# Glide-symmetric topological crystalline insulator phase in a nonprimitive
lattice

Heejae Kim1 and Shuichi Murakami1,2 1Department of Physics, Tokyo Institute of
Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo, 152-8551, Japan
2TIES, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo,
152-8551, Japan

###### Abstract

We study the topological crystalline insulator phase protected by the glide
symmetry, which is characterized by the $Z_{2}$ topological number. In the
present paper, we derive a formula for the $Z_{2}$ topological invariant
protected by glide symmetry in a nonprimitive lattice, from that in a
primitive lattice. We establish a formula for the glide-$Z_{2}$ invariant for
the space group No. $\bm{{\it 9}}$ with glide symmetry in the base-centered
lattice, by folding the Brillouin zone into that of the primitive lattice
where the formula for the glide-$Z_{2}$ invariant is known. The formula is
written in terms of integrals of the Berry curvatures and Berry phases in the
$k$-space. We also derive a formula of the glide-$Z_{2}$ invariant when the
inversion symmetry is added, and the space group becomes $\bm{{\it 15}}$. This
reduces the formula into the Fu-Kane-like formula, expressed in terms of the
irreducible representations at high-symmetry points in $k$ space. We also
construct these topological invariants by the layer-construction approach, and
the results completely agree with those from the $k$-space approach.

##  I Introduction

The importance of topology in condensed matter physics was recognized through
the discovery of quantum Hall effect [1]. Triggered by the proposals of
topological insulators (TIs) [2, 3], which emerge in the presence of time-
reversal symmetry, material realization of topological crystalline materials
is one of the hottest topics in modern condensed matter physics. To realize
topological crystalline materials, topological crystalline insulators (TCIs)
are promising candidates because of the richness and complexity of
crystallography ensuring topological properties of TCIs, such as robust
surface states as long as certain crystal symmetry is preserved. TCIs have
been theoretically proposed, and their properties and candidate materials have
been well studied [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
20, 21]. In particular, mirror-symmetric TCI phases have attracted interest
because they present the first material realization in IV-VI semiconductor
SnTe, Pb1-xSnxTe and Pb1-xSnxSe [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32,
33, 34, 35], which are supported by the ARPES (angle-resolved photoemission
spectroscopy) experiments in the same materials.

TCI phases including the nonsymmorphic symmetry represent unique topological
properties beyond topological insulating materials so far [36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49]. In particuar, in magnetic systems, the
Chern number and some other topological indices are given by integrals in
$k$-space, and such topological indices are comprehensively given by the
$K$-theory [50]. Very recently, antiferromagnetic TIs are confirmed in the
MnBi2nTe3n+1 family of materials [51, 52, 53, 54, 55, 56, 57]. Nevertheless,
there are still few proposals for magnetic topological materials even though
the first topological phenomenon in condensed matter physics was discovered in
magnetic systems, namely the quantum Hall effect.

Distinct topological phases cannot be adiabatically deformed into each other
and we can diagnose whether the system is topologically trivial or nontrivial
thanks to topological invariants, such as Chern number for quantum Hall
systems, $Z_{2}$ topological invariant for TIs, mirror Chern number for
mirror-symmetric TCIs, and so on. Namely, topological invariants which are
integrals of wavefunctions encode topology of a given system, and two systems
with a different value of topological invariants cannot be continuously
deformed to each other. One can classify TCI phases in crystalline materials
by various methods, such as $K$-theory [50], topological quantum chemistry
[58], and symmetry-based indicators [41]. Even though several expressions of
important topological invariants are given in previous works, an explicit
formula for each topological invariant of TCI phases does not follow
immediately from these theories. In Ref. 59 the authors found expressions for
symmetry-based indicators for all the space groups for systems with time-
reversal symmetry, while it does not immediately tell us how the topological
invariants in different space groups are mutually related. Furthermore,
expressions of topological invariants for systems without time-reversal
symmetry have not been fully studied. In addition, some topological invariants
such as the Chern number, the mirror Chern number and the glide-$Z_{2}$
invariant are expressed as an integral over the $k$ space, and how they are
related with symmetry-based indicators should be studied separately for the
individual cases. Thus, there is much room for further investigation for each
topological invariant to apply material realization.

In this paper, we study the $Z_{2}$ TCI phase protected by glide-symmetry for
magnetic systems in nonprimitive lattices. This phase is characterized by the
glide-$Z_{2}$ topological invariant, and its formula in primitive lattices is
discussed in Refs. 42, 43, 60. Because the glide-$Z_{2}$ invariant is given in
terms of integrals along the glide-invariant planes in momentum space,
nonprimitive lattice will alter the glide-invariant planes, and the formula of
topological invariants should be altered. We derive a formula of the
glide-$Z_{2}$ invariant in terms of integrals in nonprimitive lattice systems,
namely in the space group (SG) $\bm{{\it 9}}\ (Cc)$. Furthermore, we show that
when inversion symmetry is added and the SG becomes $\bm{{\it 15}}\ (C2/c)$,
the formula is expressed in terms of the irreducible representations at high-
symmetry points. Together with a half of the Chern number along the glide
plane, the topological invariants are $Z_{2}\times Z_{2}$, which is identical
with the results of the symmetry-based indicator discussed in Refs. 61, 62. We
also formulate a layer construction for these two space groups and find a
perfect agreement with our theory above. From this results we conclude that
the only topological invariants for these SGs are the glide-$Z_{2}$ invariant
and the Chern number. Since the glide symmetry is contained in many space
groups, this glide-symmetric TCI phase, ensured by the $Z_{2}$ topological
invariant, can exist in many space groups as well. Therefore, understanding of
this $Z_{2}$ topological phase with glide symmetry will deepen our knowledge
for topological phases in magnetic systems. Our discussion in this paper is
similarly applicable both to spinful and spinless systems. In some parts of
the text we limit our discussion to spinless cases for simplicity, and
extension to spinful systems is straightforward.

In this paper we study SGs $\bm{{\it 9}}\ (Cc)$ and $\bm{{\it 15}}\ (C2/c)$.
To be more precise, we consider the type-I magnetic SGs (MSGs), whose symbols
are the same as those for the corresponding crystallographic SGs, because they
do not contain time-reversal. The SGs $\bm{{\it 9}}$ and $\bm{{\it 15}}$ refer
to the type-I MSGs #9.37 ($Cc$) and #15.85 ($C2/c$) in the notation of Bilbao
Crystallographic Server [63]

The present paper is organized as follows. In Sec. II, we derive the formula
for the glide-$Z_{2}$ topological invariant in a nonprimitive lattice, both
without and with the inversion symmetry, corresponding to the SGs $\bm{{\it
9}}$ and $\bm{{\it 15}}$, respectively. This formula is also derived from the
layer construction in Sec. III, and the results in Sec. II and Sec. III
completely agree with each other. We conclude the paper in Sec. IV.

##  II $Z_{2}$ topological invariant for glide-symmetric magnetic systems with
a nonprimitive lattice

In this section, we focus on glide-symmetric magnetic systems in a
nonprimitive lattice. The formula of the glide-$Z_{2}$ invariant in the
previous paper [60] does not apply to systems in a nonprimitive lattice,
unlike those in a primitive lattice. Therefore, the formula for the
glide-$Z_{2}$ invariant should be altered. In this section, we briefly review
the glide-$Z_{2}$ invariant and explain our motivation. Then, we rewrite the
glide-$Z_{2}$ invariant in a nonprimitive lattice with glide symmetry only.

###  II.1 $Z_{2}$ topological invariant for magnetic glide-symmetric systems

In magnetic glide-symmetric systems, we can define the $Z_{2}$ topological
invariant to characterize a topological crystalline insulator phase ensured by
glide symmetry [42, 43]. Let us consider the glide operator

$\hat{G}_{y}=\\{M_{y}|(c/2)\hat{\bm{z}}\\},$ (1)

where $M_{y}$ is the reflection operator with respect to the $zx$ plane, and
$\hat{\bm{z}}$ is the unit vector along the $z$ axis. Here we define $a$, $b$
and $c$ to be the lattice constants along the $x$, $y$, and $z$ axes,
respectively, The SG $\bm{{\it 7}}$ ($Pc$) is generated by the glide (1) and
translations $\\{E|a\hat{\bm{x}}\\}$, $\\{E|b\hat{\bm{y}}\\}$, and
$\\{E|c\hat{\bm{z}}\\}$ in a monoclinic primitive lattice, where $E$ is the
identity operation. When we add another translation
$\\{E|\frac{1}{2}(a\hat{\bm{x}}+b\hat{\bm{y}})\\}$, the space group becomes
$\bm{{\it 9}}$ ($Cc$) in a monoclinic base-centered (nonprimitive) lattice.
Henceforth, we set the lattice constants $a$, $b$, and $c$ to be unity for
simplicity, unless otherwise specified.

In SG $\bm{{\it 7}}$, we can define the glide-$Z_{2}$ invariant $\nu$ for such
systems as [42, 43, 60]

$\displaystyle\nu$
$\displaystyle=\frac{1}{2\pi}\left[\int_{\mathcal{A}}F_{xy}dk_{x}dk_{y}+\int_{{\mathcal{B}}}F_{zx}^{-}dk_{z}dk_{x}-\int_{{\mathcal{C}}}F_{zx}^{+}dk_{z}dk_{x}\right]$
$\displaystyle\quad-\frac{1}{\pi}\left(\gamma^{+}_{A^{\prime}BA}+\gamma^{-}_{EDE^{\prime}}\right)\pmod{2},$
(2)

where $\mathcal{A},\mathcal{B},$ and $\mathcal{C}$ are the regions shown in
Fig. 1(a). The superscript $(\pm)$ indicates the glide sectors with
eigenvalues of the glide operator $g_{\pm}=\pm i^{f}e^{-ik_{z}c/2}$ in
spinless ($f=0$) and spinful ($f=1$) systems. For simplicity, we limit
ourselves to the spinless cases, and it is straightforward to extend our
discussion to spinful cases. $F_{ij}^{\pm}(\bm{k})$ is the corresponding Berry
curvatures for these subspaces

$F_{ij}^{\pm}(\bm{k})=\partial_{k_{i}}A_{j}^{\pm}(\bm{k})-\partial_{k_{j}}A_{i}^{\pm}(\bm{k}),$
(3)

and $\gamma^{\pm}_{\lambda}$ is the corresponding Berry phase within the
$g_{\pm}$-sector along the path $\lambda$,

$\gamma^{\pm}_{\lambda}(\bm{k})=\oint_{\lambda}\bm{A}^{\pm}(\bm{k})\cdot
d\bm{k},$ (4)

where $\bm{A}^{\pm}(\bm{k})$ is the corresponding Berry connections given by
the eigenstates $\ket{u^{\pm}_{n{\bm{k}}}}$:

$\bm{A}^{\pm}(\bm{k})\equiv i\sum_{n:\
\text{occ.}}\bra{u^{\pm}_{n{\bm{k}}}}\nabla_{\bm{k}}\ket{u^{\pm}_{n{\bm{k}}}},$
(5)

where $n$ runs over occupied band indices. The term
${{F}}_{ij}\equiv{{F}}_{ij}^{+}+{{F}}_{ij}^{-}$ represents the total Berry
curvature for the occupied states. We note that the Berry-phase terms
$\gamma^{+}_{A^{\prime}BA}$, $\gamma^{-}_{EDE^{\prime}}$ in Eq. (2) are along
the paths $A^{\prime}\rightarrow B\rightarrow A$, and $E\rightarrow
D\rightarrow E^{\prime}$, with their the gauge taken to be periodic along the
contours, meaning that the phases of the wavefunctions at $A$ and $A^{\prime}$
are the same, and likewise those at $E$ and $E^{\prime}$. We also note that
the definition of the glide-$Z_{2}$ invariant in Eq. (2) is changed from that
in Refs. 42, 43 for convenience in comparison with layer construction, as
discussed in detail in Ref. 60.

Figure 1: (Color online) Unit cells and Brillouin zones for the SGs $\bm{{\it
7}}$ and $\bm{{\it 9}}$. (a) Upper half of the Brillouin zone in SGs $\bm{{\it
7}}$, $\bm{{\it 13}}$ and $\bm{{\it 14}}$. $\Gamma$, A, E, Y, Z, B, C and D
denote the high-symmetry points in SG $\bm{{\it 13}}$ and SG $\bm{{\it 14}}$,
while SG $\bm{{\it 7}}$ has no high-symmetry points, but has high-symmetry
planes $k_{y}=0,\pi$. (b) Illustration of the relation between the primitive
lattice in $\bm{{\it 7}}$ and the base-centered lattice in $\bm{{\it 9}}$ in
the $xy$ plane. $U_{1}$ and $U_{2}$ denote two unit cells of the base-centered
lattice $\bm{{\it 9}}$ and $U_{1}+U_{2}$ is a unit cell of the primitive
lattice $\bm{{\it 7}}$. (c) Upper half of the Brillouin zone of the monoclinic
base-centered lattice in $\bm{{\it 9}}$ ($Cc$) and $\bm{{\it 15}}$ ($C2/c$).
$\Gamma,Y,V(V^{\prime}),A,M,$ and $L(L^{\prime})$ denote the high-symmetry
points in $\bm{{\it 15}}$. The corresponding Brillouin zone of the monoclinic
primitive lattice in SGs $\bm{{\it 7,\ 13}}$ and $\bm{{\it 14}}$ is depicted
by dotted lines. (d) Comparison between the BZ of the primitive lattice in SGs
$\bm{{\it 7}}$, $\bm{{\it 13}}$, and $\bm{{\it 14}}$ (solid line) and that of
the base-centered lattice in SGs $\bm{{\it 9}}$, and $\bm{{\it 15}}$ (broken
line).

###  II.2 Motivation

For the glide operator in Eq. (1), the Bloch Hamiltonian in momentum space
satisfies

$G_{y}\mathcal{H}(k_{x},k_{y},k_{z})G_{y}^{-1}=\mathcal{H}(k_{x},-k_{y},k_{z}),$
(6)

where $G_{y}$ is the operator representing $\hat{G}_{y}$. Thus, under the
glide operator $G_{y}$, points on the $k_{y}=0$ plane are invariant.
Meanwhile, whether or not points on the $k_{y}=\pi$ plane are invariant
depends on the lattices. In SG $\bm{{\it 7}}$, the planes $k_{y}=0$
($\mathcal{B}$) and $k_{y}=\pi$ ($\mathcal{C}$) are invariant under glide
symmetry, and therefore on these planes, gilde sectors can be defined. On the
other hand, in nonprimitive lattices like that in $\bm{{\it 9}}$ ($Cc$),
$k_{y}=\pi$ is not glide invariant unlike SG $\bm{{\it 7}}$, and one cannot
define glide sectors on this plane. Thus, it is neccessary to modify the
formula of the $Z_{2}$ topological invariant associated with the glide-
symmetry.

Let us compare $\bm{{\it 7}}$ ($Pc$) with $\bm{{\it 9}}$ ($Cc$). Both in
$\bm{{\it 7}}$ and in $\bm{{\it 9}}$, the only symmetry generator in addition
to the translations is the glide operation, Eq. (1). While $\bm{{\it 7}}$ is
defined on a primitive lattice, $\bm{{\it 9}}$ is defined on a $c$ base-
centered lattice. The primitive and reciprocal lattice vectors for $\bm{{\it
7}}$ and $\bm{{\it 9}}$ are summarized in Table 1. A half of the Brillouin
zone for $\bm{{\it 7}}$ and $\bm{{\it 9}}$ is shown in Fig. 1 (a) and (c),
respectively. In $\bm{{\it 7}}$, every $\bm{k}$ point on the $k_{y}=\pi$ plane
is invariant under the glide operation,

$\bm{k}=g_{y}\bm{k}\pmod{\bm{b}_{i}^{Pc}}.$ (7)

where $\bm{b}_{i}^{Pc}$ $(i=1,2,3)$ are the reciprocal lattice vectors for
$\bm{{\it 7}}$ ($Pc$) given in Table 1 and $g_{y}\bm{k}$ is the wavevector
transformed from $\bm{k}$ by $\hat{G}_{y}$. Therefore, the Bloch Hamiltonian
on the $k_{y}=\pi$ plane can be block-diagonalized into two blocks with glide
eigenvalues $g_{\pm}$, and eigenstates on the $k_{y}=\pi$ plane are classified
with respect to the glide sectors. On the other hand, $\bm{k}$ points in
$\bm{{\it 9}}$ on the $k_{y}=\pi$ plane are not invariant under the glide
operation, because $(0,2\pi,0)$ is not among the reciprocal vectors for
$\bm{{\it 9}}$ ($Cc$). Therefore, eigenstates on $k_{y}=\pi$ cannot be
associated with glide sectors. Thus, we need to alter the formula of the
glide-$Z_{2}$ invariant for $\bm{{\it 9}}$ due to this nonprimitive nature of
$\bm{{\it 9}}$.

Our strategy is the following. To construct a formula of the glide-$Z_{2}$
invariant for $\bm{{\it 9}}$, we double the unit cell of $\bm{{\it 9}}$ into
that of $\bm{{\it 7}}$ as shown in Fig. 1(b), leading to a folding of the BZ
of $\bm{{\it 9}}$ into that of $\bm{{\it 7}}$ (see Fig. 1(d)). Note that both
the $k_{y}=\pi$ plane and $k_{y}=-\pi$ plane in $\bm{{\it 9}}$ are projected
onto the same $k_{y}=\pi$ plane in $\bm{{\it 7}}$ by folding the BZ. While the
two points $(k_{x},\pi,k_{z})$ and $(k_{x},-\pi,k_{z})$ are distinct in
$\bm{{\it 9}}$, they are mapped to the same $\bm{k}$ point in the BZ of
$\bm{{\it 7}}$. Thus, by doubling the unit cell, we can relate the
wavefunctions between $\bm{{\it 7}}$ and $\bm{{\it 9}}$, and we address how
the terms in Eq. (2) (for SG $\bm{{\it 7}}$) on the $k_{y}=\pi$ plane are
described in terms of the wavefunctions in $\bm{{\it 9}}$. Henceforth,
$\ket{u_{n\bm{k}}}$ and $\ket{\tilde{u}_{n\bm{k}}}$ denote a wavefunction in
$\bm{{\it 9}}$ and in $\bm{{\it 7}}$, and $\mathcal{{O}}$ and
$\mathcal{\tilde{O}}$ denotes an operator in $\bm{{\it 9}}$ and in $\bm{{\it
7}}$, respectively, throughout the present section.

Table 1: Primitive lattice and reciprocal lattice vectors in monoclinic primitive and base-centered lattices. Here we set the lattice constants to be unity for simplicity. $\begin{matrix}\mathrm{space\ groups}\\\ \mathrm{(lattice)}\end{matrix}$ | $\bm{a}_{1},\bm{a}_{2},\bm{a}_{3}$ | $\bm{b}_{1},\bm{b}_{2},\bm{b}_{3}$
---|---|---
$\begin{matrix}\bm{{\it 7}}\ (Pc)\\\ \mathrm{(Monoclinic}\\\ \mathrm{primitive)}\end{matrix}$ | $(100),(010),(001)$ | $2\pi(100),2\pi(010),2\pi(001)$
$\begin{matrix}\bm{{\it 9}}\ (Cc)\\\ \mathrm{(Monoclinic}\\\ \mathrm{base}\mbox{-}\mathrm{centered)}\end{matrix}$ | $(\frac{1}{2}\frac{1}{2}0),(\bar{\frac{1}{2}}\frac{1}{2}0),(001)$ | $2\pi(110),2\pi(\bar{1}10),2\pi(001)$

###  II.3 Derivation of the formula of the $Z_{2}$ topological invariant for
$\bm{{\it 9}}$

We start by constructing the Hamiltonian for $\bm{{\it 7}}$ from that of
$\bm{{\it 9}}$. As depicted in Fig. 1(b), one unit cell of $\bm{{\it 7}}$
consists of two unit cells $U_{1}$ and $U_{2}$ of $\bm{{\it 9}}$, which are
related by a translation vector
$\bm{a}_{1}=\bm{a}_{1}^{Cc}=(\frac{1}{2},\frac{1}{2},0)$. Based on this point,
we can divide the Bloch Hamiltonian for $\bm{{\it 9}}$, $\mathcal{H}(\bm{k})$,
into two Hermitian parts:

$\mathcal{H}(\bm{k})=\mathcal{H}_{1}(\bm{k})+\mathcal{H}_{2}(\bm{k})$ (8)

The Hamiltonian $\mathcal{H}_{1}(\bm{k})$ denotes intra-unit-cell hoppings
from $U_{\alpha}$ to $U_{\alpha}$ itself where $\alpha=1,2$, while
$\mathcal{H}_{2}(\bm{k})$ denotes inter-unit-cell hoppings between $U_{1}$ and
$U_{2}$. Due to this division of the Hamiltonian, $\mathcal{H}_{1}(\bm{k})$
and
$\mathcal{H}^{\prime}_{2}(\bm{k})\equiv\mathcal{H}_{2}(\bm{k})e^{i\bm{k}\cdot\bm{a}_{1}}$
(not $\mathcal{H}_{2}(\bm{k})$ itself) satisfy the periodicity for $\bm{{\it
7}}$ i.e., $\mathcal{H}_{1}(\bm{k}+\bm{b}_{i}^{Pc})=\mathcal{H}_{1}(\bm{k})$
and
$\mathcal{H}^{\prime}_{2}(\bm{k}+\bm{b}_{i}^{Pc})=\mathcal{H}^{\prime}_{2}(\bm{k})$
$(i=1,2,3)$ Then, in terms of the SG $\bm{{\it 7}}$, the corresponding
Hamiltonian and the glide operator are given by

$\displaystyle\tilde{\mathcal{H}}(\bm{k})$
$\displaystyle=\begin{pmatrix}\mathcal{H}_{1}(\bm{k})&\mathcal{H}_{2}(\bm{k})e^{-i\bm{k}\cdot\bm{a}_{1}}\\\
\mathcal{H}_{2}(\bm{k})e^{i\bm{k}\cdot\bm{a}_{1}}&\mathcal{H}_{1}(\bm{k})\end{pmatrix},$
(9) $\displaystyle\tilde{G}_{y}(\bm{k})$
$\displaystyle=\begin{pmatrix}G_{y}(\bm{k})&\\\
&G_{y}(\bm{k})e^{-ik_{y}}\end{pmatrix},$ (10)

where $G_{y}(\bm{k})$ is the glide operator for $\bm{{\it 9}}$ satisfying
${G}_{y}(\bm{k}){\mathcal{H}}(\bm{k}){G}_{y}^{-1}(\bm{k})={\mathcal{H}}(g_{y}\bm{k})$.
Therefore, we can easily see

$\tilde{G}_{y}(\bm{k})\tilde{\mathcal{H}}(\bm{k})\tilde{G}_{y}^{-1}(\bm{k})=\tilde{\mathcal{H}}(g_{y}\bm{k}).$
(11)

Thus we have established the Hamiltonian and the glide operator for $\bm{{\it
7}}$ out of those for $\bm{{\it 9}}$.

In order to derive a formula of the glide-$Z_{2}$ invariant for $\bm{{\it
9}}$, we need to write down wavefunctions in $\bm{{\it 7}}$ in terms of those
in $\bm{{\it 9}}$. In particular, on the $k_{y}=\pi$ plane, we need
wavefunctions in each of the glide sectors. Let $\ket{u(\bm{k})}$ denote the
wavefunction in $\bm{{\it 9}}$ with the energy $E_{\bm{k}}$:
$\mathcal{H}(\bm{k})\ket{u(\bm{k})}=E_{\bm{k}}\ket{u(\bm{k})}$. Here the
wavefunction $u(\bm{k})$ is periodic with respect to the Brillouin zone of the
SG $\bm{{\it 9}}$. Then, the wavefunction for $\bm{{\it 7}}$ is given by

$\ket{\tilde{u}(\bm{k})}=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\
e^{i\bm{k}\cdot\bm{a}_{1}}\end{pmatrix}\ket{u(\bm{k})},$ (12)

and one can simply see that
$\tilde{\mathcal{H}}(\bm{k})\ket{\tilde{u}(\bm{k})}=E_{\bm{k}}\ket{\tilde{u}(\bm{k})}$.

Since in $\bm{{\it 7}}$, the $k_{y}=\pi$ plane is glide invariant, we can now
construct wavefunctions in each glide sector in $\bm{{\it 7}}$ on the
$k_{y}=\pi$ plane. The glide symmetry guarantees

$G_{y}(\bm{k})\ket{u(\bm{k})}=e^{i\chi(\bm{k})}\ket{u(g_{y}\bm{k})},$ (13)

where $\chi(\bm{k})$ is a real function. By applying $G_{y}(g_{y}\bm{k})$, we
get

$e^{-ik_{z}}=e^{i\chi(\bm{k})+i\chi(g_{y}\bm{k})},$ (14)

where we used $G_{y}(g_{y}\bm{k})G_{y}(\bm{k})=e^{-ik_{z}}$ for spinless
systems. Here, we skip a discussion on spinful systems, where similar
discussions can be straightforwardly developed by using the corresponding
relation $G_{y}(g_{y}\bm{k})G_{y}(\bm{k})=-e^{-ik_{z}}$ instead. In $\bm{{\it
9}}$, $\ket{u(k_{x},\pi,k_{z})}$ and
$G_{y}(k_{x},\pi,k_{z})\ket{u(k_{x},\pi,k_{z})}\propto\ket{u(k_{x},-\pi,k_{z})}$
reside at different wavevectors. On the other hand, when we go to $\bm{{\it
7}}$, they correspond to the same wavevector upon halving the BZ, and their
linear combinations give wavefunctions within each glide sector. By applying
$\tilde{G}_{y}(\bm{k})$ onto $\ket{\tilde{u}(\bm{k})}$ where
$\bm{k}=\bm{k}_{\pi}\equiv(k_{x},\pi,k_{z})$, we have

$\tilde{G}_{y}(\bm{k}_{\pi})\ket{\tilde{u}(\bm{k}_{\pi})}=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\
-ie^{ik_{x}/2}\end{pmatrix}e^{i\chi(\bm{k}_{\pi})}\ket{u(g_{y}\bm{k}_{\pi})}.$
(15)

Thus, from the degenerate eigenstates $\ket{\tilde{u}(\bm{k}_{\pi})}$ and
$\tilde{G}_{y}(\bm{k})\ket{\tilde{u}(\bm{k}_{\pi})}$, we can establish
eigenstates in $g_{\pm}$ sectors (i.e. $\tilde{G}_{y}=\pm\exp(-ik_{z}/2)$) by
linear combinations:

$\displaystyle\ket{\tilde{u}^{\pm}(k_{x},\pi,k_{z})}$
$\displaystyle=\frac{1}{2}\left[\begin{pmatrix}1\\\
ie^{ik_{x}/2}\end{pmatrix}\ket{u(k_{x},\pi,k_{z})}\pm\begin{pmatrix}1\\\
-ie^{ik_{x}/2}\end{pmatrix}e^{i\chi(k_{x},\pi,k_{z})+ik_{z}/2}\ket{u(k_{x},-\pi,k_{z})}\right]$
$\displaystyle\quad\times\exp\left[-i\left(\chi(\pi,\pi,k_{z})+\frac{k_{z}}{2}\right)\frac{k_{x}}{2\pi}\right]\times\begin{cases}1&(g_{+}\
\mathrm{sector})\\\ e^{-ik_{x}/2}&(g_{-}\ \mathrm{sector})\end{cases},$ (16)

by a straightforward calculation. The phase factor
$e^{-i(\chi(\pi,\pi,k_{z})+k_{z}/2)\frac{k_{x}}{2\pi}}$
$(e^{-i(\chi(\pi,\pi,k_{z})+k_{z}/2)\frac{k_{x}}{2\pi}}e^{ik_{x}/2})$ is
required to guarantee periodicity of the wavefunctions between $k_{x}=\pi$ and
$k_{x}=-\pi$ in $\bm{{\it 7}}$, which has been imposed in the Berry-phase term
in Eq. (2)

We plug in these eigenstates into the formula of the glide-$Z_{2}$ invariant
of $\bm{{\it 7}}$ in Eq. (2), with its details summarized in Appendix A. As a
consequence, the glide-$Z_{2}$ invariant for $\bm{{\it 9}}$ is written as
follows:

$\displaystyle\nu$
$\displaystyle=\frac{1}{2\pi}\left[\int_{\mathcal{A}^{\prime}}F_{xy}dk_{x}dk_{y}+\int_{\mathcal{B}^{\prime}}\left(\frac{1}{2}F_{zx}-F_{zx}^{+}\right)dk_{z}dk_{x}\right]$
$\displaystyle\quad+\frac{1}{\pi}\left(\gamma_{M_{1}M_{2}}-\gamma^{+}_{M_{1}M_{3}}\right).$
(17)

where $\mathcal{A}^{\prime}=\\{\bm{k}|k_{z}=-\pi,\ 0<k_{y},\
k_{x}+k_{y}<2\pi,\ k_{y}-k_{x}<2\pi\\}$ and
$\mathcal{B}^{\prime}=\\{\bm{k}|k_{y}=0,\ -2\pi<k_{x}<2\pi,\
-\pi<k_{z}<\pi\\}$ are depicted in Fig. 1(c) and $M_{1}(-2\pi,0,-\pi)$,
$M_{2}(0,2\pi,-\pi)$, and $M_{3}(2\pi,0,-\pi)$, Note that $\nu$ is gauge
invariant as expected.

###  II.4 Topological invariants for $\bm{{\it 15}}$

Next, we consider what happens when the inversion symmetry is added. There is
only one way to add inversion symmetry $\hat{I}=\\{I|\bm{0}\\}$ into $\bm{{\it
9}}$, and the space group becomes $\bm{{\it 15}}$. In this subsection, we
derive the formula of the glide-$Z_{2}$ invariant for $\bm{{\it 15}}$ in two
ways; one is to exploit a sewing matrix introduced in Ref. 4, and the other is
to fold the Brillouin zone of $\bm{{\it 15}}$ into that of $\bm{{\it 13}}$ and
use the formula for the glide-$Z_{2}$ invariant in $\bm{{\it 13}}$, where
$\bm{{\it 13}}$ consists of the glide symmetry and inversion symmetry in a
primitive lattice. Through these two different methods, we will show that the
glide-$Z_{2}$ invariant $\nu$ for $\bm{{\it 15}}$ is given by

$(-1)^{\nu}=\prod_{i\in\mathrm{occ}}\zeta^{+}_{i}(\Gamma)\xi_{i}(\mathrm{V})\frac{\xi_{i}(\mathrm{Y})}{\zeta^{+}_{i}(\mathrm{Y})},$
(18)

where $\zeta_{i}^{+}$ is a $C_{2}$ eigenvalue in the $g_{+}$ sector and
$\xi_{i}$ is an inversion parity for the $i$-th occupied state, and the
product runs over the occupied states. The high-symmetry points are specified
in Fig. 1(c). This formula applies both to spinless and spinful cases. In
spinless systems, this formula is rewritten as

$\nu=N_{\Gamma_{2}^{-}}(\Gamma)+N_{Y_{1}^{-}}(\mathrm{Y})+N_{V_{1}^{-}}(\mathrm{V}),$
(19)

where $N_{R}(P)$ is the number of occupied states at a high-symmetry point $P$
with an irrep $R$ of $\bm{{\it 15}}$ at $P$. The irreps of $\bm{{\it 15}}$ are
summarized in Appendix D.

In addition, we find that the Chern number $\mathcal{C}_{y}$ on the
$k_{x}$-$k_{z}$ plane defined by

$\displaystyle\mathcal{C}_{y}$
$\displaystyle=\frac{1}{2\pi}\int_{\mathcal{B}^{\prime}}F_{zx}dk_{z}dk_{x}$
(20)

can also be related with the irreps as

$\mathcal{C}_{y}\in
2\mathbb{Z},\quad(-1)^{\mathcal{C}_{y}/2}=\prod_{i\in\mathrm{occ}}\frac{\xi_{i}(\mathrm{Y})\xi_{i}(\mathrm{V})}{\xi_{i}(\mathrm{M})\xi_{i}(\mathrm{L})},$
(21)

both for spinful and spinless systems. It means that $\mathcal{C}_{y}$ is
always even, and the parity of $\mathcal{C}_{y}/2$ is known from the irreps of
the occupied states at high-symmetry points.

####  II.4.1 Approach by using sewing matrix

Let us start with symmetry consideration. Adding an inversion
$\hat{I}=\\{I|\bm{0}\\}$ with respect to the origin to $\bm{{\it 9}}$ leads
$\bm{{\it 15}}$. Then the $C_{2}$ rotational symmetry around the axis
$x=0,z=1/4$,

$\hat{C}_{2}=\hat{G}_{y}\hat{I}=\left\\{C_{2y}|\hat{\bm{z}}/2\right\\},$ (22)

is also added to symmetry operations. It satisfies the relation

$\hat{C}_{2}\hat{G}_{y}=\hat{G}_{y}\hat{C}_{2}\\{E|\hat{\bm{z}}\\}.$ (23)

We now derive a formula for the glide-$Z_{2}$ invariant for $\bm{{\it 15}}$
from that of $\bm{{\it 9}}$ in Eq. (17) with the help of the additional
symmetries. We will calculate $(-1)^{\nu}=e^{i\pi\nu}$ with $\nu$ given by Eq.
(17) term by term. We find $\int_{\mathcal{A}^{\prime}}F_{xy}dk_{x}dk_{y}=0$
from $C_{2}$ symmetry, and by using the sewing matrix [4, 60], we obtain

$e^{i\gamma_{M_{1}M_{2}}}=\prod_{i\in\mathrm{occ}}\frac{\xi_{i}(L)}{\xi_{i}(M)},$
(24)

from inversion symmetry. Since the glide sector is unchanged upon the $C_{2}$
rotation on the glide-invariant plane $k_{y}=0$, we have

$\exp\left[-\frac{i}{2}\int_{\mathcal{B}^{\prime}}F^{+}_{zx}dk_{z}dk_{x}-i\gamma^{+}_{M_{1}M_{3}}\right]=\prod_{i\in\mathrm{occ}}\frac{\zeta_{i}^{+}(\Gamma)}{\zeta_{i}^{+}(Y)},$
(25)

which is derived by folding the region $\mathcal{B}^{\prime}$ into half by the
$C_{2}$ rotation using the sewing matrix [4, 60].

Finally, we study the term of the Chern number
$\mathcal{C}_{y}=\frac{1}{2\pi}\int_{\mathcal{B}^{\prime}}F_{zx}dk_{z}dk_{x}$
in Eq. (17). The key is the fact that the Berry curvature is divergence free
in an insulating system; in other words, the divergence of the Berry curvature
can be nonzero only when the gap is closed. Thus, from the Gauss theorem, the
Chern number $\mathcal{C}_{y}$ on the $k_{y}=0$ plane is equal to the sum of
the integral of the Berry curvature over the planes
$\mathcal{D}_{1}=\\{(k-\pi,k+\pi,k_{z})|-\pi<k<\pi,-\pi<k_{z}<\pi\\}$ and
$\mathcal{D}_{2}=\\{(k+\pi,\pi-k,k_{z})|-\pi<k<\pi,-\pi<k_{z}<\pi\\}$, where
the rectangular regions $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ are depicted
in Fig. 2 with their normal vectors defined as
$\bm{n}_{1}=\frac{1}{\sqrt{2}}(-1,1,0)$ and
$\bm{n}_{2}=\frac{1}{\sqrt{2}}(1,1,0)$, respectively. Here we define the Chern
numbers $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ over the regions
$\mathcal{D}_{1}$ and $\mathcal{D}_{2}$, respectively by

$\displaystyle\mathcal{C}_{i}=\frac{1}{2\pi}\int_{\mathcal{D}_{i}}\
\bm{\Omega}_{\bm{k}}\cdot\bm{n}\ \ \ d^{2}\bm{k}\ (i=1,2),$ (26)

where $\bm{n}$ is the normal vector of the plane, and
$\bm{\Omega}_{\bm{k}}=(F_{yz},F_{zx},F_{xy})$ is the Berry curvature. Then, we
have

$\displaystyle\mathcal{C}_{y}=\frac{1}{2\pi}\int_{\mathcal{B}^{\prime}}F_{zx}dk_{x}dk_{z}$
$\displaystyle=\frac{1}{2\pi}\left(\int_{\mathcal{D}_{1}}+\int_{\mathcal{D}_{2}}\right)\bm{\Omega}_{\bm{k}}\cdot\bm{n}\
d^{2}\bm{k}$ $\displaystyle=\mathcal{C}_{1}+\mathcal{C}_{2}=2\mathcal{C}_{1},$
(27)

where we have used $\mathcal{C}_{1}=\mathcal{C}_{2}$ due to the $C_{2}$
rotational symmetry. Here, $\mathcal{C}_{1}\ (=\mathcal{C}_{2})$ is an
integer, with its parity given by

$\displaystyle e^{i\pi\mathcal{C}_{y}/2}=e^{i\pi\mathcal{C}_{1}}=$
$\displaystyle=\exp\left[\frac{i}{2}\int_{\mathcal{D}_{1}}\bm{\Omega}_{\bm{k}}\cdot\bm{n}\
d^{2}\bm{k}\right]=\prod_{i\in\mathrm{occ}}\frac{\xi_{i}(Y)\xi_{i}(V)}{\xi_{i}(M)\xi_{i}(L)},$
(28)

which is derived by folding the region $\mathcal{D}_{1}$ into half by the
inversion using the sewing matrix [4, 60]. Therefore, in the present case the
Chern number $\mathcal{C}_{y}$ on the glide plane is always an even integer.
This fact that $\mathcal{C}_{y}$ is an even integer also follows directly from
the fact that the product of the inversion parities at $A$, $M$, $Y$, and
$\Gamma$ on the $k_{y}=0$ planes being always $+1$ thanks to the compatibility
relation summarized in Table 4 in Appendix D, by noting that this product
gives $(-1)^{\mathcal{C}_{y}}$[64, 65].

Figure 2: Brillouin zone of $\bm{{\it 9}}$. (a) Upper half of the Brillouin
zone of $\bm{{\it 9}}$. (b) Schematic figure of the flux of the Berry
curvatures on the $k_{z}=(\mathrm{const.})$ plane denoted by red arrows.
$\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ denote the rectangular regions
$\mathcal{D}_{1}:\\{(k-\pi,k+\pi,k_{z})|-\pi<k<\pi,-\pi<k_{z}<\pi\\}$,
$\mathcal{D}_{2}:\\{(k+\pi,\pi-k,k_{z})|-\pi<k<\pi,-\pi<k_{z}<\pi\\}$,
respectively. Integrals of the Berry curvature on $\mathcal{D}_{1}$ and on
$\mathcal{D}_{2}$ are the same due to the twofold rotational symmetry
$C_{2z}$. Because the Berry curvature is divergence free in an insulating
system, its integral on $\mathcal{B}$ is equal to the sum of the those on
$\mathcal{D}_{1}$ and on $\mathcal{D}_{2}$.

As a consequence, in the presence of inversion symmetry, the glide-$Z_{2}$
invariant for $\bm{{\it 15}}$ is rewritten as

$\displaystyle(-1)^{\nu}$
$\displaystyle=\prod_{i\in\mathrm{occ}}\frac{\xi_{i}(L)}{\xi_{i}(M)}\frac{\zeta_{i}^{+}(\mathrm{\Gamma})}{\zeta_{i}^{+}(Y)}\frac{\xi_{i}(Y)\xi_{i}(V)}{\xi_{i}(M)\xi_{i}(L)}$
$\displaystyle=\prod_{i\in\mathrm{occ}}\zeta_{i}^{+}(\mathrm{\Gamma})\xi_{i}(V)\frac{\xi_{i}(Y)}{\zeta_{i}^{+}(Y)}.$
(29)

So far we have obtained this equation for spinless systems, and one can
similarly discuss spinful systems to get the same result Eq. (29). In
particular, in spinless systems, we have only to count the number of states
giving nontrivial contribution, whose irreps are $\Gamma_{2}^{-}$ at $\Gamma$,
$V_{1}^{-}$ at $V$, and $Y_{1}^{-}$ at $Y$. The irreps at the high-symmetry
points are summarized in Table 4 in Appendix D. Therefore, we eventually have
a formula of $\nu$ expressed in terms of irreps for $\bm{{\it 15}}$ as

$\nu=N_{\Gamma_{2}^{-}}(\Gamma)+N_{Y_{1}^{-}}(\mathrm{Y})+N_{V_{1}^{-}}(\mathrm{V}),$
(30)

which is the same as Eq. (19)

####  II.4.2 Folding of the Brillouin zone from $\bm{{\it 15}}$ onto $\bm{{\it
13}}$

Here we derive Eq. (19) for spinless cases, from the formula of the
glide-$Z_{2}$ invariant in $\bm{{\it 13}}$ we derived in Ref. 60,

$\nu=N_{\Gamma_{2}^{+}}(\Gamma_{13})+N_{Y_{2}^{+}}(Y_{13})+N_{Z_{2}^{-}}(Z_{13})+N_{C_{2}^{-}}(C_{13}),$
(31)

in which $N_{R}(P_{13})$ is the number of occupied states with an irreps $R$
for $\bm{{\it 13}}$ at a high-symmetry point $P_{13}$.

Systems in $\bm{{\it 15}}$ can be regarded as those in $\bm{{\it 13}}$ by
choosing the primitive lattice vectors to be
$\bm{a}_{1}=(1,0,0),\bm{a}_{2}=(0,1,0),\bm{a}_{3}=(0,0,1)$. Thereby, the unit
cell is doubled from that of $\bm{{\it 15}}$, and the BZ is folded by half, as
we discussed in the previous section. To distinguish the high-symmetry
$k$-points in $\bm{{\it 13}}$ from those in $\bm{{\it 15}}$, we put subscripts
“$\bm{{\it 13}}$” and “$\bm{{\it 15}}$” onto the symbols of the high-symmetry
points. On the $k_{z}=0$ plane, the high-symmetry points $\Gamma_{\it 15}$ and
$Y_{\it 15}$ in $\bm{{\it 15}}$ are projected onto $\Gamma_{\it 13}$ in
$\bm{{\it 13}}$, and $V_{\it 15}$ is projected onto $C_{\it 13}$. Similarly,
on the $k_{z}=\pi$ plane, $A_{\it 15}$ and $M_{\it 15}$ are projected onto
$B_{\it 13}$, and $L_{\it 15}$ is projected onto $E_{\it 13}$. Hence, the
irreps for $\Gamma_{\it 15}$ and $Y_{\it 15}$ ($A_{\it 15}$ and $M_{\it 15}$)
in $\bm{{\it 15}}$ should be directly associated with those for $\Gamma_{\it
13}$ ($B_{\it 13}$) in $\bm{{\it 13}}$. On the other hand, the case for
$V_{\it 15}$ ($L_{\it 15}$) is not simple because two inequivalent points
$V_{\it 15}$ and $V_{\it 15}^{\prime}$ ($L_{\it 15}$ and $L_{\it
15}^{\prime}$) in $\bm{{\it 15}}$ are mapped onto the same point in $\bm{{\it
13}}$ (Fig. 1(a)(c)).

Let us consider the state with the irrep $V_{1}^{+}$ at $V_{\it 15}$ point
$\ket{u_{V_{1}^{+}}(V_{\it 15})}$: ${I}\ket{u_{V_{1}^{+}}(V_{\it
15})}=+\ket{u_{V_{1}^{+}}(V_{\it 15})}$. By introducing a corresponding state
at $V^{\prime}_{15}$ given by $\ket{\phi(V_{\it
15}^{\prime})}\equiv{G}_{y}\ket{u_{V_{1}^{+}}(V_{\it 15})}$, we have

$\displaystyle{I}\ket{\phi(V_{\it
15}^{\prime})}={I}{G}_{y}\ket{u_{V_{1}^{+}}(V_{\it 15})}$
$\displaystyle={G}_{y}{I}\ket{u_{V_{1}^{+}}(V_{\it 15})}$
$\displaystyle={G}_{y}\ket{u_{V_{1}^{+}}(V_{\it 15})}$
$\displaystyle=\ket{\phi(V_{\it 15}^{\prime})}.$ (32)

Thus, the corresponding state at $V_{\it 15}^{\prime}$ is also characterized
by $V_{1}^{+}$. Note that $V_{\it 15}$ and $V_{\it 15}^{\prime}$ in $\bm{{\it
15}}$ are simultaneously projected onto $C_{\it 13}$ in $\bm{{\it 13}}$ in the
halved BZ and that the $C_{2}$ operation exchanges the states at $V_{\it 15}$
and that at $V_{\it 15}^{\prime}$, meaning that the character for the $C_{2}$
at $C_{\it 13}$ is zero. Therefore, the irrep $V_{1}^{+}(V_{1}^{-})$ at
$V_{\it 15}$ in $\bm{{\it 15}}$ corresponds to the irreps
$C_{1}^{+}+C_{2}^{+}\ (C_{1}^{-}+C_{2}^{-})$ at $C_{\it 13}$ in $\bm{{\it
13}}$.

In the same manner, we can find a similar relation between $L_{\it 15}$ in
$\bm{{\it 15}}$ and $E_{\it 13}$ in $\bm{{\it 13}}$. We consider a state
$\ket{u_{L_{1}^{+}}(L_{\it 15})}$ obeying the $L^{+}_{1}$ irrep at $L_{\it
15}$: ${I}\ket{u_{L_{1}^{+}}(L_{\it 15})}=+\ket{u_{L_{1}^{+}}(L_{\it 15})}$.
By introducing a corresponding state at $L^{\prime}_{15}$ given
by$\ket{\phi(L^{\prime}_{\it 15})}\equiv{G}_{y}\ket{u_{L_{1}^{+}}(L_{\it
15})}$, it follows that

${I}\ket{\phi(L_{\it 15}^{\prime})}=-{G}_{y}\ket{u_{L_{1}^{+}}(L_{\it
15})}=-\ket{\phi(L_{\it 15}^{\prime})}.$ (33)

Therefore, when the state at $L_{\it 15}$ is in the $L_{1}^{+}$ irrep, the
corresponding state at $L_{\it 15}^{\prime}$ is characterized by $L_{1}^{-}$,
and vice versa. As a result by folding the BZ by half, the states at $L_{\it
15}$ in $\bm{{\it 15}}$ are projected onto those with the 2D irrep $E_{1}$ at
$E_{\it 13}$ in $\bm{{\it 13}}$.

Next, in order to rewrite $N_{Y_{2}^{+}}(Y_{13})$ and $N_{Z_{2}^{-}}(Z_{13})$,
we investigate how the irreps at $Y_{\it 13}$ and $Z_{\it 13}$ in $\bm{{\it
13}}$ correspond to those in $\bm{{\it 15}}$. Let us start with $Y_{\it 13}$
in $\bm{{\it 13}}$. This point splits into two points $(\pm\pi,0,0)$ on the
high-symmetry plane $B_{\it 15}$ in $\bm{{\it 15}}$, which is invariant under
the glide operation but not under inversion and $C_{2}$ rotation. The irrep
$Y_{2}^{+}$ at $Y_{\it 13}$ corresponds to the irrep $B_{2}$ of $\bm{{\it
15}}$ at $B_{\it 15}$, corresponding to the $g_{-}$ sector, and we need to
find out the correponding irreps at a high-symmetry point in $\bm{{\it 15}}$.
Let us consider how it is compatible with irreps at $Y_{\it 15}$ In fact, from
a state $|u(\bm{k})\rangle$ following the irrep $B_{2}$ at
$\bm{k}=(k_{x},0,0)$, we can build another state
$|u^{\prime}(-\bm{k})\rangle|\equiv{I}|u(\bm{k})\rangle$ at $-\bm{k}$, which
is also in the same irrep $B_{2}$. Then the states
$|\psi_{\pm}\rangle=u(\bm{k})\rangle\pm{I}|u(\bm{k})$ has even (odd) parity.
Thus at $Y_{15}$, these states $|\psi_{+}\rangle$ and $|\psi_{-}\rangle$
follow the irreps $Y_{2}^{+}$ and $Y_{1}^{-}$ at $Y_{\it 15}$, respectively.
Therefore, we find

$N_{Y_{2}^{+}}(Y_{\it 13})=N_{B_{2}}(B_{\it 15})=N_{Y_{1}^{-}}(Y_{\it
15})+N_{Y_{2}^{+}}(Y_{\it 15}).$ (34)

Similarly, we obtain

$N_{Z_{2}^{-}}(Z_{\it 13})=N_{\Lambda_{2}}(\Lambda_{\it
15})=N_{Y_{2}^{+}}(Y_{\it 15})+N_{Y_{2}^{-}}(Y_{\it 15}).$ (35)

Here, We have used the fact that $Z_{\it 13}$ in $\bm{{\it 13}}$ is projected
onto $(0,\pm\pi,0)$ on the high-symmetry line $\Lambda_{\it 15}$ in $\bm{{\it
15}}$ preserving the $C_{2}$ rotational symmetry but not by inversion symmetry
and glide symmetry.

In summary, we express the glide-$Z_{2}$ invariant in $\bm{{\it 13}}$ in terms
of irreps for $\bm{{\it 15}}$ from the compatibility relations,

$\displaystyle\nu$ $\displaystyle\equiv N_{\Gamma_{2}^{+}}(\Gamma_{\it
13})+N_{Y_{2}^{+}}(Y_{\it 13})+N_{Z_{2}^{-}}(Z_{\it 13})+N_{C_{2}^{-}}(C_{\it
13})$ $\displaystyle\equiv N_{\Gamma_{2}^{+}}(\Gamma_{\it
15})+N_{Y_{2}^{+}}(\mathrm{Y}_{\it 15})+N_{Y_{1}^{-}}(\mathrm{Y}_{\it 15})$
$\displaystyle\hskip 56.9055pt+N_{Y_{2}^{-}}(\mathrm{Y}_{\it
15})+N_{V_{1}^{-}}(\mathrm{V}_{\it 15})$ $\displaystyle\equiv
N_{\Gamma_{2}^{-}}(\Gamma_{\it 15})+N_{Y_{1}^{-}}(\mathrm{Y}_{\it
15})+N_{V_{1}^{-}}(\mathrm{V}_{\it 15})\pmod{2},$ (36)

which is the same as Eq. (19). We have used the compatibility conditions for
$\bm{{\it 15}}$ summarized in the rightmost column in Table 4 in Appendix D.

###  II.5 Relationship with previous results

Both in $\bm{{\it 13}}$ and in $\bm{{\it 15}}$, the symmetry-based indicator
is ${Z}_{2}\times{Z}_{2}$ [61, 62]. By comparison with our results, we
conclude that one ${Z}_{2}$ is given by $\nu$ in Eq. (19), and the other
${Z}_{2}$ is in fact a half of the Chern number on the glide plane
$\mathcal{C}_{y}/2$, which is equal to $\mathcal{C}_{1}(=\mathcal{C}_{2})$.
Furthermore, the $E_{\infty}^{2,0}$ term of the $K$-theory classification [50]
is the ${Z}$. Since it should represent a topological invariant expressed as a
$k$-space 2D integral, this $Z$ topological invariant is
$\mathcal{C}_{y}/2(=\mathcal{C}_{1}=\mathcal{C}_{2})$.

By the way, in inversion-symmetric systems, one can define the ${Z}_{4}$
symmetry-based indicator $z_{4}$ [61, 62, 59], which is defined as a half of
the sum of the parity eigenvalues $(=\pm 1)$ over the occupied states over the
time-reversal invariant momenta. From our results on the relations between the
glide-$Z_{2}$ invariant in $\bm{{\it 13}}$ and that in $\bm{{\it 15}}$, we
immediately have [60]

$\nu\equiv\frac{z_{4}}{2}\pmod{2}.$ (37)

Here we note that $z_{4}$ is always an even integer in insulating systems [64,
65]. Therefore, since in the present paper we have assumed the system to be
insulating, the right hand side of Eq. (37) is always an integer. Thus, the
topological phase with a nontrivial glide-$Z_{2}$ invariant $\nu=1$ is also
the higher-order topological insulator phase with $z_{4}=2\pmod{4}$. We also
note that in systems with time-reversal symmetry, the formula of the $Z_{4}$
symmetry indicator has a simiar definition, just by replacing the sum of the
parity eigenvalues $(=\pm 1)$ over the occupied states by that over the
occupied Kramers pairs.

##  III Layer construction

In this section, we introduce a layer construction, which enables us to
construct topological invariants in a completely different way from the
$k$-space approach adopted in the previous section. We here construct
topological invariants for $\bm{{\it 9}}$ and $\bm{{\it 15}}$ by the layer
construction.

###  III.1 Invariants from the layer construction

One can introduce various real-space topological invariants by the LC,
similarly to Refs. 59, 60. In this LC for a SG, we introduce a set of planes,
which are located periodically to be compatible with lattice translation
symmetry. Here we consider each plane to be a 2D Chern insulator with a Chern
number equal to $+1$. Then for each of the SG operations, such as inversion,
rotation, and glide, we can distinguish whether the configurations of the
layers can be continuously trivialized or not, when the symmetry elements of
the SG except for the operation considered are ignored. Consequently, to
characterize nontrivial configurations of planes we introduce a topological
invariant for each SG operation.

To be more specific, we introduce a layer consisting of an infinite number of
planes

$\displaystyle(mnl;d)$
$\displaystyle=\left\\{\bm{r}|\bm{r}\cdot(m\bm{b}_{1}+n\bm{b}_{2}+l\bm{b}_{3})=2\pi(d+q),\
q\in\mathbb{Z}\right\\}$
$\displaystyle=\left\\{\bm{r}|\frac{mx}{a}+\frac{ny}{b}+\frac{lz}{c}=d+q,\
q\in\mathbb{Z}\right\\},$ (38)

where $\bm{b}_{i}$’s are reciprocal vectors corresponding to the primitive
vectors $\bm{a}_{1}=(a,0,0)$, $\bm{a}_{2}=(0,b,0)$, $\bm{a}_{3}=(0,0,c),$ $m$,
$n$, $l$ are integers and $0\leq d<1$. Here, in order to facilitate
understanding of the LCs for readers, we adopt the primitive vectors of the
primitive lattice throughout the section. The integer $q$ is introduced to
preserve the translation symmetry. Each plane in the layer is decorated with a
2D Chern insulator with a Chern number $+1$. Its orientation associated with
the Chern number, i.e. the orientation of the chiral edge states, is specified
by the reciprocal lattice vector $\bm{G}=m\bm{b}_{1}+n\bm{b}_{2}+l\bm{b}_{3}$,
giving the normal vector of the plane. Depending on SGs, we minimally
introduce other layers to make the set of layers to be compatible with the
given SG. Then we evaluate whether this set of layers can be trivialized while
preserving the symmetry considered is preserved. The details are shown in
Appendix B.

In $\bm{{\it 9}}$, we can find three kinds of invariants, the Chern invariants
$\delta_{\mathcal{C}_{i}},\ i=1,2,3$ associated with $\bm{a}_{i}^{Cc}$, a
glide invariant $\delta_{\mathrm{g}_{y}}$ for the glide operation
$G_{y}=\\{M_{y}|\hat{\bm{z}}/2\\}$, and the other glide invariant
$\delta_{\mathrm{g}_{n}}$ for the glide operation
$G_{n}=\\{M_{y}|(\hat{\bm{x}}+\hat{\bm{y}}+\hat{\bm{z}})/2\\}$, as presented
in Appendix B. The glide operation $G_{n}(=T_{1}G_{y})$ is known as additional
symmetry elements [66, 59], where $T_{1}$ is a translation by
$(\hat{\bm{x}}+\hat{\bm{y}})/2$. Here, one can show that
$\delta_{\mathcal{C}_{1}}=\delta_{\mathcal{C}_{2}}$,
$\delta_{\mathcal{C}_{3}}=0$,
$\delta_{g_{n}}\equiv\delta_{g_{y}}+\delta_{\mathcal{C}_{i}}\pmod{2}$, meaning
that the independent topological invariants are $\delta_{\mathcal{C}_{i}}$ and
$\delta_{g_{y}}$.

In $\bm{{\it 15}}$, we can define six kinds of invariants, the Chern
invariants $\delta_{\mathcal{C}_{i}},\ i=1,2,3$, the two glide invariants
$\delta_{\mathrm{g}_{y}},\delta_{\mathrm{g}_{n}}$, the inversion invariant
$\delta_{\text{i}}$, the $C_{2}$ invariant $\delta_{\text{r}}$, and the
$C_{2}$ screw invariant $\delta_{\mathrm{s}}$, only from geometric
configuration of layers as discussed in detail in Appendix B. The Chern
invariants $\delta_{\mathcal{C}_{i}}$ are integers, while the other invariants
$\delta_{\mathrm{g}_{y}},\delta_{\mathrm{g}_{n}},\delta_{\text{i}}$ and
$\delta_{\mathrm{s}}$ are integers defined modulo 2, and $\delta_{\text{r}}$
turns out to be always trivial: $\delta{\text{r}}=0$ Then for every LC, one
can evaluate these invariants, from which we establish relations between these
invariants:

$\displaystyle\delta_{\text{i}}\equiv\delta_{\mathrm{g}_{y}}\pmod{2},$ (39)
$\displaystyle\delta_{\mathrm{s}}\equiv\delta_{\mathcal{C}_{1}}=\delta_{\mathcal{C}_{2}}\pmod{2},$
(40)
$\displaystyle\delta_{\mathrm{i}}+\delta_{\mathrm{s}}\equiv\delta_{\mathrm{g}_{n}}\pmod{2},$
(41) $\displaystyle\delta_{\text{r}}\equiv 0\pmod{2},$ (42)
$\displaystyle\delta_{\mathcal{C}_{3}}=0$ (43)

as shown in Appendix B.

Now we can compare these invariants with the glide-$Z_{2}$ invariant $\nu$ and
the Chern number $\mathcal{C}_{y}$. One can calculate these two topological
invariants for general layers and compare them with the invariants
$\delta_{\mathcal{C}_{i}}$, $\delta_{\mathrm{g}_{y}}$,
$\delta_{\mathrm{g}_{n}}$, $\delta_{\text{i}}$, $\delta_{\text{r}}$, and
$\delta_{\text{s}}$ from the LC. With a straightforward calculation with
details presented in Appendix B, we get

$\displaystyle\frac{1}{2}\mathcal{C}_{y}=\mathcal{C}_{1}=\mathcal{C}_{2}$
$\displaystyle=\delta_{\mathcal{C}_{1}}=\delta_{\mathcal{C}_{2}},$ (44)
$\displaystyle\nu$
$\displaystyle\equiv\delta_{\text{i}}\equiv\delta_{\text{g}_{y}}\pmod{2},$
(45) $\displaystyle\frac{1}{2}\mathcal{C}_{y}$
$\displaystyle\equiv\delta_{\mathrm{s}}\pmod{2},$ (46)
$\displaystyle\nu+\frac{1}{2}\mathcal{C}_{y}$
$\displaystyle\equiv\delta_{\mathrm{g}_{n}}\pmod{2}.$ (47)

Thus the topological invariants based on the LC completely agrees with the two
topological invariants discussed in the previous section, the glide-$Z_{2}$
invariant $\nu$ and the Chern number $\mathcal{C}_{y}$.

In particular, these results in Eqs. (44)-(47) show that the independent
topological invariants are the glide-$Z_{2}$ invariant $\nu$ and the Chern
number $\mathcal{C}_{y}$. From these results we can demonstrate generators of
LCs for nontrivial values of topological invariants, i.e., minimal layer
configurations having nontrivial topological invariants. They are listed in
Table 3 in Appendix B.

###  III.2 Tight-binding models constructed from the layer construction for
$\bm{{\it 15}}$

Here, we study the two elementary LCs $(lmn;d)=(001;0)$ and $(\bar{1}10;0)$ as
summarized in Table 3 in Appendix B for $\bm{{\it 15}}$. These two LCs
correspond to the glide-symmetric $Z_{2}$ magnetic TCI and the Chern insulator
phases in $\bm{{\it 15}}$, respectively, and we construct simple tight-binding
models for them in order to investigate our scenario in this subsection. In
both models, each layer is a Chern insulator with a Chern number $+1$ for
$-2<m<0$, $-1$ for $0<m<2$ and zero otherwise, and this model shows phase
transitions at $m=0,\pm 2$. The details of the models are presented in
Appendix C.

####  III.2.1 Glide-symmetric $Z_{2}$ magnetic TCI phase: $(001;0)$

Let us begin with the glide-symmetric $Z_{2}$ magnetic TCI phase. The glide-
symmetric $Z_{2}$ magnetic TCI phase can be realized as an alternate stacking
of two 2D Chern insulator layers with opposite signs of the Chern number $\pm
1$. Suppose layers of a 2D Chern insulator with $\mathcal{C}_{z}=\pm 1$ along
the $xy$ plane are placed at $z=z_{0}+n$ in which $n$ is an integer. The glide
symmetry then requires presence of other layers of a 2D Chern insulator with
$\mathcal{C}_{z}=-1$ at $z=z_{0}+n+1/2$, because the glide operation changes
the sign of the Chern number. This system preserves the symmetry in the SG
$\bm{{\it 9}}$. This model gives a nontrivial glide-$Z_{2}$ topological
invariant $\nu$ for glide-symmetric systems by a direct calculation.

Figure 3: Layer constructions for (a) $(001;0)$, showing a glide-$Z_{2}$
topological phase (${\nu}=1$, $\mathcal{C}_{y}=0$) and (b) $(\bar{1}10;0)$,
showing a Chern insulator (${\nu}=0$, $\mathcal{C}_{y}=2$) with $\bm{{\it
15}}$. The red arrows denote the directions of the normal vector of the plane
giving a positive Chern number $+1$.

We now consider the case when inversion symmetry is added, leading to the SG
$\bm{{\it 15}}$. Inversion symmetry is preserved when we fix the value of
$z_{0}$ to be $z_{0}=0$, and the configuration allowed by symmetry is the one
with the 2D Chern insulators on $z=n$ with $\mathcal{C}_{z}=+1$ and on
$z=n+1/2$ with $\mathcal{C}_{z}=-1$, where $n$ is an integer (Fig. 3(a)).
Therefore, a representative Hamiltonian for the glide-symmetric $Z_{2}$
magnetic TCI phase with additional inversion symmetry is obtained as

$\displaystyle H^{\mathrm{LC}}_{\nu}$
$\displaystyle=\left(m+2\cos\frac{k_{x}}{2}\cos\frac{k_{y}}{2}\right)\sigma_{z}$
$\displaystyle+\sin\frac{k_{x}}{2}\cos\frac{k_{y}}{2}(\sigma_{x}-\sigma_{y})$
$\displaystyle+\cos\frac{k_{x}}{2}\sin\frac{k_{y}}{2}(\sigma_{x}+\sigma_{y})\tau_{z},$
(48)

where $\sigma_{j}$ are the Pauli matrices corresponding to an internal degree
of freedom within each layer, introduced for the purpose of realizing a Chern
insulator, and $\tau_{j}$ are the Pauli matrices acting onto the layer index
referring to the layers at $z=n$ and $z=n+\frac{1}{2}$. The corresponding
$k$-dependent glide operator and $k$-dependent inversion operator are

$\displaystyle G_{y}(k_{z})$
$\displaystyle=e^{-ik_{z}/2}\left(\cos\frac{k_{z}}{2}\tau_{x}+\sin\frac{k_{z}}{2}\tau_{y}\right),$
(49) $\displaystyle I(k_{z})$ $\displaystyle=\sigma_{z}\begin{pmatrix}1&0\\\
0&e^{-ik_{z}}\end{pmatrix}_{\tau}.$ (50)

The Hamiltonian in Eq. (48) satisfies

$\displaystyle
G_{y}(k_{z})H^{\mathrm{LC}}_{\nu}(k_{x},k_{y},k_{z})G_{y}(k_{z})^{-1}$
$\displaystyle=H^{\mathrm{LC}}_{\nu}(k_{x},-k_{y},k_{z}),$ (51) $\displaystyle
I(k_{z})H^{\mathrm{LC}}_{\nu}(\bm{k})I(k_{z})^{-1}$
$\displaystyle=H^{\mathrm{LC}}_{\nu}(-\bm{k}),$ (52)

and these operators satisfy the algebra

$G_{y}(-k_{z})I(k_{z})=e^{ik_{z}}I(k_{z})G_{y}(k_{z}).$ (53)

The Hamiltonian $H^{\mathrm{LC}}_{\nu}$ is gapped unless $m=\pm 2,0$. The gap
closes at $\Gamma$ when $m=-2$, at $V$ (and $V^{\prime}$) when $m=0$, and at
$Y$ when $m=2$. The effective Hamiltonian at the high-symmetry points
$P=\Gamma$, $Y$, and $V$ on the $k_{z}=0$ plane in Eq. (29) can be written as

$H^{\mathrm{LC}}_{\nu}(P)=\left(m+2\cos\frac{k_{x}}{2}\cos\frac{k_{y}}{2}\right)\sigma_{z}.$
(54)

The irreps of the occupied states are summarized in Table 2. By using Eq.
(19), we can show that the $Z_{2}$ topological invariant is nontrivial
(trivial) if $0<|m|<2$ ($|m|>2$). In particular, when $2<m<0$, the model
corresponds to the layer construction shown in Fig. 3 as expected.

Table 2: Irreducible representations for the two tight-binding models for the
topological phases in $\bm{{\it 15}}$ from the layer construction.
$\begin{array}[]{cccccc}\hline\cr\hline\cr&\Gamma&Y&V&M&L\\\ \hline\cr
H^{\mathrm{LC}}_{\nu}(m<-2)&\Gamma_{1}^{+}\Gamma_{2}^{+}&Y_{1}^{+}Y_{2}^{+}&2V_{1}^{+}&M_{1}&L_{1}^{+}L_{1}^{-}\\\
H^{\mathrm{LC}}_{\nu}(-2<m<0)&\Gamma_{1}^{-}\Gamma_{2}^{-}&Y_{1}^{+}Y_{2}^{+}&2V_{1}^{+}&M_{1}&L_{1}^{+}L_{1}^{-}\\\
H^{\mathrm{LC}}_{\nu}(0<m<2)&\Gamma_{1}^{-}\Gamma_{2}^{-}&Y_{1}^{+}Y_{2}^{+}&2V_{1}^{-}&M_{1}&L_{1}^{+}L_{1}^{-}\\\
H^{\mathrm{LC}}_{\nu}(2<m)&\Gamma_{1}^{-}\Gamma_{2}^{-}&Y_{1}^{-}Y_{2}^{-}&2V_{1}^{-}&M_{1}&L_{1}^{+}L_{1}^{-}\\\
\hline\cr
H^{\mathrm{LC}}_{\mathcal{C}}(m<-2)&\Gamma_{1}^{+}\Gamma_{2}^{+}&Y_{1}^{+}Y_{2}^{+}&2V_{1}^{+}&M_{1}&L_{1}^{+}L_{1}^{-}\\\
H^{\mathrm{LC}}_{\mathcal{C}}(-2<m<0)&\Gamma_{1}^{-}\Gamma_{2}^{-}&Y_{1}^{+}Y_{2}^{+}&V_{1}^{+}V_{1}^{-}&M_{1}&L_{1}^{+}L_{1}^{-}\\\
H^{\mathrm{LC}}_{\mathcal{C}}(0<m<2)&\Gamma_{1}^{-}\Gamma_{2}^{-}&Y_{1}^{-}Y_{2}^{-}&2V_{1}^{-}&M_{1}&2L_{1}^{+}\\\
H^{\mathrm{LC}}_{\mathcal{C}}(2<m)&\Gamma_{1}^{-}\Gamma_{2}^{-}&Y_{1}^{-}Y_{2}^{-}&2V_{1}^{-}&M_{1}&L_{1}^{+}L_{1}^{-}\\\
\hline\cr\hline\cr\end{array}$

The $z_{4}$ indicator is equal to two when $0<|m|<2$ and zero otherwise from
Table 2. Therefore, when $0<|m|<2$, it is a higher-order TI with the symmetry-
based indicator for $\bm{{\it 2}}$ given by
$(\mathbb{Z}_{2},\mathbb{Z}_{2},\mathbb{Z}_{2},\mathbb{Z}_{4})=(0,0,0;2)$ in
accordance with Eq. (37).

####  III.2.2 Chern insulator phase: ($\bar{1}10;0$)

Next, we construct a model with a nontrivial Chern number. In this case, we
simply stack a 2D Chern insulator with a Chern number equal to $+1$ along
$(\bar{1}10)$ and $(110)$ directions (Fig. 3(b)). Then, we have a
representative Hamiltonian

$H^{\mathrm{LC}}_{\mathcal{C}}(\bm{k})=\begin{pmatrix}H^{(+)}(\bm{k})&\\\
&H^{(-)}(\bm{k})\end{pmatrix}_{\tau},$ (55)

where

$\displaystyle H^{(\pm)}(\bm{k})$ $\displaystyle=\left(m+\cos
k_{z}+\cos\frac{k_{x}\pm k_{y}}{2}\right)\sigma_{z}$ $\displaystyle+\sin
k_{z}\sigma_{x}+\sin\frac{k_{x}\pm k_{y}}{2}\sigma_{y}.$ (56)

The Hamiltonian $H^{\mathrm{LC}}_{\mathcal{C}}$ indeed satisfies the relations
in Eqs. (51) and (52) under the corresponding operators given by Eqs. (49) and
(50).

The irreps of the occupied states are summarized in Table 2. By using Eq.
(19), we can show that the $Z_{2}$ topological invariant is nontrivial
(trivial) if $0<|m|<2$ ($|m|>2$). Because each layer has a Chern number $+1$
for $-2<m<0$, $-1$ for $0<m<2$ and zero otherwise, it is a trivial insulator
when $|m|>2$ and a Chern insulator with $\mathcal{C}_{y}=+2$ at $-2<m<0$, and
$\mathcal{C}_{y}=-2$ at $0<m<2$. Meanwhile, the $Z_{2}$ topological invariant
(29) is always trivial, as we have expected. In particular, when $2<m<0$, the
model corresponds to the layer construction shown in Fig. 3 as expected.

##  IV Conclusion

In the present paper, we find the formula for the topological invariant
protected by glide symmetry in a nonprimitive lattice. We establish a formula
for the glide-$Z_{2}$ invariant for the space group $\bm{{\it 9}}$ with glide
symmetry in the base-centered lattice, by folding its Brillouin zone into that
of the primitive lattice where the formula for the glide-$Z_{2}$ invariant is
known. The formula is written in terms of the integrals of the Berry
curvatures and Berry phases in the $k$-space. We then calculate the formula of
the glide-$Z_{2}$ invariant when the inversion symmetry is added, and the
space group becomes $\bm{{\it 15}}$. This reduces the formula into the Fu-
Kane-like formula, expressed in terms of the irreducible representations at
high-symmetry points in $k$ space. This result gives an interpretation to the
previous results on topological indices, such as symmetry-based indicators and
K-theory. In particular, the symmetry-based indicator for the SG $\bm{{\it
15}}$ is ${Z}_{2}\times{Z}_{2}$, and the results in our paper show that one
${Z}_{2}$ is the glide-$Z_{2}$ topological invariant and the other is the
parity of the half of the Chern number along the glide invariant plane
$k_{y}=0$.

We also construct toplogical invariants by the layer-construction approach,
and the results completely agree with those from the $k$-space approach in
this paper. Based on this we find two elementary layer constructions for
$\bm{{\it 15}}$ in a form of tight-binding models.

In the previous paper [60] and in the present paper, we discussed the fate of
the glide-$Z_{2}$ invariant when the inversion symmetry is added. There are
only three possible minimal space groups consisting of inversion symmetry and
glide symmetry: the space groups $\bm{{\it 13}}$, $\bm{{\it 14}}$ and
$\bm{{\it 15}}$. The former two ($\bm{{\it 13}}$ and $\bm{{\it 14}}$) are on a
primitive lattice studied in the previous paper [60], and the last one
($\bm{{\it 15}}$) is on a nonprimitive lattice studied in the present paper.
The former two cases are distinguished by whether the inversion center is on a
glide-mirror plane ($\bm{{\it 13}}$) or not ($\bm{{\it 14}}$). We found that
in the absence of inversion symmetry ($\bm{{\it 7}}$ and $\bm{{\it 9}}$), the
glide-$Z_{2}$ invariant is expressed in terms of $k$-space integrals, while by
adding inversion symmetry, the glide-$Z_{2}$ invariant is related with the
irreps at high-symmetry points in $k$-space, as shown in the present paper and
Ref. 60. One might think that further analysis is needed for many other space
groups with inversion and glide symmetries, but it is not the case. Since we
have established that the glide-$Z_{2}$ invariant is related to the irreps at
high-symmetry points in $k$-space in the space groups $\bm{{\it 13}}$,
$\bm{{\it 14}}$ and $\bm{{\it 15}}$, it is now straightforward to apply these
formulae to other space groups with inversion and glide symmetries. Thus, the
topological invariants obtained in this paper should be useful for
characterizing and designing the glide-$Z_{2}$ phase in systems with various
kinds of particles.

###### Acknowledgements.

This work was supported by Japan Society for the Promotion of Science (JSPS)
KAKENHI Grant No. JP18H03678 and JP20H04633. H. K. is supported by Japan
Society for the Promotion of Science (JSPS) KAKENHI Grant-in-Aid for JSPS
Fellows Grant No. JP17J10672.

##  Appendix A Details of the derivation of the glide-$Z_{2}$ invariant in SG
$\bm{{\it 9}}$

In this Appendix, we derive the formula of the glide-$Z_{2}$ invariant for the
SG $\bm{{\it 9}}$ from that for $\bm{{\it 7}}$ given in Eq. (2). We plug in
the eigenstates of $\bm{{\it 7}}$ in Eq. (16) constructed from those in
$\bm{{\it 9}}$ into the formula of the glide $Z_{2}$ invariant of $\bm{{\it
7}}$ in Eq. (2). In particular, we can recast the third integral in Eq. (2)
into

$\displaystyle\int_{\mathcal{C}}=\int_{-\pi}^{\pi}dk_{z}\int_{-\pi}^{\pi}dk_{x}{\tilde{F}}_{zx}^{+}(k_{x},\pi,k_{z})$
$\displaystyle=\frac{1}{2}\left(\int_{-\pi}^{\pi}dk_{z}\int_{-\pi}^{\pi}dk_{x}{F}_{zx}(k_{x},\pi,k_{z})\right.$
$\displaystyle+\left.\int_{-\pi}^{\pi}dk_{z}\int_{-\pi}^{\pi}dk_{x}{F}_{zx}(k_{x},-\pi,k_{z})\right)$
$\displaystyle\quad+\frac{1}{4}\int_{-\pi}^{\pi}dk_{x}\sum_{i\in\text{occ.}}\left(i\bra{u_{i}(\bm{q})}{G}_{y}(\bm{q})\ket{u_{i}(\bm{q})}+\mathrm{c.c.}\right),$
(57)

where $\bm{q}=(k_{x},\pi,-\pi)$. Here we define $\tilde{F}_{ij}^{\pm}(\bm{k})$
and $\tilde{\bm{A}}^{\pm}(\bm{k})$ to be the Berry cuvatures and the Berry
connections within the glide $g_{\pm}$ subspaces defined in terms of the
wavefunctions $\tilde{u}$ in SG $\bm{7}$, similar to Eqs. (3) and (5). The
Berry phase term along
$\text{E}\rightarrow\text{D}\rightarrow\text{E}^{\prime}$ is expressed as

$\displaystyle-2\gamma_{EDE^{\prime}}^{-}=2\oint_{-\pi}^{\pi}dk_{x}\tilde{A}_{x}^{-}(k_{x},\pi,-\pi)$
$\displaystyle=\oint_{-\pi}^{\pi}dk_{x}{A}_{x}(k_{x},\pi,-\pi)+\oint_{-\pi}^{\pi}dk_{x}{A}_{x}(k_{x},-\pi,-\pi)$
$\displaystyle+\sum_{i\in\text{occ.}}(\chi_{i}(\pi,\pi,-\pi)+\chi_{i}(-\pi,\pi,-\pi))$
$\displaystyle\quad+\frac{1}{4}\int_{-\pi}^{\pi}dk_{x}\sum_{i\in\text{occ.}}\left(i\bra{u_{i}(\bm{q})}{G}_{y}(\bm{q})\ket{u_{i}(\bm{q})}+\mathrm{c.c.}\right).$
(58)

Next, we consider the $k_{z}=-\pi$ plane. The term on this plane for $\bm{{\it
9}}$ is rewritten as

$\displaystyle\int_{\mathcal{A}}=\int_{0}^{\pi}dk_{y}\int_{-\pi}^{\pi}dk_{x}{\tilde{F}}_{xy}(k_{x},k_{y},-\pi)$
$\displaystyle=\int_{0}^{2\pi}dk_{y}\int_{k_{y}-2\pi}^{-k_{y}+2\pi}dk_{x}{F}_{xy}(k_{x},k_{y},-\pi)$
$\displaystyle+\left(\oint_{\text{C}_{a}}+\oint_{\text{C}_{b}}\right)d\bm{k}\cdot\bm{A}(\bm{k})$
$\displaystyle-\oint_{-\pi}^{\pi}dk_{x}{A}_{x}(k_{x},\pi,-\pi)-\oint_{-\pi}^{\pi}dk_{x}{A}_{x}(k_{x},-\pi,-\pi),$
(59)

where the paths for the Berry-phase terms are
$\text{C}_{a}=\\{\bm{k}|k_{z}=-\pi,\ k_{y}=k_{x}+2\pi,\ -2\pi<k_{x}<0\\}$
$\text{C}_{b}=\\{\bm{k}|k_{z}=-\pi,\ k_{y}=-k_{x}+2\pi,\ 0<k_{x}<2\pi\\}$,
depicted in Fig. 4(a). The term
${\tilde{F}}_{ij}\equiv{\tilde{F}}_{ij}^{+}+{\tilde{F}}_{ij}^{-}$ represents
the total Berry curvature for the occupied states in SG $\bm{7}$.

Therefore, to summarize up to this point we have

$\displaystyle\int_{\mathcal{A}}-\int_{\mathcal{C}}-2\gamma_{EDE^{\prime}}^{-}$
(60)
$\displaystyle=-\frac{1}{2}\left(\int_{-\pi}^{\pi}dk_{z}\int_{-\pi}^{\pi}dk_{x}{F}_{zx}(k_{x},\pi,k_{z})\right.$
$\displaystyle+\left.\int_{-\pi}^{\pi}dk_{z}\int_{-\pi}^{\pi}dk_{x}{F}_{zx}(k_{x},-\pi,k_{z})\right)$
$\displaystyle+\sum_{i\in\text{occ.}}(\chi_{i}(\pi,\pi,-\pi)+\chi_{i}(-\pi,\pi,-\pi))$
$\displaystyle+\int_{0}^{2\pi}dk_{y}\int_{k_{y}-2\pi}^{-k_{y}+2\pi}dk_{x}{F}_{xy}(k_{x},k_{y},-\pi)$
$\displaystyle+\left(\oint_{\text{C}_{a}}+\oint_{\text{C}_{b}}\right)d\bm{k}\cdot\bm{A}(\bm{k}).$
(61)

In order to proceed, we have to clarify properties of the function
$\chi_{i}(\bm{k})$. Owing to the periodicity of BZ, we can write

$\displaystyle\chi_{i}(\bm{k}+\bm{b}_{1}^{Cc})$
$\displaystyle=\chi_{i}(\bm{k})+2\pi N_{i},$ (62)
$\displaystyle\chi_{i}(\bm{k}+\bm{b}_{2}^{Cc})$
$\displaystyle=\chi_{i}(\bm{k})+2\pi N_{i}^{\prime},$ (63)

where $N_{i}$ and $N_{i}^{\prime}$ are integers. By substituting
$\bm{k}\rightarrow\bm{k}+\bm{b}_{1}^{Cc}$ in Eq. (65), we get
$N_{i}=N_{i}^{\prime}$. Then we obtain an expression of $\chi_{i}(\bm{k})$ as

$\chi_{i}(\bm{k})=N_{i}k_{y}+\theta_{i}(k_{x},k_{y},k_{z}),$ (64)

where $N_{i}$ is an integer and $\theta_{i}(k_{x},k_{y},k_{z})$ is a periodic
function of $k_{x}$ and $k_{y}$ in the BZ of SG $\bm{{\it 9}}$:
$\theta_{i}(\bm{k}+\bm{b}_{j}^{Cc})=\theta_{i}(\bm{k})$ ($j=1,2$). Next, in
Eq. (14) we can write

$\chi_{i}(\bm{k})+\chi_{i}(g_{y}\bm{k})=-k_{z}+2\pi Q_{i},$ (65)

where $Q_{i}$ is an integer. Therefore, by summing over all the occupied
states, we get

$\displaystyle\sum_{i\in\text{occ.}}\left(\chi_{i}(\pi,\pi,-\pi)+\chi_{i}(-\pi,\pi,-\pi)\right)$
$\displaystyle=\sum_{i\in\text{occ.}}\left(2\pi
N_{i}+\theta_{i}(\pi,\pi,-\pi)+\theta_{i}(-\pi,\pi,-\pi)\right)$
$\displaystyle=\sum_{i\in\text{occ.}}\left(2\pi
N_{i}+\theta_{i}(\pi,\pi,-\pi)+\theta_{i}(\pi,-\pi,-\pi)\right)$
$\displaystyle=\sum_{i\in\text{occ.}}\left(2\pi N_{i}+2\pi Q_{i}+\pi\right).$
(66)

To further evaluate this quantity, we put $k_{y}=0$ in Eq. (65), which leads
to $2\chi_{i}(k_{x},0,k_{z})=-k_{z}+2\pi Q_{i}$. On the other hand, on the
glide-invariant $k_{y}=0$ plane, the states are classified into the $g_{\pm}$
sectors with $G_{y}=\pm e^{-ik_{z}/2}$, meaning that

$\displaystyle\chi_{i}(k_{x},0,k_{z})\equiv-k_{z}/2\pmod{2\pi}\ \rightarrow\
Q_{i}=\text{even}\ :\ g_{+}\text{sector}$
$\displaystyle\chi_{i}(k_{x},0,k_{z})\equiv\pi-k_{z}/2\pmod{2\pi}\
\rightarrow\ Q_{i}=\text{odd}\ :\ g_{-}\text{sector}$

Since these two sectors switch when $k_{z}$ increases by $2\pi$, meaning that
the numbers of occupied states in the $g_{+}$ sector and that in the $g_{-}$
sector are the same. Therefore, we finally conclude from Eq. (66) that

$\displaystyle\sum_{i\in\text{occ.}}\left(\chi_{i}(\pi,\pi,-\pi)+\chi_{i}(-\pi,\pi,-\pi)\right)$
$\displaystyle\hskip 42.67912pt\equiv\sum_{i\in\text{occ.}}2\pi
N_{i}\pmod{4\pi}.$ (67)

Second, we investigate the Berry phase terms in Eq. (61). Due to the
periodicity of the BZ, we have

$\displaystyle\oint_{\text{C}_{b}}d\bm{k}\cdot\bm{A}(\bm{k})$
$\displaystyle=\sum_{i\in\text{occ.}}2\pi N_{i}+\
\oint_{\text{C}_{a}}d\bm{k}\cdot\bm{A}(\bm{k}),$ (68)

Thus in the sum of Eqs. (67) and (68), the $N_{i}$ terms becomes $4\pi$ times
an integer.

On the other hand, the terms on the glide-invariant plane $k_{y}=0$ in
$\bm{{\it 9}}$ are given by

$\displaystyle\int_{\mathcal{B}}=\int_{-\pi}^{\pi}dk_{z}\int_{-\pi}^{\pi}dk_{x}{\tilde{F}}_{zx}^{-}(k_{x},0,k_{z})$
$\displaystyle=\int_{-\pi}^{\pi}dk_{z}\int_{-2\pi}^{2\pi}dk_{x}{F}_{zx}^{-}(k_{x},0,k_{z}),$
(69)

and

$\gamma_{A^{\prime}BA}^{+}=\oint_{-\pi}^{\pi}dk_{x}{\tilde{A}}_{x}^{+}(k_{x},0,-\pi)=\oint_{-2\pi}^{2\pi}dk_{x}{A}_{x}^{+}(k_{x},0,-\pi).$
(70)

Therefore, by combining these results, the glide-$Z_{2}$ invariant for
$\bm{{\it 9}}$ recast as follows:

$\displaystyle\nu$
$\displaystyle=\frac{1}{2\pi}\left[2\oint_{\text{C}_{a}}d\bm{k}\cdot\bm{A}(\bm{k})\right.$
$\displaystyle-\left.2\oint_{-2\pi}^{2\pi}dk_{x}{A}_{x}^{+}(k_{x},0,-\pi)\right.$
$\displaystyle+\int_{0}^{2\pi}dk_{y}\int_{k_{y}-2\pi}^{-k_{y}+2\pi}dk_{x}{F}_{xy}(k_{x},k_{y},-\pi)$
$\displaystyle+\int_{-\pi}^{\pi}dk_{z}\int_{-2\pi}^{2\pi}dk_{x}{F}_{zx}^{-}(k_{x},0,k_{z})$
$\displaystyle-\frac{1}{2}\left(\int_{-\pi}^{\pi}dk_{z}\int_{-\pi}^{\pi}dk_{x}{F}_{zx}(k_{x},\pi,k_{z})\right.$
$\displaystyle+\left.\left.\int_{-\pi}^{\pi}dk_{z}\int_{-\pi}^{\pi}dk_{x}{F}_{zx}(k_{x},-\pi,k_{z})\right)\right].$
(71)

Because we are assuming an insulating system, the vector field of the Berry
curvature is divergence free. Therefore, by noticing that the summation of the
last two terms in Eq. (71) is equal to the Chern number on the glide-invariant
plane $k_{y}=0$ (see Fig. 4(b)), we have an alternative expression for $\nu$,

$\displaystyle\nu$
$\displaystyle=\frac{1}{2\pi}\left[2\oint_{\text{C}_{a}}d\bm{k}\cdot\bm{A}(\bm{k})\right.$
$\displaystyle\left.-2\oint_{-2\pi}^{2\pi}dk_{x}{A}_{x}^{+}(k_{x},0,-\pi)\right.$
$\displaystyle\left.+\int_{0}^{2\pi}dk_{y}\int_{k_{y}-2\pi}^{-k_{y}+2\pi}dk_{x}{F}_{xy}(k_{x},k_{y},-\pi)\right.$
$\displaystyle\left.+\int_{-\pi}^{\pi}dk_{z}\int_{-2\pi}^{2\pi}dk_{x}\frac{1}{2}({F}_{zx}(k_{x},0,k_{z})-{F}_{zx}^{+}(k_{x},0,k_{z}))\right],$
(72)

which is the same as Eq. (17). We have used the relation
${F}_{ij}(\bm{k})={F}_{ij}^{+}(\bm{k})+{F}_{ij}^{-}(\bm{k})$.

So far we studied spinless cases. Meanwhile, Eq. (72) applies also to spinful
cases as well. Instead of Eq. (13), in spinful cases the glide operator
satisties $G_{y}(g_{y}\bm{k})G_{y}(\bm{k})=-e^{-ik_{z}}$ and
$-e^{-ik_{z}}=e^{i\chi(\bm{k})+i\chi(g_{y}\bm{k})}$ instead of Eq. (14).
Therefore, instead of Eq. (16) the wavefunctions in each glide sector of
$\bm{{\it 9}}$ is given by

$\displaystyle\ket{\tilde{u}^{\pm}(k_{x},\pi,k_{z})}$
$\displaystyle=\frac{1}{2}\left[\begin{pmatrix}1\\\
ie^{ik_{x}/2}\end{pmatrix}\ket{u(k_{x},\pi,k_{z})}\mp i\begin{pmatrix}1\\\
-ie^{ik_{x}/2}\end{pmatrix}e^{i\chi(k_{x},\pi,k_{z})+ik_{z}/2}\ket{u(k_{x},-\pi,k_{z})}\right]$
$\displaystyle\quad\times\exp\left[-i\left(\chi(\pi,\pi,k_{z})+\frac{k_{z}}{2}\right)\frac{k_{x}}{2\pi}\right]\times\begin{cases}e^{ik_{x}/4}&(g_{+}\
\mathrm{sector})\\\ e^{-ik_{x}/4}&(g_{-}\ \mathrm{sector})\end{cases}.$ (73)

From this equation, the expression
$\chi_{i}(\pi,\pi,-\pi)+\chi_{i}(-\pi,\pi,-\pi)$ in this Appendix acquires an
additional $-\pi$ term. Nonetheless, Eq. (65) is replaced by

$\chi_{i}(\bm{k})+\chi_{i}(g_{y}\bm{k})=\pi-k_{z}+2\pi Q_{i},$ (74)

which also has an additional $\pi$. As a result, Eq. (66) becomes

$\displaystyle\sum_{i\in\text{occ.}}\left(\chi_{i}(\pi,\pi,-\pi)+\chi_{i}(-\pi,\pi,-\pi)-\pi\right)$
$\displaystyle=\sum_{i\in\text{occ.}}\left(2\pi
N_{i}+\theta_{i}(\pi,\pi,-\pi)+\theta_{i}(-\pi,\pi,-\pi)-\pi\right)$
$\displaystyle=\sum_{i\in\text{occ.}}\left(2\pi
N_{i}+\theta_{i}(\pi,\pi,-\pi)+\theta_{i}(\pi,-\pi,-\pi)-\pi\right)$
$\displaystyle=\sum_{i\in\text{occ.}}\left(2\pi N_{i}+2\pi Q_{i}+\pi\right),$
(75)

and its result is the same as the spinless cases. Thus, the final expression
of the glide-$Z_{2}$ invariant in spinful cases is the same as that in
spinless cases in Eq. (72).

Figure 4: Domain of integration in the derivation of the formula of the
glide-$Z_{2}$ invariant in $\bm{{\it 9}}$. (a) The paths for the Berry phase
terms in Eq. (61) on $k_{z}=-\pi$. (b) Domains of integration of the Berry
curvature shown on a $k_{z}=(\text{const.})$ plane. In Eq. (71) the Berry
curvature term is over the two regions shown as red and yellow lines. Because
of the Brillouin zone periodicity, the integral over the blue region is the
same as that over the red one, and by adding the term from the yellow region,
it equals to the integral over the green region along $k_{y}=0$.

##  Appendix B Layer construction for space groups $\bm{{\it 9}}$ and
$\bm{{\it 15}}$

In this appendix, we argue a layer construction (LC) in class A for $\bm{{\it
9}}$ and $\bm{{\it 15}}$. The formalism for the LC in class A is the same as
that in Ref. 60.

###  B.1 Setup

Let us start with a layer $(mnl;d)$ which is a set of planes described by the
Miller indices $(mnl)$ and displaced from the origin by $d$ $(0\leq d<1)$,

$\displaystyle(mnl;d)$
$\displaystyle=\left\\{\bm{r}|\bm{r}\cdot(m\bm{b}_{1}+n\bm{b}_{2}+l\bm{b}_{3})=2\pi(d+q),\
q\in\mathbb{Z}\right\\}$
$\displaystyle=\left\\{\bm{r}|\frac{mx}{a}+\frac{ny}{b}+\frac{lz}{c}=d+q,\
q\in\mathbb{Z}\right\\},$ (76)

where $\bm{b}_{i}$’s $(i=1,2,3)$ are reciprocal vectors corresponding to
lattice vectors $\bm{a}_{1}=a(100),\bm{a}_{2}=b(010),\bm{a}_{3}=c(001)$ of the
primitive lattice, which are $\bm{a}_{i}^{Pc}$’s in our main text. We take
this choice for convenience.

###  B.2 Topological invariants in glide-symmetric systems

Here we summarize the results of the topological invariants associated with
the layer construction comprising 2D Chern insulator layers. The detailed
explanations are in Ref. 60, and we show its outline here.

####  B.2.1 Chern invariant

For a reciprocal plane $\mathcal{D}_{i}$ normal to one of the primitive
lattice vectors $\bm{a}_{i}$, the Chern number is defined as

$\delta_{\mathcal{C}_{i}}=\frac{1}{2\pi}\int_{\mathcal{D}_{i}}d^{2}\bm{k}\
\bm{n}_{i}\cdot\bm{\Omega}_{\bm{k}},$ (77)

where $\bm{\Omega}_{\bm{k}}=(F_{yz},F_{zx},F_{xy})$ is the Berry curvature,
$\bm{n}_{i}\equiv\bm{a}_{i}/|\bm{a}_{i}|$ is a unit vector along $\bm{a}_{i}$,
and the integral is taken over the 2D BZ $\mathcal{D}_{i}$ on the plane
perpendicular to $\bm{n}_{i}$.

These Chern numbers are associated with the primitive lattice vectors
$\bm{a}_{i}$, and we can calculate them by counting intersections of the LC
$L=(mnl;d)$ with an axis parallel to the $\bm{a}_{i}$ within the unit cell.
Therefore, we have

$\delta_{\mathcal{C}_{i}}=\frac{1}{2\pi}\sum_{L\in
E}\bm{\mathrm{g}}_{L}\cdot\bm{a}_{i},$ (78)

where $E$ is a set of LCs compatible with the space group considered, and
$\bm{\mathrm{g}}_{L}=m\bm{b}_{1}+n\bm{b}_{2}+l\bm{b}_{3}$ for the layer $L\in
E$. We note that even though the Miller indices for the layer, the symmetry
operations, and the other topological invariants are given by conventional
lattice vectors, these Chern numbers are defined with respect to the primitive
cell. Namely, in the SG $\bm{{\it 9}}$ and $\bm{{\it 15}}$ we take
$\bm{a}_{1}^{Cc}=(\frac{1}{2},\frac{1}{2},0)$,
$\bm{a}_{2}^{Cc}=(\bar{\frac{1}{2}},\frac{1}{2},0)$,
$\bm{a}_{3}^{Cc}=(0,0,1)$.

####  B.2.2 Glide invariant $\delta_{\mathrm{g}}$

In the presence of glide symmetry, we can find a $Z_{2}$ invariant for glide-
symmetric systems[42, 43]. We can express this invariant based on the LC in
the similar way as in Ref. 59. We first choose one of the glide planes, and we
fix this choice throughout our theory. Then, the glide invariant
$\delta_{\mathrm{g}}$ is expressed as

$\displaystyle\delta_{\mathrm{g}}=\sum_{L\in
E}(N^{\mathrm{o}}_{m,\bm{\mathrm{t}}_{\parallel}}(L)+N^{\mathrm{s}}_{m,\bm{\mathrm{t}}_{\parallel}}(L))\pmod{2}$
(79) $\displaystyle
N^{\mathrm{o}}_{m,\bm{\mathrm{t}}_{\parallel}}(L)=\begin{cases}1&\mathrm{if}\
m\in L\\\ 0&\mathrm{otherwise}\end{cases},$ (80) $\displaystyle
N^{\mathrm{s}}_{m,\bm{\mathrm{t}}_{\parallel}}(L)=\frac{1}{2\pi}|\bm{\mathrm{t}}_{\parallel}\cdot\bm{\mathrm{g}}_{L}|,$
(81)

where $m$ is the glide plane of our choice, $m\in L$ means the layer $L$
occupies the glide plane $m$, and $\bm{\mathrm{t}}_{\parallel}$ is a glide
vector.

####  B.2.3 Inversion invariant $\delta_{\mathrm{i}}$

In the presence of inversion symmetry, we can define the inversion invariant
$\delta_{\mathrm{i}}$. We first choose one out of the eight inversion centers
in a unit cell, and we fix the choice throughout our theory. When a layer
occupies the inversion center, this layer cannot be trivialized as long as
inversion symmetry is preserved. Meanwhile, one can trivialize the
configuration where two layers occupy the inversion center, rendering it into
a trivial atomic insulator. Therefore, the inversion invariant
$\delta_{\mathrm{i}}$ is expressed as

$\displaystyle\delta_{\mathrm{i}}(E)=\sum_{L\in
E}N^{\mathrm{o}}_{\mathrm{i}}(L)\pmod{2},$ (82) $\displaystyle
N^{\mathrm{o}}_{\mathrm{i}}(L)=\begin{cases}1&\mathrm{if}\ i\in L\\\
0&\mathrm{otherwise}\end{cases},$ (83)

where $i$ is the inversion center of our choice out of the eight different
inversion centers, $i\in L$ means that the layer $L$ occupies the inversion
center $i$.

####  B.2.4 $C_{2}$ rotation invariant $\delta_{\mathrm{r}}$

In the presence of $C_{2}$ rotation symmetry, we obtain the $C_{2}$ rotation
invariant $\delta_{\mathrm{r}}$ in a similar manner. In fact, this invariant
is always trivial as long as a layer is decorated by a 2D Chern insulator
[60]. Therefore, we have

$\delta_{\mathrm{r}}(E)=0.$ (84)

####  B.2.5 $C_{2}$ screw invariant $\delta_{\mathrm{s}}$

In the presence of $C_{2}$ screw symmetry, we can find the $C_{2}$ screw
invariant $\delta_{\mathrm{s}}$ [60] as

$\displaystyle\delta_{\mathrm{s}}(E)=\sum_{L\in
E}N^{\mathrm{s}}_{s,\bm{\mathrm{t}}_{\parallel}}(L)\pmod{2},$ (85)
$\displaystyle
N^{\mathrm{s}}_{s,\bm{\mathrm{t}}_{\parallel}}(L)=\frac{1}{2\pi}|\bm{\mathrm{t}}_{\parallel}\cdot\bm{\mathrm{g}}_{L}|,$
(86)

where $\bm{\mathrm{t}}_{\parallel}$ is a screw vector.

###  B.3 Invariants based on LCs

####  B.3.1 Space group $\bm{{\it 9}}$

In $\bm{{\it 9}}$, we can find the Chern invariants
$\delta_{\mathcal{C}_{i=1,2,3}}$, a glide invariant $\delta_{\mathrm{g}_{y}}$
defined by the glide operation $G_{y}=\\{M_{y}|\hat{\bm{z}}/2\\}$, and the
other glide invariant $\delta_{\mathrm{g}_{n}}$ defined by the glide operation
$G_{n}=\\{M_{y}|(\hat{\bm{x}}+\hat{\bm{y}}+\hat{\bm{z}})/2\\}$. The glide
operation $G_{n}(=T_{1}G_{y})$ is known as an additional symmetry element
[66], where $T_{1}=\\{E|(\hat{\bm{x}}+\hat{\bm{y}})/2\\}$, and the two glide
operations lead to different glide invariants $\delta_{g_{y}}$ and
$\delta_{g_{n}}$.

Let us investigate the invariants in $\bm{{\it 9}}$. The glide operations
$G_{y}$ and $G_{n}$ transform the layer $L=(mnl;d)$ to

$\displaystyle G_{y}L$
$\displaystyle=\left(\bar{m}n\bar{l}\Big{|}-\frac{l}{2}-d\right),$ (87)
$\displaystyle G_{n}L$
$\displaystyle=\left(\bar{m}n\bar{l}\Big{|}-\frac{1}{2}(m-n+l)-d\right),$ (88)

and

$G_{n}G_{y}^{-1}L=T_{1}L=\left(mnl;d+\frac{1}{2}(m+n)\right).$ (89)

Note that $G_{y}L$ is not written as $(m\bar{n}l;d-\frac{l}{2})$ because the
2D Chern insulator has its own orientation corresponding to the direction of
the chiral edge states. Therefore, the Chern invariants for the LC $L(mnl;d)$
in $\bm{{\it 9}}$ are given as

$\displaystyle\delta_{\mathcal{C}_{1}}=\delta_{\mathcal{C}_{2}},$ (90)
$\displaystyle=\left\\{\begin{array}[]{ll}n/2&(m=l=0,n=\mathrm{even},d=0,1/2)\\\
n&(m=l=0,n=\mathrm{odd},4d\in\mathbb{Z}\\\ &\ \mathrm{or}\
m=l=0,n=\mathrm{even},d\neq 0,1/2\\\ &\ \mathrm{or}\
(m,l)\neq(0,0),m+n=\mathrm{even})\\\
2n&(\mathrm{otherwise})\end{array}\right.,$ (95)
$\displaystyle\delta_{\mathcal{C}_{3}}=0.$ (96)

We emphasize that the Chern invariants $\delta_{\mathcal{C}_{i=1,2,3}}$ are
associated with the primitive lattice vectors for $\bm{{\it 9}}$,
$\bm{a}_{i}^{Cc}$ shown in Table 1.

The glide invariant $\delta_{\mathrm{g}_{y}}$ is given by

$\delta_{\mathrm{g}_{y}}\equiv\left\\{\begin{array}[]{ll}1&(m=l=0,d=0\
\text{or}\\\ &\ \ m=l=0,n=\text{odd},d=1/2)\\\
l&((m,l)\neq(0,0),m+n=\mathrm{even})\\\
0&(\mathrm{otherwise})\end{array}\right.$ (97)

modulo 2. The glide plane is chosen to be $y=0$, and not $y=1/2$. Similarly,
the other glide invariant $\delta_{\mathrm{g}_{n}}$ is given by

$\delta_{\mathrm{g}_{n}}\equiv\left\\{\begin{array}[]{ll}1&(m=l=0,n\in
4\mathbb{Z},d=0\ \mathrm{or}\\\ &\ m=l=0,n=\mathrm{odd},d=1/4,3/4\
\mathrm{or}\\\ &\ m=l=0,n=2\times\mathrm{odd},d=1/2)\\\
m+l&((m,l)\neq(0,0),m+n=\mathrm{even})\\\
0&(\mathrm{otherwise})\end{array}\right.$ (98)

modulo 2. Here we choose the glide plane to be $y=1/4$, and not $y=3/4$. One
can directly show that
$\delta_{g_{n}}\equiv\delta_{g_{y}}+\delta_{\mathcal{C}_{1}}\pmod{2}$.

####  B.3.2 Space group $\bm{{\it 15}}$

In $\bm{{\it 15}}$, we can define six kinds of invariants, the Chern
invariants $\delta_{\mathcal{C}_{i=1,2,3}}$, the two different types of glide
invariant $\delta_{\mathrm{g}_{y}}$, $\delta_{\mathrm{g}_{n}}$, the inversion
invariant $\delta_{\mathrm{i}}$, the $C_{2}$ rotation invariant
$\delta_{\mathrm{r}}$, and the $C_{2}$ screw invariant $\delta_{\mathrm{s}}$
for $\\{C_{2y}|(\hat{\bm{x}}+\hat{\bm{y}}+\hat{\bm{z}})/2\\}$. The Chern
invariant $\delta_{\mathcal{C}_{i}}$ takes an integer value, while the other
invariants $\delta_{\mathrm{g}_{y}}$, $\delta_{\mathrm{g}_{n}}$,
$\delta_{\mathrm{i}}$, and $\delta_{\mathrm{s}}$ are defined modulo 2, and
$\delta_{\mathrm{r}}$ turns out to be always zero. The inversion operator
transforms the layer $(mnl;d)$ into $(mnl;-d)$, and this leads to doubling to
the number of layers to be eight, if the additional layers are not identical
with the original ones. In this system, the Chern invariants are obtained as
follows:

$\displaystyle\delta_{\mathcal{C}_{1}}=\delta_{\mathcal{C}_{2}}$
$\displaystyle=\left\\{\begin{array}[]{ll}n/2&(m=l=0,n=\mathrm{even},\\\
&d=0,1/2)\\\ n&(m=l=0,n=\mathrm{odd},\\\ &d=0,1/4,1/2,3/4\\\
n&(m=l=0,n=\mathrm{even},d\neq 0,1/2)\\\
n&((m,l)\neq(0,0),m+n=\mathrm{even},d=0,1/2)\\\
2n&((m,l)\neq(0,0),m+n=\mathrm{even},\ d\neq 0,1/2\\\ &\ \mathrm{or}\
(m,l)\neq(0,0),m+n=\mathrm{odd},4d\in\mathbb{Z}\\\ &\ \mathrm{or}\
m=l=0,n=\mathrm{odd},d\neq 0,1/4,1/2,3/4)\\\
4n&(\mathrm{otherwise})\end{array}\right.,$ (109)
$\displaystyle\delta_{\mathcal{C}_{3}}=0.$ (110)

The glide invariant $\delta_{\mathrm{g}_{y}}$ and the inversion invariant
$\delta_{\mathrm{i}}$ are calculated, and they turn out to be completely
equal;

$\displaystyle\delta_{\mathrm{g}_{y}}$
$\displaystyle=\delta_{\mathrm{i}}\equiv\left\\{\begin{array}[]{ll}1&(m=l=0,d=0\\\
&\ \ \ \text{or}\ m=0,n=\text{odd},l=0,d=\frac{1}{2}))\\\
l&((m,l)\neq(0,0),m+n=\mathrm{even},\\\ &\ \ \ \ \ d=0,1/2)\\\
0&\mathrm{otherwise}\end{array}\right.$ (116)
$\displaystyle\delta_{\mathrm{g}_{y}}$ $\displaystyle=\delta_{\mathrm{i}}$
(117)

modulo 2. The other glide invariant $\delta_{\mathrm{g}_{n}}$ is given by

$\delta_{\mathrm{g}_{n}}\equiv\left\\{\begin{array}[]{ll}1&(m=l=0,n\in
4\mathbb{Z},d=0\ \mathrm{or}\\\ &\ m=l=0,n=\mathrm{odd},d=1/4,3/4\
\mathrm{or}\\\ &\ m=l=0,n/2=\mathrm{odd},d=\frac{1}{2})\\\
m+l&(m+n=\mathrm{even},d=0,1/2)\\\ 0&(\mathrm{otherwise})\end{array}\right.$
(118)

modulo 2. The $C_{2}$ screw invariant $\delta_{\mathrm{s}}$ is given by

$\delta_{\mathrm{s}}\equiv\delta_{\mathcal{C}_{1}}=\delta_{\mathcal{C}_{2}}$
(119)

modulo 2. They are related by the following equations:

$\displaystyle\delta_{\mathrm{i}}$
$\displaystyle\equiv\delta_{\mathrm{g}_{y}}\pmod{2},$ (120)
$\displaystyle\delta_{\mathrm{s}}$
$\displaystyle\equiv\frac{1}{2}\delta_{\mathcal{C}_{y}}\pmod{2},$ (121)
$\displaystyle\delta_{\mathrm{i}}+\delta_{\mathrm{s}}$
$\displaystyle\equiv\delta_{\mathrm{g}_{n}}\pmod{2},$ (122)

in which we define
$\delta_{\mathcal{C}_{y}}=\delta_{\mathcal{C}_{1}}+\delta_{\mathcal{C}_{2}}=2\delta_{\mathcal{C}_{1}}$.

Let us verify the relations between these invariants and the glide-$Z_{2}$
invariant $\nu$ and the Chern number $\mathcal{C}_{y}$. We can calculate $\nu$
and $\mathcal{C}_{y}$ for general layers and compare them with the invariants
from the LC. We then get

$\displaystyle\delta_{\mathcal{C}_{y}}$ $\displaystyle=\mathcal{C}_{y}\in
2\mathbb{Z},$ (123)
$\displaystyle\delta_{\mathrm{i}}\equiv\delta_{\mathrm{g}_{y}}$
$\displaystyle\equiv\nu\pmod{2},$ (124) $\displaystyle\delta_{\mathrm{s}}$
$\displaystyle\equiv\frac{1}{2}\mathcal{C}_{y}\pmod{2},$ (125)
$\displaystyle\delta_{\mathrm{g}_{n}}$
$\displaystyle\equiv\nu+\frac{1}{2}\mathcal{C}_{y}\pmod{2}.$ (126)

The topological invariants calculated from the LC hence totally agree with the
known topological invariants, the glide-$Z_{2}$ invariant $\nu$ and the Chern
number $\mathcal{C}_{y}$. We will show elementary layer constructions, showing
minimal layer configurations with nontrivial topological invariants from these
results shown in Table 3.

###  B.4 Elementary layer constructions

Following the argument in Ref. 59, we can find eLCs in $\bm{{\it 9}}$ and
$\bm{{\it 15}}$ summarized in Table 3. All SGs listed here have two
independent topological invariants as we have seen in this paper, Therefore,
each SG has two eLCs.

Table 3: Summary of elementary layer construction, symmetry-based indicators,
and topological invariants for $\bm{{\it 9}}$ and $\bm{{\it 15}}$.
$\delta_{\mathcal{C}_{i}}$ are Chern indices. In addition, $\delta_{g_{y}}$,
$\delta_{i}$, $\delta_{s}$ and $\delta_{g_{n}}$ are topological indices
associated with the glide $\\{M_{y}|\hat{\bm{z}}/2\\}$, inversion, $C_{2}$
screw $\\{C_{2y}|(\hat{\bm{x}}+\hat{\bm{y}}+\hat{\bm{z}})/2\\}$, and the glide
$\\{M_{y}|(\hat{\bm{x}}+\hat{\bm{y}}+\hat{\bm{z}})/2\\}$, respectively. In the
notation of Ref. 59, these four symmetry operations are written as
$\mathrm{g}^{010}_{00\frac{1}{2}}$, $i$,
$2^{010}_{\frac{1}{2}\frac{1}{2}\frac{1}{2}}$, and
$\mathrm{g}^{010}_{\frac{1}{2}\frac{1}{2}\frac{1}{2}}$.  $\begin{array}[]{c c
c
ccccc}\hline\cr\hline\cr\mathrm{SG}&\mathrm{eLC}&\mathbb{Z}_{2,2,2,4}&\lx@intercol\hfil\mathrm{Invariants}\hfil\lx@intercol\\\
\hline\cr&(mnl;d)&&\delta_{\mathcal{C}_{i}}&\delta_{g_{y}}&&&\delta_{g_{n}}\\\
\hline\cr\bm{{\it 9}}\ \ (Cc)&001;d_{0}&\mathrm{N}/\mathrm{A}&000&1&&&1\\\
&\bar{1}10;d_{0}&&110&0&&&1\\\
\hline\cr&(mnl;d)&&\delta_{\mathcal{C}_{i}}&\delta_{g_{y}}&\delta_{i}&\delta_{s}&\delta_{g_{n}}\\\
\hline\cr\bm{{\it 15}}\ (C2/c)&001;0&0002&000&1&1&0&1\\\
&\bar{1}10;0&1100&110&0&0&1&1\\\ \hline\cr\hline\cr\end{array}$

##  Appendix C Tight-binding models for the elementary layer constructions of
$\bm{{\it 15}}$

Let us consider the system with the SG $\bm{{\it 15}}$ with the glide symmetry
and inversion symmetry given by

$G_{y}=\left\\{M_{y}|\hat{\bm{z}}/2\right\\},\quad I=\\{I|0\\}.$ (127)

As we presented in Appendix B, there are two eLCs, $(001;0)$ and
$(\bar{1}10;0)$. We discuss these two eLCs separately.

###  C.1 $\nu=1,\ \mathcal{C}_{y}=0\ (L=(001;0))$

This eLC is given by putting a Chern insulator layer with $\mathcal{C}=1$ on
the $xy$ plane at $z=0$ with the inversion symmetry, and by generating other
layers by symmetry as shown in Fig 3(a). The model Hamiltonian for the layer
on the $z=0$ plane is written as

$\displaystyle\left(\sum_{x,y\in\mathbb{Z}}+\sum_{x,y\in\mathbb{Z}+\frac{1}{2}}\right)\left[\psi^{\dagger}(x+\frac{1}{2},y+\frac{1}{2})\frac{\sigma_{z}+i\sigma_{x}}{2}\psi(x,y)\right.$
$\displaystyle+\psi^{\dagger}(x-\frac{1}{2},y+\frac{1}{2})\frac{\sigma_{z}+i\sigma_{y}}{2}\psi(x,y)+h.c.$
$\displaystyle+\left.m\psi^{\dagger}(x,y)\sigma_{z}\psi(x,y)\right],$

satisfying

$\hat{I}\psi^{\dagger}(x,y)\hat{I}^{-1}=\psi^{\dagger}(-x,-y)\sigma_{z}.$

Making copies by the glide transformation

$\hat{G}_{y}\psi^{\dagger}(x,y,z)\hat{G}_{y}^{-1}=\psi^{\dagger}(x,-y,z+\frac{1}{2}).$
(128)

The Hamiltonian reads

$\displaystyle\hat{H}$
$\displaystyle=\left(\sum_{x,y,z\in\mathbb{Z}}+\sum_{x,y\in\mathbb{Z}+\frac{1}{2},z\in\mathbb{Z}}\right)\left[\psi^{\dagger}(x+\frac{1}{2},y+\frac{1}{2},z)\frac{\sigma_{z}+i\sigma_{x}}{2}\psi(x,y,z)+\psi^{\dagger}(x-\frac{1}{2},y+\frac{1}{2},z)\frac{\sigma_{z}+i\sigma_{y}}{2}\psi(x,y,z)\right.$
$\displaystyle+\psi^{\dagger}(x+\frac{1}{2},-y-\frac{1}{2},z+\frac{1}{2})\frac{\sigma_{z}+i\sigma_{x}}{2}\psi(x,-y,z+\frac{1}{2})+\psi^{\dagger}(x-\frac{1}{2},-y-\frac{1}{2},z+\frac{1}{2})\frac{\sigma_{z}+i\sigma_{y}}{2}\psi(x,-y,z+\frac{1}{2})+h.c.$
$\displaystyle+m\psi^{\dagger}(x,y,z)\sigma_{z}\psi(x,y,z)+\left.m\psi^{\dagger}(x,-y,z+\frac{1}{2})\sigma_{z}\psi(x,-y,z+\frac{1}{2})\right]$
$\displaystyle=\left(\sum_{x,y,z\in\mathbb{Z}}+\sum_{x,y\in\mathbb{Z}+\frac{1}{2},z\in\mathbb{Z}}\right)\left[\psi^{\dagger}(x+\frac{1}{2},y+\frac{1}{2},z)\frac{\sigma_{z}+i\sigma_{x}}{2}\psi(x,y,z)+\psi^{\dagger}(x-\frac{1}{2},y+\frac{1}{2},z)\frac{\sigma_{z}+i\sigma_{y}}{2}\psi(x,y,z)\right.$
$\displaystyle+\psi^{\dagger}(x+\frac{1}{2},y-\frac{1}{2},z+\frac{1}{2})\frac{\sigma_{z}+i\sigma_{x}}{2}\psi(x,y,z+\frac{1}{2})+\psi^{\dagger}(x-\frac{1}{2},y-\frac{1}{2},z+\frac{1}{2})\frac{\sigma_{z}+i\sigma_{y}}{2}\psi(x,y,z+\frac{1}{2})+h.c.$
$\displaystyle+m\psi^{\dagger}(x,y,z)\sigma_{z}\psi(x,y,z)+\left.m\psi^{\dagger}(x,y,z+\frac{1}{2})\sigma_{z}\psi(x,y,z+\frac{1}{2})\right]$

Let us introduce $k$-space basis

$\Psi^{\dagger}(k_{x},k_{y},k_{z})\equiv\left(\sum_{x,y,z\in\mathbb{Z}}+\sum_{x,y\in\mathbb{Z}+\frac{1}{2},z\in\mathbb{Z}}\right)\left(\psi^{\dagger}(x,y,z),\psi^{\dagger}(x,y,z+\frac{1}{2})\right)e^{i(k_{x}x+k_{y}y+k_{z}z)}.$
(129)

The glide and inversion are replaced as

$\displaystyle\hat{G}_{y}\Psi^{\dagger}(k_{x},k_{y},k_{z})\hat{G}_{y}^{-1}$
$\displaystyle=\left(\sum_{x,y,z\in\mathbb{Z}}+\sum_{x,y\in\mathbb{Z}+\frac{1}{2},z\in\mathbb{Z}}\right)\left(\psi^{\dagger}(x,-y,z+\frac{1}{2}),\psi^{\dagger}(x,-y,z+1)\right)e^{i(k_{x}x+k_{y}y+k_{z}z)}$
$\displaystyle=\Psi^{\dagger}(k_{x},-k_{y},k_{z})\begin{pmatrix}0&e^{-ik_{z}}\\\
1&0\end{pmatrix}_{\tau},$
$\displaystyle\hat{I}\Psi^{\dagger}(k_{x},k_{y},k_{z})\hat{I}^{-1}$
$\displaystyle=\left(\sum_{x,y,z\in\mathbb{Z}}+\sum_{x,y\in\mathbb{Z}+\frac{1}{2},z\in\mathbb{Z}}\right)\left(\psi^{\dagger}(-x,-y,-z),\psi^{\dagger}(-x,-y,-z-\frac{1}{2})\right)e^{i(k_{x}x+k_{y}y+k_{z}z)}\sigma_{z}$
$\displaystyle=\Psi^{\dagger}(-k_{x},-k_{y},-k_{z})\sigma_{z}\begin{pmatrix}1&0\\\
0&e^{-ik_{z}}\end{pmatrix}_{\tau}.$

To summarize, the momentum space Hamiltonian and symmetry operator are

$\displaystyle H(k_{x},k_{y},k_{z})$
$\displaystyle=\left(m+\cos\frac{k_{x}+k_{y}}{2}+\cos\frac{-k_{x}+k_{y}}{2}\right)\sigma_{z}\tau_{0}+\sin\frac{k_{x}+k_{y}}{2}\begin{pmatrix}\sigma_{x}&0\\\
0&-\sigma_{y}\end{pmatrix}+\sin\frac{k_{x}-k_{y}}{2}\begin{pmatrix}-\sigma_{y}&0\\\
0&\sigma_{x}\end{pmatrix}$
$\displaystyle=\left(m+2\cos\frac{k_{x}}{2}\cos\frac{k_{y}}{2}\right)\sigma_{z}\tau_{0}+\sin\frac{k_{x}}{2}\cos\frac{k_{y}}{2}(\sigma_{x}-\sigma_{y})\tau_{0}+\cos\frac{k_{x}}{2}\sin\frac{k_{y}}{2}(\sigma_{x}+\sigma_{y})\tau_{z},$
$\displaystyle G_{y}(k_{x},k_{y},k_{z})=\begin{pmatrix}0&e^{-ik_{z}}\\\
1&0\end{pmatrix}_{\tau},\quad
I(k_{x},k_{y},k_{z})=\sigma_{z}\begin{pmatrix}1&0\\\
0&e^{-ik_{z}}\end{pmatrix}_{\tau}.$

Let us compute the indicator for $-2<m<0$. The $C_{2}$ rotation around the
axis $x=0$, $z=1/4$ is given by

$C_{2}(k_{x},k_{y},k_{z})\equiv
G_{y}(-k_{x},-k_{y},-k_{z})I(k_{x},k_{y},k_{z})=\sigma_{z}\tau_{x}$ (130)

At high-symmetry points, the Hamiltonians and symmetry operators within the
occupied states are given by the following:

$\begin{array}[]{c|cccc}P&H(P)&C_{2}(P)|_{\mathrm{occ}}&G_{y}(P)|_{\mathrm{occ}}&I(P)|_{\mathrm{occ}}\\\
\hline\cr\Gamma&(m+2)\sigma_{z}&\sigma_{z}\tau_{x}&\tau_{x}&\sigma_{z}\\\
Y&(m-2)\sigma_{z}&\sigma_{z}\tau_{x}&\tau_{x}&\sigma_{z}\\\
V&m\sigma_{z}&\sigma_{z}\tau_{x}&\tau_{x}&\sigma_{z}\end{array}$

Thus the irreps for various values of $m$ in insulating cases ($m\neq 0,\pm
2$) are summarized in Table 2. Therefore, we find that the glide-$Z_{2}$
invariant is $\nu=1$ when $0<|m|<2$ and $\nu=0$ when $|m|>2$, in agreement
with the expected results.

###  C.2 $\nu=0,\ \mathcal{C}_{y}=2\ (L=(\bar{1}10;0))$

We here calculate the topological invariants for the LC $(\bar{1}10;0)$ as
shown in Fig. 3(b). This layer is accompanied by the layer $(110;0)$ by
$C_{2z}$ symmetry. We consider a Chern insulator with $\mathcal{C}=1$ on a
$[\bar{1}10]$ plane including the origin. Its Hamiltonian and the inversion
operation is written as

$\displaystyle\left(\sum_{x,y,z\in\mathbb{Z}}+\sum_{x,y\in\mathbb{Z}+\frac{1}{2},z\in\mathbb{Z}}\right)\left[\psi^{\dagger}(x,y,z+1)\frac{\sigma_{z}+i\sigma_{x}}{2}\psi(x,y,z)\right.$
$\displaystyle+\psi^{\dagger}(x+\frac{1}{2},y+\frac{1}{2},z)\frac{\sigma_{z}+i\sigma_{y}}{2}\psi(x,y,z)+h.c.$
$\displaystyle+\left.m\psi^{\dagger}(x,y,z)\sigma_{z}\psi(x,y,z)\right],$
(131)
$\hat{I}\psi^{\dagger}(x,y,z)\hat{I}^{-1}=\psi^{\dagger}(-x,-y,-z)\sigma_{z}.$
(132)

We then make its copies by the glide transformation

$\hat{G}_{y}\psi^{\dagger}(x,y,z)\hat{G}_{y}^{-1}=\psi(x,-y,z+\frac{1}{2}).$
(133)

Then the total Hamiltonian reads

$\displaystyle\hat{H}$
$\displaystyle=\left(\sum_{x,y,z\in\mathbb{Z}}+\sum_{x,y\in\mathbb{Z}+\frac{1}{2},z\in\mathbb{Z}}\right)\left[\psi^{\dagger}(x,y,z+1)\frac{\sigma_{z}+i\sigma_{x}}{2}\psi(x,y,z)+\psi^{\dagger}(x+\frac{1}{2},y+\frac{1}{2},z)\frac{\sigma_{z}+i\sigma_{y}}{2}\psi(x,y,z)\right.$
$\displaystyle+\psi^{\dagger}(x,-y,z+\frac{3}{2})\frac{\sigma_{z}+i\sigma_{x}}{2}\psi(x,-y,z+\frac{1}{2})+\psi^{\dagger}(x+\frac{1}{2},-y-\frac{1}{2},z+\frac{1}{2})\frac{\sigma_{z}+i\sigma_{y}}{2}\psi(x,-y,z+\frac{1}{2})+h.c.$
$\displaystyle+m\psi^{\dagger}(x,y,z)\sigma_{z}\psi(x,y,z)+\left.m\psi^{\dagger}(x,-y,z+\frac{1}{2})\sigma_{z}\psi(x,-y,z+\frac{1}{2})\right]$
$\displaystyle=\left(\sum_{x,y,z\in\mathbb{Z}}+\sum_{x,y\in\mathbb{Z}+\frac{1}{2},z\in\mathbb{Z}}\right)\left[\psi^{\dagger}(x,y,z+1)\frac{\sigma_{z}+i\sigma_{x}}{2}\psi(x,y,z)+\psi^{\dagger}(x+\frac{1}{2},y+\frac{1}{2},z)\frac{\sigma_{z}+i\sigma_{y}}{2}\psi(x,y,z)\right.$
$\displaystyle+\psi^{\dagger}(x,y,z+\frac{3}{2})\frac{\sigma_{z}+i\sigma_{x}}{2}\psi(x,y,z+\frac{1}{2})+\psi^{\dagger}(x+\frac{1}{2},y-\frac{1}{2},z+\frac{1}{2})\frac{\sigma_{z}+i\sigma_{y}}{2}\psi(x,y,z+\frac{1}{2})+h.c.$
$\displaystyle+m\psi^{\dagger}(x,y,z)\sigma_{z}\psi(x,y,z)+\left.m\psi^{\dagger}(x,y,z+\frac{1}{2})\sigma_{z}\psi(x,y,z+\frac{1}{2})\right].$

Let us introduce the $k$-space basis

$\displaystyle\Psi^{\dagger}(k_{x},k_{y},k_{z})=\left(\sum_{x,y,z\in\mathbb{Z}}+\sum_{x,y\in\mathbb{Z}+\frac{1}{2},z\in\mathbb{Z}}\right)$
$\displaystyle\ \ \
(\Psi^{\dagger}(x,y,z),\Psi^{\dagger}(x,y,z+\frac{1}{2}))e^{i(k_{x}x+k_{y}y+k_{z}z)}.$
(134)

The momentum space Hamiltonian and symmetry operators are represented as

$\displaystyle H(\bm{k})=\begin{pmatrix}H^{(+)}(\bm{k})&\\\
&H^{(-)}(\bm{k})\end{pmatrix}_{\tau}$ $\displaystyle
H^{(\pm)}(\bm{k})=\left(m+\cos k_{z}+\cos\frac{k_{x}\pm
k_{y}}{2}\right)\sigma_{z}$ $\displaystyle+\sin
k_{z}\sigma_{x}+\sin\frac{k_{x}\pm k_{y}}{2}\sigma_{y},$

and

$\displaystyle G_{y}(k_{x},k_{y},k_{z})=\begin{pmatrix}0&e^{-ik_{z}}\\\
1&0\end{pmatrix}_{\tau},$ $\displaystyle
I(k_{x},k_{y},k_{z})=\sigma_{z}\begin{pmatrix}1&0\\\
0&e^{-ik_{z}}\end{pmatrix}_{\tau}.$

The $C_{2}$ operation is

$C_{2}(k_{x},k_{y},k_{z})\equiv
G_{y}(-k_{x},-k_{y},-k_{z})I(k_{x},k_{y},k_{z})=\sigma_{z}\tau_{x}.$ (135)

Let us compute the indicator for $-2<m<0$. At the high-symmetry points, the
Hamiltonian and the symmetry operators within the occupied states are given by
the following:

$\begin{array}[]{c|ccc}P&H(P)&G_{y}|_{\mathrm{occ}}&I|_{\mathrm{occ}}\\\
\hline\cr\Gamma&(m+2)\sigma_{z}&\tau_{x}&\sigma_{z}\\\
Y&m\sigma_{z}&\tau_{x}&\sigma_{z}\\\
V(V^{\prime})&(m+1)\sigma_{z}\mp\sigma_{z}\tau_{z}&&\sigma_{z}\\\
M&(m-2)\sigma_{z}&-i\tau_{y}&\sigma_{z}\tau_{z}\\\
L(L^{\prime})&(m-1)\sigma_{z}\mp\sigma_{z}\tau_{z}&&\sigma_{z}\tau_{z}\\\
\end{array}$

Thus the irreps for various values of $m$ in insulating cases ($m\neq 0,\pm
2$) are summarized in Table 2. It confirms that $\mathcal{C}_{y}/2=1\pmod{2}$
when $|m|<2$ and $\mathcal{C}_{y}/2=0\pmod{2}$ when $|m|>2$, whereas $\nu$ is
trivial on both sides, in agreement with the expected results.

##  Appendix D Table for the irreps of the SG $\bm{{\it 15}}$

Here we summarize the irreps of the SG $\bm{{\it 15}}$ used throughout the
present paper in Table 4.

Table 4: Summary of irreducible representations, correspondences to characters
denoted in the main text and the numbers of states in the irreps for $\bm{{\it
15}}$ where $a,b,m,x,y$, and $z$ are integers. The number of states in this
table represents the compatibility relations. The positions of the high-
symmetry points and lines are $\Gamma=(0,0,0)$, $Y=(0,2\pi,0)$,
$V=(\pi,\pi,0)$, $A=(0,0,\pi)$, $M=(0,2\pi,\pi)$, $L=(\pi,\pi,\pi)$,
$\Lambda=(0,v,0)$, and $B=(u,0,w)$. $\begin{array}[]{c c c c c
c}\hline\cr\hline\cr\mathrm{Seitz}&\\{1|t_{1},t_{2},t_{3}\\}&\\{2_{010}|0,0,1/2\\}&\\{\bar{1}|0,0,0\\}&\\{m_{010}|0,0,1/2\\}&\begin{matrix}\mathrm{number}\\\
\mathrm{of}\\\ \mathrm{states}\end{matrix}\\\
\hline\cr\begin{matrix}\mathrm{Matrix}\\\
\mathrm{presentation}\end{matrix}&\begin{pmatrix}1&0&0&t_{1}\\\ 0&1&0&t_{2}\\\
0&0&1&t_{3}\end{pmatrix}&\begin{pmatrix}-1&0&0&0\\\ 0&1&0&0\\\
0&0&-1&1/2\end{pmatrix}&\begin{pmatrix}-1&0&0&0\\\ 0&-1&0&0\\\
0&0&-1&0\end{pmatrix}&\begin{pmatrix}1&0&0&0\\\ 0&-1&0&0\\\
0&0&1&1/2\end{pmatrix}&\\\ \hline\cr\Gamma^{+}_{1}&1&1&1&1&a\\\
\Gamma^{-}_{1}&1&1&-1&-1&b+m\\\ \Gamma^{+}_{2}&1&-1&1&-1&a-m\\\
\Gamma^{-}_{2}&1&-1&-1&1&b\\\\[4.0pt] \mathrm{Y}^{+}_{1}&e^{i2\pi
t_{2}}&1&1&1&a+x\\\ \mathrm{Y}^{-}_{1}&e^{i2\pi t_{2}}&1&-1&-1&b+m-x\\\
\mathrm{Y}^{+}_{2}&e^{i2\pi t_{2}}&-1&1&-1&a-m+x\\\
\mathrm{Y}^{-}_{2}&e^{i2\pi t_{2}}&-1&-1&1&b-x\\\\[4.0pt]
\mathrm{V}^{+}_{1}&e^{i\pi(t_{1}+t_{2})}&&1&&a+b+y\\\
\mathrm{V}^{-}_{1}&e^{i\pi(t_{1}+t_{2})}&&-1&&a+b-y\\\\[4.0pt]
\mathrm{A}_{1}&e^{i\pi
t_{3}}\sigma_{0}&\sigma_{x}&\sigma_{z}&-i\sigma_{y}&2(a+b)\\\\[4.0pt]
\mathrm{M}_{1}&e^{i\pi(2t_{2}+t_{3})}\sigma_{0}&\sigma_{x}&\sigma_{z}&-i\sigma_{y}&2(a+b)\\\\[4.0pt]
\mathrm{L}^{+}_{1}&e^{i\pi(t_{1}+t_{2}+t_{3})}&&1&&a+b+z\\\
\mathrm{L}^{-}_{1}&e^{i\pi(t_{1}+t_{2}+t_{3})}&&-1&&a+b-z\\\\[4.0pt]
\Lambda_{1}&e^{i2\pi t_{2}v}&1&&&a+b+m\\\ \Lambda_{1}&e^{i2\pi
t_{2}v}&-1&&&a+b-m\\\\[4.0pt]
\mathrm{B}_{1}&e^{i2\pi(t_{1}u+t_{3}w)}&&&e^{i\pi w}&a+b\\\
\mathrm{B}_{2}&e^{i2\pi(t_{1}u+t_{3}w)}&&&e^{i\pi(1+w)}&a+b\\\\[4.0pt]
\hline\cr\hline\cr\end{array}$

## References

  * Klitzing  _et al._ [1980] K. v. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980).
  * Hasan and Kane [2010] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
  * Qi and Zhang [2011] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).
  * Fang  _et al._ [2012] C. Fang, M. J. Gilbert, and B. A. Bernevig, Phys. Rev. B 86, 115112 (2012).
  * Slager  _et al._ [2013] R.-J. Slager, A. Mesaros, V. Juricic, and J. Zaanen, Nat. Phys. 9, 98 (2013).
  * Fang  _et al._ [2013] C. Fang, M. J. Gilbert, and B. A. Bernevig, Phys. Rev. B 87, 035119 (2013).
  * Chiu  _et al._ [2013] C.-K. Chiu, H. Yao, and S. Ryu, Phys. Rev. B 88, 075142 (2013).
  * Zhang  _et al._ [2013] F. Zhang, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 111, 056403 (2013).
  * Morimoto and Furusaki [2013] T. Morimoto and A. Furusaki, Phys. Rev. B 88, 125129 (2013).
  * Jadaun  _et al._ [2013] P. Jadaun, D. Xiao, Q. Niu, and S. K. Banerjee, Phys. Rev. B 88, 085110 (2013).
  * Ueno  _et al._ [2013] Y. Ueno, A. Yamakage, Y. Tanaka, and M. Sato, Phys. Rev. Lett. 111, 087002 (2013).
  * Alexandradinata  _et al._ [2014] A. Alexandradinata, C. Fang, M. J. Gilbert, and B. A. Bernevig, Phys. Rev. Lett. 113, 116403 (2014).
  * Shiozaki and Sato [2014] K. Shiozaki and M. Sato, Phys. Rev. B 90, 165114 (2014).
  * Kargarian and Fiete [2013] M. Kargarian and G. A. Fiete, Phys. Rev. Lett. 110, 156403 (2013).
  * Kindermann [2015] M. Kindermann, Phys. Rev. Lett. 114, 226802 (2015).
  * Weng  _et al._ [2014] H. Weng, J. Zhao, Z. Wang, Z. Fang, and X. Dai, Phys. Rev. Lett. 112, 016403 (2014).
  * Fulga  _et al._ [2014] I. C. Fulga, B. van Heck, J. M. Edge, and A. R. Akhmerov, Phys. Rev. B 89, 155424 (2014).
  * Hsieh  _et al._ [2014] T. H. Hsieh, J. Liu, and L. Fu, Phys. Rev. B 90, 081112(R) (2014).
  * Tang and Fu [2014] E. Tang and L. Fu, Nat. Phys. 10, 964 (2014).
  * Ando and Fu [2015] Y. Ando and L. Fu, Annual Review of Condensed Matter Physics 6, 361 (2015).
  * Chiu  _et al._ [2016] C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Rev. Mod. Phys. 88, 035005 (2016).
  * Hsieh  _et al._ [2012] T. H. Hsieh, H. Lin, J. Liu, W. Duan, A. Bansil, and L. Fu, Nat. Commun. 3, 982 (2012).
  * Dziawa  _et al._ [2012] P. Dziawa, B. J. Kowalski, K. Dybko, R. Buczko, A. Szczerbakow, M. Szot, E. Łusakowska, T. Balasubramanian, B. M. Wojek, M. H. Berntsen, O. Tjernberg, and T. Story, Nat. Mater. 11, 1023 (2012).
  * Tanaka  _et al._ [2012] Y. Tanaka, Z. Ren, T. Sato, K. Nakayama, S. Souma, T. Takahashi, K. Segawa, and Y. Ando, Nat. Phys. 8, 800 (2012).
  * Xu  _et al._ [2012] S.-Y. Xu, C. Liu, N. Alidoust, M. Neupane, D. Qian, I. Belopolski, J. Denlinger, Y. Wang, H. Lin, L. Wray, G. Landolt, B. Slomski, J. Dil, A. Marcinkova, E. Morosan, Q. Gibson, R. Sankar, F. Chou, R. Cava, A. Bansil, and M. Hasan, Nat. Commun. 3, 1192 (2012).
  * Okada  _et al._ [2013] Y. Okada, M. Serbyn, H. Lin, D. Walkup, W. Zhou, C. Dhital, M. Neupane, S. Xu, Y. J. Wang, R. Sankar, _et al._ , Science 341, 1496 (2013).
  * Liang  _et al._ [2013] T. Liang, Q. Gibson, J. Xiong, M. Hirschberger, S. P. Koduvayur, R. Cava, and N. Ong, Nat. Commun. 4, 2696 (2013).
  * Liu  _et al._ [2013] J. Liu, W. Duan, and L. Fu, Phys. Rev. B 88, 241303(R) (2013).
  * Sun  _et al._ [2013] Y. Sun, Z. Zhong, T. Shirakawa, C. Franchini, D. Li, Y. Li, S. Yunoki, and X.-Q. Chen, Phys. Rev. B 88, 235122 (2013).
  * Tang  _et al._ [2014] P. Tang, B. Yan, W. Cao, S.-C. Wu, C. Felser, and W. Duan, Phys. Rev. B 89, 041409(R) (2014).
  * Liu  _et al._ [2014a] J. Liu, T. H. Hsieh, P. Wei, W. Duan, J. Moodera, and L. Fu, Nat. Mater. 13, 178 (2014a).
  * Ozawa  _et al._ [2014] H. Ozawa, A. Yamakage, M. Sato, and Y. Tanaka, Phys. Rev. B 90, 045309 (2014).
  * Wrasse and Schmidt [2014] E. O. Wrasse and T. M. Schmidt, Nano letters 14, 5717 (2014).
  * Liu  _et al._ [2015] J. Liu, X. Qian, and L. Fu, Nano Lett. 15, 2657 (2015).
  * Gyenis  _et al._ [2013] A. Gyenis, I. K. Drozdov, S. Nadj-Perge, O. B. Jeong, J. Seo, I. Pletikosi$\acute{c}$, T. Valla, G. D. Gu, and A. Yazdani, Phys. Rev. B 88, 125414 (2013).
  * Liu  _et al._ [2014b] C.-X. Liu, R.-X. Zhang, and B. K. VanLeeuwen, Phys. Rev. B 90, 085304 (2014b).
  * Parameswaran  _et al._ [2013] S. A. Parameswaran, A. M. Turner, D. P. Arovas, and A. Vishwanath, Nat. Phys. 9, 299 (2013).
  * Young and Kane [2015] S. M. Young and C. L. Kane, Phys. Rev. Lett. 115, 126803 (2015).
  * Varjas  _et al._ [2015] D. Varjas, F. de Juan, and Y.-M. Lu, Phys. Rev. B 92, 195116 (2015).
  * Watanabe  _et al._ [2015] H. Watanabe, H. C. Po, A. Vishwanath, and M. Zaletel, Proceedings of the National Academy of Sciences 112, 14551 (2015).
  * Po  _et al._ [2016] H. C. Po, H. Watanabe, M. P. Zaletel, and A. Vishwanath, Sci. Adv. 2, e1501782 (2016).
  * Fang and Fu [2015] C. Fang and L. Fu, Phys. Rev. B 91, 161105(R) (2015).
  * Shiozaki  _et al._ [2015] K. Shiozaki, M. Sato, and K. Gomi, Phys. Rev. B 91, 155120 (2015).
  * Dong and Liu [2016] X.-Y. Dong and C.-X. Liu, Phys. Rev. B 93, 045429 (2016).
  * Chen  _et al._ [2016] Y. Chen, H.-S. Kim, and H.-Y. Kee, Phys. Rev. B 93, 155140 (2016).
  * Kim and Murakami [2016] H. Kim and S. Murakami, Phys. Rev. B 93, 195138 (2016).
  * Shiozaki  _et al._ [2016] K. Shiozaki, M. Sato, and K. Gomi, Phys. Rev. B 93, 195413 (2016).
  * Zhao and Schnyder [2016] Y. X. Zhao and A. P. Schnyder, Phys. Rev. B 94, 195109 (2016).
  * Chang  _et al._ [2017] P.-Y. Chang, O. Erten, and P. Coleman, Nat. Phys. 13, 794 (2017).
  * Shiozaki  _et al._ [2018] K. Shiozaki, M. Sato, and K. Gomi, arXiv preprint arXiv:1802.06694 (2018).
  * Zhang  _et al._ [2019] D. Zhang, M. Shi, T. Zhu, D. Xing, H. Zhang, and J. Wang, Phys. Rev. Lett. 122, 206401 (2019).
  * Li  _et al._ [2019a] J. Li, Y. Li, S. Du, Z. Wang, B.-L. Gu, S.-C. Zhang, K. He, W. Duan, and Y. Xu, Science Advances 5 (2019a).
  * Vidal  _et al._ [2019] R. C. Vidal, H. Bentmann, T. R. F. Peixoto, A. Zeugner, S. Moser, C.-H. Min, S. Schatz, K. Kißner, M. Ünzelmann, C. I. Fornari, H. B. Vasili, M. Valvidares, K. Sakamoto, D. Mondal, J. Fujii, I. Vobornik, S. Jung, C. Cacho, T. K. Kim, R. J. Koch, C. Jozwiak, A. Bostwick, J. D. Denlinger, E. Rotenberg, J. Buck, M. Hoesch, F. Diekmann, S. Rohlf, M. Kalläne, K. Rossnagel, M. M. Otrokov, E. V. Chulkov, M. Ruck, A. Isaeva, and F. Reinert, Phys. Rev. B 100, 121104(R) (2019).
  * Wu  _et al._ [2019] J. Wu, F. Liu, M. Sasase, K. Ienaga, Y. Obata, R. Yukawa, K. Horiba, H. Kumigashira, S. Okuma, T. Inoshita, and H. Hosono, Sci. Adv. 5 (2019).
  * Li  _et al._ [2019b] H. Li, S.-Y. Gao, S.-F. Duan, Y.-F. Xu, K.-J. Zhu, S.-J. Tian, J.-C. Gao, W.-H. Fan, Z.-C. Rao, J.-R. Huang, J.-J. Li, D.-Y. Yan, Z.-T. Liu, W.-L. Liu, Y.-B. Huang, Y.-L. Li, Y. Liu, G.-B. Zhang, P. Zhang, T. Kondo, S. Shin, H.-C. Lei, Y.-G. Shi, W.-T. Zhang, H.-M. Weng, T. Qian, and H. Ding, Phys. Rev. X 9, 041039 (2019b).
  * Hao  _et al._ [2019] Y.-J. Hao, P. Liu, Y. Feng, X.-M. Ma, E. F. Schwier, M. Arita, S. Kumar, C. Hu, R. Lu, M. Zeng, Y. Wang, Z. Hao, H.-Y. Sun, K. Zhang, J. Mei, N. Ni, L. Wu, K. Shimada, C. Chen, Q. Liu, and C. Liu, Phys. Rev. X 9, 041038 (2019).
  * Chen  _et al._ [2019] Y. J. Chen, L. X. Xu, J. H. Li, Y. W. Li, H. Y. Wang, C. F. Zhang, H. Li, Y. Wu, A. J. Liang, C. Chen, S. W. Jung, C. Cacho, Y. H. Mao, S. Liu, M. X. Wang, Y. F. Guo, Y. Xu, Z. K. Liu, L. X. Yang, and Y. L. Chen, Phys. Rev. X 9, 041040 (2019).
  * Bradlyn  _et al._ [2017] B. Bradlyn, L. Elcoro, J. Cano, M. Vergniory, Z. Wang, C. Felser, M. Aroyo, and B. A. Bernevig, Nature 547, 298 (2017).
  * Song  _et al._ [2018] Z. Song, T. Zhang, Z. Fang, and C. Fang, Nat. Commun. 9, 3530 (2018).
  * Kim  _et al._ [2019] H. Kim, K. Shiozaki, and S. Murakami, Phys. Rev. B 100, 165202 (2019).
  * Po  _et al._ [2017] H. C. Po, A. Vishwanath, and H. Watanabe, Nat. Commun. 8, 50 (2017).
  * Watanabe  _et al._ [2018] H. Watanabe, H. C. Po, and A. Vishwanath, Sci. Adv. 4, eaat8685 (2018).
  * Aroyo  _et al._ [2006] M. I. Aroyo, J. M. Perez-Mato, C. Capillas, E. Kroumova, S. Ivantchev, G. Madariaga, A. Kirov, and H. Wondratschek, Zeitschrift für Kristallographie-Crystalline Materials 221, 15 (2006).
  * Hughes  _et al._ [2011] T. L. Hughes, E. Prodan, and B. A. Bernevig, Phys. Rev. B 83, 245132 (2011).
  * Ono and Watanabe [2018] S. Ono and H. Watanabe, Phys. Rev. B 98, 115150 (2018).
  * Hahn [2002] T. Hahn,  _International Tables for Crystallography, Volume A: Space Group Symmetry 5th Edition_ (Springer, 2002).