††thanks: These authors contributed equally to this study.††thanks: These
authors contributed equally to this study.

# Topological classification and diagnosis in magnetically ordered electronic
materials

Bingrui Peng  Beijing National Laboratory for Condensed Matter Physics and
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
University of Chinese Academy of Sciences, Beijing 100049, China  Yi Jiang
Beijing National Laboratory for Condensed Matter Physics and Institute of
Physics, Chinese Academy of Sciences, Beijing 100190, China  University of
Chinese Academy of Sciences, Beijing 100049, China  Zhong Fang  Beijing
National Laboratory for Condensed Matter Physics and Institute of Physics,
Chinese Academy of Sciences, Beijing 100190, China  Hongming Weng  Beijing
National Laboratory for Condensed Matter Physics and Institute of Physics,
Chinese Academy of Sciences, Beijing 100190, China  Songshan Lake Materials
Laboratory, Dongguan, Guangdong 523808, China  Chen Fang<EMAIL_ADDRESS>Beijing National Laboratory for Condensed Matter Physics and Institute of
Physics, Chinese Academy of Sciences, Beijing 100190, China  Songshan Lake
Materials Laboratory, Dongguan, Guangdong 523808, China  Kavli Institute for
Theoretical Sciences, Chinese Academy of Sciences, Beijing 100190, China

###### Abstract

We show that compositions of time-reversal and spatial symmetries, also known
as the magnetic-space-group symmetries, protect topological invariants as well
as surface states that are distinct from those of all preceding topological
states. We obtain, by explicit and exhaustive construction, the topological
classification of electronic band insulators that are magnetically ordered for
each one of the 1421 magnetic space groups in three dimensions. We have also
computed the symmetry-based indicators for each nontrivial class, and, by
doing so, establish the complete mapping from symmetry representations to
topological invariants.

##  I Introduction

Magnetic space groups (MSGs)Bradley and Davies (1968) describe the symmetry of
lattices where spins are magnetically ordered. Magnetic ordering necessarily
breaks time-reversal symmetry, and, as the order parameter is a vector,
usually also breaks some point-group symmetries (rotation and reflection). In
many magnetically ordered materials, specially those having
antiferromagnetism, there are a special type of composite symmetries: a group
element $m=g\cdot{T}$ is the composition of a space-group (SG) symmetry $g$
and time-reversal symmetry $T$. Consider for example an anti-ferromagnetic
Neel order polarized along $z$ with propagation vector
$\mathbf{Q}=(\pi/a,0,0)$, where $a$ is the lattice constant. The lattice
translation by one unit cell along $x$, $\\{E|100\\}$, is broken, and time
reversal, $T$, is also broken, while their composition $\\{E|100\\}\cdot{T}$
remains a symmetry. (Here we use $E$ to represent the identity $3\times 3$
matrix, representing the trivial element of $O(3)$.) According to the types of
$g$ in $m=g\cdot{T}$, the MSGs are classified into four types: if $g$ does not
exist (so that $m$ does not exist), the magnetic group is type-I; if $g$ is
identity, $\\{E|000\\}$, type-II; if $g$ involves a nontrivial point group
operation, type-III; and if $g$ is a pure lattice translation, $\\{E|klm\\}$,
the group is type-IV. By this definition, type-II MSGs contain time reversal,
and hence describe nonmagnetic materials. In this work, we focus on type-I,
III, IV MSGs111the “MSGs” we use in this work means type-1, 3, and 4 MSGs, and
the non-magnetic type-2 MSGs are excluded..

MSGs are also the symmetries of the effective Hamiltonians describing
elementary excitations, such as magnons and electrons, that move within a
magnetically ordered lattice. In this work, we focus on magnetic materials in
which coherent quasiparticle fermion excitations form band(s) within a finite
range of the Fermi energy, and study the band topology of these fermions. Our
theory can in principle be applied to any magnetic materials where the notion
of electron-like quasiparticles is valid, at least near the Fermi energy.
Itinerant magnetsMoriya and Takahashi (1984), heavy-fermion metalsStewart
(1984), and doped Mott insulatorsLee et al. (2006) that maintain a
magnetically ordered state, are considered to belong to this large class of
materials.

The interplay between symmetry and topology has been a focus of modern
condensed-matter researchHasan and Kane (2010); Qi and Zhang (2011); Chiu et
al. (2016); Armitage et al. (2018). For a given symmetry group and a nonzero
gap, all band Hamiltonians are grouped into _equivalence classes_ , where two
Hamiltonians in the same (different) class(es) can(not) be smoothly deformed
into each other, while maintaining both the gap and the symmetry. Each
equivalence class is denoted by a unique set of integers called the
_topological invariants_ Thouless et al. (1982); Haldane (1983); Affleck et
al. (1987); Wen (1990); Kitaev (2001); Kane and Mele (2005); Moore and Balents
(2007); Fu and Kane (2007), the forms and types of which only depend on the
symmetry group and dimensionality. How many distinct equivalence classes exist
for a given symmetry group in a given dimension, and what are the topological
invariants for each class? This is the question we call the problem of
_topological classification_. The theory of topological classification for
time-reversal and particle-hole symmetries has been done in all dimensions
using the K-theorySchnyder et al. (2008); Kitaev (2009); Ryu et al. (2010);
the classification problem for a single spatial symmetry plus time reversal in
three dimensions has been solved, usually heuristically, for several
symmetriesFu et al. (2007); Hsieh et al. (2012); Teo et al. (2008); Wang et
al. (2016a); Shiozaki and Sato (2014); Song et al. (2017); Fang and Fu (2019);
Turner et al. (2010); Hughes et al. (2011); Song et al. (2018a); Khalaf et al.
(2018); and the classification problem for arbitrary SGs plus time reversal in
three dimensions have been attempted using either the “real-space recipe”
argumentSong et al. (2018a, 2019, 2020) or the “double-strong-topological-
insulator construction”Fang and Fu (2019); Khalaf et al. (2018) argument more
recently.

The classification problem for MSGs begins with the theory of axion insulators
protected by space-inversion symmetry without time reversalQi et al. (2008);
Li et al. (2010); Hughes et al. (2011); Qi and Zhang (2011); Turner et al.
(2012); Yue et al. (2019); Zhang et al. (2019a); Otrokov et al. (2019); Rienks
et al. (2019), followed by the theory of antiferromagnetic topological
insulatorsMong et al. (2010); Fang et al. (2013); Liu (2013); Zhang and Liu
(2015), again followed by the discovery of several topological invariants
protected by wallpaper groupsLiu et al. (2014); Shiozaki et al. (2017). A more
systematic attempt is made in Ref.Elcoro et al. (2020), where the layer-
construction (LC) method reveals a number of new topological states. In this
work, we use the _real-space recipe_ method developed in Song et al. (2019,
2020). Effectively, this method converts the problem of topological
classification into a “LEGO puzzle”, where one tries to find distinct ways one
can build an “edgeless” construction using some given pieces. This “LEGO
puzzle” is then further transformed into finding all independent integer
solutions of a set of linear equations on a $\mathbb{Z}_{n}$ ring. This method
not only yields a complete topological classification of gapped bands in each
of the 1421 groups, but also gives us, for each nontrivial equivalence class,
one explicit and microscopic construction that we call the “topological
crystal” (TC)Song et al. (2019). We emphasize that our method finds more
topological classes than the layer-construction method, because LCs are a type
of TCs, while not all TCs are LCs. (In fact, we show that at least 553 of the
1421 MSGs have at least one topological state that cannot be layer-
constructed.)

If the classification gives us the labels (topological invariants) for the
equivalence classes into which gapped states are put in, the _topological
diagnosis_ then tells us to which equivalence class a specific, given material
(Hamiltonian) belongs. Ideally, a diagnosis scheme computes the topological
invariants of that Hamiltonian, and by comparing these values with the labels
on the equivalence classes, one puts the Hamiltonian into the right one.
However, topological invariants are notoriously difficult to computeFang and
Fu (2015); Wang et al. (2016a), and for some, we do not even have the explicit
expressions in terms of the wave functions of the bandsSong et al. (2017);
Fang and Fu (2019). Fortunately, if we relax the requirement of ideal
diagnosis to approximate diagnosis, the story is completely changed. An
approximate diagnosis uses partial information on the wave function, and in
return gives us partial information on the topological invariants, not
invariants themselves. For example, an approximate diagnosis for systems with
$n$-fold rotation symmetry yields the invariant (Chern number) modulo $n$ by
using only the rotation eigenvalues at several high-symmetry momentaHughes et
al. (2011); Fang et al. (2012). Recently, the theory of symmetry-based
indicators (SIs)Po et al. (2017) and that of topological quantum
chemistryBradlyn et al. (2017), enhanced by the full mapping from indicators
to topological invariantsSong et al. (2018a), give birth to a fast approximate
diagnosis scheme. This fast-diagnosis scheme has been applied to a large
number of non-magnetic materialsZhang et al. (2019b); Vergniory et al. (2019);
Tang et al. (2019).

In this work, we extend the above diagnosis scheme to magnetic materials that
can be well-characterized in terms of band structures. We find the explicit
formulas for all SIs in terms of band representations (which in part differ
from previous worksOno and Watanabe (2018); Elcoro et al. (2020)); and we
calculate the values of indicators for each TC in every MSG. Since each gapped
state can be adiabatically continued to a TC, this result in fact yields the
mapping from invariants to indicators. In this calculation, we find that
certain indicator values are never taken in any TC, and hence can only
indicate nodal band structures Fang et al. (2003); Wan et al. (2011); Xu et
al. (2011); Wang et al. (2016b); Kübler and Felser (2016); Chang et al.
(2016); Yang et al. (2017); Liu et al. (2018); Wang et al. (2018); Belopolski
et al. (2019); Morali et al. (2019); Liu et al. (2019); Nie et al. (2017);
Tang et al. (2016); Hua et al. (2018); Kim et al. (2018); Nie et al. (2019);
Zou et al. (2019). These nodes are “evasive” as they are away from any high-
symmetry points or lines. In fact, we show that all these nodal indicators
indicate Weyl nodes at generic momenta. In a numerical diagnosis, the
indicators are by far easier to obtain than the invariants, because the former
only depend on the band representations at fewer than or equal to eight
momenta and the latter depend on the valence-band wave functions in the entire
Brillouin zone (BZ). Therefore, an inverse mapping from indicators to
invariants/nodes is generated using a script, provided along with the paper at
https://github.com/yjiang-iop/MSG_mapping.

##  II Classification

###  II.1 General scheme

The basic idea of the real-space recipe is that all topological crystalline
insulators (TCIs) can be adiabatically deformed into a special form of real-
space constructions called the TCs, such that classifying TCIs is equivalent
to classifying TCs. TCs are real-space patterns built from topological pieces
in lower dimensions, which were first applied to non-magnetic SGs in order to
obtain the full classifications of non-magnetic TCIsSong et al. (2019). In
this work, we apply the real-space recipe for the construction and the
classification of all gapped topological states protected by MSGs.

To start with, we build a structure of cell complex by using MSG symmetries,
including both unitary and anti-unitary ones, and partition the 3D space into
finite 3D regions called asymmetric units (AUs) that fill the whole 3D space
without overlaps. In fact, all AUs are symmetry-related and can be generated
by choosing one AU and then copying it using MSG operations. AUs are also
called 3-cells, and the 2D faces where they meet are 2-cells. Similarly, 1D
lines where 2-cells intersect are 1-cells, and the endpoints of 1-cells are
0-cells.

To construct TCs in 3D, we should take account of all $d$-dimensional
topological building blocks with $d\leq 3$. For each cell, its local symmetry
group is defined as the collection of symmetries that keep every point of the
cell unchanged. The local symmetry group of a cell determines the onsite
symmetry class (Altland-Zirnbauer classAltland and Zirnbauer (1997)) of the
Hamiltonian on that cell. For MSGs, the effective symmetry class of a cell is
always class A or class AI. Note that although some cells have mirror plane as
onsite symmetry, the states on them can be divided into two sectors by mirror
eigenvalues, each of which belongs to class A. A 2-cell may have local
symmetry group generated by $M\cdot T$, and a 1-cell local symmetry group
generated by $C_{2}\cdot T$. Since $(M\cdot T)^{2}=(C_{2}\cdot T)^{2}=+1$,
those cells belong to symmetry class AI. According to the tenfold way
resultsRyu et al. (2010), systems of class AI have trivial classification in
all dimension $\leq 3$, and systems of class A have nontrivial classification
in 2D and trivial classification in 0D, 1D, and 3D, which means only 2D
topological building blocks need to be considered in the construction of TCs
for MSG. Furthermore, there are two types of 2-cells depending on whether they
coincide with mirror planes. If coincide, they can be decorated with mirror
Chern insulators characterized by two $\mathbb{Z}$ numbers, i.e., two mirror
Chern numbers for $\pm i$ mirror sectors222Due to the absence of TRS in MSGs,
the “mirror Chern insulator” we use here do not require two mirror Chern
numbers to be opposite with each other, i.e., $C^{+}_{m}\neq C^{-}_{m}$
generically, which are different with the mirror Chern insulators in non-
magnetic SGs with $C^{+}_{m}=-C^{-}_{m}$Teo et al. (2008); Hsieh et al.
(2012)., and if not, with Chern insulators characterized by one $\mathbb{Z}$
number, i.e., the Chern number. Therefore, there are only two types of
building blocks for our real-space recipe, i.e., Chern insulators and mirror
Chern insulators.

Having building blocks in hand, we next enumerate all topological inequivalent
decorations on the cell complexes in MSGs. Before proceeding, note that the
building blocks themselves form a finitely generated Abelian group, e.g.,
$\mathbb{Z}$ for Chern insulators and $\mathbb{Z}^{2}$ for mirror Chern
insulators. Therefore, the TCs built from them form a linear space with
integer coefficients, such that two TCs can be “added” to obtain another TC,
and there exists a maximal set of linearly independent TCs (the generators)
for each MSG. As a result, one just needs to obtain the generators to describe
the full set of TCs. As TCs are supposed to be fully gapped topological
states, all the boundary states contributed by 2D building blocks (Chern
insulators and mirror Chern insulators) should cancel with each other on each
1-cell, leading to fully gapped states inside the bulk, a condition known as
the “gluing condition”Song et al. (2019) or “no-open-edge condition”Song et
al. (2020). After this procedure, we obtain a set of generators that form an
Abelian group $\mathbb{Z}^{n}$. However, this is generally not the final
classification, because some generators will reduce from $\mathbb{Z}$-type to
$\mathbb{Z}_{2}$-type after a process of subtracting topological trivial
elements called “bubble-equivalence”Song et al. (2019, 2020). The final
classification can be expressed as a quotient group Ker/Img, a structure
resembling group (co)homology, where Ker stands for the linear space of TCs
satisfying no-open-edge condition and Img for the space of bubble-equivalence.
These final classifications of MSGs have the form
$\mathbb{Z}^{n}\times\mathbb{Z}_{2}^{l}$ and can be found in Appendix.N.

Lastly, we compare the real-space constructions for non-magnetic SGs and MSGs.
First, observe that their building blocks are different. In non-magnetic SGs,
due to the time-reversal symmetry (TRS), the building blocks are two-
dimensional topological insulators and mirror Chern insulators with
$C^{+}_{m}=-C^{-}_{m}$, while in MSGs, the building blocks are Chern
insulators and mirror Chern insulators with independent $C^{+}_{m}$ and
$C^{-}_{m}$333in the following context we use the notation $C^{\pm}_{m}$ for
both real-space mirror Chern numbers and momentum-space mirror Chern numbers,
and the specific meaning of $C^{\pm}_{m}$ can be inferred from the context.
More significantly, for non-magnetic SGs, one can obtain most of the TCs by
LCs, which involve only layered 2D TIs and mirror Chern insulators as building
blocksSong et al. (2018a), with only 12 non-magnetic SGs having states beyond
LCsSong et al. (2019). However, non-layer constructions (non-LCs) exist widely
in MSGs, and constructing with layers only may lose a number of TCs. By
contrast, the real-space recipe employing the structure of the cell complex
automatically includes both LCs and non-LCs, giving the complete collection of
TCs, hence the complete classification of gapped topological states.

Figure 1: Surface states of symmetries with non-trivial topological invariants
in MSGs, with the surface terminations preserving corresponding symmetries.
These surface states can be 1D chiral hinge modes and 2D surface modes, with
2D modes being either slope-like chiral surface modes in (1),(6),(10), or
Dirac cones in (2),(3),(4),(5). More details can be found in Appendix.C.

###  II.2 Topological invariants

TCIs are characterized by crystalline-symmetry-protected topological
invariants. Previous works have looked into some of the topological invariants
protected by magnetic crystalline symmetries, with the earliest one being the
axion insulators with inversion invariantTurner et al. (2012); Hughes et al.
(2011), followed by the anti-ferromagnetic topological insulators protected by
anti-unitary translationsMong et al. (2010); Fang et al. (2013). In this work,
we exhaustively enumerate all topological invariants in MSGs.

Formally, having nontrivial symmetry-protected topological invariants means a
topological state cannot be smoothly deformed into a trivial state when the
symmetry is preserved. Specially, if a TCI can not be adiabatically connected
to an atomic insulator with the symmetry operation $g$ preserved, we say it
has nontrivial $g$-invariant. As all TCIs are adiabatically connected to TCs,
we can utilize TCs to derive all invariants protected by MSGs symmetries. In
our real-space recipe, given a symmetry operation $g$ alone, if nontrivial TCs
compatible with $g$ can be constructed, then we say $g$ can protect nontrivial
topological invariants. More specifically, each independent TC corresponds to
an independent invariant, and they have the same group structure, e.g., a
$\mathbb{Z}_{2}$ TC owns a $\mathbb{Z}_{2}$ invariant.

To find which symmetry operations in MSGs can protect nontrivial invariants
and what kinds of the invariants they protect, we take account of all of MSG
symmetries one by one, including translation, rotation, inversion, mirror,
rotoinversion ($S_{n}$), screw, glide, and those combined with TRS such as
$C_{n}\cdot T$, etc. We consider when each of them is present alone, what TCs
can be constructed. If no TC exists, this symmetry has only trivial invariant;
if there exist TCs, this symmetry must protect nontrivial topological
invariants, and we further decide how many independent TCs and whether they
are $\mathbb{Z}$-type or $\mathbb{Z}_{n}$ type by checking if the TCs can be
smoothly deformed into trivial states after multiplying $n$ times. In this
way, we know all the MSG symmetries that can singly protect nontrivial
topological invariants, with each symmetry hosting at most one invariant,
being either $\mathbb{Z}_{2}$-type or $\mathbb{Z}$-type. We summarize the
results in Table.1, and classify these invariants into three types:

  * •

Trivial invariants: $C_{n}$ rotations and anti-unitary improper point group
symmetries in MSGs cannot host nontrivial decorations. Take $C_{2}$ for
example. When there is only a $C_{2}$ symmetry alone, consider a 2D plane that
passes the $C_{2}$-axis, which is partitioned into two 2-cells by the axis.
Chern insulators cannot be decorated on the two 2-cells, because their chiral
edge modes on the $C_{2}$-axis are $C_{2}$-related, and as such are in the
same direction and cannot gap out each other. This is different from the non-
magnetic case where $C_{2}$ has a $\mathbb{Z}_{2}$ invariantSong et al. (2019,
2018a), as the building blocks in non-magnetic SGs are 2D TIs and two
$C_{2}$-related helical edge states can cancel with each other.

  * •

$\mathbb{Z}$ invariants: Unlike non-magnetic MSGs where only mirror symmetries
have $\mathbb{Z}$ invariant, here in MSGs, as the building blocks are both
$\mathbb{Z}$-type, we have many other $\mathbb{Z}$ invariants, including the
invariant of unitary translation, unitary screw, and anti-unitary glide.

  * •

$\mathbb{Z}_{2}$ invariants: A large proportion of MSG symmetries protect
$\mathbb{Z}_{2}$ invariants, with many of them unique to MSGs, such as anti-
unitary translations/rotations/screws. In Appendix.D, we define a special type
of decoration called “$Z_{2}$ decoration” which has zero weak invariants and
all $\mathbb{Z}_{2}$ invariants bond together and equal to 1 (for mirror Chern
numbers, we can define $C_{m}^{+}+C_{m}^{-}\bmod 2$ as a $\mathbb{Z}_{2}$
invariant), and can be seen as an axion insulator with the axion angle
$\theta=\pi$. For $Z_{2}$ decorations, all these $\mathbb{Z}_{2}$ invariants
merge into one $\mathbb{Z}_{2}$ invariant, i.e., the “axion invariant”, whose
definition can be taken as the 3D magnetoelectric polarization $P_{3}$
according to Ref.Qi et al. (2008); Qi and Zhang (2011); Fang et al. (2012).

MSG Symmetries | Invariant type
---|---
unitary rotation $C_{n}$ | Trivial
anti-unitary improper point group symmetries $P\cdot T$, $M\cdot T$,
$S_{n}\cdot T$
unitary improper $S_{n}$, $P$, $\\{M|\frac{1}{2}\\}$ | $\mathbb{Z}_{2}$
anti-unitary translation $\\{E|t\\}\cdot T$
anti-unitary proper $C_{n}\cdot T$, $\\{C_{n}|t\\}\cdot T$
unitary translation $\\{E|R\\}$ | $\mathbb{Z}$
unitary mirror $M$
unitary screw $\\{C_{n}|t\\}$
anti-unitary glide $\\{M|\frac{1}{2}\\}\cdot T$
Table 1:  MSG symmetries and their corresponding invariant types. We use the
Seitz Symbol $\\{O|t\\}$ to represent symmetries with nonzero translations,
where $R$ denotes a lattice translation and $t$ a fractional translation. The
invariants of the unitary screw and anti-unitary glide are bound to the
translation invariant, with their values being $\frac{1}{n}$ and $\frac{1}{2}$
of the translation invariant, respectively.

Each nontrivial invariant has its distinct anomalous surface state due to the
bulk-boundary correspondence. Because the topological invariants together with
their surface states can be superimposed, we only need to derive the surface
state for every single invariant, and the surface states of all TCIs can be
readily known from their invariants. Among these surface states in MSGs, some
have already been discussed in previous works, including those protected by
$C_{2}\cdot T$ Shiozaki and Sato (2014); Fang and Fu (2015); Ahn and Yang
(2019), $C_{4}\cdot T$Schindler et al. (2018), glideFang and Fu (2015);
Shiozaki et al. (2015, 2016); Kim et al. (2019); Kim and Murakami (2020), and
anti-unitary translationMong et al. (2010); Fang et al. (2013), etc. (for
spinless invariants including $C_{n}\cdot T$ and anti-unitary translation, see
Ref.Liu (2013); Zhang and Liu (2015)), while some are first proposed in this
work, such as the one protected by anti-unitary glide. We plot these surface
states in Fig.1, and more details can be found in Appendix.C. As an example
shown in Fig.1(3), on a cylinder geometry, the surface states protected by
$C_{n}\cdot T,n=2,4,6$ have a single Dirac cone on the top surface, and $n$
chiral hinge modes related to each other under $C_{n}\cdot T$ on the side
surface.

Although we find all the topological invariants protected by MSG symmetries,
we have not derived their explicit formulas in k-space, which could be
complicated. Instead, we compute these topological invariants in real-space
for the TCs we built using a unified method as followsSong et al. (2019).
Given an MSG $M$ and an element $g\in M$, first choose a generic point $r$
inside an arbitrary AU, then draw a path connecting $r$ and its image point
$g\cdot r$, under the only constraint that the path does not cross any 1-cells
and 0-cells. The corresponding invariant $\delta(g)$ is determined by the
decorated 2-cells that the path crosses, i.e., the total Chern number or
mirror Chern number accumulated through the path. The invariants thus
calculated are well-defined and do not depend on the choice of the generic
point $r$ or the path. The full listing of topological invariants of the TCs
will prove to be useful in the second part of this work, where we use SIs to
diagnose topological states, and TCs function as an intermediate to connect
SIs and invariants.

We take one example to show the correspondence between invariants and surface
states for a given MSG. The type-III MSG 3.3 $P2^{\prime}$ has three
independent generators of TCs. One of them is protected by $C_{2y}\cdot T$
with nonzero invariant $\delta(C_{2y}\cdot T)=1$, while the other two are
protected by the translation symmetries in $x$ and $z$ directions, with
nonzero weak invariant $\delta(\\{E|100\\})=1$ and $\delta{\\{E|001\\}}=1$,
respectively. For the first decoration, the surface state protected by
$C_{2y}\cdot T$ has been described before, as shown in Fig.1(3), while the
other two translation decorations have one chiral surface mode on the 2D
surface preserving the translation symmetry in $x/z$ direction, as shown in
Fig.1(1).

###  II.3 Non-layer constructions

The abundance of non-LCs distinguishes the TCs in MSGs from those in non-
magnetic SGs, where most of the decorations are LCs. In fact, LCs are just a
special type of TCs where 2D planes are uniformly decorated, while non-LCs
contain non-uniform decorations or incomplete 2D planes.

We use type-1 MSG $Pmmm$ as a representative to show the characters of non-
LCs. $Pmmm$ has three orthogonal mirrors $M_{x},M_{y}$, and $M_{z}$, which do
not commute with each other due to the spin rotation. Placing mirror Chern
insulator layers with $(C^{+}_{m},C^{-}_{m})=(1,-1)$ on any of the six mirror
planes, i.e., $x,y,z=0,\frac{1}{2}$, forms 6 independent LCs. However, one can
still construct a distinct non-layer decoration falling outside the linear
space of these 6 LCs. As shown in Fig.2, this non-layer decoration is
constructed by sewing small patches of mirror Chern insulators with
$(C^{+}_{m},C^{-}_{m})=(1,0)$ or $(0,-1)$ on each mirror 2-cell, with adjacent
patches having opposite mirror Chern numbers, making each mirror plane non-
uniform. Each 1-cell is shared by 4 patches, with patches in different
directions contributing opposite chiral edge modes, canceling with each other
and satisfying the no-open-edge condition. This non-LC is distinct from LCs,
where each mirror plane is decorated with a uniform, infinite-sized mirror
Chern insulator. The decorated 2-cells of this non-LC are all pinned on the
mirror planes, preventing it to be deformed into an LC.

Figure 2: The non-LC of $Pmmm$, where the arrows denote the directions of the
chiral edge modes on the 2-cells. Note we only plot the 2-cells around the
origin point inside the unit cell, and omit the six side-surfaces for
simplicity, which are in fact all decorated. Assume the 8 inversion centers
have coordinate
$\bm{r}=\frac{1}{2}\mathbf{a}_{1}\delta_{1}+\frac{1}{2}\mathbf{a}_{2}\delta_{2}+\frac{1}{2}\mathbf{a}_{3}\delta_{3}$,
where $\delta_{i}=0,1$ and $\mathbf{a}_{i}$ is the lattice vector. The 2-cells
around an inversion center have the same decoration as shown in the figure if
$\sum_{i}\delta_{i}$ is even, while the 2-cells have opposite directional edge
modes if $\sum_{i}\delta_{i}$ is odd.

Although this non-LC seems complicated, we observe that it can be connected to
the time-reversal strong TI (STI) from the perspective of surface states. On
the one hand, this non-LC has mirror Chern numbers
$(C_{m,k_{i}=0}^{+},C_{m,k_{i}=0}^{-},C_{m,k_{i}=\pi}^{+},C_{m,k_{i}=\pi}^{-})=(1,-1,0,0),i=x,y,z$,
which lead to a nontrivial mirror-protected surface state with a single Dirac
cone on $k_{i}=0$ in the 2D surface BZ. On the other hand, an STI put on
$Pmmm$ lattice also has a single Dirac cone on each surface. By the principle
of bulk-edge correspondence, we conclude that the STI with $Pmmm$ symmetry can
be adiabatically deformed into this non-LC by breaking the TRS while
preserving crystalline symmetries. Note that for $Pmmm$, despite the absence
of TRS, the three mirrors anti-commuting with each other also enforce the
energy bands to form Kramer-like pairs, i.e., twofold degeneracy with the same
parity and opposite mirror eigenvalues. Therefore, the three mirror symmetries
still pin the surface Dirac cone on the mirror invariant lines when the TRS is
broken. As a result, we can use the tight-binding models of STIs, for example,
the 3D Bernevig-Hughes-Zhang (BHZ) modelBernevig et al. (2006), to describe
this non-LC.

As mentioned before, unlike non-magnetic SGs where only 12 SGs have non-
LCsSong et al. (2019), non-LCs exist widely in MSGs. For instance, all the
supergroups of MSG 47.249 $Pmmm$ can host a non-LC as one of its TCI
classification generators, and similar for MSG 25.57 $Pmm2$, 84.51 $P4_{2}/m$,
10.44 $P2^{\prime}/m$, 75.3 $P4^{\prime}$, 6.21 $P_{a}m$, and 75.5 $P_{C}4$,
thus the total number of MSGs that have non-LCs is at least 553, i.e., the
number of the supergroups of these MSGs, including MSGs with different Bravais
lattices. As argued in Ref.Song et al. (2019), STI is compatible with all
crystalline symmetries, thus the abundance of non-LCs in MSGs can be
understood in a way that they can be obtained by inducing magnetism in STIs in
different lattices.

We remark that the non-LCs in these MSGs, when being a “$Z_{2}$ decoration”,
are all axion insulators. As STIs are also axion insulators, when the TRS is
broken, there still exist other unitary improper or anti-unitary proper
symmetries that have non-trivial $\mathbb{Z}_{2}$ invariants, preserving the
quantized $\pi$ axion angle.

##  III Diagnosis

Symmetry-based indicator is a powerful tool for diagnosing topological
statesBradlyn et al. (2017); Po et al. (2017); Kruthoff et al. (2017) and has
been applied in both non-magnetic SGsSong et al. (2018a, b); Khalaf et al.
(2018) and MSGsOno and Watanabe (2018); Watanabe et al. (2018); Elcoro et al.
(2020); Bouhon et al. (2020), which leads to the discovery of a significant
number of new topological materialsZhang et al. (2019b); Vergniory et al.
(2019); Tang et al. (2019); Xu et al. (2020). SI theory in MSGs has been
partly tackled in previous works, with the SI group structures and part of the
SI expressions given.

However, the mappings between SIs to topological invariants in MSGs have not
been fully investigated. In this work, we derive the explicit formulas of all
indicators, and their quantitative mappings to topological invariants if
correspond to gapped states, and possible Weyl point configurations if
correspond to gapless states. These two different correspondences to gapped
and gapless states are found thanks to the TCI classifications, which include
all the gapped TCI states, and states having nonzero SI while not belonging to
any gapped state must be Weyl semimetals.

###  III.1 Explicit expression of indicators

Among 1421 MSGs, 688 of them have nontrivial SIs. Despite the seemingly large
number, SIs in all MSGs can be induced from 16 generating MSGs, which we list
in Table.2, with only 10 corner cases to be discussed later.

MSG | $\bm{X}_{\text{BS}}$ | SI
---|---|---
$P\overline{1}$ | $\mathbb{Z}_{2,2,2,4}$ | $z_{2P,1},z_{2P,2},z_{2P,3},z_{4P}$
$Pmmm$ | $\mathbb{Z}_{2,2,2,4}$ | $z_{2P,1}^{\prime},z_{2P,2}^{\prime},z_{2P,3}^{\prime},z_{4P}^{\prime}$
$Pn,n=2,3,4,6$ | $\mathbb{Z}_{n}$ | $z_{nC}$
$Pn/m,n=2,3,4,6$ | $\mathbb{Z}_{n,n,n}$ | $z_{nm,0}^{+},z_{nm,0}^{-},z_{nm,\pi}^{+}$
$P\overline{4}$ | $\mathbb{Z}_{2,2,4}$ | $z_{2,S_{4}},z_{2,\text{Weyl}},z_{4C}$
$P4/mmm$ | $\mathbb{Z}_{2,4,8}$ | $z_{2P,1}^{\prime},z_{4m,\pi}^{+},z_{8}$
$P6/mmm$ | $\mathbb{Z}_{6,12}$ | $z_{6m,\pi}^{+},z_{12}$
$Pnc^{\prime}c^{\prime},n=2,4,6$ | $\mathbb{Z}_{n}$ | $z_{nC}^{\prime}$
Table 2: Generating SIs and generating MSGs in MSGs. We use a simplified
notation to represent the SI group, i.e.,
$\mathbb{Z}_{n_{1},n_{2},\cdots}=\mathbb{Z}_{n_{1}}\times\mathbb{Z}_{n_{2}}\times\cdots$.
Among these generating MSGs, only $Pnc^{\prime}c^{\prime}$ are type-3 MSGs,
while all the others are type-1. The definition and interpretation of these
SIs can be found in Appendix.G.

Most of the SIs can be expressed in terms of some topological invariants. In
fact, the idea of diagnosing nontrivial topology using only the symmetry data
on high-symmetry points (HSPs) in the BZ stems from the Fu-Kane formulaFu and
Kane (2007) for non-magnetic SGs with inversion symmetry, and in the following
worksHughes et al. (2011); Turner et al. (2012); Fang et al. (2012) $C_{n}$
rotation eigenvalues are used to calculate the Chern number modulo $n$, i.e.,
the $z_{nC}$ indicator in Table.2. When there exist mirror planes
perpendicular to the rotation axis, the $z_{nC}$ indicator can be applied to
two mirror sectors which give the $z_{nm,0/\pi}^{\pm}$ indicator, where $\pm$
represents two mirror sectors. There are also three type-3 generating MSGs
$Pnc^{\prime}c^{\prime},n=2,4,6$ which have anti-unitary glide symmetries that
make all irreducible corepresentations (coirreps) twofold degenerate (with the
same $C_{n}$ eigenvalues) on the $k_{z}=\pi$ plane, allowing the definition of
a new set of SIs $z_{nC}^{\prime}$ using the number of degenerate pairs, the
value of which corresponds to the Chern number divided by two modulo n of the
2D BZ. The interpretations of other generating SIs are left in Appendix.G.

The word “generating” means the SIs in other MSGs can be expressed in terms of
these generating SIs. Some times, the “generation” involves the reduction of
the indicator group, e.g. from $\mathbb{Z}_{4}$ to $\mathbb{Z}_{2}$ by taking
only the even numbers in $\mathbb{Z}_{4}$. Moreover, the interpretation of the
SI, generated from the same indicator, can be different in different MSGs. For
example, the $z_{nC}$ indicator originally represents the Berry phase of a
closed loop in the 2D BZ as shown in Appendix.K, and it becomes the Chern
number when the state is gapped. However, as shown later, in some other
groups, this indicator if nonzero indicates gapless topological states, and
its value represents the number of Weyl points. For another example, the
$z_{4P}^{\prime}$ indicator is adopted in some MSGs where the coirreps are not
doubly degenerate with the same parity. In these cases, the interpretation of
these indicators may change which takes case-by-case examination, and the SI
expressions are only effective and may become invalid when, for example, the
origin point is changed. Thus we fix the coordinate system to avoid these
problems by adopting the Bilbao conventionAroyo et al. (2006a, b, 2011) when
calculating the indicators.

To sum up, the indicator formulas in generating SIs can be used to express all
SIs in other MSGs, except 10 corner-case MSGs which have SI formulas given
independently using coirreps in Appendix.H. Here we show an example of MSG
42.222 $Fm^{\prime}m^{\prime}2$, which has a $\mathbb{Z}_{2}$ indicator with
expression:

$z_{2,42.222}=N(\overline{\Gamma}_{3})-N(\overline{A}_{3})\bmod 2$ (1)

where $N(\bar{K}_{i})$ represents the number of coirrep $\bar{K}_{i}$. This
MSG has TCI classification $\mathbb{Z}$, the generator of which has weak
invariants $\delta_{w}=(1,1,0)$ and all other invariants equal zero. The SI
$z_{2,42.222}=1$ represents this generator as well as all the odd number
copies of it. Note that this indicator is different from the choice in
Ref.Elcoro et al. (2020).

###  III.2 Computation of indicators for topological crystals

In this section, we show how one can calculate the SIs for a given TC. As most
of the SIs correspond to some topological invariants, including
$z_{2P,i},z_{4P},z_{2P}^{\prime},z_{nC},z_{nm,0/\pi}^{\pm},z_{2,S_{4}}$, and
$z_{nC}^{\prime}$, their values can be determined directly. For example, the
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{4}$
indicators defined in $P\bar{1}$ have straightforward meanings:

$z_{2P,i}=\delta_{w,i}\bmod 2,\ \ z_{4P}=2\delta(P)$ (2)

where $\delta_{w,i}(i=1,2,3)$ is the weak invariant and $\delta(P)$ is the
inversion invariant. Note that $z_{4P}=1,3$ represent Weyl semimetalsHughes et
al. (2011); Turner et al. (2012).

However, there also exist many SIs that cannot be read directly from the
invariants, including $z_{4P}^{\prime}$, $z_{8}$, and $z_{12}$. To determine
their value for a TC, we need to find a compatible set of coirreps for the TC
and then calculate the SIs of the coirrep set. The result does not, however,
depend on which particular set we choose as long as it is compatible with the
TC. This is because a given set of invariants, that is, a given TC, can only
correspond to one possible set of SIs. Therefore, we only need to find one
compatible set of coirreps for the TC and calculate its SI.

Finding compatible coirreps for an LC is simple, but to a non-LC TC can be
challenging. Fortunately, as shown before that many of the non-LCs in MSGs are
adiabatically connected to STIs, the BHZ model describing STIs can be used,
which has symmetries of SG 221 plus the TRS, a supergroup to many MSGs. As a
result, we can adjust the model parameters such that the BHZ model and the
non-LCs share the same set of invariants, and then compute the indicators for
the BHZ model, with details in Appendix.L.

The mappings between invariants and SIs only need to be derived for the
generating MSGs shown in Table.2. Other MSGs, except the corner cases and Weyl
states, either are the supergroups of these generating MSGs or have a
different Bravais lattice, and their mappings can be induced from the mappings
of the generating MSGs. For corner cases, we derive their mappings case-by-
case in Appendix.H.

###  III.3 The mapping from indicators to invariants

So far we have obtained the mapping from invariants to indicators for
topological gapped states protected by all MSGs, listed in Appendix.O. A
practically more useful mapping is its inverse, one from indicators to
invariants. This is because, on one hand, the indicators only depend on the
coirreps at HSPs, and as such are far easier to obtain in first-principles
calculations. On the other hand, the invariants, as shown in the previous
section, are directly related to the topological surface states that may be
detected experimentally. A mapping from indicator to invariants hence links
first-principles calculation to experimental observables. These mappings from
SIs to invariants are “one-to-many” in general, in which a given SI set may be
mapped to multiple invariant sets.

Assume an MSG has $m$ linearly independent TCs, which have invariant sets
$V_{1},V_{2},...,V_{m}$ and SI sets $S_{1},S_{2},...,S_{m}$. Given an
arbitrary SI set $S$ from the SI group, to find all the possible invariant
sets corresponding to it, one needs to solve the linear equation
$\sum_{i=1}^{m}x_{i}S_{i}=S$, where $x_{i}$ are coefficients to be determined.
The invariant sets corresponding to $S$ can be obtained using the solved
$x_{i}$. This inhomogeneous linear equation can be solved by first finding a
special solution and then adding all the general solutions to its homogeneous
counterpart, i.e., the solutions of the corresponding homogeneous linear
equation, $\sum_{i=1}^{m}x_{i}S_{i}=0$, which gives the invariant sets that
have zero SIs.

Following this line, we have developed an algorithm that automatically
computes all possible sets of invariants for a given set of indicators. The
code and the full results which have been entered into a large table may be
downloaded from https://github.com/yjiang-iop/MSG_mapping.

###  III.4 Weyl semimetal indicators

Unlike non-magnetic spinful SGsSong et al. (2018a) where all SIs indicate
topological gapped states, we find many SIs in MSGs correspond to gapless Weyl
semimetals. As the full mapping from TCs to SIs has been found for each MSG,
indicator values that do not belong to the image of this mapping necessarily
correspond to gapless states. Although these Weyl semimetals have Weyl points
at generic momenta in the BZ, the symmetries of MSGs require their creation or
annihilation to happen at HSPs, which makes it possible to detect them using
SIs. These Weyl semimetals can be classified into two types, with one being
inter-plane Weyl points that lie between $k_{i}=0$ and $k_{i}=\pi$ plane, and
the other being in-plane Weyl points that lie on $k_{i}=0$ or $\pi$ plane,
where $i$ denotes the main rotation axis of the MSG. We leave the full
discussion of Weyl semimetals to Appendix.I, and use one example of type-4 MSG
81.37 $P_{C}\overline{4}$ to illustrate the basic idea.

MSG 81.37 can host a multiple of 4 Weyl points on both $k_{z}=0/\pi$ plane,
with Weyl points connected by $S_{4}$ symmetry having opposite chirality, as
shown in Fig.3. This MSG has TCI classification $\mathbb{Z}_{2}$ and SI group
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$. The SI group is larger than the TCI
group. One of the $\mathbb{Z}_{2}$ indicator can be chosen as $z_{2,S_{4}}$,
the odd value of which corresponds to the decoration with $S_{4}$ invariant
$\delta(S_{4})=1$, while the other $\mathbb{Z}_{2}$ indicator can be chosen as
$z_{4C}/2$, the odd value of which represents the Weyl states. Notice that the
Weyl points are $S_{4}$-symmetric, and they can only be annihilated at HSPs on
$k_{z}=0/\pi$ plane. Moreover, $z_{4C,k_{z}=0}=z_{4C,k_{z}=\pi}=0$ or $2$
always holds, which means the Weyl points will appear simultaneously on
$k_{z}=0,\pi$ planes. The in-plane nature of this type of Weyl states is
enforced by the $C_{2}\cdot T$ symmetry.

Figure 3: The Weyl point configuration of MSG 81.37, where the blue and orange
dots represent Weyl points of opposite chiralities, the red line represents
the $S_{4}$ axis, and two yellow planes represent $k_{z}=0,\pi$ planes.

## References

  * Bradley and Davies (1968) C. J. Bradley and B. L. Davies, Rev. Mod. Phys. 40, 359 (1968), URL https://link.aps.org/doi/10.1103/RevModPhys.40.359.
  * Note (1) Note1, the “MSGs” we use in this work means type-1, 3, and 4 MSGs, and the non-magnetic type-2 MSGs are excluded.
  * Moriya and Takahashi (1984) T. Moriya and Y. Takahashi, Annual Review of Materials Science 14, 1 (1984), eprint https://doi.org/10.1146/annurev.ms.14.080184.000245, URL https://doi.org/10.1146/annurev.ms.14.080184.000245.
  * Stewart (1984) G. R. Stewart, Rev. Mod. Phys. 56, 755 (1984), URL https://link.aps.org/doi/10.1103/RevModPhys.56.755.
  * Lee et al. (2006) P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys. 78, 17 (2006), URL https://link.aps.org/doi/10.1103/RevModPhys.78.17.
  * Hasan and Kane (2010) M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010), URL https://link.aps.org/doi/10.1103/RevModPhys.82.3045.
  * Qi and Zhang (2011) X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011), URL https://link.aps.org/doi/10.1103/RevModPhys.83.1057.
  * Chiu et al. (2016) C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Rev. Mod. Phys. 88, 035005 (2016), URL https://link.aps.org/doi/10.1103/RevModPhys.88.035005.
  * Armitage et al. (2018) N. P. Armitage, E. J. Mele, and A. Vishwanath, Rev. Mod. Phys. 90, 015001 (2018), URL https://link.aps.org/doi/10.1103/RevModPhys.90.015001.
  * Thouless et al. (1982) D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982), URL https://link.aps.org/doi/10.1103/PhysRevLett.49.405.
  * Haldane (1983) F. D. M. Haldane, Phys. Rev. Lett. 50, 1153 (1983), URL https://link.aps.org/doi/10.1103/PhysRevLett.50.1153.
  * Affleck et al. (1987) I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Phys. Rev. Lett. 59, 799 (1987), URL https://link.aps.org/doi/10.1103/PhysRevLett.59.799.
  * Wen (1990) X.-G. Wen, International Journal of Modern Physics B 4, 239 (1990), URL https://www.worldscientific.com/doi/abs/10.1142/S0217979290000139.
  * Kitaev (2001) A. Y. Kitaev, Physics-Uspekhi 44, 131 (2001), URL https://iopscience.iop.org/article/10.1070/1063-7869/44/10S/S29.
  * Kane and Mele (2005) C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802 (2005), URL https://link.aps.org/doi/10.1103/PhysRevLett.95.146802.
  * Moore and Balents (2007) J. E. Moore and L. Balents, Phys. Rev. B 75, 121306 (2007), URL https://link.aps.org/doi/10.1103/PhysRevB.75.121306.
  * Fu and Kane (2007) L. Fu and C. L. Kane, Phys. Rev. B 76, 045302 (2007), URL https://link.aps.org/doi/10.1103/PhysRevB.76.045302.
  * Schnyder et al. (2008) A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, Phys. Rev. B 78, 195125 (2008), URL https://link.aps.org/doi/10.1103/PhysRevB.78.195125.
  * Kitaev (2009) A. Kitaev, in _AIP conference proceedings_ (American Institute of Physics, 2009), vol. 1134, pp. 22–30, URL https://aip.scitation.org/doi/abs/10.1063/1.3149495.
  * Ryu et al. (2010) S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. Ludwig, New Journal of Physics 12, 065010 (2010), URL https://iopscience.iop.org/article/10.1088/1367-2630/12/6/065010/meta.
  * Fu et al. (2007) L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 98, 106803 (2007), URL https://link.aps.org/doi/10.1103/PhysRevLett.98.106803.
  * Hsieh et al. (2012) T. H. Hsieh, H. Lin, J. Liu, W. Duan, A. Bansil, and L. Fu, Nature communications 3, 1 (2012), URL https://www.nature.com/articles/ncomms1969.
  * Teo et al. (2008) J. C. Y. Teo, L. Fu, and C. L. Kane, Phys. Rev. B 78, 045426 (2008), URL https://link.aps.org/doi/10.1103/PhysRevB.78.045426.
  * Wang et al. (2016a) Z. Wang, A. Alexandradinata, R. J. Cava, and B. A. Bernevig, Nature 532, 189 (2016a), URL https://doi.org/10.1038/nature17410.
  * Shiozaki and Sato (2014) K. Shiozaki and M. Sato, Phys. Rev. B 90, 165114 (2014), URL https://link.aps.org/doi/10.1103/PhysRevB.90.165114.
  * Song et al. (2017) Z. Song, Z. Fang, and C. Fang, Phys. Rev. Lett. 119, 246402 (2017), URL https://link.aps.org/doi/10.1103/PhysRevLett.119.246402.
  * Fang and Fu (2019) C. Fang and L. Fu, Science advances 5, eaat2374 (2019), URL https://advances.sciencemag.org/content/5/12/eaat2374.
  * Turner et al. (2010) A. M. Turner, Y. Zhang, and A. Vishwanath, Phys. Rev. B 82, 241102 (2010), URL https://link.aps.org/doi/10.1103/PhysRevB.82.241102.
  * Hughes et al. (2011) T. L. Hughes, E. Prodan, and B. A. Bernevig, Phys. Rev. B 83, 245132 (2011), URL https://link.aps.org/doi/10.1103/PhysRevB.83.245132.
  * Song et al. (2018a) Z. Song, T. Zhang, Z. Fang, and C. Fang, Nature communications 9, 1 (2018a), URL https://www.nature.com/articles/s41467-018-06010-w.
  * Khalaf et al. (2018) E. Khalaf, H. C. Po, A. Vishwanath, and H. Watanabe, Phys. Rev. X 8, 031070 (2018), URL https://link.aps.org/doi/10.1103/PhysRevX.8.031070.
  * Song et al. (2019) Z. Song, S.-J. Huang, Y. Qi, C. Fang, and M. Hermele, Science advances 5, eaax2007 (2019), URL https://advances.sciencemag.org/content/5/12/eaax2007.
  * Song et al. (2020) Z. Song, C. Fang, and Y. Qi, Nature communications 11, 1 (2020), URL https://www.nature.com/articles/s41467-020-17685-5.
  * Qi et al. (2008) X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B 78, 195424 (2008), URL https://link.aps.org/doi/10.1103/PhysRevB.78.195424.
  * Li et al. (2010) R. Li, J. Wang, X.-L. Qi, and S.-C. Zhang, Nature Physics 6, 284 (2010), URL https://www.nature.com/articles/nphys1534.
  * Turner et al. (2012) A. M. Turner, Y. Zhang, R. S. K. Mong, and A. Vishwanath, Phys. Rev. B 85, 165120 (2012), URL https://link.aps.org/doi/10.1103/PhysRevB.85.165120.
  * Yue et al. (2019) C. Yue, Y. Xu, Z. Song, H. Weng, Y.-M. Lu, C. Fang, and X. Dai, Nature Physics 15, 577 (2019), URL https://www.nature.com/articles/s41567-019-0457-0.
  * Zhang et al. (2019a) D. Zhang, M. Shi, T. Zhu, D. Xing, H. Zhang, and J. Wang, Phys. Rev. Lett. 122, 206401 (2019a), URL https://link.aps.org/doi/10.1103/PhysRevLett.122.206401.
  * Otrokov et al. (2019) M. M. Otrokov, I. I. Klimovskikh, H. Bentmann, D. Estyunin, A. Zeugner, Z. S. Aliev, S. Gaß, A. Wolter, A. Koroleva, A. M. Shikin, et al., Nature 576, 416 (2019), URL https://doi.org/10.1038/s41586-019-1840-9.
  * Rienks et al. (2019) E. D. Rienks, S. Wimmer, et al., Nature 576, 423 (2019), URL https://doi.org/10.1038/s41586-019-1826-7.
  * Mong et al. (2010) R. S. K. Mong, A. M. Essin, and J. E. Moore, Phys. Rev. B 81, 245209 (2010), URL https://link.aps.org/doi/10.1103/PhysRevB.81.245209.
  * Fang et al. (2013) C. Fang, M. J. Gilbert, and B. A. Bernevig, Phys. Rev. B 88, 085406 (2013), URL https://link.aps.org/doi/10.1103/PhysRevB.88.085406.
  * Liu (2013) C.-X. Liu, arXiv preprint arXiv:1304.6455 (2013), URL https://arxiv.org/abs/1304.6455.
  * Zhang and Liu (2015) R.-X. Zhang and C.-X. Liu, Phys. Rev. B 91, 115317 (2015), URL https://link.aps.org/doi/10.1103/PhysRevB.91.115317.
  * Liu et al. (2014) C.-X. Liu, R.-X. Zhang, and B. K. VanLeeuwen, Phys. Rev. B 90, 085304 (2014), URL https://link.aps.org/doi/10.1103/PhysRevB.90.085304.
  * Shiozaki et al. (2017) K. Shiozaki, M. Sato, and K. Gomi, Phys. Rev. B 95, 235425 (2017), URL https://link.aps.org/doi/10.1103/PhysRevB.95.235425.
  * Elcoro et al. (2020) L. Elcoro, B. J. Wieder, Z. Song, Y. Xu, B. Bradlyn, and B. A. Bernevig, arXiv preprint arXiv:2010.00598 (2020), URL https://arxiv.org/abs/2010.00598.
  * Fang and Fu (2015) C. Fang and L. Fu, Phys. Rev. B 91, 161105 (2015), URL https://link.aps.org/doi/10.1103/PhysRevB.91.161105.
  * Fang et al. (2012) C. Fang, M. J. Gilbert, and B. A. Bernevig, Phys. Rev. B 86, 115112 (2012), URL https://link.aps.org/doi/10.1103/PhysRevB.86.115112.
  * Po et al. (2017) H. C. Po, A. Vishwanath, and H. Watanabe, Nature communications 8, 1 (2017), URL https://www.nature.com/articles/s41467-017-00133-2.
  * 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), URL https://www.nature.com/articles/nature23268.
  * Zhang et al. (2019b) T. Zhang, Y. Jiang, Z. Song, H. Huang, Y. He, Z. Fang, H. Weng, and C. Fang, Nature 566, 475 (2019b), URL https://www.nature.com/articles/s41586-019-0944-6.
  * Vergniory et al. (2019) M. Vergniory, L. Elcoro, C. Felser, N. Regnault, B. A. Bernevig, and Z. Wang, Nature 566, 480 (2019), URL https://www.nature.com/articles/s41586-019-0954-4.
  * Tang et al. (2019) F. Tang, H. C. Po, A. Vishwanath, and X. Wan, Nature 566, 486 (2019), URL https://www.nature.com/articles/s41586-019-0937-5.
  * Ono and Watanabe (2018) S. Ono and H. Watanabe, Phys. Rev. B 98, 115150 (2018), URL https://link.aps.org/doi/10.1103/PhysRevB.98.115150.
  * Fang et al. (2003) Z. Fang, N. Nagaosa, K. S. Takahashi, A. Asamitsu, R. Mathieu, T. Ogasawara, H. Yamada, M. Kawasaki, Y. Tokura, and K. Terakura, Science 302, 92 (2003), URL https://science.sciencemag.org/content/302/5642/92.abstract.
  * Wan et al. (2011) X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, Phys. Rev. B 83, 205101 (2011), URL https://link.aps.org/doi/10.1103/PhysRevB.83.205101.
  * Xu et al. (2011) G. Xu, H. Weng, Z. Wang, X. Dai, and Z. Fang, Phys. Rev. Lett. 107, 186806 (2011), URL https://link.aps.org/doi/10.1103/PhysRevLett.107.186806.
  * Wang et al. (2016b) Z. Wang, M. G. Vergniory, S. Kushwaha, M. Hirschberger, E. V. Chulkov, A. Ernst, N. P. Ong, R. J. Cava, and B. A. Bernevig, Phys. Rev. Lett. 117, 236401 (2016b), URL https://link.aps.org/doi/10.1103/PhysRevLett.117.236401.
  * Kübler and Felser (2016) J. Kübler and C. Felser, EPL (Europhysics Letters) 114, 47005 (2016), URL https://iopscience.iop.org/article/10.1209/0295-5075/114/47005/meta.
  * Chang et al. (2016) G. Chang, S.-Y. Xu, H. Zheng, B. Singh, C.-H. Hsu, G. Bian, N. Alidoust, I. Belopolski, D. S. Sanchez, S. Zhang, et al., Scientific reports 6, 1 (2016), URL https://doi.org/10.1038/srep38839.
  * Yang et al. (2017) H. Yang, Y. Sun, Y. Zhang, W.-J. Shi, S. S. Parkin, and B. Yan, New Journal of Physics 19, 015008 (2017), URL https://iopscience.iop.org/article/10.1088/1367-2630/aa5487/meta.
  * Liu et al. (2018) E. Liu, Y. Sun, N. Kumar, L. Muechler, A. Sun, L. Jiao, S.-Y. Yang, D. Liu, A. Liang, Q. Xu, et al., Nature physics 14, 1125 (2018), URL https://doi.org/10.1038/s41567-018-0234-5.
  * Wang et al. (2018) Q. Wang, Y. Xu, R. Lou, Z. Liu, M. Li, Y. Huang, D. Shen, H. Weng, S. Wang, and H. Lei, Nature communications 9, 1 (2018), URL https://doi.org/10.1038/s41467-018-06088-2.
  * Belopolski et al. (2019) I. Belopolski, K. Manna, D. S. Sanchez, G. Chang, B. Ernst, J. Yin, S. S. Zhang, T. Cochran, N. Shumiya, H. Zheng, et al., Science 365, 1278 (2019), URL https://science.sciencemag.org/content/365/6459/1278.abstract.
  * Morali et al. (2019) N. Morali, R. Batabyal, P. K. Nag, E. Liu, Q. Xu, Y. Sun, B. Yan, C. Felser, N. Avraham, and H. Beidenkopf, Science 365, 1286 (2019), URL https://science.sciencemag.org/content/365/6459/1286.abstract.
  * Liu et al. (2019) D. Liu, A. Liang, E. Liu, Q. Xu, Y. Li, C. Chen, D. Pei, W. Shi, S. Mo, P. Dudin, et al., Science 365, 1282 (2019), URL https://science.sciencemag.org/content/365/6459/1282.abstract.
  * Nie et al. (2017) S. Nie, G. Xu, F. B. Prinz, and S.-c. Zhang, Proceedings of the National Academy of Sciences 114, 10596 (2017).
  * Tang et al. (2016) P. Tang, Q. Zhou, G. Xu, and S.-C. Zhang, Nature Physics 12, 1100 (2016), URL https://doi.org/10.1038/nphys3839.
  * Hua et al. (2018) G. Hua, S. Nie, Z. Song, R. Yu, G. Xu, and K. Yao, Phys. Rev. B 98, 201116 (2018), URL https://link.aps.org/doi/10.1103/PhysRevB.98.201116.
  * Kim et al. (2018) K. Kim, J. Seo, E. Lee, K.-T. Ko, B. Kim, B. G. Jang, J. M. Ok, J. Lee, Y. J. Jo, W. Kang, et al., Nature materials 17, 794 (2018), URL https://doi.org/10.1038/s41563-018-0132-3.
  * Nie et al. (2019) S. Nie, H. Weng, and F. B. Prinz, Phys. Rev. B 99, 035125 (2019), URL https://link.aps.org/doi/10.1103/PhysRevB.99.035125.
  * Zou et al. (2019) J. Zou, Z. He, and G. Xu, npj Computational Materials 5, 1 (2019), URL https://doi.org/10.1038/s41524-019-0237-5.
  * Altland and Zirnbauer (1997) A. Altland and M. R. Zirnbauer, Phys. Rev. B 55, 1142 (1997), URL https://link.aps.org/doi/10.1103/PhysRevB.55.1142.
  * Note (2) Note2, due to the absence of TRS in MSGs, the “mirror Chern insulator” we use here do not require two mirror Chern numbers to be opposite with each other, i.e., $C^{+}_{m}\not=C^{-}_{m}$ generically, which are different with the mirror Chern insulators in non-magnetic SGs with $C^{+}_{m}=-C^{-}_{m}$Teo et al. (2008); Hsieh et al. (2012).
  * Note (3) Note3, in the following context we use the notation $C^{\pm}_{m}$ for both real-space mirror Chern numbers and momentum-space mirror Chern numbers, and the specific meaning of $C^{\pm}_{m}$ can be inferred from the context.
  * Ahn and Yang (2019) J. Ahn and B.-J. Yang, Phys. Rev. B 99, 235125 (2019), URL https://link.aps.org/doi/10.1103/PhysRevB.99.235125.
  * Schindler et al. (2018) F. Schindler, A. M. Cook, M. G. Vergniory, Z. Wang, S. S. P. Parkin, B. A. Bernevig, and T. Neupert, Science Advances 4 (2018), URL https://advances.sciencemag.org/content/4/6/eaat0346.
  * Shiozaki et al. (2015) K. Shiozaki, M. Sato, and K. Gomi, Phys. Rev. B 91, 155120 (2015), URL https://link.aps.org/doi/10.1103/PhysRevB.91.155120.
  * Shiozaki et al. (2016) K. Shiozaki, M. Sato, and K. Gomi, Phys. Rev. B 93, 195413 (2016), URL https://link.aps.org/doi/10.1103/PhysRevB.93.195413.
  * Kim et al. (2019) H. Kim, K. Shiozaki, and S. Murakami, Phys. Rev. B 100, 165202 (2019), URL https://link.aps.org/doi/10.1103/PhysRevB.100.165202.
  * Kim and Murakami (2020) H. Kim and S. Murakami, Phys. Rev. B 102, 195202 (2020), URL https://link.aps.org/doi/10.1103/PhysRevB.102.195202.
  * Bernevig et al. (2006) B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, science 314, 1757 (2006), URL https://science.sciencemag.org/content/314/5806/1757.
  * Kruthoff et al. (2017) J. Kruthoff, J. de Boer, J. van Wezel, C. L. Kane, and R.-J. Slager, Phys. Rev. X 7, 041069 (2017), URL https://link.aps.org/doi/10.1103/PhysRevX.7.041069.
  * Song et al. (2018b) Z. Song, T. Zhang, and C. Fang, Phys. Rev. X 8, 031069 (2018b), URL https://link.aps.org/doi/10.1103/PhysRevX.8.031069.
  * Watanabe et al. (2018) H. Watanabe, H. C. Po, and A. Vishwanath, Science advances 4, eaat8685 (2018), URL https://advances.sciencemag.org/content/4/8/eaat8685.
  * Bouhon et al. (2020) A. Bouhon, G. F. Lange, and R.-J. Slager, arXiv preprint arXiv:2010.10536 (2020), URL https://arxiv.org/abs/2010.10536.
  * Xu et al. (2020) Y. Xu, L. Elcoro, Z.-D. Song, B. J. Wieder, M. Vergniory, N. Regnault, Y. Chen, C. Felser, and B. A. Bernevig, Nature 586, 702 (2020), URL https://www.nature.com/articles/s41586-020-2837-0.
  * Aroyo et al. (2006a) 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 (2006a), URL https://www.degruyter.com/document/doi/10.1524/zkri.2006.221.1.15/html.
  * Aroyo et al. (2006b) M. I. Aroyo, A. Kirov, C. Capillas, J. Perez-Mato, and H. Wondratschek, Acta Crystallographica Section A: Foundations of Crystallography 62, 115 (2006b), URL http://scripts.iucr.org/cgi-bin/paper?S0108767305040286.
  * Aroyo et al. (2011) M. I. Aroyo, J. Perez-Mato, D. Orobengoa, E. Tasci, G. de la Flor, and A. Kirov, Bulg. Chem. Commun 43, 183 (2011), URL http://bcc.bas.bg/BCC_Volumes/Volume_43_Number_2_2011/Volume_43_Number_2_2011_PDF/2011_43_2_1.pdf.
  * Fang et al. (2016) C. Fang, L. Lu, J. Liu, and L. Fu, Nature Physics 12, 936 (2016), URL https://www.nature.com/articles/nphys3782.
  * Qi et al. (2006) X.-L. Qi, Y.-S. Wu, and S.-C. Zhang, Physical Review B 74, 085308 (2006), URL https://journals.aps.org/prb/abstract/10.1103/PhysRevB.74.085308.
  * Cano et al. (2018) J. Cano, B. Bradlyn, Z. Wang, L. Elcoro, M. G. Vergniory, C. Felser, M. I. Aroyo, and B. A. Bernevig, Phys. Rev. B 97, 035139 (2018), URL https://link.aps.org/doi/10.1103/PhysRevB.97.035139.

###### Contents

  1. I Introduction
  2. II Classification
    1. II.1 General scheme
    2. II.2 Topological invariants
    3. II.3 Non-layer constructions
  3. III Diagnosis
    1. III.1 Explicit expression of indicators
    2. III.2 Computation of indicators for topological crystals
    3. III.3 The mapping from indicators to invariants
    4. III.4 Weyl semimetal indicators
  4. A Topological crystals, construction and classification
    1. A.1 Cell Complex
    2. A.2 2D building Blocks
    3. A.3 No-open-edge condition
    4. A.4 Bubble equivalence
    5. A.5 Decorations, first encounter
    6. A.6 Final TCI classification
  5. B Topological invariants
  6. C Surface states
  7. D Decorations, second encounter
  8. E Layer and non-layer constructions
  9. F A brief review of the symmetry-based indicator theory
  10. G Generating SIs in MSGs
    1. G.1 Generating SIs in type-1 MSG
      1. G.1.1 MSG 2.4 $P\overline{1}$, $X_{BS}=\mathbb{Z}_{2,2,2,4}$, Classification=$\mathbb{Z}^{3}\times\mathbb{Z}_{2}$
      2. G.1.2 $Pn,\ X_{BS}=\mathbb{Z}_{n}$, Classification=$\mathbb{Z}$
      3. G.1.3 $Pn/m,\ X_{BS}=\mathbb{Z}_{n,n,n}$, Classification=$\mathbb{Z}^{3}$
      4. G.1.4 MSG 81.33 $P\overline{4},\ X_{BS}=\mathbb{Z}_{2,2,4}$, Classification=$\mathbb{Z}\times\mathbb{Z}_{2}$
      5. G.1.5 MSG 47.249 $Pmmm,\ X_{BS}=\mathbb{Z}_{2,2,2,4}$, Classification=$\mathbb{Z}^{6}$
      6. G.1.6 MSG 123.339 $P4/mmm,\ X_{BS}=\mathbb{Z}_{2,4,8}$, Classification=$\mathbb{Z}^{5}$
      7. G.1.7 MSG 191.233 $P6/mmm$, $X_{BS}=\mathbb{Z}_{6,12}$, Classification=$\mathbb{Z}^{4}$
      8. G.1.8 MSG 147.13 $P\overline{3}$, $X_{BS}=\mathbb{Z}_{2,12}$, Classification=$\mathbb{Z}\times\mathbb{Z}_{2}$
    2. G.2 Generating MSG in type-2 MSGs
    3. G.3 Generating MSGs in type-3 MSGs
      1. G.3.1 $Pnc^{\prime}c^{\prime}$, $X_{BS}=\mathbb{Z}_{n}$, Classification=$\mathbb{Z}$
    4. G.4 Induce SIs in other MSGs from generating MSGs
      1. G.4.1 8 special MSGs with $z_{4P}^{\prime}$ indicator
      2. G.4.2 The BZ folding process
  11. H Corner cases of SI
    1. H.1 Corner cases in type-1 MSGs
    2. H.2 Corner case in type-2 MSGs
    3. H.3 Corner cases in type-3 MSGs
    4. H.4 Corner cases in type-4 MSGs
    5. H.5 Interpretation of the type-3 corner case indicator formulas
  12. I Weyl semimetals in MSGs
    1. I.1 Type-1 MSGs
    2. I.2 Type-4 MSGs
    3. I.3 Type-3 MSGs
  13. J Summary of generating SIs in MSGs
    1. J.1 Generating SIs in type-1 & 2 MSGs
    2. J.2 Generating SIs in type-3 MSG
    3. J.3 Corner cases SI
  14. K The Berry phase of closed loops in 2D BZ
  15. L The BHZ model for describing non-layer constructions
    1. L.1 2D Chern insulator model
    2. L.2 3D BHZ model
  16. M The construction of atomic insulator basis in MSGs
    1. M.1 General procedures
    2. M.2 Modification of the procedures for anti-unitary symmetries
  17. N Table: TCI classifications of MSGs
  18. O Table: Invariants and SIs of TCI classification generators for MSGs with non-trivial SI group
    1. O.1 Notations in the table of mapping
  19. P Table: Invariants of TCI classification generators for MSGs with trivial SI group

##  Appendix A Topological crystals, construction and classification

In this section, we discuss in detail the complete procedure of constructing
topological crystals (TCs) protected by magnetic space group (MSG) symmetries
using our real-space recipe, including cell-complex, building blocks, no-open-
edge condition, and bubble equivalence.

###  A.1 Cell Complex

To construct a TC, one first needs to find the geometric structure called
cell-complex, which is obtained by using the elements of crystalline symmetry
group to cut the 3D space into small regions and lower-dimensional pieces,
i.e., 3, 2, 1, and 0-dimensional cells:

  * •

3D regions are called 3-cells. A single 3-cell is also called an asymmetric
unit (AU), which fills the whole space without overlap after copied by all
symmetry group operations;

  * •

2D planes where two 3-cells meet are called 2-cells;

  * •

1D lines where 2-cells intersect are called 1-cells;

  * •

0D end-points of 1-cells are called 0-cells.

One thing worth noting is that, for any d-cell with $d=0,1,2,3$, two different
points inside it cannot be related by any crystalline symmetries, which means
a single d-cell owns no crystalline symmetries itself. However, some
symmetries can keep every point of the d-cell invariant, forming its on-site
symmetry group. The property that a single d-cell having no crystalline
symmetries is significant, as one can only place topological states protected
by onsite symmetries on a d-cell, whose classifications are already known from
the ten-fold way resultsRyu et al. (2010).

The cell complex for MSGs is similar to that for space groups (SGs) discussed
comprehensively in Ref.Song et al. (2019), with the only difference being that
MSGs can have anti-unitary symmetries, including magnetic half translations,
which makes the cell complex more complicated.

We show in Fig.4 the cell-complex of type-1 MSG 75.1 $P4$. The unit cell is
divided into four cuboids by $C_{4}$ symmetry. The AU, or the symmetry-
independent 3-cell, could be chosen as any one of the four cuboids. We choose
the 3 colored faces as symmetry-independent 2-cells and the 5 blue lines as
symmetry-independent 1-cells.

Figure 4: The cell complex of $P4$, where the red number 1-3 mark three
2-cells and blue number 1-5 mark five 1-cells.

###  A.2 2D building Blocks

Next, we consider what topological states can be decorated on the cell
complex. To begin with, we have to decide the effective symmetry classes,
i.e., Altland-Zirnbauer classesAltland and Zirnbauer (1997), of d-cells in the
cell-complex, which are determined by their on-site symmetry groups. We notice
that for all d-cells with $d=0,1,2,3$ protected by MSG symmetries, their
effective symmetry classes are either class A or class AI. Although some
d-cells can have onsite unitary symmetry, denoted by $g$, the states on them
can be diagonalized according to the eigenvalues of $g$ into different
sectors, with each sector belonging to class A. Some d-cells belong to class
AI since they have onsite anti-unitary symmetries with square $+1$, e.g., the
combination of mirror and time-reversal symmetry (TRS), $M\cdot T$. Thanks to
the well-know result of ten-fold way classifications, the classification of
class A is trivial when $d=0,1,3$ and $\mathbb{Z}$ when $d=2$, while class AI
have trivial classifications for $d\leq 3$. Therefore, we only need to
consider nontrivial decorations on 2-cells, which can be further divided into
two cases:

  1. 1.

When the 2-cell does not coincide with any mirror planes, the effective
symmetry class is just class A with $\mathbb{Z}$ classification in 2D and the
topological state to be decorated on it is nothing but the Chern insulator.

  2. 2.

When the 2-cell coincides with a mirror plane, we can divide the states into
two mirror $\pm i$ sectors, with each sector’s effective symmetry class being
A and having $\mathbb{Z}$ classification, which corresponds to the mirror
Chern numberHsieh et al. (2012); Teo et al. (2008). In this case, the
topological state decorated on it is the mirror Chern insulator. Note that
here TRS is absent, so the two mirror Chern numbers $C_{m}^{+}$ and
$C_{m}^{-}$ are not necessarily opposite, and the total Chern number
$C_{m}^{+}+C_{m}^{-}$ are generically nonzero.

In summary, there are two kinds of 2D building blocks for constructing TCs
protected by MSG symmetries: (i) Chern insulators and (ii) mirror Chern
insulators.

###  A.3 No-open-edge condition

Given a certain MSG, finding all independent magnetic TCs is equivalent to
enumerating all the topologically distinct ways of decorating the cell
complex. As topological states we construct are fully gapped in the bulk, all
the boundary states contributed by the 2-cells must cancel with each other on
the 1-cells they meet, a condition known as the “non-open-edge condition”.

We discuss the no-open-edge condition for the two types of building blocks
separately:

  1. 1.

For non-mirror 2-cells, the no-open-edge condition requires an equal number of
opposite directional 1D edge modes such that they cancel with each other and
result in a fully gapped state on the 1-cell.

Below we show an example of a 2D non-mirror plane sliced by $C_{2}$ rotation
into two 2-cells. As shown in Fig.5(a), when the 2-cells are parallel to the
$C_{2}$ axis, the $C_{2}$-related edge modes run in the same direction and
cannot be canceled. However, when the 2-cells are vertical to the $C_{2}$ axis
shown in Fig.5(b), the edge modes on the intersecting 1-cell have opposite
directions and can be canceled. The no-open-edge condition fails for the first
case while holds for the second case.

Figure 5: No-open-edge conditions. (a). Two $C_{2}$-related 2-cells that pass
the $C_{2}$ axis fail the no-open-edge condition. (b). Two $C_{2}$-related
2-cells vertical to the $C_{2}$ axis satisfy the no-open-edge condition. (c).
A mirror plane decorated with mirror Chern insulators that passes the $C_{2}$
axis fails the no-open-edge condition. (d) A mirror plane decorated with
mirror Chern insulators that passes the $C_{2}\cdot T$ axis satisfies the no-
open-edge condition.

  2. 2.

For mirror 2-cells, the no-open-edge condition is more complicated because not
only the chiralities but also the mirror eigenvalues of the edge modes need to
be considered, i.e., the total chiralities of edge modes should be zero for
both mirror $\pm i$ sectors. In another world, two edge modes can be gapped
out if and only if they have the same mirror eigenvalue and opposite
chiralities.

For example, see a mirror plane passes the $C_{2}$ or $C_{2}\cdot T$ axis, as
shown in Fig.5(c)(d). For the $C_{2}$ case, note $C_{2}$ preserves the
direction of edge mode and inverts the mirror sector, as the $C_{2}$ and
mirror axes are perpendicular which leads to $C_{2}M=-MC_{2}$. As a result,
the two edge states on the 1-cell have opposite mirror eigenvalues and the
same chirality, which can not be gapped out.

For the $C_{2}\cdot T$ case, note that $T$ also inverts mirror sector as
$[M,T]=0$ and $T(\pm i)=\mp i$. The combination of $C_{2}$ and $T$ keeps the
mirror sector unchanged. $T$ also inverts the chirality. Thus the two edge
modes on the 1-cell have the same mirror eigenvalue but opposite chiralities,
which can be gapped out.

Practically, we define a “boundary map” which maps symmetry-independent
2-cells to symmetry-independent 1-cells. This boundary map decides the way how
boundary states are contributed on each 1-cell by all 2-cells attached to it,
written in a matrix form of dimension $N_{1\text{-cells}}\times
N_{2\text{-cells}}$. By calculating the kernel space of this matrix, one
obtains the generators of decorations that satisfy the no-open-edge condition.

We use the example of MSG 75.1 $P4$ to illustrate this procedure. As shown in
Fig.4, the cell complex of $P4$ contains 3 independent 2-cells labeled as
$c_{1},c_{2},c_{3}$ and five independent 1-cells labeled as
$b_{1},b_{2},b_{3},b_{4},b_{5}$. The matrix of boundary map is calculated as

$\displaystyle\begin{pmatrix}0&0&0\\\ 0&0&0\\\ 0&4&0\\\ 0&2&-2\\\
0&0&4\end{pmatrix}$ (3)

where the $i$-th rows represent 1-cell $b_{i}$ and the $j$-th column represent
2-cell $c_{j}$. The $(i,j)$-th entry tells us how many edge modes will be
contributed on the $i$-th 1-cell if one puts a unit decoration, i.e., a Chern
insulator with Chern number $C=1$, on the $j$-th 2-cell. The kernel space of
this matrix is spanned by only one vector, $(1,0,0)$, which means there is
only one generator of decorations satisfying the no-open-edge condition. The
vector $(1,0,0)$ means only the 2-cell $c_{1}$ can be decorated with Chern
insulators, and after copying $c_{1}$ using $C_{4}$ and translation
symmetries, this decoration turns out to be a 3D quantum anomalous Hall
insulator (QAHI), i.e., Chern insulators stacked along $z$ direction.

###  A.4 Bubble equivalence

By solving the no-open-edge condition for MSGs, one gets the generators of TCs
forming a $\mathbb{Z}^{n}$ group. However, the states generated by
superimposing these generators are not all necessarily nontrivial, and one
needs to consider the so-called “bubble equivalence” process to remove the
trivial states and arrive at the finial TCI classification. Sometimes an even
number of copies of a generator could be trivialized, resulting in a
$\mathbb{Z}_{2}$ classification.

Specifically, a “bubble” represents a topological trivial state in 3D space
which has a 2D surface with non-vanishing Chern number, as shown in Fig.6(a).
It can be created at a generic point inside an AU and grow larger until it
coincides with the AU. Despite its surface with nonzero Chern number, a bubble
itself is intrinsically trivial since it can smoothly shrink back into a point
without breaking any symmetries.

When a bubble is created in an AU, there will be symmetry-related bubbles
created simultaneously inside all the symmetry-related 3-cells, and the
relative signs of surface Chern numbers of these bubbles are also determined
by symmetries. On the 2-cell where two bubbles meet, the Chern number can be
changed by either $\pm 2$ or $0$, which can be decided by considering how
Berry curvature transforms under the symmetry operation relating these two
bubbles.

Similar to the discussion of the no-open-edge condition, we discuss bubble
equivalence for non-mirror 2-cells and mirror 2-cells separately.

  1. 1.

For non-mirror 2-cells, we only have to consider the change of Chern number on
it by bubble equivalence, which could be either $\pm 2$ or $0$. When the
change of Chern number is $\pm 2$, the TCI classification is affected by
bubble equivalence.

For example, we consider the type-4 MSG 1.3, $P_{S}1$, which has an anti-
unitary half translation in the $z$ direction aside from three unitary integer
translations. Note that the layer generated by anti-unitary half translation
has the opposite Chern number from the original layer. As shown in Fig.6(b),
the decoration with even Chern numbers on each layer can be trivialized by
bubble equivalence, which leads to the $\mathbb{Z}_{2}$ TCI classification of
MSG 1.3, whose generator can be chosen as a layer construction with $C=1$ on
integer planes and $C=-1$ on half-integer planes.

Figure 6: Bubble equivalence. (a). A simple bubble with surfaces Chern number
$C=\pm 1$, where we only plot the upper and lower surfaces, and the side
surfaces are in fact all have $\pm 1$ Chern numbers. (b). A bubble equivalence
process that transforms a $C=\pm 2$ decoration into a trivial decoration. (c).
When two mirror-related bubbles coincide with a mirror plane, they change the
mirror Chern number by $(C_{m}^{+},C_{m}^{-})=\pm(1,1)$.

  2. 2.

For mirror 2-cells, one has to take into account the mirror eigenvalues aside
from Chern numbers. We show that a minimal bubble equivalence contribute
mirror Chern numbers $(C_{m}^{+},C_{m}^{-})=\pm(1,1)$ on the 2-cell where they
meet. This property is significant and will be used frequently in the
following context.

Without loss of generality, we set the mirror in $z$ direction. Note the Berry
curvature defined on x-y plane,
$(\nabla\times\vec{A})_{z}=\partial_{x}A_{y}-\partial_{y}A_{x}$, is a pseudo
vector which is unchanged under $M_{z}$. Thus two mirror-symmetric bubbles
always modify the total Chern number on the mirror plane by an even number,
i.e., the $M_{z}$-related surfaces of two bubbles have the same Chern number.

Next, we decide the mirror eigenvalues of two $M_{z}$-related bubbles.
Denoting the states on the two bubbles as $\psi_{1}$ and $\psi_{2}$. They
transform to each other under $M_{z}$ as: $\psi_{1}\rightarrow\zeta\psi_{2}$,
$\psi_{2}\rightarrow\zeta^{\prime}\psi_{1}$, where
$\zeta\cdot\zeta^{\prime}=-1$ since $M_{z}^{2}=-1$. Thus the matrix
representation of $M_{z}$ is $\begin{pmatrix}0&\zeta\\\
\zeta^{\prime}&0\end{pmatrix}$, which has two eigenvalues $\pm i$. One can
construct two linear combinations of $\psi_{1}$ and $\psi_{2}$, i.e.,
$\frac{1}{\sqrt{1+|\zeta|^{2}}}(i\psi_{1}+\zeta\psi_{2})$ and
$\frac{1}{\sqrt{1+|\zeta^{\prime}|^{2}}}(\zeta^{\prime}\psi_{1}-i\psi_{2})$,
as the two eigenstates with mirror eigenvalue $+i$ and $-i$, respectively. As
a result, a minimal bubble equivalence changes the mirror Chern numbers by
$(C_{m}^{+},C_{m}^{-})=\pm(1,1)$, as shown in Fig.6(c).

###  A.5 Decorations, first encounter

According to previous discussions, there are two types of building blocks,
i.e., Chern insulators and mirror Chern insulators, according to their
coincidence with mirror planes or not. Nevertheless, we can also classify the
building blocks in another way for the convenience of the further procedure,
that is, whether the net Chern numbers they host are zero or not. The
decorations with zero net Chern number are called M-building blocks, and if
nonzero, C-building blocks.

M-building blocks are only decorated on mirror 2-cells and are actually mirror
Chern insulators with zero net Chern numbers, i.e., $C^{+}_{m}+C^{-}_{m}=0$.
By contrast, C-building blocks can be decorated on both mirror and non-mirror
2-cells, in both cases with nonzero net Chern numbers. In another word,
C-building blocks are either Chern insulators or mirror Chern insulators with
$C^{+}_{m}\neq-C^{-}_{m}$.

##### Decorations by M-building blocks

In practice, for MSGs with mirror symmetries, one does not have to deal with
no-open-edge condition for two mirror sectors directly, which could be
technically complicated as some symmetry operations can invert the direction
of a mirror plane thus convert mirror $+i$ sector to mirror $-i$ sector or
vise versa.

As discussed in Ref.Song et al. (2019), when decorated with M-building blocks,
all 2-cells on the mirror plane can always be “glued” together and extend to
the whole mirror plane, which means the non-open-edge condition does not need
to be considered explicitly, as well as the bubble equivalence which only
effects decorations by C-building blocks. Instead, the number of independent
decorations is obtained by enumerating all symmetry-independent mirror planes.
This step, i.e., decorating M-building blocks, results in a $\mathbb{Z}^{m}$
group as part of the final classification, where $m$ is the number of
independent mirror planes.

In a word, decorations by M-building blocks can be seen as the whole
collection of decorations with zero net Chern numbers on all 2-cells.

##### Decorations by C-building blocks

To arrive at the final classification, the next step is to find the complement
of decorations by M-building blocks, i.e., all the decorations with (at least
some) 2-cells hosting nonzero net Chern numbers, which are exactly the
decorations by C-building blocks.

In this step, even for mirror 2-cells, we only have to assign one Chern
number, i.e., the total Chern number. The reason is as following. Note we
already have the decorations by M-building blocks, which can be freely added
or subtracted now as they can be regarded as trivial elements in the space of
decorations by C-building blocks. More specifically, if we assign two mirror
Chern numbers, $(C_{m}^{+},C_{m}^{-})$, on a mirror 2-cell, we can add an
M-building block with $(C_{m}^{-},-C_{m}^{-})$ to it, resulting in a state of
$(C_{m}^{+}+C_{m}^{-},0)$, which can be represented by its total Chern number.
With this simplification, we effectively reduce the degree of freedom on each
mirror 2-cell from 2 to 1 and only have to deal with the boundary maps
together with the bubble equivalence involving only one Chern number (the
total Chern number) for each 2-cell.

In practice, the result of this step can be calculated by the quotient space
Ker/Img, where Ker stands for the kernel space of no-open-edge condition and
Img stands for the space defined by bubble equivalence. After this step, we
get a set of generators like $\mathbb{Z}^{l}\times\mathbb{Z}_{2}^{n}$ by
decorating C-building blocks, where $n=0,1$, which means there is at most one
$\mathbb{Z}_{2}$ generator and will be explained later.

###  A.6 Final TCI classification

Finally, we combine the results obtained by decorating M-building blocks and
C-building blocks into
$\mathbb{Z}^{m}\times\mathbb{Z}^{l}\times\mathbb{Z}_{2}^{n}$. However, this
could still not be the final classification, because when there are mirror
planes, the $\mathbb{Z}_{2}$ generator obtained in the second step can be
absorbed into $\mathbb{Z}^{m}$ and results in
$\mathbb{Z}^{m}\times\mathbb{Z}^{l}$. This is a simple case of group extension
that, after doubling and a process of bubble equivalence, the $\mathbb{Z}_{2}$
generator becomes a decoration by M-building blocks with each mirror plane
having $C^{+}_{m}=-C^{-}_{m}$, i.e., an element in $Z^{m}$, which will be
explained more specifically later.

Among our final results , we remark two points about the TCI classifications
in type-4 MSGs are worth mentioning:

  * •

when the MSG has no mirror symmetry, the classification is always
$\mathbb{Z}_{2}$;

  * •

when the MSG has mirror symmetries, the classification is always
$\mathbb{Z}^{m}$, since the $\mathbb{Z}_{2}$ generator is absorbed into mirror
decorations, i.e., decorations by M-building blocks.

This means that the $\mathbb{Z}_{2}$ generator always exists for type-4 MSGs,
i.e., no type-4 MSG has trivial classification. The above two conclusions,
although not proved rigorously, could be understood from the following
arguments:

  * •

$\mathbb{Z}^{l}$ generators (which correspond to translation $\mathbb{Z}$
decorations, as discussed below) are prohibited by anti-unitary translations
in type-4 MSGs;

  * •

The simplest type-4 MSG $P_{S}1$, which has anti-unitary half-integer
translation in one direction and unitary integer translations in other two
directions, has a $\mathbb{Z}_{2}$ generator of TCI classification
(corresponding to a $Z_{2}$ decoration, as discussed below) which has a
nonzero invariant of the anti-unitary half translation. For an arbitrary
type-4 MSG, its $\mathbb{Z}_{2}$ generator can be seen as a complication of
the $\mathbb{Z}_{2}$ generator in $P_{S}1$, as other crystalline symmetries
will add more decorated 2-cells, but cannot trivialize the original $Z_{2}$
decoration in $P_{S}1$.

##  Appendix B Topological invariants

Topological invariants quantitatively distinguish TCIs from trivial
insulators. Formally, if a state can not be adiabatically deformed into a
trivial state preserving symmetry $g$, it has a nontrivial topological
invariant protected by $g$.

To find all MSG symmetries that can independently protect topological
invariants, we consider when a single generator of MSG symmetry operation $g$
is present, what TC can be constructed. More specifically, we first construct
the cell complex with respect to symmetry $g$, and then decorate possible 2D
building blocks on 2-cells under the no-open-edge condition. Lastly, we check
the effect of bubble equivalence on the decorations. We say $g$ has (i)
trivial invariant if no TC can be constructed, (ii) $\mathbb{Z}$ invariant if
there exists a non-trivial TC and can not be trivialized by copying arbitrary
times, or (iii) $\mathbb{Z}_{n}$ invariant if there exists a non-trivial TC
and can be trivialized by copying $n$ times.

MSG Symmetries | Invariant type
---|---
unitary rotation $C_{n}$ | Trivial
anti-unitary improper point group symmetries $P\cdot T$, $M\cdot T$,
$S_{n}\cdot T$
unitary improper $S_{n}$, $P$, $\\{M|\frac{1}{2}\\}$ | $\mathbb{Z}_{2}$
anti-unitary translation $\\{E|t\\}\cdot T$
anti-unitary proper $C_{n}\cdot T$, $\\{C_{n}|t\\}\cdot T$
unitary translation $\\{E|R\\}$ | $\mathbb{Z}$
unitary mirror $M$
unitary screw $\\{C_{n}|t\\}$
anti-unitary glide $\\{M|\frac{1}{2}\\}\cdot T$
Table 3:  MSG symmetries and their corresponding invariant types. The
invariants of the unitary screw and anti-unitary glide are bound to the
translation invariant. Figure 7: The minimal decoration of the corresponding
topological invariant.

We enumerate all topological invariants protected by MSG symmetries in
Table.3, which are either $\mathbb{Z}$-type or $\mathbb{Z}_{2}$-type, and
their corresponding minimal nontrivial decoration, i.e., with invariant
$\delta(g)=1$, in Fig.7.

  1. 1.

Unitary integer translation. Its corresponding invariant is nothing but the
weak invariant $\delta_{w,i}$. A minimal decoration with $\delta_{w,i}=1$ is a
simple LC by placing $C=1$ layers on integer planes in the $\bm{a}_{i}$
direction. After doubling $n$ times, this LC is still nontrivial with layers
having $C=n$, which confirms the $\mathbb{Z}$-type of weak invariants.

  2. 2.

Anti-unitary translation. A minimal decoration with
$\delta(\\{E|\frac{1}{2}\bm{a}_{i}\\}\cdot T)=1$ is a LC with $C=1$ layers on
integer planes and $C=-1$ layers on half-integer planes in the $\bm{a}_{i}$
direction. Its doubled state can be trivialized by the bubble equivalence,
confirming its $\mathbb{Z}_{2}$-type.

  3. 3.

Anti-unitary rotations. As shown in Fig.7(3), for $C_{2}\cdot T$ and
$C_{6}\cdot T$, their minimal decorations have 1 and 3 layers passing the
rotation axis, respectively, while for $C_{4}\cdot T$, there are 4 half-layers
that intersect at the rotation axis, forming a non-LC. Their doubled states
can all be trivialized, validating their $\mathbb{Z}_{2}$-type.

  4. 4.

Mirror. With only mirror symmetry, the 3D space is separated by a single
mirror plane (the only 2-cell) into two semi-infinite regions (the two
3-cells), and we need to consider decorations on the mirror plane and the
effect of 3D bubbles. Without TRS, the two real-space mirror Chern numbers
$(C^{+}_{m},C^{-}_{m})$ on the mirror plane can take independent values, and
they seem to serve as two generators of decorations, i.e.,
$(C^{+}_{m},C^{-}_{m})=(1,0)$ and $(0,1)$, forming a
$\mathbb{Z}\times\mathbb{Z}$ group. However, as discussed in Sec.A.4, a
decoration with $(C^{+}_{m},C^{-}_{m})=\pm(1,1)$ can be trivialized by the
bubble equivalence. Therefore, there is only one independent generator, which
can be chosen as either $(1,0)$ or $(0,1)$, forming a
$\mathbb{Z}\times\mathbb{Z}/\\{z_{1}=z_{2}\\}=\mathbb{Z}$ group. We plot a
minimal decoration of mirror by placing a $(C_{m}^{+},C_{m}^{-})=(1,0)$ state
on the mirror plane, as shown in Fig.7(4).

We can recombine the two mirror Chern numbers into
$C_{\text{diff}}=C^{+}_{m}-C^{-}_{m}$ and
$C_{\text{total}}=C^{+}_{m}+C^{-}_{m}$. Note the bubble equivalence has no
effect on $C_{\text{diff}}$, but reduces $C_{\text{total}}$ from
$\mathbb{Z}$-valued to $\mathbb{Z}_{2}$-valued, so the two redefined mirror
Chern numbers seem to form a $\mathbb{Z}\times\mathbb{Z}_{2}$ group. However,
the two generators of $\mathbb{Z}$ and $\mathbb{Z}_{2}$ are not independent,
because the $\mathbb{Z}_{2}$ generator can also be the generator of
$\mathbb{Z}$, thus can be absorbed into $\mathbb{Z}$, leading to a
$\mathbb{Z}$ group as the final result.

  5. 5.

Unitary glide. A minimal decoration with
$\delta(\\{M|\frac{1}{2}\bm{a}_{i}\\})=1$ is shown in Fig.7(5), which has two
layers perpendicular to the glide plane with $C=\pm 1$ and connected by the
glide half translation in a unit cell. As
$(\\{M|\frac{1}{2}\bm{a}_{i}\\})^{2}=\\{E|\bm{a}_{i}\\}$, where $\bm{a}_{i}$
is a lattice vector inside the glide plane, the existence of this non-
symmorphic glide symmetry indicates the presence of translation symmetry
$\bm{a}_{i}$. This decoration is similar to the anti-unitary translation
decoration, which confirms the $\mathbb{Z}_{2}$-type of the glide invariant.
We remark that this LC of horizontal layers can be deformed into another LC of
vertical layers preserving the glide symmetry.

Figure 8: (a)-(c). The process of deforming horizontal layers into vertical
layers preserving the glide symmetry.

  6. 6.

Anti-unitary glide. Its minimal decoration has a similar configuration with
the unitary glide, but the Chern number of the two layers in the unit cell are
both $C=1$. As $(\\{M|\frac{1}{2}\bm{a}_{i}\\}\cdot
T)^{2}=\\{E|\bm{a}_{i}\\}$, the existence of the anti-unitary glide also
implies the translation symmetry, with their invariants bound together and
both being $\mathbb{Z}$-type, i.e., $\delta(\\{M|\frac{1}{2}\bm{a}_{i}\\}\cdot
T)=\frac{1}{2}\delta_{w,i}$, which enforces $\delta_{w,i}$ to take even
numbers. For this minimal decoration, we have $\delta_{w,i}=2$ and
$\delta(\\{M|\frac{1}{2}\bm{a}_{i}\\}\cdot T)=1$.

  7. 7.

Anti-unitary screws. The minimal decorations of $C_{n}\cdot T$ screws are
similar to those of $C_{n}\cdot T$ rotations. Moreover, these decorations can
be transformed into layers vertical to the screw axis with alternating Chern
numbers preserving the anti-unitary screw symmetry, as shown in Fig.9. We plot
the transformation process for $\\{C_{2}|\frac{1}{2}\bm{a}_{3}\\}\cdot T$ in
Fig.9(4). These invariants are also $\mathbb{Z}_{2}$-type, as their doubled
decorations can be trivialized.

Figure 9: (1)-(3). Minimal decorations of anti-unitary screws
$\\{C_{n}|\frac{1}{n}\bm{a}_{i}\\}\cdot T,n=2,4,6$, which are equivalent to
those in Fig.7(7). (4). The process of deforming two horizontal layers into
vertical layers, preserving the $\\{C_{2}|\frac{1}{2}\bm{a}_{3}\\}\cdot T$
symmetry.

  8. 8.

Inversion. A minimal decoration with $\delta(P)=1$ is constructed by placing a
$C=1$ layer that passes the inversion center. Its doubled state, i.e., a $C=2$
layer, can be trivialized by the bubble equivalence, or equivalently, split
into two $C=1$ layers and move to infinity in an inversion-symmetric way. Thus
the doubled state has $\delta(P)=0$ and confirms the $\mathbb{Z}_{2}$-type of
$\delta(P)$.

  9. 9.

$S_{4}$. A minimal decoration with $\delta(S_{4})=1$ can be a non-LC with four
half-layers similar to the decoration of $C_{4}\cdot T$, as shown in Fig.7(9).
But unlike $C_{4}\cdot T$, this non-LC can be transformed into a single layer
that passes the $S_{4}$-center preserving $S_{4}$ symmetry. $\delta(S_{4})$ is
$\mathbb{Z}_{2}$-type as its doubled state can be trivialized.

Note there are four rotoinverisons $S_{n},n=2,3,4,6$, among which
$S_{2}=M,S_{3}=C_{3}\cdot P,S_{6}=C_{3}\cdot M$, and $S_{4}$ is the only
independent symmetry in the sense that it cannot be decompose into a direct
product of smaller point groups. As a result, we only discuss the minimal
decoration $S_{4}$.

  10. 10.

Unitary screws. In Fig.7(10), we plot minimal decorations of
$\\{C_{n}|\frac{1}{n}\bm{a}_{i}\\}$, which have $n$ layers with $C=1$
connected by the screw in a unit cell, and can be seen as a $n$-time copy of
the translation decoration.

In general, a unitary screw can be $\\{C_{n}|\frac{m}{l}\bm{a}_{i}\\}$, where
$l,m$ are coprime numbers, e.g., $\frac{3}{4}$ and $\frac{5}{6}$. In these
cases, there are also $l$ layers in a unit cell, which has $\delta_{w,i}=l$
and $\delta(\\{C_{n}|\frac{m}{l}\bm{a}_{i}\\})=m$.

As discussed in the main text, these topological invariants can be calculated
for TCs in an intuitive way. For a given symmetry $g$, first, choose a generic
point $r$ inside an arbitrary AU, and then use $g$ to transform it to its
image point $g\cdot r$. Draw a path connecting these two points without
touching any 1- or 0-cells. The invariant $\delta(g)$ is determined by the
decorated 2-cells through the path:

  * •

For every symmetry $g$, $\delta(g)$ can be the net Chern number accumulated
through the path. Even if $g$ is a mirror symmetry, we can ignore the two
mirror sectors and count only the net (total) Chern number. We call this type
of invariant defined by the net Chern number “C-invariants” and denote them as
$\delta_{C}(g)$.

  * •

For a mirror symmetry $M$, we can also consider the Chern numbers for each
mirror sectors, i.e., the real-space mirror Chern numbers
$(C_{m}^{+},C_{m}^{-})$, which are obtained by counting the mirror Chern
insulators decorated on the mirror plane. The momentum space mirror Chern
numbers can be calculated from the real space mirror Chern numbers, with
procedure similar to that for non-magnetic case shown in Ref.Song et al.
(2018a).

From the above algorithm for calculating C-invariants for TCs, a homomorphism
between the symmetry operations and their invariants naturally arises, as the
Chern number accumulated through the path can be superimposed:

$\delta_{C}(g_{1})+\delta_{C}(g_{2})=\delta_{C}(g_{1}\cdot g_{2})$ (4)

With this homomorphism, one only needs to calculate the C-invariants for the
generators of an MSG in order to obtain all the invariants. However, we remark
that as some C-invariants are $\mathbb{Z}$-type while others are
$\mathbb{Z}_{2}$-type, it is the original value calculated from the path
$r\rightarrow g\cdot r$ before taking module that should be adopted when
applying this homomorphism.

In the following text, we sometimes use “invariants” to indicate
“C-invariants” for simplicity, since in most cases they are completely the
same except for mirrors. The specific meaning of “invariants” can be inferred
from the context.

Interestingly, we notice the types of C-invariants are closely related to the
types of corresponding symmetry operations, especially for those protected by
magnetic point group (MPG) symmetries. As can been from Table.3, proper
unitary symmetries have trivial C-invariants, while anti-unitary proper
symmetries have $\mathbb{Z}_{2}$ C-invariants (here mirror is special, as
strictly speaking its invariant should be $\mathbb{Z}$-type, nevertheless we
could regard the total Chern number as its $\mathbb{Z}_{2}$ C-invariant);
unitary improper symmetries have $\mathbb{Z}_{2}$ C-invariants, while anti-
unitary improper symmetries have trivial C-invariants.

##  Appendix C Surface states

Surface states are in one-to-one correspondence with the topological
invariants. In other words, a nontrivial invariant has a corresponding
nontrivial surface state, and different surface states can be superimposed. We
enumerate all surface states protected by topological invariants in MSGs,
which can be seen from their minimal decorations. Note when constructing the
minimal decorations of invariants in Fig.7, we ignore the three lattice
translations. However, when constructing the surface states of these
invariants, we sometimes need to restore the lattice translations and require
irrelevant invariants to be zero.

Figure 10: Surface states in MSGs, with the corresponding symmetries marked
above the plots. The 2D surface modes can be either sloped-like chiral surface
modes in (1),(6),(10) or Dirac cones in (2),(3),(4),(5).

  1. 1.

Unitary integer translation $\\{E|\bm{a}_{i=1,2,3}\\}$. The corresponding
surface state is known as the 3D quantum anomalous Hall effect. There are
$Z=\delta_{w,i}$ chiral surface modes on $\bm{a}_{i}$-preserving planes, with
the surface modes look like a slope in the 2D surface Brillouin zone (BZ).

  2. 2.

Anti-unitary translation $\\{E|\frac{1}{2}\bm{a}_{i=1,2,3}\\}\cdot T$. As
discussed in Ref.Mong et al. (2010); Fang et al. (2013), there are an odd
number of Dirac cones on $\bm{a}_{i}$-preserving planes, where $\bm{a}_{i}$ is
the direction of anti-unitary translation. These Dirac cones can only appear
at the 4 TRIMs in the 2D surface BZ.

  3. 3.

Anti-unitary rotation $C_{n}\cdot T,n=2,4,6$. Surface states protected by
$C_{2}\cdot T$ and $C_{4}\cdot T$ have been previously discussed in Ref.Fang
and Fu (2015); Shiozaki and Sato (2014); Ahn and Yang (2019) and Ref.Schindler
et al. (2018), respectively.

In general, the surface states for $C_{n}\cdot T,n=2,4,6$ have 2, 4, and 6
chiral hinge modes on the side surface, respectively, while a single Dirac
cone on the top surface. For $C_{2}\cdot T$, the position of the Dirac cone is
unpinnedFang and Fu (2015), while for $C_{n=4,6}\cdot T$, the Dirac cone is
pinned at $\Gamma$ (or a $C_{n=4,6}\cdot T$-invariant point) in the surface
BZ. Note for $C_{6}\cdot T$, the single Dirac cone can split into three
$C_{6}\cdot T$-symmetric Dirac cones by adding trivial bands.

These surface Dirac cones can be analyzed from the surface Hamiltonian as
follows. Consider a minimal Dirac surface Hamiltonian

$H(k)=v_{x}k_{x}\sigma_{1}+v_{y}k_{y}\sigma_{2}$ (5)

which is invariant under $C_{n}\cdot T$ symmetry, i.e.,

$(C_{n}\cdot T)H(k)(C_{n}\cdot T)^{-1}=H((C_{n}\cdot T)^{-1}k)$ (6)

where

$\displaystyle
C_{n}=e^{i\frac{\pi}{n}\sigma_{3}},\;\;T=i\sigma_{2}\mathcal{K}$ (7)

and $\sigma_{i,i=1,2,3}$ are the three Pauli matrices. Mass term proportional
to $\sigma_{3}$ is forbidden by $C_{n}\cdot T$, i.e., one can not add mass
term to gap the Hamiltonian $H(k)$. However, the doubled Hamiltonian
$\tau_{0}\otimes H(k)$ can be gapped by adding mass term proportional to
$\tau_{2}\otimes\sigma_{3}$, which means a single Dirac cone is stable while
two Dirac cones can be gapped. Therefore, the classification of this surface
state is $\mathbb{Z}_{2}$, in accordance with the $\mathbb{Z}_{2}$ invariant.

  4. 4.

Mirror. As shown in Fig.10(4), when a mirror-preserving termination is made,
the mirror-invariant planes in the 3D BZ now become mirror-invariant lines in
the 2D surface BZ. Surface states protected by mirror symmetries are
determined by the momentum space mirror Chern numbers on each mirror-invariant
lines in the surface BZ. In the following, we discuss the surface states on a
specific mirror-invariant line of the surface BZ, with $(C_{m}^{+},C_{m}^{-})$
denoting the momentum space mirror Chern numbers on it. If $C^{\pm}_{m}>0$,
there are $C^{\pm}_{m}$ right-moving surface modes with mirror eigenvalue $\pm
i$, while for $C^{\pm}_{m}<0$ there are $|C^{\pm}_{m}|$ left-moving modes with
mirror eigenvalue $\pm i$ (the direction of surface modes is convention-
dependent and can be exchanged). As a result,

    1. (a)

when $C_{m}^{+}=-C_{m}^{-}$, there are
$|C_{m}|=\frac{1}{2}|C^{+}_{m}-C^{-}_{m}|$ Dirac cones, as plotted in
Fig.10(4).

    2. (b)

when $C_{m}^{+}\neq-C_{m}^{-}$:

       * •

If $C_{m}^{+}C_{m}^{-}<0$, there are $N=\min\\{|C_{m}^{+}|,|C_{m}^{-}|\\}$
Dirac cones and $||C_{m}^{+}|-|C_{m}^{-}||$ chiral surface modes.

       * •

If $C_{m}^{+}C_{m}^{-}\geq 0$, there are $|C_{m}^{+}|+|C_{m}^{-}|$ chiral
surface modes.

Note that the chiral surface modes are protected by not only mirror but also
the (total) Chern number of the system. In fact, even if the mirror symmetry
is broken, the chiral modes are still preserved due to nonzero Chern number.

Lastly, we remark that there exist hinge modes when a mirror-preserving hinge
is made.

  5. 5.

Unitary glide $\\{M|\frac{1}{2}\bm{a}_{i}\\}$. Its surface state has been
discussed in Ref.Fang and Fu (2015); Shiozaki et al. (2015, 2016), and more
recently in Ref.Kim et al. (2019); Kim and Murakami (2020), which has an odd
number of Dirac cones on the glide-symmetric plane. This surface state can
also be understood from the minimal decoration of
$\\{M|\frac{1}{2}\bm{a}_{i}\\}$, which resembles that of the anti-unitary
translation. Thus they should share the same type of surface states. These
Dirac cones can appear at the glide-invariant lines in the surface BZ, and an
even number of Dirac cones can annihilated with each other.

We remark that there exist chiral hinge modes when a glide-preserving hinge is
made.

  6. 6.

Anti-unitary glide $\\{M|\frac{1}{2}\bm{a}_{i}\\}\cdot T$. As
$(\\{M|\frac{1}{2}\bm{a}_{i}\\}\cdot T)^{2}=\\{E|\bm{a}_{i}\\}$, the weak
invariant $\delta_{w,i}=2Z$ when the invariant
$\delta(\\{M|\frac{1}{2}\bm{a}_{i}\\}\cdot T)=Z$, which means there are $2Z$
chiral surface modes on a glide-symmetric plane.

  7. 7.

Anti-unitary screw $\\{C_{n}|t\\}\cdot T,n=2,4,6$. On a typical cylinder
termination, there are 2, 4, and 6 $C_{n}$-symmetric chiral edge modes on the
side surface, respectively. Note the top surface breaks the
$\\{C_{n}|t\\}\cdot T$ symmetry and thus has no gapless modes in general.

  8. 8.

Inversion. There is one inversion-symmetric chiral hinge mode on a surface
termination that preserves the inversion symmetry. States with $\delta(P)=1$
have non-trivial axion angle $\theta=\pi$ and are usually referred to as
“axion insulators”. The axion angle $\theta$ is quantized to $0$ or $\pi$ when
there exist anti-unitary proper symmetries (including TRS) or unitary improper
symmetries, and systems with $\theta=\pi$ can be seen as axion insulators in a
generalized sense.

  9. 9.

$S_{4}$. There is one $S_{4}$-symmetric chiral hinge mode on a surface
termination that preserves the $S_{4}$ symmetry.

  10. 10.

Unitary screw $\\{C_{n}|t\\},n=2,3,4,6$. In the surface states shown in
Fig.10(10), we set $t=\frac{1}{n}\bm{a}_{i}$ for simplicity, and there are $n$
chiral surface modes related by $\\{C_{n}|\frac{1}{n}\bm{a}_{i}\\}$ on the
side surface when $\delta(\\{C_{n}|\frac{1}{n}\bm{a}_{i}\\})=1$.

Generally, the screw vector could be $t=\frac{m}{l}\bm{a}_{i}$, where $m,l$
are coprime integers. On the side surface, there are $\delta_{w,i}$ chiral
surface modes related by $\\{C_{n}|\frac{m}{l}\bm{a}_{i}\\}$ per lattice
vector, which can be seen from the invariant. As both
$\delta(\\{C_{n}|\frac{m}{l}\bm{a}_{i}\\})$ and $\delta_{w,i}$ are integers
and $\\{C_{n}|\frac{m}{l}\bm{a}_{i}\\}^{l}=\\{(C_{n})^{l}|m\bm{a}_{i}\\}$,
$\delta_{w,i}$ should be a multiply of $l$ and
$\delta(\\{C_{n}|\frac{m}{l}\bm{a}_{i}\\})=\frac{m}{l}\delta_{w,i}$. Thus
$\\{C_{n}|t\\}$ shares the same surface state with the unitary translation,
similar to the anti-unitary glide.

##  Appendix D Decorations, second encounter

In this section, we classify the TCs into three types and discuss them in
terms of their invariants. As can be seen from the general form of TCI
classifications, i.e., $\mathbb{Z}^{m}\times\mathbb{Z}^{l}$ or
$\mathbb{Z}^{l}\times\mathbb{Z}_{2}$, there are three types of decorations of
TCs:

  1. 1.

Mirror Decorations. Mirror Chern insulators with $(C^{+}_{m},C^{-}_{m})$ =
$(n,-n)$, i.e., zero net Chern numbers, decorated on mirror planes, while all
non-mirror 2-cells are not decorated, with an example shown in Fig.11(a).

  2. 2.

(Translation) $\mathbb{Z}$ decorations. Decorations with nonzero weak
(translation) invariants, which means the Chern number per unit cell is
nonzero, as shown in Fig.11(b).

  3. 3.

$Z_{2}$ decorations. Decorations that are not protected by (unitary) integer
translation symmetries (but some are protected by half magnetic translations).
They have zero net Chern number per unit cell, i.e., zero weak invariants, and
nontrivial $\mathbb{Z}_{2}$ invariants. As discussed in the main text, these
$Z_{2}$ decorations are axion insulators, characterized by the $\pi$ axion
angle. We plot a $Z_{2}$ decoration of MSG $Pmm2$ in Fig.11(c).

$Z_{2}$ decorations can also be further divided into two types depending on
whether they have mirror symmetries. When there are no mirror planes, the
$Z_{2}$ decorations become trivial after doubling, while when there are mirror
planes, the doubling of a $Z_{2}$ decoration followed by bubble equivalence
becomes a mirror decoration, where all mirror planes are decorated with
$(C_{m}^{+},C_{m}^{-})=\pm(1,-1)$. Caution that we still call it a $Z_{2}$
decoration for simplicity, which is an abuse of notation, as the decoration
actually does not become trivial but transforms into another type of
decoration after doubling.

When mirror planes are present, this $Z_{2}$ decoration must be chosen as a
generator of classifications, though its original contribution to the total
classification, i.e., the $\mathbb{Z}_{2}$ factor, is absorbed into
$\mathbb{Z}^{m}$ in a way that $2\bar{Z}_{2}\sim\sum^{m}_{i=1}\bar{M}_{i}$,
where $\bar{M}_{i}$’s are generators of mirror decorations and $\sim$ means up
to bubble equivalence. More specifically, if there are $m$ mirror planes, the
$m$ generators for the classification $\mathbb{Z}^{m}$ should be chosen as
$\\{\bar{Z}_{2},\bar{M}_{1},\bar{M}_{2},\cdots,\bar{M}_{m-1}\\}$.

Figure 11: Three types of decorations. (a). Mirror decoration. (b). $Z$ or
translation decorations. (c). $Z_{2}$ decorations.

The three types of decorations can be well characterized by their invariants:

  1. 1.

Mirror Decorations. As discussed above, this type of decorations are mirror
Chern insulators with $C^{+}_{m}=-C^{-}_{m}$ on all mirror planes. Since the
net Chern number is zero for all 2-cells, all invariants except mirror Chern
numbers are zero.

This type of decorations can be defined as those with zero net Chern number on
every 2-cell, which is enough to distinguish them from the other two types of
decorations.

  2. 2.

(Translation) $\mathbb{Z}$ decorations. As mentioned before, this type of
decorations are protected by (unitary) integer translation symmetries, thus
they must have nonzero weak invariants. Besides, other invariants, including
the mirror Chern numbers and $\mathbb{Z}_{2}$ invariants, can also be nonzero.

When mirror planes are present, $\mathbb{Z}$ decorations can have mirror Chern
insulators with $C^{+}_{m}\neq-C^{-}_{m}$ on mirror planes, which have nonzero
net Chern number and do not belong to the mirror decorations. Moreover, when
the center of a symmetry operation with $\mathbb{Z}_{2}$ invariant, e.g.,
$S_{4}$, is passed by a decorated layer with odd Chern number, the
$\mathbb{Z}_{2}$ invariant is nonzero for this $\mathbb{Z}$ decoration. In
fact, a $Z$ decoration just becomes another $\mathbb{Z}$ decoration when added
with a mirror or $Z_{2}$ decoration.

In a word, $\mathbb{Z}$ or translation decorations are those with nonzero net
Chern number per unit cell.

  3. 3.

$Z_{2}$ decorations. Contrary to the $\mathbb{Z}$ decorations, this type of
decorations must have zero weak invariants. Note the “weak” invariants refer
to those protected by unitary integer translations, but not magnetic half
translations in type-4 MSGs.

For this type of decorations, all $\mathbb{Z}$-type invariants (except mirror
Chern numbers), including weak invariants together with unitary screw
invariants and anti-unitary glide invariants, are all zero, while all
$\mathbb{Z}_{2}$ invariants must be nonzero, i.e.
$\delta(g)_{\mathbb{Z}_{2}}=1$.

This can be seen with the help of the homomorphism between symmetries and
invariants, i.e., Eq.(4). Firstly, a $Z_{2}$ decoration must have at least one
nonzero $\mathbb{Z}_{2}$ invariant of some symmetry $g_{1}$, i.e.,
$\delta(g_{1})=1$. Then suppose there is another symmetry $g_{2}$ having
$\mathbb{Z}_{2}$ invariant. Their product, denoted as $g_{3}=g_{1}\cdot
g_{2}$, must have $\mathbb{Z}$ or trivial invariant. This is because $g_{1}$
and $g_{2}$, which have $\mathbb{Z}_{2}$ invariant, must be unitary improper
or anti-unitary proper, and their product $g_{3}$ must be unitary proper or
anti-unitary improper, and these two symmetry operations have $\mathbb{Z}$ or
trivial invariants, as shown in Table.3. If $\delta(g_{2})=0$, then
$\delta(g_{3})=\delta(g_{1})+\delta(g_{2})=1+0=1$, which is forbidden in
$Z_{2}$ decorations no matter $g_{3}$ has $\mathbb{Z}$ or trivial invariant.
Thus we must have $\delta(g_{2})=1$. Similarly, for any other $g$ with
$\mathbb{Z}_{2}$ invariant, we have $\delta(g)=1$.

As a result, the values of all $\mathbb{Z}_{2}$ invariants are bonded and
equal to 1 for the $Z_{2}$ decoration, which can be called as the “axion
invariant” and take the form of 3D magnetoelectric polarization $P_{3}$Fang et
al. (2012). This dictates that for any MSG, there is either one independent
$Z_{2}$ decoration (when $\delta(g)_{\mathbb{Z}_{2}}=1$), i.e., an axion
insulator, or no $Z_{2}$ decoration (when $\delta(g)_{\mathbb{Z}_{2}}=0$).

Figure 12: The $Z_{2}$ decoration becomes a mirror decoration with all mirror
planes decorated when doubled and transformed into an LC using the bubble
equivalence.

There is another interesting and significant property for the $Z_{2}$
decorations as mentioned previously. When there are mirror planes, the
doubling of a $Z_{2}$ decoration followed by the bubble equivalence becomes a
mirror decoration with all mirror planes decorated. This can also be
understood from the invariants. As demonstrated before, the $Z_{2}$ decoration
has all $\mathbb{Z}_{2}$ invariants nonzero, and mirror symmetries can have
the net Chern number as their corresponding $\mathbb{Z}_{2}$ C-invariants.
This indicates that for a $Z_{2}$ decoration, each mirror 2-cell is decorated
with $C_{\text{total}}=C^{+}_{m}+C^{-}_{m}=\pm 1$ state, i.e., state with
$(C^{+}_{m},C^{-}_{m})=(\pm 1,0)$ or $(0,\pm 1)$. As shown in Fig.12, after
doubling, each mirror 2-cell has $(C^{+}_{m},C^{-}_{m})=(\pm 2,0)$ or $(0,\pm
2)$. Bubble equivalence changes $(C^{+}_{m},C^{-}_{m})$ by $\pm(1,1)$,
resulting in $(C^{+}_{m},C^{-}_{m})=\pm(1,-1)$ on each mirror 2-cell, which is
a mirror decoration with all mirror planes decorated.

To summarise, $Z_{2}$ decorations are those with zero net Chern number per
unit cell but nonzero net Chern number on some 2-cells.

##  Appendix E Layer and non-layer constructions

Recall that an overwhelming majority of TCs in non-magnetic SGs are layer
constructions(LCs), with exceptions in only 12 SGs, which are called non-layer
constructions (non-LCs)Song et al. (2019, 2018a). Those non-LCs built by 2D
TIs have 2D planes that are only partly decorated, namely, the 2-cells
decorated with 2D TIs do not extend to the entire 2D plane. On the contrary,
there is a vast number of non-LCs in (type-1,3,4) MSGs. It is difficult and
also not necessary to exhaustively find all non-LCs in all MSGs, and here we
give three representatives for demonstration.

We remark an interesting property of non-LCs in MSGs. Unlike the non-LCs in
non-magnetic SGs which have incomplete 2D decorated planes, here in MSGs all
the non-LCs we know have complete 2D decorated planes, which split into small
2D (mirror) Chern insulator pieces with different signs of (mirror) Chern
number.

Figure 13: Three examples of non-layer constructions(non-LCs). (1). The non-LC
in $Pmm2$, which has $C_{2z},M_{x}$, and $M_{y}$. (2). The non-LC in
$P2^{\prime}/m$, which has $C_{2}\cdot T$. (3). The non-LC in $P_{C}4$, which
has $C_{4z}$ and anti-unitary half translation $T\cdot t_{110}$, where
$t_{110}=\\{E|\frac{1}{2}\frac{1}{2}0\\}$. Note in (1) and (2), the side
surfaces of the unit cell are also decorated, which we omit for simplicity.

  * •

The first representative non-LC is the $Z_{2}$ decoration of type-1 MSG
$Pmm2$, which has $C_{2z}$ and two orthogonal mirrors $M_{x}$ and $M_{y}$
anti-commuting with each other.

Two 2-cells related by $C_{2}$ symmetry on the same mirror plane have the same
directional chiral edge modes on the 1-cell they intersect, i.e., the two
red/green curves in Fig.13(1). Thus decorating only one mirror plane with
C-building blocks breaks the no-open-edge condition, and the other mirror
plane must also be decorated to compensate the edge modes on the $C_{2}$-axis.
These decorated 2-cells are fixed on mirror planes and cannot be moved
together to trivialize. This decoration could be seen as the simplest one
among all non-LCs in MSGs, where only planes in the $x$ and $y$ directions are
decorated non-uniformly, with adjacent pieces on a 2D plane having opposite
Chern numbers.

  * •

The second representative non-LC is the $Z_{2}$ decoration of type-3 MSG
$P2^{\prime}/m$ ($P4^{\prime}/m$ and $P6^{\prime}/m$ have similar non-LCs),
where the crucial symmetry operation that makes it a non-LC is the anti-
unitary rotation $C_{2y}\cdot T$. The mirror planes defined by $M_{y}$ is
split by $C_{2y}\cdot T$ into two 2-cells, where the two edge modes related by
$C_{2y}\cdot T$ have the same direction and can not cancel with each other on
the 1-cell they meet. This forces the 2D plane vertical to mirror plane to be
decorated as well, in order to satisfy the no-open-edge condition.

  * •

The third representative non-LC is the $Z_{2}$ decoration of type-4 MSG 3.6
$P_{C}2$. In this case there is no mirror symmetry, but the magnetic
translation $T\cdot\\{E|\frac{1}{2}\frac{1}{2}0\\}$ in the diagonal direction
makes the decoration a non-LC.

##  Appendix F A brief review of the symmetry-based indicator theory

Symmetry-based indicator (SI) theory uses the symmetry data at high-symmetry-
points (HSPs) in the BZ to diagnose the band topology of a system. The
symmetry data composes of the irreducible representations (irreps) of the
valence bands at HSPs in SGs, or irreducible co-representations (coirreps) in
MSGs, and the collection of coirreps at all HSPs is denoted as the band
representation (BR). SI theory compares the difference between two linear
spaces, i.e., the space of all atomic insulators (AIs) and the space of BRs
that satisfy the compatibility relations between all HSPs. A BR having zero SI
can be decomposed into an integer combination of AIs, while a BR with nonzero
SI cannot.

AIs are constructed by placing non-interacting atoms in real space, which have
flat dispersions. For a specific MSG, the BR of its AIs form a finitely
generated Abelian group $\mathbb{Z}^{l}$, denoted as $\\{\text{AI}\\}$, where
$l$ is the number of generators of this group. We can construct
$\\{\text{AI}\\}$ by first choosing specific Wyckoff positions in the real
space and then placing orbits on them, where the orbits have symmetries of the
site symmetry group of the Wyckoff positions and can be represented by a
coirreps of the site symmetry group, with details shown in Appendix.M.

Similarly, the linear space $\\{\text{BS}\\}$ composes of the BRs that satisfy
the compatibility relations between all HSPs, which ensures no (symmetry-
protected) band crossings at high-symmetry lines or planes connecting HSPs,
although gapless points could exist at generic momenta. For a given MSG, these
two Abelian groups have the same dimensionWatanabe et al. (2018), and the SI
group is defined as the quotient group
$X_{\text{BS}}=\\{\text{BS}\\}/\\{\text{AI}\\}=\mathbb{Z}_{n_{1}}\times\mathbb{Z}_{n_{2}}\times\dots\mathbb{Z}_{n_{l}}$,
which is a finite Abelian group.

Practically, $\\{\text{BS}\\}$ can be derived from $\\{\text{AI}\\}$ and does
not need to be calculated explicitly. For a given MSG, arrange its
$\\{\text{AI}\\}$ bases $\\{a_{1},a_{2},\dots,a_{n}\\}$ into a matrix $A$ and
do the Smith normal form decomposition:

$\displaystyle A_{n\times m}$ $\displaystyle=\left(\begin{matrix}a_{1}\\\
a_{2}\\\ \vdots\\\ a_{n}\\\ \end{matrix}\right)=L_{n\times n}M_{n\times
m}R_{m\times m}$ (8) $\displaystyle=L\times\left(\begin{matrix}d_{1}&&&0\\\
&d_{2}&&\\\ &&\ddots&\\\ 0&&&d_{l}&\\\
\vdots&0&0&\vdots\end{matrix}\right)\left(\begin{matrix}b_{1}\\\ b_{2}\\\
\vdots\\\ b_{n}\\\ \vdots\\\ b_{m}\\\ \end{matrix}\right)$
$\displaystyle\Rightarrow L^{-1}\left(\begin{matrix}a_{1}\\\ a_{2}\\\
\vdots\\\ a_{n}\\\ \end{matrix}\right)=\left(\begin{matrix}d_{1}b_{1}\\\
d_{2}b_{2}\\\ \vdots\\\ d_{n}b_{n}\\\ \end{matrix}\right)$

where the nonzero diagonal terms $\\{d_{1},d_{2},\dots,d_{l}\\}$ of $M$ are
arranged in ascending order and $d_{i}|d_{i+1}$, with the terms greater than 1
corresponding to the SI group
$\mathbb{Z}_{d_{k}}\times\mathbb{Z}_{d_{k+1}}\times\cdots\mathbb{Z}_{d_{l}}$.
Each row $b_{i}$ of the right matrix $R$ is a basis of $\\{\text{BS}\\}$, and
$a_{i}^{\prime}=d_{i}b_{i}$ gives the independent basis of $\\{\text{AI}\\}$.
The rows in $R$ that correspond to the SI group, i.e.,
$b_{k},b_{k+1},\dots,b_{l}$, are called nontrivial BSs, and must have
nontrivial indicators values which generate the SI group.

In the next section, we derive the formulas of SIs in MSGs using physical
bases such that their values correspond to some topological invariants.
However, we remark here that the formulas of SIs do not have a unique choice,
and Smith form decomposition naturally gives a set of SI formula, i.e., the
$i$-th column of $R^{-1}$ gives rise to the corresponding $z_{i}$ indicator,
as shown below.

Given an arbitrary band representation $B$, first decompose it using the
$\\{\text{AI}\\}$ bases:

$B=c\cdot AI=\tilde{c}\cdot M\cdot R,\ \ (\tilde{c}=cL)$ (9)

The indicators are then given by

$z_{i}=(B\cdot R^{-1})_{i}\bmod n_{i}$ (10)

which means the $i$-th column of $R^{-1}$ gives the $z_{i}$ indicator, because
$z_{i}$ corresponds to the numerator of the fractional number in the
decomposition of $B$.

##  Appendix G Generating SIs in MSGs

In Table.4, we list the generating SIs in 1651 MSGs, and their explicit
formulas and correspondence to TCI classifications as well as topological
invariants are derived in the following subsections. The word “generating” we
adopt here means the SIs in all other MSGs can be generated using the
combinations of these SIs, with only 10 exceptions denoted as “corner cases”,
which will be introduced in Appendix.H. We also include the type-2 generating
MSG $P\overline{4}1^{\prime}$ here for completeness, which is omitted in the
main text where only type-1,3,4 MSGs are considered.

MSG | MSG type | $\bm{X}_{\text{BS}}$ | SI
---|---|---|---
$P\overline{1}$ | 1 | $\mathbb{Z}_{2,2,2,4}$ | $z_{2P,1},z_{2P,2},z_{2P,3},z_{4P}$
$Pn,n=2,3,4,6$ | 1 | $\mathbb{Z}_{n}$ | $z_{nC}$
$Pn/m,n=2,3,4,6$ | 1 | $\mathbb{Z}_{n,n,n}$ | $z_{nm,0}^{+},z_{nm,0}^{-},z_{nm,\pi}^{+}$
$P\overline{4}$ | 1 | $\mathbb{Z}_{2,2,4}$ | $z_{2,S_{4}},z_{2,\text{Weyl}},z_{4C}$
$Pmmm$ | 1 | $\mathbb{Z}_{2,2,2,4}$ | $z_{2P,1}^{\prime},z_{2P,2}^{\prime},z_{2P,3}^{\prime},z_{4P}^{\prime}$
$P4/mmm$ | 1 | $\mathbb{Z}_{2,4,8}$ | $z_{2P,1}^{\prime},z_{4m,\pi}^{+},z_{8}$
$P6/mmm$ | 1 | $\mathbb{Z}_{6,12}$ | $z_{6m,\pi}^{+},z_{12}$
$P\overline{4}1^{\prime}$ | 2 | $\mathbb{Z}_{2}$ | $z_{2,S_{4}}^{\prime}$
$Pnc^{\prime}c^{\prime},n=2,4,6$ | 3 | $\mathbb{Z}_{n}$ | $z_{nC}^{\prime}$
Table 4: Generating SIs and their corresponding MSGs. We use a simplified
notation to represent the SI group, i.e.,
$\mathbb{Z}_{n_{1},n_{2},\dots}=\mathbb{Z}_{n_{1}}\times\mathbb{Z}_{n_{2}}\times\dots$.

###  G.1 Generating SIs in type-1 MSG

In Ref.Ono and Watanabe (2018), Ono and Watanabe derived SIs in a few type-1
key MSGs. Here we do a more detailed investigation, including not only the SI
formulas but also their correspondence to topological crystals.

####  G.1.1 MSG 2.4 $P\overline{1}$, $X_{BS}=\mathbb{Z}_{2,2,2,4}$,
Classification=$\mathbb{Z}^{3}\times\mathbb{Z}_{2}$

MSG 2.4 $P\overline{1}$ has inversion symmetry $P$, and its SIs can be defined
using parities at 8 time-reversal invariant momenta (TRIMs):

$\displaystyle z_{2P,1}=$
$\displaystyle\sum_{\mathbf{k}\in\text{TRIM},k_{1}=\pi}\frac{1}{2}(N_{\mathbf{k}}^{-}-N_{\mathbf{k}}^{+})\bmod
2$ (11) $\displaystyle z_{2P,2}=$
$\displaystyle\sum_{\mathbf{k}\in\text{TRIM},k_{2}=\pi}\frac{1}{2}(N_{\mathbf{k}}^{-}-N_{\mathbf{k}}^{+})\bmod
2$ $\displaystyle z_{2P,3}=$
$\displaystyle\sum_{\mathbf{k}\in\text{TRIM},k_{3}=\pi}\frac{1}{2}(N_{\mathbf{k}}^{-}-N_{\mathbf{k}}^{+})\bmod
2$ $\displaystyle z_{4P}=$
$\displaystyle\sum_{\mathbf{k}\in\text{TRIM}}\frac{1}{2}(N_{\mathbf{k}}^{-}-N_{\mathbf{k}}^{+})\bmod
4$

where $N_{\mathbf{k}}^{\pm}$ is the number of valence bands having positive
(negative) parity.

The first three $z_{2P,i}$ indicators diagnose the Chern number mod 2 on
$k_{i}=0,\pi$ planes, while the $z_{4P}$ indicator can be seen as the sum of
Chern number on $k_{z}=0,\pi$ planes.

When $\mathbb{Z}_{4}=1,3$, the Chern number on $k_{z}=0$ and $k_{z}=\pi$ plane
must differ by an odd number, which indicates an odd number of Weyl points
exist between $k_{z}=0$ and $k_{z}=\pi$. An equal number of Weyl points of
opposite chirality exist between $k_{z}=0,-\pi$ planes, enforced by the
inversion symmetry.

The 4 nontrivial BSs that corresponds to $\mathbb{Z}_{2,2,2,4}$ are

$\displaystyle b_{1}$
$\displaystyle=\overline{V}_{2}-\overline{V}_{3}+\overline{X}_{2}-\overline{X}_{3}+\overline{Y}_{2}-\overline{Y}_{3}-\overline{Z}_{2}+\overline{Z}_{3}$
(12) $\displaystyle b_{2}$
$\displaystyle=\overline{Y}_{2}-\overline{Y}_{3}-\overline{Z}_{2}+\overline{Z}_{3}$
$\displaystyle b_{3}$
$\displaystyle=\overline{X}_{2}-\overline{X}_{3}-\overline{Z}_{2}+\overline{Z}_{3}$
$\displaystyle b_{4}$ $\displaystyle=\overline{Z}_{2}-\overline{Z}_{3}$

Their indicators are calculated to be (0012), (0110), (1010), (0013), which
indeed generate the group $\mathbb{Z}_{2,2,2,4}$. The minus sign in the BSs
may be confusing, and one can add proper AIs to them to make the coefficients
all positive, which does not change SIs as AIs have zero SIs.

The TCI classification of $P\bar{1}$ is $\mathbb{Z}^{3}\times\mathbb{Z}_{2}$,
where the three $\mathbb{Z}$ indexes correspond to translation decorations,
while the $\mathbb{Z}_{2}$ index is the $Z_{2}$ decoration protected by
inversion. The generators of these decorations have weak and inversion
invariants
$(\delta_{w,1},\delta_{w,2},\delta_{w,3},\delta(P))=(100,1),(010,1),(001,0),(000,1)$,
respectively. We can attach compatible irreps to these decorations, as shown
in Fig.14(a), (b), (c), and (d). Their SIs are calculated to be (1002),
(0102), (0010), and (0002). The mappings between invariants and SIs are
straightforward:

$z_{2P,i}=\delta_{w,i}\bmod 2,\ \ z_{4P}=2\delta(P)$ (13)

which means $z_{4P}=2$ states have $\delta(P)=1$, i.e., they have 1D chiral
hinge modes on inversion-preserving surfaces. Note that $z_{4P}=1,3$ are
gapless Weyl states and cannot be constructed using the real-space recipe.

Figure 14:  (a)-(e). Layer constructions and their compatible sets of irreps
as well as the corresponding SIs in $P\overline{1}$. (f). The Weyl state and a
compatible set of irreps in $P\overline{1}$.

####  G.1.2 $Pn,\ X_{BS}=\mathbb{Z}_{n}$, Classification=$\mathbb{Z}$

$Pn,n=2,3,4,6$ have $C_{n}$ rotations as generators. Because the $C_{n}$
eigenvalues must be the same along the rotation axis, the Chern number on
$k_{i}=0$ and $k_{i}=\pi$ plane must be the same, and we take the Chern number
on $k_{i}=0$ plane as the $\mathbb{Z}_{n}$ indicator, where $i$ denotes the
rotation direction. The classification group $\mathbb{Z}$ is generated by a LC
with $C=1$ layers on $x_{i}=n$ planes, which has weak invariant
$\delta_{w,i}=1$.

The formulas for calculating Chern numbers (mod n) using $C_{n}$ eigenvalues
are derived in Ref.Fang et al. (2012):

$\displaystyle C_{2}:$ $\displaystyle\ \
(-1)^{C}=\prod_{l\in\mathrm{occ}.}B_{C_{2}}^{l}(\Gamma)B_{C_{2}}^{l}(X)B_{C_{2}}^{l}(Y)B_{C_{2}}^{l}(M)$
(14) $\displaystyle C_{4}:$ $\displaystyle\ \ e^{i\pi
C/2}=\prod_{l\in\mathrm{occ}.}(-1)B_{C_{4}}^{l}(\Gamma)B_{C_{4}}^{l}(M)B_{C_{2}}^{l}(Y)$
$\displaystyle C_{3}:$ $\displaystyle\ \ e^{i2\pi
C/3}=\prod_{l\in\mathrm{occ}.}(-1)B_{C_{3}}^{l}(\Gamma)B_{C_{3}}^{l}(K)B_{C_{3}}^{l}(K^{\prime})$
$\displaystyle C_{6}:$ $\displaystyle\ \ e^{i\pi
C/3}=\prod_{l\in\mathrm{occ}.}(-1)B_{C_{6}}^{l}(\Gamma)B_{C_{3}}^{l}(K)B_{C_{2}}^{l}(M)$

where $B_{C_{n}}^{l}(k)$ represent the $C_{n}$ eigenvalue of the $l$-th band
at $C_{n}$ -invariant point $k$, and the HSPs used in these formulas are
plotted in Figure 15. We can take logarithm to each side of the equation to
turn them into the more familiar summation and mod n form, and denote them as
$z_{nC}$:

$\displaystyle z_{2C}$ $\displaystyle=\sum_{l\in
occ}\ln(B_{C_{2}}^{l}(\Gamma)B_{C_{2}}^{l}(X)B_{C_{2}}^{l}(Y)B_{C_{2}}^{l}(M))/(i\pi)\bmod
2$ (15)
$\displaystyle=\sum_{\mathbf{k}\in\Gamma,M,X,Y}\frac{1}{2}(N_{\mathbf{k}}^{-}-N_{\mathbf{k}}^{+})\bmod
2$ $\displaystyle z_{4C}$ $\displaystyle=\sum_{l\in
occ}\ln(-B_{C_{4}}^{l}(\Gamma)B_{C_{4}}^{l}(M)B_{C_{2}}^{l}(Y))/(i\frac{\pi}{2})\bmod
4$ $\displaystyle z_{3C}$ $\displaystyle=\sum_{l\in
occ}\ln(-B_{C_{3}}^{l}(\Gamma)B_{C_{3}}^{l}(K)B_{C_{3}}^{l}(K^{\prime}))/(i\frac{2\pi}{3})\bmod
3$ $\displaystyle z_{6C}$ $\displaystyle=\sum_{l\in
occ}\ln(-B_{C_{6}}^{l}(\Gamma)B_{C_{3}}^{l}(K)B_{C_{2}}^{l}(M))/(i\frac{\pi}{3})\bmod
6$

These $z_{nC}$ indicators correspond to the weak invariant:

$z_{nC}=\delta_{w,i}\bmod n$ (16)

Figure 15: HSPs used to calculate Chern number mod n on 2D BZ

In $P\overline{4}$ and $P\overline{3}$, as $S_{4}=C_{4}^{-1}$ and
$S_{3}=C_{6}^{-1}$ on $k_{z}=0,\pi$ planes, they can also be used to calculate
the Chern number. The modified formulas are

$\displaystyle S_{4}$ $\displaystyle:\ z_{4C}=\sum_{l\in
occ}\ln(-B_{S_{4}^{-1}}^{l}(\Gamma)B_{S_{4}^{-1}}^{l}(M)B_{C_{2}}^{l}(Y))/(i\frac{\pi}{2})\bmod
4$ (17) $\displaystyle S_{3}$ $\displaystyle:\ z_{6C}=\sum_{l\in
occ}\ln(B_{S_{3}^{-1}}^{l}(\Gamma)B_{C_{3}}^{l}(K)B_{P}^{l}(M))/(i\frac{\pi}{3})\bmod
6$

In practice, when calculating these $z_{nC}$ indicators using coirreps of
double MSGs, we need to choose the proper representation matrices s.t. the
SU(2) part of $C_{n}$ rotations are aligned, i.e.,

$\displaystyle U(C_{4})^{2}=U(C_{2}),\ U(C_{4}^{-1})^{2}=-U(C_{2}),$ (18)
$\displaystyle U(C_{6})^{2}=U(C_{3}),\ U(C_{6}^{-1})^{2}=U(C_{3}^{-1}),$
$\displaystyle U(C_{6})^{3}=U(C_{2}),\ U(C_{6}^{-1})^{3}=-U(C_{2})$

where $U$ denotes the SU(2) matrix. For $S_{3},S_{4}$, these relations become

$U(S_{4}^{-1})^{2}=U(C_{2}),U(S_{3}^{-1})^{2}=U(C_{3}),U(S_{3}^{-1})^{3}=U(P)$
(19)

Note for $S_{3}$, there is no $-1$ in the $z_{6C}$ formula, because we are
using $P$ instead of $C_{2}$, which satisfies $P^{-1}=P$ instead of
$C_{2}^{-1}=-C_{2}$ when SU(2) matrices are considered.

####  G.1.3 $Pn/m,\ X_{BS}=\mathbb{Z}_{n,n,n}$,
Classification=$\mathbb{Z}^{3}$

$Pn/m$ represents four MSGs: $P2/m$, $P4/m$, $P\overline{6}(=P3/m)$, and
$P6/m$. All these MSGs have mirror planes vertical to the $C_{n}$ rotation
axis. Assume the rotation is along $x_{i}$ direction. There are two mirror
planes in the BZ, i.e., $k_{i}=0,\pi$, which allows us to define four Chern
numbers: $C_{m,0}^{\pm}$ and $C_{m,\pi}^{\pm}$, where $\pm$ means the Chern
number defined on the mirror $\pm i$ sector, respectively. It can be proven
that only three of them are independent, which correspond to the three
$\mathbb{Z}$ indexes in the TCI classification.

The indicators can be chosen as any three independent mirror Chern numbers mod
n, and here we take the first three:
$z_{nm,0}^{+},z_{nm,0}^{-},z_{nm,\pi}^{+}$. The mapping between invariants and
indicators are simple:

$z_{nm,0/\pi}^{\pm}=C_{m,0/\pi}^{\pm}\bmod n$ (20)

##### Example: $P4/m$

We take $P4/m$ as an example, which has three nontrivial BSs as:

$\displaystyle b_{1}=$
$\displaystyle-\overline{\Gamma}_{10}+\overline{\Gamma}_{12}-\overline{M}_{8}+\overline{M}_{12}+\overline{R}_{4}-\overline{R}_{5}-\overline{X}_{5}$
(21)
$\displaystyle+\overline{X}_{6}+2\overline{Z}_{6}+3\overline{Z}_{7}-3\overline{Z}_{10}-3\overline{Z}_{11}+\overline{Z}_{12}$
$\displaystyle b_{2}=$
$\displaystyle\overline{\Gamma}_{10}-\overline{\Gamma}_{12}-\overline{M}_{6}+\overline{M}_{10}-\overline{R}_{4}+\overline{R}_{5}+\overline{X}_{5}$
$\displaystyle-\overline{X}_{6}-\overline{Z}_{6}-2\overline{Z}_{7}+2\overline{Z}_{10}+2\overline{Z}_{11}-\overline{Z}_{12}$
$\displaystyle b_{3}=$
$\displaystyle\overline{Z}_{6}+\overline{Z}_{7}-\overline{Z}_{10}-\overline{Z}_{11}$

Their SIs are calculated to be
$\\{z_{4m,0}^{+},z_{4m,0}^{-},z_{4m,\pi}^{+}\\}=(3,0,3),(0,1,0),(0,0,3)$,
which indeed generate the $\mathbb{Z}_{4,4,4}$ group.

##### Example: $P4_{2}/m$, $X_{BS}=\mathbb{Z}_{2,4}$

$P4_{2}/m$ has a screw symmetry $\\{C_{4z}|0,0,1/2\\}$ and SI group
$\mathbb{Z}_{2,4}$. Its TCI classification is $\mathbb{Z}^{2}$, whose
generators can be taken as a translation decoration (placing
$(C_{m}^{+},C_{m}^{-})=(1,0)$ states on $z=1/4,3/4$ planes) and a $Z_{2}$
decoration, which is a non-LC similar to the non-LC in $Pmmm$, as shown later.

The SIs can be taken as $C_{4m,0}^{+},C_{2m,\pi}^{+}$. The Chern number on
$k_{z}=\pi$ plane can only be calculated by $C_{2z}$, but not $C_{4z}$-screw,
as $C_{4z}$-screw does not commute with $M_{z}$. The nontrivial BSs are

$\displaystyle b_{1}$
$\displaystyle=2\overline{M}_{6}+2\overline{M}_{8}-2\overline{M}_{10}-2\overline{M}_{12}+\overline{R}_{4}-\overline{R}_{5}+\overline{X}_{4}-\overline{X}_{5}$
(22) $\displaystyle b_{2}$
$\displaystyle=\overline{M}_{6}+\overline{M}_{8}-\overline{M}_{10}-\overline{M}_{12}$

which has SI=(1,0) and (0,1).

Similar argument hold for $P2_{1}/m$ and $P6_{3}/m$. $P6_{3}/m$ has indicator
group $\mathbb{Z}_{3,6}$ and can be taken as $C_{6m,0}^{+},C_{3m,\pi}^{+}$,
while $P2_{1}/m$ has indicator group $\mathbb{Z}_{2}$, and can be taken as
$z_{2P}^{\prime}$.

####  G.1.4 MSG 81.33 $P\overline{4},\ X_{BS}=\mathbb{Z}_{2,2,4}$,
Classification=$\mathbb{Z}\times\mathbb{Z}_{2}$

$P\overline{4}$ has $S_{4}$ symmetry. Its TCI classification is
$\mathbb{Z}\times\mathbb{Z}_{2}$, which correspond to a translation and a
$S_{4}$ decoration, respectively. The translation decoration is placing 2D
layers with Chern number $C$ on $z=n+1/2$ planes, while the $S_{4}$ decoration
is composed of 2D layers with Chern number 1 on $z=n$ planes and layers with
Chern number -1 on $z=n+\frac{1}{2}$ planes.

The SI group of $P\overline{4}$ is $\mathbb{Z}_{2,2,4}$. Because $S_{4}$