# Theoretical analysis of glide-$Z_{2}$ magnetic topological photonic crystals

Heejae Kim  1 Hengbin Cheng  2 Ling Lu  2  and Shuichi Murakami1,3
1Department of Physics, Tokyo Institute of Technology, 2-12-1 Ookayama,
Meguro-ku, Tokyo, 152-8551, Japan
2Institute of Physics, Chinese Academy of Sciences/Beijing National Laboratory
for Condensed Matter Physics, Beijing 100190, China
3TIES, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo,
152-8551, Japan<EMAIL_ADDRESS>
###### Abstract

Gapped systems with glide symmetry can be characterized by a $Z_{2}$
topological invariant. We study the magnetic photonic crystal with a gap
between the second and third lowest bands, which is characterized by the
nontrivial glide-$Z_{2}$ topological invariant that can be determined by
symmetry-based indicators. We show that under the space group No. $\bm{{\it
230}}$ ($Ia\bar{3}d$), the topological invariant is equal to a half of the
number of photonic bands below the gap, and therefore, the band gap between
the second and third lowest bands is always topologically nontrivial, and to
realize the topological phase, we need to open a gap for the Dirac point at
the $P$ point by breaking time-reversal symmetry. With staggered
magnetization, the photonic bands are gapped and the photonic crystal becomes
topological, whereas with uniform magnetization, a gap does not open, which
can be attributed to the minimal band connectivity exceeding two in this case.
By introducing the notion of Wyckoff positions, we show how the topological
characteristics are determined from the structure of the photonic crystals.

††articletype: Research Article

##  1 Introduction

Manipulations of magnetic topological materials are an intriguing and
promising topic in condensed matter physics. In particular, a glide-symmetric
topological crystalline insulator (TCI) exhibits a $Z_{2}$ topological phase
in class A hosting a single surface Dirac cone topological phase with the
nontrivial $Z_{2}$ topopogical invariant [1, 2, 3, 4]. One of our targets for
material realizations is bosonic systems. In photonics, a new type of
topological band-crossing points beyond the Weyl and Dirac points emerges. For
example, at a generalized Dirac point [5], the bands are four-fold degenerate
and they split into three or four bands along any direction, in contrast with
a Dirac points, which generally splits into two sets of double degenerate
bands along any direction. When the system is perturbed, this generalized
Dirac point becomes line nodes, Weyl points, or opens a band gap [6, 5]. As an
example, in the photonic crystal having a structure in the first blue phase of
liquid crystals (BPI) [5] shown in Fig. 1, by introducing magnetization to
break time-reversal symmetry (TRS) the generalized Dirac point opens a band
gap and the system goes into topological phases protected by glide symmetry.
On the other hand, in the double gyroid (DG) photonic crystal [6] (Fig. 1(b))
belonging to the same space group $\bm{{\it 230}}$ (henceforth, we call a
space group by its number in bold italic following in Ref. [7]) with the BPI
photonic crystal, the system has Weyl points [8] between the fourth and fifth
lowest bands in the absence of TRS. Nonetheless, how the band structures and
topological properties are determined from the structure of the photonic
crystals needs to be clarified.

In the present paper, we show how this topological phase can be manipulated,
based on the relationship between space group representations and band
structures. In the BPI photonic crystal, the gap opening at non-equivalent $P$
and $P^{\prime}$ points in the Brillouin zone (BZ) (Fig. 1(c)) makes the
system gapped and topologically nontrivial. However, the reason why this
structure leads to the topolgical phase, and how the eigenmodes at these $P$
and $P^{\prime}$ points affect the topological phase is not yet clear. In this
paper, we clarify these points, by using the formula of the glide-$Z_{2}$
invariant which we derived in our previous paper [9]. In particular, in the
space group $\bm{{\it 230}}$, which the photonic crystals in the previous
works [6, 5] belong to, the topological invariant is equal to a half of the
number of bands below the gap in terms of modulo 2, and therefore, the gap
between the second and third lowest bands is always topologically nontrivial.
On the other hand, in the presence of the TRS, since all modes at the $P$
points are fourfold degenerate, the band gap between the second and the third
lowest modes is closed at the $P$ point. Therefore, we conclude that when this
gap is open by breaking the TRS, the photonic crystal is a topological one
protected by the glide symmetry, as has been realized in the BPI photonic
crystal [5] with staggered magnetization. We also show that it automatically
becomes a higher-order topological insulators [10, 11, 12, 13, 14, 15]. On the
other hand, in a uniform magnetization, a gap does not open between the second
and third lowest bands, which is explained by the notion of the minimal band
connectivity [16]. Then, we propose a way to design such a topological
photonic crystal by introducing the notion of Wyckoff positions into the
design of photonic crystals.

This paper is organized as follows. Section 2 is devoted to reviewing the
previous works necessary for the discussion of the main part of this paper. In
Sec. 3, we analyze the irreducible representations of the photonic crystals
considered, and discuss which cases will realize topological photonic
crystals. In Sec. 4 we introduce the notion of the Wyckoff positions into the
design of the photonic crystals, and discuss the relationship between the
positions of the dielectrics and topological phases. We conclude the paper in
Sec. 5.

Figure 1: The BPI and DG photonic crystals in the body-centered cubic (BCC)
lattice and their Brillouin zone (BZ) of the BCC lattice. (a) The BPI photonic
crystal, shown in the cubic unit cell of length $a$. It consists of four
identical dielectric rods oriented along the BCC lattice vectors along (111)
(red), $(11\bar{1})$ (yellow), $(1\bar{1}1)$ (blue) and $(\bar{1}11)$ (green),
and they go through the points (0,0,0)$a$, (0,0.5,0)$a$, (0.5,0,0)$a$ and
(0,0,0.5)$a$, respectively. (b) The DG photonic crystal whose dielectric
regions are given by $g(\bm{r})>\lambda_{\mathrm{iso}}=1.1$ (blue) and
$g(-\bm{r})>\lambda_{\mathrm{iso}}=1.1$ (red) of Eq. (2). (c) The BZ of the
BCC lattice and its high-symmetry points $\Gamma$, $P$, $N$ and H.

##  2 $Z_{2}$ topological photonic crystal with glide symmetry

In this section, we first review the BPI photonic crystal, which is proposed
to be in the topological phase with glide symmetry in Ref. [17], and then we
analyze its characteristics from our viewpoint.

###  2.1 Review on the BPI topological photonic crystal

We consider a 3D photonic crystal in a body-centered cubic (BCC) lattice,
composed of four identical dielectric rods. A schematic illustration of the
system is shown in Fig. 1(a). We take the $x$, $y$, $z$ axes to be parallel to
the edges of the cubic unit cell. The four dielectric rods are along the BCC
primitive vectors along (111), $(11\bar{1})$, $(1\bar{1}1)$ and $(\bar{1}11)$,
and they go through the points (0,0,0)$a$, (0,0.5,0)$a$, (0.5,0,0)$a$ and
(0,0,0.5)$a$, respectively, where $a$ is the lattice constant of the cubic
cell. The space group is $\bm{{\it 230}}$ ($Ia\bar{3}d$), which is
nonsymmorphic, containing glide reflections and inversion.

When we set the dielectric constant of the rods to be $\varepsilon=11$ and the
radius of the rods to be $r=0.13a$, the fourfold generalized Dirac point
appears at two non-equivalent $P$ points in the BZ formed by the bands 1, 2, 3
and 4 counted from the lowest band (Fig. 2(b)), because of the cubic and time-
reversal (TR) symmetries [5]. If the magnetization is present in these rods
with the glide symmetry preserved, the band structure acquires a band gap
between bands 2 and 3 by breaking the TRS and it becomes the topological phase
with the nontrivial $Z_{2}$ topopogical invariant [1, 2, 3] protected by glide
symmetry [5]. To be more specific, in the dielectric tensor of the rod
$\bm{\varepsilon}$, we add off-diagonal imaginary terms $\kappa$, which breaks
TRS,

$\bm{\varepsilon}=\begin{pmatrix}\varepsilon_{\parallel}&\kappa&0\\\
-\kappa&\varepsilon_{\parallel}&0\\\ 0&0&\varepsilon_{33}\end{pmatrix},$ (1)

where $\varepsilon_{\parallel}\ (>0)$ and $\varepsilon_{33}\ (>0)$ are
dielectric constants with
$\varepsilon_{\parallel}^{2}-|\kappa|^{2}=\varepsilon_{33}^{2}$ [5, 6], and
$\kappa$ is a non-zero imaginary number. These off-diagonal imaginary terms
lead to gyroelectric response of materials [18]. Similarly, gyromagnetic
response in ferrites is generated by off-diagonal terms of the permeability
tensor [19]. In the calculation, the off-diagonal terms are introduced in a
staggered way as $\kappa=\pm 10i,\pm 10i,\mp 10i,\mp 10i$ with
$\varepsilon_{33}=11$ for the dielectric rods along (111) (red), $(11\bar{1})$
(yellow), $(1\bar{1}1)$ (blue) and $(\bar{1}11)$ (green) shown in Fig. 1(a).
In Fig. 2(a) and (c), we show the band structures of the BPI photonic crystal
with these two patterns of the sign of $\kappa$, with the rod along the (111)
direction having $\kappa_{(111)}=\pm 10i$, respectively. This value of
$\kappa$ does not reflect that in real materials, but is only for
demonstration of the behaviors of the gap. At $\kappa_{(111)}=0$ preserving
TRS, a four-fold degenerate generalized Dirac point exists at the $P$ point
(Fig. 2(b)). Then by breaking the TRS via nonzero $\kappa_{(111)}$, a band gap
opens between the bands 2 and 3 at the $P$ points, and both of these cases
with $\kappa_{(111)}=\pm 10i$ are found to be topological in Ref. [5].

Figure 2: Band structures for (a)-(c) the BPI photonic crystal. The band
structures with magnetization where (a) $\varepsilon_{33}=11$ and
$\kappa=10i,10i,-10i,-10i$ and (c) with $\varepsilon_{33}=11$ and
$\kappa=-10i,-10i,10i,10i$ for the red, yellow, green and blue rods shown in
Fig. 1(a), respectively, are exactly the same, because of the TRS, and both
sides are topological.

###  2.2 Discussion on symmetry and topology of the topological photonic
crystal

Thus, regardless of the sign of $\kappa_{(111)}$, the band gap opens at the
$P$ point and this gap becomes topological. This is related with the fact that
the irreducible representations (irreps) at the $P$ points do not affect the
values of the topological invariant; namely, the gap at $P$ is inverted
between $\kappa_{(111)}=+10i$ and $\kappa_{(111)}=-10i$, but it does not
affect the topological invariant. This situation is similar to the Kane-Mele
model, which is a well-known model for the two-dimensional topological
insulator (TI) phase with TRS [20, 21]. The Kane-Mele model is topological,
regardless of the sign of the spin-orbit coupling $\lambda_{\mathrm{so}}$,
which opens the gap at the $K$ point. Thus both in the BPI photonic crystal
and in the Kane-Mele model, the topological invariant is independent of the
irreps at $P$ or $K$ points, and regardless of the sign of the parameter
($\kappa_{(111)}$ or $\lambda_{\mathrm{so}}$) to open a gap, the system
becomes topological as long as the parameter is nonzero. More details are
presented in Sec. S1 in the Supplemental Document.

##  3 Symmetry consideration for band theory of glide-symmetric $Z_{2}$
magnetic topological photonic crystal

In this section, we discuss what determines the topological invariant in the
photonic crystal from the symmetry viewpoint. We first investigate the irreps
for the glide-symmetric $Z_{2}$ magnetic topological photonic crystals, i.e.
the BPI [5], and DG [6] photonic crystals. Then, we calculate the
glide-$Z_{2}$ invariant by using our new formula derived in Ref. [9].

###  3.1 Representations at the high-symmetry points from symmetry
considerations

We first explain irreps at high-symmetry points in the BZ in the space group
No. $\bm{{\it 230}}$. There are four kinds of high-symmetry points $\Gamma$,
$H$, $N$, and $P$ in the BZ (see Fig. 4 (c)). Among these four high-symmetry
points, the points $\Gamma$, $H$, and $N$ are TRIMs in BZ, but $P$ is not.
Even though $P$ is not a TRIM, the bands at the $P$ point should be degenerate
in the presence of TRS. This double degeneracy comes from combined symmetry
$\mathcal{T}^{\prime}=\mathcal{T}G_{y}$ where $\mathcal{T}$ is time-reversal
operation, and $G_{y}$ is the glide symmetry
$G_{y}=\\{M_{y}|\frac{a}{2}\hat{z}\\}$, where $\hat{\bm{x}},\hat{\bm{y}}$ and
$\hat{\bm{z}}$ denote the unit vectors along $\bm{x},\bm{y},$ and $\bm{z}$
directions, respectively. As $\mathcal{T}^{\prime}$ is an antiunitary
symmetry, and it satisfies
$\left(\mathcal{T}^{\prime}\right)^{2}=G_{y}^{2}=T_{z}=-1$ at the $P$ point,
where $T_{z}=\\{E|a\hat{z}\\}$, and $E$ is an identity operation, the bands
are always doubly degenerate at the $P$ point.

Our strategy is the following. We first start with a vacuum having a
homogeneous dielectric constant $\varepsilon=1$. Thereby, the eigenmodes
behave as electromagnetic plane waves with a linear polarization in free
space. From symmetry considerations for these plane-wave basis functions, we
can identify representations at high-symmetry points which characterize the
eigenmodes at $\varepsilon=1$ (Fig. 3(a)). Next, we gradually increase the
dielectric constant from $\varepsilon=1$ within the dielectrics to describe a
photonic crystal. Then, the eigenmodes split into irreps at the high-symmetry
points in the BZ, from which we discuss the topological invariant.

Table 1: Eigenmodes with plane waves at the $H^{(0)}$ point.
$\bm{\hat{x}},\bm{\hat{y}}$, and $\bm{\hat{z}}$ denote the unit vectors along
$\bm{x},\bm{y}$, and $\bm{z}$ directions, respectively. We put $\omega=e^{2\pi
i/3}$ here. $\begin{array}[]{c|c}\hline\cr\hline\cr
H_{1}&\begin{matrix}\psi^{1}_{H_{1}}=\sin\left(\frac{2\pi}{a}x\right)\bm{\hat{y}}+\sin\left(\frac{2\pi}{a}y\right)\bm{\hat{z}}+\sin\left(\frac{2\pi}{a}z\right)\bm{\hat{x}}\\\
\psi^{2}_{H_{1}}=\cos\left(\frac{2\pi}{a}y\right)\bm{\hat{x}}+\cos\left(\frac{2\pi}{a}x\right)\bm{\hat{z}}+\cos\left(\frac{2\pi}{a}z\right)\bm{\hat{y}}\end{matrix}\\\
\hline\cr
H_{2}&\begin{matrix}\psi^{1}_{H_{2}}=\sin\left(\frac{2\pi}{a}x\right)\bm{\hat{y}}+\omega\sin\left(\frac{2\pi}{a}y\right)\bm{\hat{z}}+\omega^{2}\sin\left(\frac{2\pi}{a}z\right)\bm{\hat{x}}\\\
\psi^{2}_{H_{2}}=\cos\left(\frac{2\pi}{a}y\right)\bm{\hat{x}}+\omega\cos\left(\frac{2\pi}{a}x\right)\bm{\hat{z}}+\omega^{2}\cos\left(\frac{2\pi}{a}z\right)\bm{\hat{y}}\end{matrix}\\\
\hline\cr
H_{3}&\begin{matrix}\psi^{1}_{H_{3}}=\sin\left(\frac{2\pi}{a}x\right)\bm{\hat{y}}+\omega^{2}\sin\left(\frac{2\pi}{a}y\right)\bm{\hat{z}}+\omega\sin\left(\frac{2\pi}{a}z\right)\bm{\hat{x}}\\\
\psi^{2}_{H_{3}}=\cos\left(\frac{2\pi}{a}y\right)\bm{\hat{x}}+\omega^{2}\cos\left(\frac{2\pi}{a}x\right)\bm{\hat{z}}+\omega\cos\left(\frac{2\pi}{a}z\right)\bm{\hat{y}}\end{matrix}\\\
\hline\cr
H_{4}&\begin{matrix}\psi_{H_{4}}^{1}=\cos\left(\frac{2\pi}{a}z\right)\bm{\hat{y}},&\psi_{H_{4}}^{2}=\cos\left(\frac{2\pi}{a}x\right)\bm{\hat{z}}\\\
\psi_{H_{4}}^{3}=\cos\left(\frac{2\pi}{a}y\right)\bm{\hat{x}},&\psi_{H_{4}}^{4}=\sin\left(\frac{2\pi}{a}z\right)\bm{\hat{x}}\\\
\psi_{H_{4}}^{5}=\sin\left(\frac{2\pi}{a}y\right)\bm{\hat{z}},&\psi_{H_{4}}^{6}=-\sin\left(\frac{2\pi}{a}x\right)\bm{\hat{y}}\end{matrix}\\\
\hline\cr\hline\cr\end{array}$

As an example, let us consider the six $H$ points $\bm{k}=\pm b\hat{\bm{i}}\
(i=x,y,z)$, $b=2\pi/a$, and we call them $H^{(0)}$ points. They are the
smallest wavevectors among the $H$ points in $k$-space. At these points the
basis functions are formed by the 12 plane waves in free space for the six
wavevectors $\bm{k}=\pm b\hat{\bm{i}}\ (i=x,y,z)$ having two polarizations
each. These basis functions are decomposed into irreps as summarized in Table
1, by using the character table for the irreps of the $H$ point summarized in
Table S3 in Supplemental Document. Therefore, the eigenmodes with the lowest
frequency at the $H^{(0)}$ points are decomposed into irreps
$H_{1}+H_{2}H_{3}+H_{4}$, where $H_{2}$ and $H_{3}$ are degenerate in the
presence of TRS.

###  3.2 Representations at the high-symmetry points in the BPI and DG
photonic crystals

This result based on the plane waves matches with our numerical calculations
for the BPI and the DG photonic crystals with various values of the dielectric
constant $\varepsilon$, as shown in Fig. 3. The lowest four modes for the
vacuum in Fig. 3(a) splits into $H_{1}+H_{2}H_{3}+H_{4}$ in both the BPI and
the DG photonic crystals with increasing the dielectric constants of the
dielectrics. In the same manner, one can classify the irreps for the several
lowest frequencies at the high-symmetry points such as $H$, $P$ and $N$. These
results are summarized in Table S1 in Supplemental Document.

Figure 3: Band structures for BPI and DG photonic crystals. Band structures
for (a) $\varepsilon=1$, which is an air band structure embedded into the
Brillouin zone of the body-centered cubic (BCC) photonic crystal. Band
structures are shown for the BPI photonic crystal (b) at $\varepsilon=4$, (c)
at $\varepsilon=11$, and for the DG photonic crystal (d) at $\varepsilon=4$,
(e) at $\varepsilon=9$, (f) at $\varepsilon=16$. In (b) and (d) we show the
irreps for some of the lowest bands at $H$, $P$ and $N$ points mentioned in
the text. There is (a) a twelve-fold, (b), (c) two-fold ($H_{1}$ irrep), and
(d)-(f) four-fold degeneracy ($H_{2}H_{3}$ irreps) in the lowest bands at the
$H$ point.

Let us discuss the change of the band structure and the irreps in the BPI
photonic crystal, when the dielectric constant $\varepsilon$ in photonic
crystal is varied from 1 (Fig. 3(a)) via 4 (Fig. 3(b)) to 11 (Fig. 3(c)). The
degeneracies at the high-symmetry points such as $H$, $N$ and $P$ at
$\varepsilon=1$ are split into irreps in Table S1 when $\varepsilon$ is
increased. In this case, the irreps of the lowest bands at high-symmetry
points are given by $H_{1}$, $N_{1}$, and $P_{3}$ summarized in Table. 2.
These irreps remain the lowest bands even up to $\varepsilon=11$.

On the other hand, behaviors of the DG photonic crystal are different from
that in the BPI photonic crystal. The DG photonic crystal is shown in Fig.
1(b) whose space group is $\bm{{\it 230}}$, being the same as the BPI photonic
crystal. The single gyroid surface is given by an isosurface of

$\displaystyle g(\bm{r})=\sin(2\pi x/a)\cos(2\pi y/a)+\sin(2\pi y/a)\cos(2\pi
z/a)$ $\displaystyle+\sin(2\pi z/a)\cos(2\pi x/a),$ (2)

where $a$ is the lattice constant in the BCC lattice [22], and the DG surfaces
are given by isosurfaces of $g(\bm{r})$ and of its counterpart by space
inversion, $g(-\bm{r})$. We set the dielectric regions as
$g(\bm{r})>\lambda_{\mathrm{iso}}$ and $g(-\bm{r})>\lambda_{\mathrm{iso}}$
with $\lambda_{\mathrm{iso}}=1.1$, shown in blue and red, respectively in Fig.
1(b). From our results on the band structures when the dielectric constant
$\varepsilon$ is changed from 1 (Fig. 3(a)) via 4 (Fig. 3(d)) and 9 (Fig.
3(e)) to 16 (Fig. 3(f)), the irreps of the four lowest bands at high-symmetry
points are given by $H_{2}H_{3}$, $N_{1}$, and $P_{1}P_{2}$ which are
summarized in Table 2. This set of the irreps are different from the BPI
photonic crystals.

Table 2: A set of irreducible representations at high-symmetry points $H$,
$N$, and $P$ for the lowest bands in BPI and DG photonic crystals. The numbers
in the parenthesis show the dimensions of the irreps.
$\begin{array}[]{cccc}\hline\cr\hline\cr\mathrm{high\mbox{-}symmetry\
point}&{H}&{P}&{N}\\\ \hline\cr\mathrm{BPI\ photonic\
crystal}&H_{1}(2)&P_{3}(4)&N_{1}(2)\\\ \mathrm{DG\ photonic\
crystal}&H_{2}H_{3}(4)&P_{1}P_{2}(4)&N_{1}(2)\\\
\hline\cr\hline\cr\end{array}$

###  3.3 Photonic waves at the $\Gamma$ point in photonic bands

We have been focusing on the glide-$Z_{2}$ invariant for the lowest two bands
of this photonic crystal. Here, although the electromagnetic waves are
singular at $\omega=0=|\bm{k}|$ as mentioned in Ref. [16], one can identify
their symmetry properties near $\omega=0=|\bm{k}|$, which are needed for
calculation of the $Z_{2}$ topological invariant. First, the electromagnetic
waves in a homogeneous isotropic medium are written analytically. Then, as
remarked in Ref. [16], even in photonic crystals, the electromagnetic waves
near $\omega=0=|\bm{k}|$ behave similarly with those in a homogeneous
isotropic medium, because in the longwavelength limit, the spatial variation
of the dielectric constants in the photonic crystal will become irrelevant. In
the next subsection, we show how we calculate the symmetry property of the
waves around the singularity at $\omega=0$ within our theory.

###  3.4 Glide-$Z_{2}$ invariant for the photonic crystal

In magnetic glide-symmetric systems, the glide-$Z_{2}$ topological invariant
can be defined, which characterizes a topological crystalline insulator phase
[1, 2, 3]. In Ref. [5], the BPI photonic crystal is proposed to be in the
topological phase ensured by this glide-$Z_{2}$ topological invariant, but a
physical reason why this structure leads to the topological phase is unknown.
Here we show this reason by directly studying the formula of the glide-$Z_{2}$
topological invariant.

The glide-$Z_{2}$ topological invariant $\nu$ is defined in a crystal with
glide symmetry, and its formula is expressed as a sum of line and surface
integrals in $k$-space [1, 2, 3], which is not easy to evaluate numerically.
On the other hand, when inversion symmetry is preserved, like the photonic
crystals of our interest, its formula is drastically simplified, and it is
expressed in terms of the irreps at high-symmetry points[9] in the form of a
symmetry-based indicator [23, 24]; thus it is much easier to evaluate. In the
space group $\bm{{\it 230}}$, which our photonic crystals belong to, we obtain
a formula for the glide-$Z_{2}$ topological invariant $\nu$, associated with a
certain band gap, as

$\displaystyle(-1)^{\nu}=\prod_{i}\zeta^{+}_{i}(\Gamma)\xi_{i}(N)\frac{\xi_{i}(H)}{\zeta_{i}^{+}(H)},$
(3)

where the product is taken over the bands below the gap considered,
$\zeta_{i}^{+}$ is a $C_{2}$ eigenvalue in the glide sector with a glide
eigenvalue $g_{+}=e^{-ik_{z}/2}$, $\xi_{i}$ is an inversion parity for the
$i$-th occupied state, and the product runs over the bands below the gap
considered. The detailed derivation of this formula is presented in Section S2
in the Supplemental Document. The value of $\nu$ is either $\nu=0$ or $\nu=1$,
corresponding to the topologically trivial and nontrivial phases,
respectively. The high-symmetry points in the Brillouin zones for $\bm{{\it
230}}$ are shown in Fig. 4. We note that from Eq. (3), the irreps at the $P$
point in $\bm{{\it 230}}$ do not contribute to the glide-$Z_{2}$ invariant, as
we mentioned in the last part of the Section 2.

Remarkably, in $\bm{{\it 230}}$, the formula (3) for the $Z_{2}$ topological
invariant is further simplified drastically, by evaluating Eq. (3) term by
term. In this calculation, we focus on the band gap between the second and
third bands, when TRS is broken by applying a magnetic field. First, the $H$
point has three possible physically irreducible representations, $H_{1}$,
$H_{2}H_{3}$, and $H_{4}$, and all of them yield
$\prod_{i}\frac{\xi_{i}(H)}{\zeta_{i}^{+}(H)}=1$, by a direct calculation from
Table S4 in Supplemental Document. Second, the $N$ point has two possible
physically irreducible representations, $N_{1}$ and $N_{2}$. Both of them
contributreare to the product in Eq. (3) by a factor $\prod_{i\in
N_{a}}\xi_{i}(N)=-1$ $(a=1,2)$. Therefore, the total contribution from the $N$
point is $(-1)^{n/2}$, where $n$ is the number of bands below the gap
considered. Third, as noted earlier, the eigenmodes with $\omega=0$ at the
$\Gamma$ point are special in any photonic crystals because of the
transversality condition of the electromagnetic wave, and they do not follow
the ten irreps at the $\Gamma$ point. Through a direct calculation, we show
that it contributes trivially (i.e. by a factor $+1$) to the product in Eq.
(3). To summarize, the glide-$Z_{2}$ invariant $\nu$ for $\bm{{\it 230}}$ is
calculated as

$\displaystyle(-1)^{\nu}=(-1)^{n/2}\ \rightarrow\ \nu=n/2\ (\text{mod}\ 2).$
(4)

It means that when the number of bands $n$ below the gap is $n=4m+2$ ($m$:
integer), the photonic crystal is topologically nontrivial, and when it is
$n=4m$ ($m$: integer), it is topologically trivial, provided the gap is open
everywhere in the Brillouin zone.

Figure 4: (Color online) Upper half of the Brillouin zone of the monoclinic
base-centered lattice in $\bm{{\it 230}}$ ($Ia\bar{3}d$). $\Gamma$, $P$, $H$,
and $N$ denote the high-symmetry points in SG $\bm{{\it 230}}$. Here, all the
lattice constants are set to be unity.

At the $P$ point, there are two possible physically irreps under the TRS,
$P_{1}P_{2}$ and $P_{3}$, both of which are four-dimensional. Therefore, when
$n=4m+2$ (i.e. $\nu=1$), there is no gap at the $P$ point when the TRS is
preserved. Therefore we need to open a gap at $P$ by breaking TRS to make it
topologically nontrivial. On the other hand, when $n=4m$ (i.e. $\nu=0$), the
$P$ point is gapped. Here, the $P_{1}P_{2}$ and $P_{3}$ irreps lead to the
doubly degenerate Dirac point, and the generalized Dirac point, respectively,
and in both cases, quad-helicoid surface states are realized on the (001)
surface, thanks to the combination of the two glides and the time-reversal
symmetries [25, 26], and another $Z_{2}$ topological invariant, which we call
a helicoid-$Z_{2}$ invariant, could be defined to represent nontrivial winding
of the helicoid surface states. How the helicoid-$Z_{2}$ and glide-$Z_{2}$ are
mutually related is an open question.

Thus far, we have understood how to make the glide-$Z_{2}$ topological
invariant $\nu$ nontrivial. To make the photonic crystal topological, it is
important how we break the TRS, i.e. how we introduce magnetization into
dielectrics. Here we introduce two types of magnetizations employed in the
previous works, and we call these types I and II. The type I is realized in
the BPI photonic crystal with the two dielectric rods (red and yellow in Fig.
1) having $+z$ magnetization, and the other two (blue and green in Fig. 1)
having $-z$ magnetization, as employed in Ref. [5]. It is classified as the
magnetic space group (MSG) $\bm{{\it 142.565}}$, whose list is in the Bilbao
Crystallographic Server [27]. The type II is the DG photonic crystal with the
uniform magnetization along the $+z$ direction employed in Ref. [6], which
belongs to the MSG $\bm{{\it 142.567}}$. As we have seen in this paper, the
photonic crystal is in the topological phase, if a gap is open between the
second and the third lowest bands. From an argument based on band topology, we
conclude that in the type I the photonic crystal opens a gap between the
second and third lowest bands everywhere in $k$-space, while in the type II,
the photonic crystal does not open a gap. This is explained by extending the
notion of minimal band connectivity $M$ for a nonmagnetic photonic crystal in
Ref. [16] into magnetic photonic crystals, whose details are shown in Section
S2 in the Supplemental Document. This conclusion fully agrees with the results
in Refs. [5, 6], saying that the BPI photonic crystal is in a topological
insulator phase ensured by glide symmetry for the gap between the second and
the third bands, while the DG photonic crystal does not open a gap.

We note that in general, some other symmetries apart from glide symmetries can
give topological surface modes. For example, mirror symmetry can give us
topological phases associated with a mirror Chern number, leading to
topological surface modes in the gap [28]. Meanwhile, the space group
$\bm{{\it 230}}$, which we study in this paper, contains various symmetries
such as threefold rotations and twofold screw rotations other than glide
symmetries, but none of the other symmetries except for glide symmetries are
associated with topological surface modes. Thus in this space group $\bm{{\it
230}}$, the band gap topology solely depends on the glide symmetry.

##  4 Design of topological photonic crystals with glide symmetry

So far we have seen how the topological properties of these photonic bands are
understood in terms of the irreps at high-symmetry points in $k$-space. In
this section, we propose how topological photonic crystals are designed, based
on the representation theory and Wyckoff positions. In particular, we focus on
the BPI and DG photonic crystals as two characterisitic examples, and how the
photonic bands of these photonic crystals result from their structure in real
space. To show this we focus on the irreps at the $H$ point, and as we noted,
in the BPI photonic crystal, the irrep $H_{1}$ is the lowest, while in the DG
photonic crystal, the irreps $H_{2}H_{3}$ are the lowest.

###  4.1 Perturbation theory

As we argued in the previous section, the irreps for the lowest bands at the
$H$ point for the BPI photonic crystal and DG photonic crystal are unchanged
by a change of the dielectric constant from unity to a larger value. We need
to see how the level splitting occurs at the $H$ point by increasing
$\varepsilon$. In order to see the level splitting at the $H$ point for a
value of $\varepsilon$ close to unity, we can employ the perturbation theory
in the dielectric function. When we add a small perturbation of the dielectric
function $\varepsilon=1\ \rightarrow\
\varepsilon=1+\delta\varepsilon(\bm{r})$, the frequency shift $\delta\omega$
is given to the first order in the perturbation $\delta\varepsilon(\bm{r})$ as
follows [29]:

$\delta\omega=-\frac{\omega}{2}\frac{\int d^{3}\bm{r}\
\delta\varepsilon(\bm{r})|\bm{\mathrm{E}}(\bm{r})|^{2}}{\int d^{3}\bm{r}\
\varepsilon(\bm{r})|\bm{\mathrm{E}}(\bm{r})|^{2}}.$ (5)

This relation implies that when the electric field is more concentrated within
the dielectrics with having nonzero $\delta\varepsilon(\bm{r})$, the frequency
shift increases.

We apply this formula to the lowest bands at the $H$ point. To this end we use
the 12 plane wave basis functions at the lowest frequency at the $H^{(0)}$
points, decomposed into a set of eigenstates following the irreps $H_{1}$,
$H_{2}$, $H_{3}$, and $H_{4}$, as summarized in Table 1. Then, we can
calculate the frequency shift by using Eq. (5), if we know the spatial
distribution of $\delta\varepsilon(\bm{r})$ corresponding to a photonic
crystal. As a result, within the first-order perturbation theory in Eq. (5),
one cannot determine the difference of the frequency shifts for the irreps
$H_{1},H_{2}$, and $H_{3}$ in Table 1, because the distribution of
$|\bm{\mathrm{E}}|^{2}$ in Eq. (5) is identically the same for $H_{1},H_{2}$,
and $H_{3}$. Here, the reason why the degeneracy is not lifted in the first
order in $\delta\varepsilon(\bm{r})$ is because of the high symmetry of the
cubic space group. We expect that in the higher order in the perturbation
$\delta\varepsilon(\bm{r})$, this degeneracy will be lifted, as required from
the space-group symmetry. In order to see how this degeneracy is lifted, we
perform numerical calculations in the next subsection, instead of developing
the higher-order perturbation theory, which is lengthy and complicated.

###  4.2 Photonic band structures based on Wyckoff positions

So far we have seen that in the present case it is not easy to see
relationships between the structure of the photonic crystal and the band
structure in an analytic way. Therefore, we approach this problem with
numerical calculations. Here, we introduce the notion of Wyckoff positions,
often used in the context of electronic systems. In electronic systems,
Wyckoff positions classify spatial locations consistent with a given space
group. By putting orbitals with various symmetry properties at the Wyckoff
positions, one can exhaust all the band structures of atomic insulators.
Inspired by this, we adopt the concept of the Wyckoff positions into our
theoretical analysis on photonic crystals.

Table 3: Wyckoff positions in $\bm{{\it 230}}$, their site symmetries, and the
irreducible representations (irreps) for the lowest bands with
$\varepsilon=12$ at the $H$ point, when dielectric spheres are put at the
specified Wyckoff positions. If more than one bands have almost the same
frequency, they are shown altogether as the lowest band. Detailed coordinates
of each Wyckoff position are summarized in Table S5 in Supplemental Document.
The site symmetry $3$ for $32e$ is a subgroup of $\bar{3}$ for $16a$ and $32$
for $16b$, and the site symmetry $2$ for $48f$ and $48g$ is a subgroup of $32$
for $16b$, $222$ for $24c$, and $\bar{4}$ for $24d$.
$\begin{array}[]{ccc}\hline\cr\hline\cr\mathrm{Wyckoff\
positions}&\mathrm{Site\ symmetry}&\begin{matrix}\mathrm{Irreps\ for\ the}\\\
\mathrm{lowest\ band\ at}\\\ \mathrm{the\ }{H}\mathrm{\
point}\end{matrix}\\\\[4.0pt] \hline\cr 16a&\bar{3}(C_{3i})&H_{1},\
H_{2}H_{3},\ H_{4}\\\ 16b&32(D_{3})&H_{2}H_{3}\\\ 24c&222(D_{2})&H_{2}H_{3}\\\
24d&\bar{4}(S_{4})&H_{1},\ H_{2}H_{3}\\\ 32e&3(C_{3})&H_{1}\\\
48f&2(C_{2})&H_{1},\ H_{2}H_{3}\\\ 48g&2(C_{2})&H_{2}H_{3}\\\
\hline\cr\hline\cr\end{array}$

There are 8 kinds of Wyckoff positions in $\bm{{\it 230}}$ and their site
symmetries are summarized in Table 3 [7]. Their positions for $16a$, $16b$ and
$32e$ are in Table 4, and other Wyckoff positions are summarized in Table S5
in Supplemental Document. Our objective is to find which Wyckoff positions
correspond to the band structure of the BPI and to that of the DG photonic
crystals. To this end, we numerically calculate the band structure by putting
dielectric spheres on a given Wyckoff position. When dielectric spheres are
absent, the light propagates in vacuum whose band structure is the same in
Fig. 3(a), but if the dielectric spheres have a finite radius, the band
structure changes, depending on the positions of the dielectric spheres.

First, we put dielectric spheres at the Wyckoff positions labeled $16a$, which
are located at $(\frac{n_{1}}{2},\frac{n_{2}}{2},\frac{n_{3}}{2})$ and
$(\frac{n_{1}}{2}+\frac{1}{4},\frac{n_{2}}{2}+\frac{1}{4},\frac{n_{3}}{2}+\frac{1}{4})$
with $n_{1},n_{2},n_{3}$ being integers, according to Table 4. We here set the
dielectric constant of the dielectric spheres to be $\varepsilon=16$. We set
the radius of the dielectric spheres to be the maximum one, i.e the radius
when they touch each other. In this case, the band structure hardly changes
from that for vacuum as seen from Fig. 5(a). Recall that Eq. (5) implies the
degree of concentration of the electric fields in the dielectric regions.
Since the splitting of frequencies among the eigenmodes with
$H_{1},H_{2}H_{3},$ and $H_{4}$ in the case of the Wyckoff position $16a$ (in
Fig. 5(a)) are not conspicuous, the electromagnetic waves in these eigenmodes
are localized on the dielectric spheres at the Wyckoff position $16a$ to the
same degree.

In the similar manner, we can calculate the corresponding band structures for
dielectric spheres at other Wyckoff positions in $\bm{{\it 230}}$, listed in
Table 3. As a consequence, we find that dielectric spheres at the Wyckoff
positions labeled by $16b,24c$, and $48g$ generate band structure, with
$H_{2}H_{3}$ being the lowest bands at $H$, similar to that from the DG
photonic crystal, and the Wyckoff position labeled by $32e$ generates band
structure with $H_{1}$ being the lowest bands at $H$, similar to that from the
BPI photonic crystal. For the Wyckoff position $24d$, the eigenmodes for
$H_{1}$ and $H_{2}H_{3}$ are also very close to each other, while they are
away from those for $H_{4}$. The lowest eight bands at the $N$ point and those
at the $P$ point hardly split. For the Wyckoff position $48f$, the band
structure hardly changes from vacuum. We skip the most general Wyckoff
position $96h$, because no special feature from symmetry is expected.
Remarkable cases are depicted in Fig. 5, where we set the dielectric constant
as $\varepsilon=12$ and the radius of dielectric spheres as $r=\sqrt{3}/8$ for
$16a$ (Fig. 5(a)) and $r=0.18$ for $16b$ (Fig. 5(b)) with the lattice constant
being unity.

Figure 5: The photonic band structures at $\varepsilon=12$ by putting
dielectric spheres on the Wyckoff positions (a) $16a$ and (b) $16b$. (a) Band
structure with dielectric spheres with the radius $\sqrt{3}/8$ at the Wyckoff
position $16a$. It is hardly changed from an air band structure of a body-
centered cubic photonic crystal. (b) Band structure with dielectric spheres
with the radius $0.18$ at the Wyckoff position $16b$, not touching each other.
The band structure is similar to that of the DG photonic crystal. Here we set
the lattice constant to be unity.

Let us compare these results with the BPI and DG photonic crystals in the
related previous works. In the BPI photonic crystals, the dielectric rods lie
along the $(111),(\bar{1}11),(1\bar{1}1),$ and $(11\bar{1})$ directions, which
corresponds to the Wyckoff position $32e$, with changing the free parameter
$x$ in Table 4. Indeed, the photonic crystal with dielectric spheres at the
Wyckoff position $32e$ was shown to have $H_{1}$ as the lowest band (see Table
3), similar to the BPI photonic crystal.

Next we discuss the DG photonic crystal with dielectrics located at
$g(\bm{r})>\lambda_{\mathrm{iso}}$ and $g(-\bm{r})>\lambda_{\mathrm{iso}}$
where $g(\bm{r})$ is defined in Eq. (2). The isosurface function $g(\bm{r})$
has a maximum value $1.5$ when $\bm{r}=(x,y,z)$ is equal to the Wyckoff
position $16b$. By changing $\lambda_{\mathrm{iso}}$ smaller from
$\lambda_{\mathrm{iso}}=1.5$, the dielectric regions for the DG photonic
crystal broaden from the sites of $16b$ along $48g$ toward $24c$, i.e., along
the planes perpendicular to the four $C_{3}$ rotation axes. Indeed, in the
band structures for $16b$, $48g$ and $24c$, the lowest bands follow the
$H_{2}H_{3}$ irreps, and their band structures are simiar to that of the DG
photonic crystal.

Table 4: Summary of Wyckoff positions $32e$, $16b$ and $16a$ in $\bm{{\it
230}}$. The first column denotes the multiplicity and the Wyckoff letter, the
second column denotes site symmetry, and the coordinates with two sets
$(0,0,0)+$ and $(\frac{1}{2},\frac{1}{2},\frac{1}{2})+$. The lattice constant
is set to be unity. $\begin{array}[]{cc cccccccc}\hline\cr\hline\cr 32e&.\ 3\
.&\lx@intercol\hfil
x,x,x\hfil\lx@intercol&\lx@intercol\hfil\bar{x}+\frac{1}{2},\bar{x},x+\frac{1}{2}\hfil\lx@intercol&\lx@intercol\hfil\bar{x},x+\frac{1}{2},\bar{x}+\frac{1}{2}\hfil\lx@intercol&\lx@intercol\hfil
<EMAIL_ADDRESS>&&\lx@intercol\hfil
x+\frac{3}{4},x+\frac{1}{4},\bar{x}+\frac{1}{4}\hfil\lx@intercol&\lx@intercol\hfil\bar{x}+\frac{3}{4},\bar{x}+\frac{3}{4},\bar{x}+\frac{3}{4}\hfil\lx@intercol&\lx@intercol\hfil
x+\frac{1}{4},\bar{x}+\frac{1}{4},x+\frac{3}{4}\hfil\lx@intercol&\lx<EMAIL_ADDRESS>&&\lx@intercol\hfil\bar{x},\bar{x},\bar{x}\hfil\lx@intercol&\lx@intercol\hfil
x+\frac{1}{2},x,\bar{x}+\frac{1}{2}\hfil\lx@intercol&\lx@intercol\hfil
x,\bar{x}+\frac{1}{2},x+\frac{1}{2}\hfil\lx@intercol&\lx<EMAIL_ADDRESS>&&\lx@intercol\hfil\bar{x}+\frac{1}{4},\bar{x}+\frac{3}{4},x+\frac{3}{4}\hfil\lx@intercol&\lx@intercol\hfil
x+\frac{1}{4},x+\frac{1}{4},x+\frac{1}{4}\hfil\lx@intercol&\lx@intercol\hfil\bar{x}+\frac{3}{4},x+\frac{3}{4},\bar{x}+\frac{1}{4}\hfil\lx@intercol&\lx@intercol\hfil
<EMAIL_ADDRESS>16b&.\ 3\
2&\frac{1}{8},\frac{1}{8},\frac{1}{8}&\frac{3}{8},\frac{7}{8},\frac{5}{8}&\frac{7}{8},\frac{5}{8},\frac{3}{8}&\frac{5}{8},\frac{3}{8},\frac{7}{8}&\frac{7}{8},\frac{7}{8},\frac{7}{8}&\frac{5}{8},\frac{1}{8},\frac{3}{8}&\frac{1}{8},\frac{3}{8},\frac{5}{8}&\frac{3}{8},\frac{5}{8},\frac{1}{8}\\\\[8.0pt]
16a&.\ \bar{3}\
.&0,0,0&\frac{1}{2},0,\frac{1}{2}&0,\frac{1}{2},\frac{1}{2}&\frac{1}{2},\frac{1}{2},0&\frac{3}{4},\frac{1}{4},\frac{1}{4}&\frac{3}{4},\frac{3}{4},\frac{3}{4}&\frac{1}{4},\frac{1}{4},\frac{3}{4}&\frac{1}{4},\frac{3}{4},\frac{1}{4}\\\\[4.0pt]
\hline\cr\hline\cr\end{array}$

###  4.3 Design of topological photonic crystals based on Wyckoff positions

Figure 6: Photonic crystals with dielectric spheres at the Wyckoff positions
$16a$ and $16b$ and the corresponding band structures with time-reversal
symmetry (TRS). (a) Photonic crystal with the dielectric spheres with the
radius $\sqrt{3}/16$ at the Wyckoff positions $16a$ and $16b$. (b) The band
structure is qualitatively similar to that of the BPI photonic crystal. (c)
Photonic crystal with the dielectric spheres at the Wyckoff position $16b$
with the radii $5\sqrt{3}/48$ (blue) and those at the Wyckoff position $16a$
with the radii $\sqrt{3}/48$ (red). (d) The band structure is similar to that
of the DG photonic crystal.

We have shown that dielectric spheres located at the Wyckoff position $32e$ in
$\bm{{\it 230}}$ generates band structures, similar to the BPI photonic
crystal, while those at $16b$ in $\bm{{\it 230}}$ generates band structures
similar to the DG photonic crystal. Now we discuss how these two cases are
related if we gradually change the structure of the photonic crystal between
the two cases. To this end, we focus on the Wyckoff positions, $16a$ and
$16b$, which are regarded as special cases of the Wyckoff position 32$e$ by
setting $x=0$ and $x=1/8$, repsectively (see Table 4)

Figure 7: Configurations of photonic crystals with dielectric cylinders and
the corresponding band structures with time-reversal symmetry (TRS). (a) When
the dielectric cylinders at the Wyckoff position $32e$ are dominant, the band
structures in (b) is similar to those of the BPI photonic crystal. (c) If the
dielectric cylinders at the Wyckoff position $16b$ are dominant, the band
structures in (d) is similar to those of the DG photonic crystal.

First, we examine band structures for photonic crystals with dielectric
spheres both at $16a$ and at $16b$ for various values of their radii. Let
$r_{a}$ and $r_{b}$ denote the radii of dielectric spheres at the Wyckoff
positions $16a$ and $16b$, respectively, and we change $r_{a}$ and $r_{b}$,
with keeping the spheres touching each other, which leads to a constraint
$r_{a}+r_{b}=\sqrt{3}/8$. Henceforth, we set the lattice constant $a$ to be
unity. Let us start with $r_{a}=r_{b}=\sqrt{3}/16$. The configuration is
depicted in Fig. 6(a) where the gray cuboid represents a conventional unit
cell $0\leq x,y,z<1$, and the band structure is shown in Figs. 6(b). The band
structure qualitatively agrees with that of the BPI photonic crystal, having
the rods along the Wyckoff position $32e$(Fig. 2(b)), because the structure is
similar to that of the BPI photonic crystal with dielectrics at $32e$, as can
be seen in from Fig. 6(a). Next, we make the system similar to the photonic
crystal with spheres at $16b$, by putting $r_{a}=\sqrt{3}/48$ and
$r_{b}=5\sqrt{3}/48$ depicted in Fig. 6(c). Then the band structure is given
in Figs. 6(d). In this case, the band structure qualitatively agrees with that
of the DG photonic crystal having dielectrics mainly at 16$b$.

As another example, we also consider photonic crystals with dielectric
cylinders, in order to see how the band structures changes between the two
cases. We combine two species of cylinders, one along $32e$ and the other
centered at $16b$ with sharing their axes in common, and we consider the
dielectrics located inside the union of these sets of cylinders. We fix the
height and radius of dielectric cylinders at $16b$ as $h_{b}=\sqrt{3}/12$ and
$r_{b}=0.15$, and change the radius $r_{e}$ of dielectric cylinders with an
infinite height along $32e$. In the case of $r_{e}=0.1$ shown in Fig. 7(a),
where the photonic crystal is almost identical with the BPI photonic crystal,
the band structure with TRS in Fig. 7(b) is similar to that of the BPI
photonic crystal, as we intuitively expected, with the lowest bands identical
with those for the BPI photonic crystal in Table 2. On the other hand, in the
photonic crystal with $r_{e}=0.02$ in Fig. 7(c), where the dielectrics are
localized around the Wyckoff position $16b$, the band structure is shown in
Fig. 7(d), and is similar to that of the DG photonic crystal, as listed in
Table 2. These results show that by classifying the photonic crystals in terms
of the Wyckoff positions of the dielectrics, one can predict the ordering of
irreps at high-symmetry points (see Table 3). Thus, photonic crystals with
dielectrics at various Wyckoff positions can serve as building blocks to find
ones with desired band ordering at high-symmetry positions, and it can be a
way to “design” photonic crystals with desired properties.

Now we discuss universality and limitation of the method of designing
topological photonic crystals based on Wyckoff positions. We consider that our
analysis in terms of Wyckoff positions works well in the perturbative regime
discussed in Sec. 4.1, where the dielectric constant of the dielectrics is
close to that of vacuum. Indeed, in the present paper, we show that the
ordering of the irreps of the bands can be predicted from the Wyckoff
positions of the dielectrics, as summarized in Table 3, and it universally
holds as long as the dielectric constant is close to that of vacuum. If the
dielectric constant of the dielectrics become larger, various bands with
different irreps can be inverted and the band ordering may differ from that in
Table 3.

##  5 Conclusion and discussion

In the present paper, by investigating the relationship between band
structures and space group representations, we study how to manipulate
topological photonic crystals ensured by glide symmetry. In particular, for
the BPI photonic crystal without time-reversal symmetry, which is proposed to
realize the glide-$Z_{2}$ magnetic topological crystalline insulator phase in
the previous paper, we explain the physical reason of the topological phase by
using our new formula of the glide-$Z_{2}$ invariant in terms of irreducible
representations. By comparing band structures and irreps at high-symmetry
points in our numerical calculation, we have also figured out that the
photonic crystal with $\bm{{\it 230}}$ realizes the glide-$Z_{2}$ magnetic
topological phase only from symmetry considerations. In such photonic crystal
designed in this way, by opening the gap between the second and third bands by
breaking time-reversal symmetry in a staggered way (type I in the main text),
we always obtain the glide-$Z_{2}$ topological crystalline insulator.

Moreover, because the glide-$Z_{2}$ topological crystalline insulator phase is
automatically the higher-order topological insulator in the presence of
inversion symmetry, such glide-$Z_{2}$ topological photonic crystals discussed
in this paper are higher-order topological insulators, and one can expect
topological hinge states [3]. Under the glide operation
$\hat{G}_{y}:(x,y,z)\rightarrow(x,-y,z+(c/2))$, the $(100)$ surface preserves
glide symmetry and the topological surface states of the glide-$Z_{2}$
topological phase emerge on this surface (Fig. 8(a)). On the other hand, if
one consider a photonic crystal with an inversion-symmetric shape without
glide-symmetric surfaces, the topological surface states do not appear, while
hinge states appear because inversion symmetry is still preserved (Fig. 8(b))
[30, 17, 31]. This can be demonstrated by using a simple tight-binding model
in Supplemental Document. In electronic systems, it has been proposed that
MnBi2nTe3n+1 and ${\mathrm{EuIn}}_{2}{\mathrm{As}}_{2}$ support gapless
surface states ensured by glide symmetry and hinge states ensured by inversion
symmetry [32, 33, 34].

Figure 8: Illustration of (a) the surface states (blue planes) of the
glide-$Z_{2}$ topological crystalline insulators for system with glide-
preserving surfaces, and (b) the hinge states (red lines) of the higher-order
topological insulator ensured by inversion symmetry.

There remain several issues. First, it is generaly difficult to open a common
gap throughout the whole Brillouin zone, because the gap induced by breaking
time-reversal symmetry is small in general. If the dielectric constant in the
dielectrics is perturbatively introduced, $\varepsilon=1+\delta\varepsilon$,
$\delta\varepsilon\ll 1$, the lowest frequency at the $H$ point is always
higher than that at the $P$ point, because the $H$-$\Gamma$ distance is larger
than the $P$-$\Gamma$ distance in $k$ space. Even when the dielectric constant
in the dielectrics becomes larger, the frequencies of the lowest bands at $P$
and $H$ are different, which means that a large TRS-breaking term is required
to open a large gap.

In this paper, we focused on the topological photonic crystals protected by
glide symmetry by breaking TRS. On the other hand, magnons in a magnet form
band structure without TRS, and it can also be described by the same
topological invariant. Therefore, we can use Eq. (3) to identify the
glide-$Z_{2}$ topological invariant when inversion symmetry is present. As an
example, we can apply our theory to magnons in yttrium iron garnet (YIG), a
ferromagnetic material belonging to the same space group $\bm{{\it 230}}$.
Nonetheless, so far the band structure of YIG has not been investigated enough
for our purpose. According to Ref. [35], the degree of degeneracy for the
lowest bands at the $H$ point in the magnon band structure looks higher than
four, which means that the irreps are not fully resolved, partially because of
numerical difficulty due to the complex lattice structure having many atoms
per unit cell. Thus, topological characterization of magnons in YIG remains a
future work.

Funding Japan Society for the Promotion of Science (JP17J10672, JP18H03678,
JP20H04633); National Key R&D Program of China (2017YFA0303800,
2016YFA0302400); Natural Science Foundation of China (12025409, 11721404,
11974415); Strategic Priority Research Program (XDB33000000); international
partnership program (112111KYSB20200024); Beijing Natural Science Foundation
(Z200008).

Acknowledgments H. K. is supported by Japan Society for the Promotion of
Science (JSPS) KAKENHI Grant-in-Aid for JSPS Fellows Grant No. JP17J10672.
This work was supported by JSPS KAKENHI Grant No. JP18H03678 and JP20H04633.
Ling Lu was supported by the National Key R&D Program of China
(2017YFA0303800, 2016YFA0302400), by Natural Science Foundation of China
(12025409, 11721404, 11974415), by the Strategic Priority Research Program
(XDB33000000) and the international partnership program (112111KYSB20200024)
between the Chinese Academy of Sciences and the Croucher Foundation, and by
Beijing Natural Science Foundation (Z200008).

Disclosures The authors declare no conflicts of interest.

Data availability statement Data underlying the results presented in this
paper are not publicly available at this time but may be obtained from the
authors upon reasonable request.

Supplemental document See Supplement 1 for supporting content.

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