alpha_{1}+\alpha_{2}+\alpha_{3}$
$\displaystyle\equiv\alpha_{1}+\alpha_{2}+\alpha_{3}\ (\text{mod}\ 3),$ (59)
$\displaystyle\frac{1}{2}[M_{1}^{(2)}]$ $\displaystyle=-\alpha_{5}+\alpha_{6}$
$\displaystyle\equiv\alpha_{5}+\alpha_{6}\ (\text{mod}\ 2).$ (60)

Therefore, $n^{(e)}_{2b}$ (mod 3) and $n^{(e)}_{3c}$ (mod 2) can be determined
from $[\Pi_{p}^{(n)}]$ as follows:

$\displaystyle n^{(e)}_{2b}$ $\displaystyle=[K_{1}^{(3)}]\ (\text{mod}\ 3),$
(61) $\displaystyle n^{(e)}_{3c}$ $\displaystyle=\frac{1}{2}[M_{1}^{(2)}]\
(\text{mod}\ 2).$ (62)

Then, $n^{(e)}_{1a}=\nu-2n^{(e)}_{2b}-3n^{(e)}_{3c}$ is also determined from
$\nu$ and $[\Pi_{p}^{(n)}]$. Finally, the corner charge formula is summarized
as follows:

$\displaystyle Q_{c\\_1a}^{(6)}$
$\displaystyle\equiv\frac{|e|}{6}\Big{(}n_{1a}^{(\text{ion})}-\nu+2[K_{1}^{(3)}]+\frac{3}{2}[M_{1}^{(2)}]\Big{)}\
(\text{mod}\ e).$ (63)

##  Appendix B Relationship between the topological invariants
$[\Pi_{p}^{(n)}]$ and the $n^{(e)}_{mX}$ in class AI systems

In class AI systems, as compared to class A systems, several Wannier
configurations are connected by the time-reversal symmetry $\Theta$
($\Theta^{2}=1$). Then, the Wannier configurations are changed as shown in
Tables 4, 5 and 6 for $C_{3}$, $C_{4}$ and $C_{6}$ symmetric systems,
respectively.

###  B.1 $C_{3}$-symmetry

Label | WC | invariants
---|---|---
$j$ |  | $[K^{(3)}_{1}]$ | $[K^{(3)}_{2}]$ | $[K^{\prime(3)}_{1}]$ | $[K^{\prime(3)}_{2}]$
1 | $W_{1b|_{0}}^{(3)}$ | $-1$ | 1 | $-1$ | 0
2 | $W_{1b|_{2\pi/3}}^{(3)}\oplus W_{1b|_{4\pi/3}}^{(3)}$ | 1 | $-1$ | 1 | 0
3 | $W_{1c|_{0}}^{(3)}$ | $-1$ | 0 | $-1$ | 1
4 | $W_{1c|_{2\pi/3}}^{(3)}\oplus W_{1c|_{4\pi/3}}^{(3)}$ | 1 | 0 | 1 | $-1$
Table 4:  Wannier configurations (WC) with labels and their invariants with
$C_{3}$ symmetry in class AI systems. The first column is the label for the
WCs. The Wannier configuration belonging to Wyckoff position $1a$ is omitted
in the table because all topological invariants are 0.

In $C_{3}$-symmetric systems, we need to consider four sets of Wannier
configurations in Table 4. Table 4 is summarized as follows:

$\displaystyle\begin{pmatrix}[K^{(3)}_{1}]\\\ [K^{(3)}_{2}]\\\
[K^{\prime(3)}_{1}]\\\
[K^{\prime(3)}_{2}]\end{pmatrix}=\begin{pmatrix}-1&1&-1&1\\\ 1&-1&0&0\\\
-1&1&-1&1\\\ 0&0&1&-1\end{pmatrix}\begin{pmatrix}\alpha_{1}\\\ \alpha_{2}\\\
\alpha_{3}\\\ \alpha_{4}\end{pmatrix}.$ (64)

From the definition, $n^{(e)}_{1b}=\alpha_{1}\\!+\\!2\alpha_{2}$ and
$n^{(e)}_{1c}=\alpha_{3}\\!+\\!2\alpha_{4}$. From Eq. (64), the following
relations hold:

$\displaystyle[K^{(3)}_{1}]$ $\displaystyle=[K^{\prime(3)}_{1}],$ (65)
$\displaystyle[K^{(3)}_{2}]+[K^{\prime(3)}_{2}]$
$\displaystyle=-[K^{(3)}_{1}],$ (66) $\displaystyle[K^{\prime(3)}_{2}]$
$\displaystyle=\alpha_{1}-\alpha_{2}$
$\displaystyle\equiv\alpha_{1}+2\alpha_{2}\ (\text{mod}\ 3),$ (67)
$\displaystyle[K^{(3)}_{2}]$ $\displaystyle=\alpha_{3}-\alpha_{4}$
$\displaystyle\equiv\alpha_{3}+2\alpha_{4}\ (\text{mod}\ 3).$ (68)

Therefore, $n^{(e)}_{1b}$ and $n^{(e)}_{1c}$ (mod $3$) can be determined from
$\bm{\chi}$ as follows:

$\displaystyle n^{(e)}_{1b}=[K^{\prime(3)}_{2}]\ (\text{mod}\ 3),$ (69)
$\displaystyle n^{(e)}_{1c}=[K^{(3)}_{2}]\ (\text{mod}\ 3).$ (70)

Then,
$n^{(e)}_{1a}=\nu-([K^{(3)}_{2}]+[K^{\prime(3)}_{2}])=\nu+[K^{(3)}_{1}]$. The
corner charge formulas are represented as follows:

$\displaystyle Q_{c\\_1a}^{(3)}$
$\displaystyle\equiv\frac{|e|}{3}\Big{(}n_{1a}^{(\text{ion})}-\nu-[K^{(3)}_{1}]\Big{)}\
(\text{mod}\ e),$ (71) $\displaystyle Q_{c\\_1b}^{(3)}$
$\displaystyle\equiv\frac{|e|}{3}\Big{(}n_{1b}^{(\text{ion})}-[K^{\prime(3)}_{2}]\Big{)}\
(\text{mod}\ e),$ (72) $\displaystyle Q_{c\\_1c}^{(3)}$
$\displaystyle\equiv\frac{|e|}{3}\Big{(}n_{1c}^{(\text{ion})}-[K^{(3)}_{2}]\Big{)}\
(\text{mod}\ e).$ (73)

Our result in Eq. (71) is identical with Eq. (17) in the main text. In
particular, Eq. (17) (Eq. (71)) is reduced to the result of Ref.
[PhysRevB.99.245151, ] (i.e., Eq. (20) in the main text) provided (i) all the
ions are located at $1a$. i.e., $n_{1a}^{(\text{ion})}=\nu$, and (ii) the bulk
electric polarization is zero. Under these assumptions, from Eq. (35) and
$n_{1b}^{(\text{ion})}=n_{1c}^{(\text{ion})}=0$,

$\displaystyle q_{b}+2q_{c}=-|e|(n^{(e)}_{1b}+2n^{(e)}_{1c})\equiv 0\
(\text{mod}\ 3e).$ (74)

This means that $n^{(e)}_{1b}\equiv n^{(e)}_{1c}$ (mod 3), i.e.,

$\displaystyle[K^{\prime(3)}_{2}]\equiv[K^{(3)}_{2}]\ (\text{mod}\ 3).$ (75)

Then, from Eq. (66),

$\displaystyle[K^{(3)}_{2}]\equiv[K^{(3)}_{1}]\ (\text{mod}\ 3).$ (76)

Therefore, Eq. (17) is reduced to Eq. (20) under the assumption that
$n_{1a}^{(\text{ion})}=\nu$ and vanishing polarization.

###  B.2 $C_{4}$-symmetry

Label | WC | invariants
---|---|---
$j$ |  | $[X^{(2)}_{1}]$ | $[M^{(4)}_{1}]$ | $[M^{(4)}_{2}]$ | $[M^{(4)}_{3}]$
1 | $W_{1b|_{0}}^{(4)}$ | $-1$ | $-1$ | 0 | 1
2 | $W_{1b|_{\pi}}^{(4)}$ | $-1$ | 1 | 0 | $-1$
3 | $W_{1b|_{\pi/2}}^{(4)}\oplus W_{1b|_{-\pi/2}}^{(4)}$ | 2 | 0 | 0 | 0
4 | $W_{2c|_{0}}^{(4)}$ | $-1$ | $-1$ | 1 | $-1$
5 | $W_{2c|_{\pi}}^{(4)}$ | 1 | 1 | $-1$ | 1
Table 5:  Wannier configurations (WC) with labels and their invariants with
$C_{4}$ symmetry in class AI systems. The first column is the label for the
WCs. The Wannier configuration belonging to Wyckoff position $1a$ is omitted
in the table because all topological invariants are 0.

In $C_{4}$-symmetric systems, we need to consider five sets of Wannier
configurations in Table 5. Table 5 is summarized as follows:

$\displaystyle\begin{pmatrix}[X^{(2)}_{1}]\\\ [M^{(4)}_{1}]\\\
[M^{(4)}_{2}]\\\ [M^{(4)}_{3}]\end{pmatrix}=\begin{pmatrix}-1&-1&2&-1&1\\\
-1&1&0&-1&1\\\ 0&0&0&1&-1\\\
1&-1&0&-1&1\end{pmatrix}\begin{pmatrix}\alpha_{1}\\\ \alpha_{2}\\\ \vdots\\\
\alpha_{5}\end{pmatrix}.$ (77)

From the definition, $n^{(e)}_{1b}=\alpha_{1}+\alpha_{2}+2\alpha_{3}$ and
$n^{(e)}_{2c}=\alpha_{4}+\alpha_{5}$. From Eq. (77), the following relation
holds:

$\displaystyle[X^{(2)}_{1}]-2[M^{(4)}_{1}]-[M^{(4)}_{2}]$
$\displaystyle=\alpha_{1}-3\alpha_{2}+2\alpha_{3}$
$\displaystyle\equiv\alpha_{1}+\alpha_{2}+2\alpha_{3}\ (\text{mod}\ 4),$ (78)
$\displaystyle[M^{(4)}_{2}]$ $\displaystyle=\alpha_{4}-\alpha_{5}$
$\displaystyle\equiv\alpha_{4}+\alpha_{5}\ (\text{mod}\ 2).$ (79)

Therefore, $n^{(e)}_{1b}$ (mod $4$) and $n^{(e)}_{1c}$ (mod $2$) can be
determined from $[\Pi_{p}^{(n)}]$ as follows:

$\displaystyle n^{(e)}_{1b}=[X^{(2)}_{1}]-2[M^{(4)}_{1}]-[M^{(4)}_{2}]\
(\text{mod}\ 4),$ (80) $\displaystyle n^{(e)}_{2c}=[M_{2}^{(4)}]\ (\text{mod}\
2).$ (81)

Then, $n^{(e)}_{1a}=\nu-n^{(e)}_{1b}-2n^{(e)}_{2c}$ is also determined from
$\nu$ and $[\Pi_{p}^{(n)}]$. Finally, the corner charge formulas are
represented as follows:

$\displaystyle Q_{c\\_1a}^{(4)}$
$\displaystyle\equiv\frac{|e|}{4}\Big{(}n_{1a}^{(\text{ion})}-\nu+[X_{1}^{(2)}]-2[M_{1}^{(4)}]+[M_{2}^{(4)}]\Big{)}$
$\displaystyle\
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\
(\text{mod}\ e),$ (82) $\displaystyle Q_{c\\_1b}^{(4)}$
$\displaystyle\equiv\frac{|e|}{4}\Big{(}n_{1b}^{(\text{ion})}-[X^{(2)}_{1}]+2[M^{(4)}_{1}]+[M^{(4)}_{2}]\Big{)}$
$\displaystyle\
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\
(\text{mod}\ e).$ (83)

This Eq. (82) is the same as Eq. (18) in the main text. In particular, when
$n_{1a}^{(\text{ion})}=\nu$ and vanishing polarization, Eq. (18) is reduced to
Eq. (21). Under these assumptions, from Eq. (33) and
$n_{1b}^{(\text{ion})}=n_{2c}^{(\text{ion})}=0$,

$\displaystyle q_{b}+q_{c}=-|e|(n^{(e)}_{1b}+n^{(e)}_{2c})\equiv 0\
(\text{mod}\ 2e).$ (84)

This means that $n^{(e)}_{1b}+n^{(e)}_{2c}\equiv 0$ (mod 2), i.e.,

$\displaystyle[X^{(2)}_{1}]-2[M^{(4)}_{1}]\equiv[X^{(2)}_{1}]\equiv 0\
(\text{mod}\ 2).$ (85)

Therefore, Eq. (18) is reduced to Eq. (21) under the assumption that
$n_{1a}^{(\text{ion})}=\nu$ and vanishing polarization.

###  B.3 $C_{6}$-symmetry

Label | WC | invariants
---|---|---
$j$ |  | $[M^{(2)}_{1}]$ | $[K^{(3)}_{1}]$ | $[K^{(3)}_{2}]$
1 | $W_{2b|_{0}}^{(6)}$ | 0 | $-2$ | 1
2 | $W_{2b|_{2\pi/3}}^{(6)}\oplus W_{2b|_{-2\pi/3}}^{(6)}$ | 0 | 2 | $-1$
3 | $W_{3c|_{0}}^{(6)}$ | $-2$ | 0 | 0
4 | $W_{3c|_{\pi}}^{(6)}$ | 2 | 0 | 0
Table 6:  Wannier configurations (WC) with labels and their invariants with
$C_{6}$ symmetry in class AI systems. The first column is the label for the
WCs. The Wannier configuration belonging to Wyckoff position $1a$ is omitted
in the table because all topological invariants are 0.

In $C_{6}$-symmetric systems, we need to consider four sets of Wannier
configurations in Table 6. Table 6 is summarized as follows:

$\displaystyle\begin{pmatrix}[M^{(2)}_{1}]\\\ [K^{(3)}_{1}]\\\
[K^{(3)}_{2}]\end{pmatrix}=\begin{pmatrix}0&0&-2&2\\\ -2&2&0&0\\\
1&-1&0&0\end{pmatrix}\begin{pmatrix}\alpha_{1}\\\ \alpha_{2}\\\ \alpha_{3}\\\
\alpha_{4}\end{pmatrix}.$ (86)

From the definition, $n^{(e)}_{2b}=\alpha_{1}+2\alpha_{2}$ and
$n^{(e)}_{3c}=\alpha_{3}+\alpha_{4}$. From Eq. (86), the following relations
hold:

$\displaystyle[K_{1}^{(3)}]$ $\displaystyle=-2\alpha_{1}+2\alpha_{2}$
$\displaystyle\equiv\alpha_{1}+2\alpha_{2}\ (\text{mod}\ 3),$ (87)
$\displaystyle\frac{1}{2}[M_{1}^{(2)}]$ $\displaystyle=-\alpha_{3}+\alpha_{4}$
$\displaystyle\equiv\alpha_{3}+\alpha_{4}\ (\text{mod}\ 2).$ (88)

Therefore, $n^{(e)}_{2b}$ (mod 3) and $n^{(e)}_{3c}$ (mod 2) can be determined
from $[\Pi_{p}^{(n)}]$ as follows:

$\displaystyle n^{(e)}_{2b}$ $\displaystyle=[K_{1}^{(3)}]\ (\text{mod}\ 3),$
(89) $\displaystyle n^{(e)}_{3c}$ $\displaystyle=\frac{1}{2}[M_{1}^{(2)}]\
(\text{mod}\ 2).$ (90)

Then, $n^{(e)}_{1a}=\nu-2n^{(e)}_{2b}-3n^{(e)}_{3c}$ is also determined from
$\nu$ and $[\Pi_{p}^{(n)}]$. Finally, the corner charge formula is summarized
as follows:

$\displaystyle Q_{c\\_1a}^{(6)}$
$\displaystyle\equiv\frac{|e|}{6}\Big{(}n_{1a}^{(\text{ion})}-\nu+2[K_{1}^{(3)}]+\frac{3}{2}[M_{1}^{(2)}]\Big{)}\
(\text{mod}\ e),$ (91)

which is the same as Eq. (19) in the main text.

Because both $[M^{(2)}_{1}]$ and $[K^{(3)}_{1}]$ are even numbers from Eq.
(86), Eq. (19) is equivalent to Eq. (22), provided $n^{(e)}_{1a}=\nu$.

##  Appendix C Relationship between the topological invariants
$[\Pi_{p}^{(n)}]$ and the $n^{(e)}_{mX}$ in class AII systems

In class AII systems, several Wannier configurations are connected by the
time-reversal symmetry $\Theta$ ($\Theta^{2}=-1$). Wannier configurations
acquire double degeneracy due to the Kramers theorem. Then, the Wannier
configurations are changed as shown in Tables 7, 8, 9.

Label | WC | invariants
---|---|---
$j$ |  | $[K^{(3)}_{1}]$ | $[K^{(3)}_{2}]$ | $[K^{\prime(3)}_{1}]$ | $[K^{\prime(3)}_{2}]$
1 | $2\times W_{1b|_{\pi}}^{(3)}$ | 0 | $-2$ | 2 | $-2$
2 | $W_{1b|_{\pi/3}}^{(3)}\oplus W_{1b|_{-\pi/3}}^{(3)}$ | 0 | 1 | $-1$ | 1
3 | $2\times W_{1c|_{\pi}}^{(3)}$ | 2 | $-2$ | 0 | $-2$
4 | $W_{1c|_{\pi/3}}^{(3)}\oplus W_{1c|_{-\pi/3}}^{(3)}$ | $-1$ | 1 | 0 | 1
Table 7:  Wannier configurations (WC) with labels and their invariants with
$C_{3}$ symmetry in class AII systems. The first column is the label for the
WCs. The Wannier configuration belonging to Wyckoff position $1a$ is omitted
in the table because all topological invariants are 0.

In $C_{3}$-symmetric systems, we need to consider four sets of Wannier
configurations in Table 7. Table 7 is summarized as follows:

$\displaystyle\begin{pmatrix}[K^{(3)}_{1}]\\\ [K^{(3)}_{2}]\\\
[K^{\prime(3)}_{1}]\\\
[K^{\prime(3)}_{2}]\end{pmatrix}=\begin{pmatrix}2&-1&0&0\\\ -2&1&-2&1\\\
0&0&2&-1\\\ -2&1&-2&1\end{pmatrix}\begin{pmatrix}\alpha_{1}\\\ \alpha_{2}\\\
\alpha_{3}\\\ \alpha_{4}\end{pmatrix}.$ (92)

From the definition, $n^{(e)}_{1b}=2(\alpha_{1}\\!+\\!\alpha_{2})$ and
$n^{(e)}_{1c}=2(\alpha_{3}\\!+\\!\alpha_{4})$. From Eq. (92), the following
relations hold:

$\displaystyle[K^{(3)}_{2}]$ $\displaystyle=[K^{\prime(3)}_{2}],$ (93)
$\displaystyle[K^{(3)}_{1}]+[K^{\prime(3)}_{1}]$
$\displaystyle=-[K^{(3)}_{2}],$ (94) $\displaystyle[K^{(3)}_{1}]$
$\displaystyle=2\alpha_{1}-\alpha_{2}$ $\displaystyle\equiv
2(\alpha_{1}+\alpha_{2})\ (\text{mod}\ 3),$ (95)
$\displaystyle[K^{\prime(3)}_{1}]$ $\displaystyle=2\alpha_{3}-\alpha_{4}$
$\displaystyle\equiv 2(\alpha_{3}+\alpha_{4})\ (\text{mod}\ 3).$ (96)

Therefore, $n^{(e)}_{1b}$ and $n^{(e)}_{1c}$ (mod $3$) can be determined from
$\bm{\chi}$ as follows:

$\displaystyle n^{(e)}_{1b}=[K^{(3)}_{1}]\ (\text{mod}\ 3),$ (97)
$\displaystyle n^{(e)}_{1c}=[K^{\prime(3)}_{1}]\ (\text{mod}\ 3).$ (98)

Then,
$n^{(e)}_{1a}=\nu-([K^{(3)}_{1}]+[K^{\prime(3)}_{1}])=\nu+[K^{(3)}_{2}]$. The
corner charge formulas are represented as follows:

$\displaystyle Q_{c\\_1a}^{(3)}$
$\displaystyle\equiv\frac{|e|}{3}\Big{(}n_{1a}^{(\text{ion})}-\nu-[K^{(3)}_{2}]\Big{)}\
(\text{mod}\ e),$ (99) $\displaystyle Q_{c\\_1b}^{(3)}$
$\displaystyle\equiv\frac{|e|}{3}\Big{(}n_{1b}^{(\text{ion})}-[K^{(3)}_{1}]\Big{)}\
(\text{mod}\ e),$ (100) $\displaystyle Q_{c\\_1c}^{(3)}$
$\displaystyle\equiv\frac{|e|}{3}\Big{(}n_{1c}^{(\text{ion})}-[K^{\prime(3)}_{1}]\Big{)}\
(\text{mod}\ e).$ (101)

Label | WC | invariants
---|---|---
$j$ |  | $[X^{(2)}_{1}]$ | $[M^{(4)}_{1}]$ | $[M^{(4)}_{2}]$ | $[M^{(4)}_{3}]$
1 | $W_{1b|_{\pi/4}}^{(4)}\oplus W_{1b|_{-\pi/4}}^{(4)}$ | 0 | $-1$ | 1 | 1
2 | $W_{1b|_{3\pi/4}}^{(4)}\oplus W_{1b|_{-3\pi/4}}^{(4)}$ | 0 | 1 | $-1$ | $-1$
Table 8:  Wannier configurations (WC) with labels and their invariants with
$C_{4}$ symmetry in class AII systems. The first column is the label for the
WCs. The Wannier configurations belonging to Wyckoff positions $1a$ and $2c$
are omitted in the table because all topological invariants are 0.

In $C_{4}$-symmetric systems, we need to consider two sets of Wannier
configurations in Table 8. Table 8 is summarized as follows:

$\displaystyle\begin{pmatrix}[X^{(2)}_{1}]\\\ [M^{(4)}_{1}]\\\
[M^{(4)}_{2}]\\\ [M^{(4)}_{3}]\end{pmatrix}=\begin{pmatrix}0&0\\\ -1&1\\\
1&-1\\\ 1&-1\end{pmatrix}\begin{pmatrix}\alpha_{1}\\\
\alpha_{2}\end{pmatrix}.$ (102)

From the definition of $\alpha_{j}$,
$n^{(e)}_{1b}=2(\alpha_{1}\\!+\\!\alpha_{2})$ and $n^{(e)}_{2c}\equiv 0$ (mod
4). From Eq. (102), the following relations hold:

$\displaystyle 2[M^{(4)}_{1}]=2(-\alpha_{1}+\alpha_{2})$ $\displaystyle\equiv
2(\alpha_{1}+\alpha_{2})\ (\text{mod}\ 4),$ (103) $\displaystyle n^{(e)}_{1b}$
$\displaystyle\equiv 2[M^{(4)}_{1}]\ (\text{mod}\ 4).$ (104)

The corner charge formulas are summarized as follows:

$\displaystyle Q_{c\\_1a}^{(4)}$
$\displaystyle\equiv\frac{|e|}{4}\Big{(}n_{1a}^{(\text{ion})}-\nu+2[M^{(4)}_{1}]\Big{)}\
(\text{mod}\ e),$ (105) $\displaystyle Q_{c\\_1b}^{(4)}$
$\displaystyle\equiv\frac{|e|}{4}\Big{(}n_{1b}^{(\text{ion})}-2[M^{(4)}_{1}]\Big{)}\
(\text{mod}\ e).$ (106)

In $C_{6}$-symmetric systems, we need to consider two sets of Wannier
configurations in Table 9. Table 9 is summarized as follows:

$\displaystyle\begin{pmatrix}[M^{(2)}_{1}]\\\ [K^{(3)}_{1}]\\\
[K^{(3)}_{2}]\end{pmatrix}=\begin{pmatrix}0&0\\\ 2&-1\\\
-4&2\end{pmatrix}\begin{pmatrix}\alpha_{1}\\\ \alpha_{2}\end{pmatrix}.$ (107)

From the definition, $n^{(e)}_{2b}=2(\alpha_{1}+\alpha_{2})$. From Eq. (107),
the following relations hold:

$\displaystyle[K^{(3)}_{1}]=2\alpha_{1}-\alpha_{2}$ $\displaystyle\equiv
2(\alpha_{1}+\alpha_{2})\ (\text{mod}\ 3),$ (108) $\displaystyle n^{(e)}_{2b}$
$\displaystyle\equiv[K^{(3)}_{1}]\ (\text{mod}\ 3).$ (109)

The corner charge formula is summarized as follows:

$\displaystyle Q_{c\\_1a}^{(6)}$
$\displaystyle\equiv\frac{|e|}{6}\Big{(}n_{1a}^{(\text{ion})}-\nu+2[K^{(3)}_{1}]\Big{)}\
(\text{mod}\ e).$ (110)

Label | WC | invariants
---|---|---
$j$ |  | $[M^{(2)}_{1}]$ | $[K^{(3)}_{1}]$ | $[K^{(3)}_{2}]$
1 | $2\times W_{2b|_{\pi}}^{(6)}$ | 0 | 2 | $-4$
2 | $W_{2b|_{\pi/3}}^{(6)}\oplus W_{2b|_{-\pi/3}}^{(6)}$ | 0 | $-1$ | 2
Table 9:  Wannier configurations (WC) with labels and their invariants with
$C_{6}$ symmetry in class AII systems. The first column is the label for the
WCs. The Wannier configurations belonging to Wyckoff positions $1a$ and $3c$
are omitted in the table because all topological invariants are 0.

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