# Homotopic classification of band structures: Stable, fragile, delicate, and
stable representation-protected topology

Piet W. Brouwer  Vatsal Dwivedi  Dahlem Center for Complex Quantum Systems and
Physics Department, Freie Universität Berlin, Arnimallee 14, 14195 Berlin,
Germany

###### Abstract

The topological classification of gapped band structures depends on the
particular definition of topological equivalence. For translation-invariant
systems, stable equivalence is defined by a lack of restrictions on the
numbers of occupied and unoccupied bands, while imposing restrictions on one
or both leads to “fragile” and “delicate” topology, respectively. In this
article, we describe a homotopic classification of band structures — which
captures the topology beyond the stable equivalence — in the presence of
additional lattice symmetries. As examples, we present complete homotopic
classifications for spinless band structures with twofold rotation, fourfold
rotation and fourfold dihedral symmetries, both in presence and absence of
time-reversal symmetry. Whereas the rules of delicate and fragile topology do
not admit a bulk-boundary correspondence, we identify a version of stable
topology, which restricts the representations of bands, but not their numbers,
which does allow for anomalous states at symmetry-preserving boundaries, which
are associated with nontrivial bulk topology.

##  I Introduction

The concept of topological equivalence, combined with symmetry constraints,
has been an important tool to classify gapped band structures of non-
interacting fermions [1, 2, 3, 4]. Topological gapped band structures include
the quantized Hall effect, the quantum spin-Hall effect, topological
insulators, and, if crystalline symmetries are imposed, topological
crystalline insulators. A nontrivial topology of the band structure implies
the existence of anomalous, gapless states on the sample boundary and vice
versa, provided the boundary respects the symmetry constraints that are
imposed on the bulk.

Various notions of topological equivalence of band structures have been
employed in the literature, which differ in the possible transformations that
are allowed to deform two topologically equivalent Hamiltonians into each
other. The standard definition in condensed matter physics has been the
“stable equivalence” [5], which allows addition of an arbitrary number of
“trivial” bands which respect the symmetry requirements. This definition
ensures that the topology realized in mathematical models, often consisting of
only a few bands, is realized in physical systems consisting of atoms with
infinitely many orbitals. Indeed, a plethora of stable topological phases [1,
2, 3, 4], that were originally discovered theoretically, have been realized
experimentally, such as the quantum Hall effect, the quantum spin Hall
insulator and the three dimensional topological insulators.

A conceptually simpler, although physically less useful, definition of
topological equivalence does away with the freedom to add of extra bands,
resulting in “delicate” topology [6, 7]. The paradigmatic example of such a
band structure is the so-called Hopf insulator [8, 9, 6, 10], realized by two-
band models in three dimensions without any additional symmetries. A third
intermediate possibility, which involves the freedom to add unoccupied bands
while keeping the occupied subspace unchanged, results in “fragile” topology
[11, 12, 13, 14]. Nontrivial fragile topology has been invoked to explain,
e.g., the obstruction to the construction of real-space models for the lowest
two bands of magic-angle twisted bilayer graphene [15, 16, 17, 18, 19]. In
general, topology that is trivial according to the rules of stable topology,
but nontrivial following the delicate or fragile schemes is referred to as
“unstable’

The classification of stable topological phases with only local symmetries
leads to the well-known “periodic table of topological insulators and
superconductors” [5, 20, 21]. Stable stable topological insulators with
additional crystalline symmetries have been comprehensively classified for
many crystalline symmetries [22, 23, 24, 25, 26]. However, delicate and
fragile band structures have only been identified systematically in the
absence of crystalline symmetries [27] or, with crystalline symetries, using
partial classifications or symmetry-specific arguments [11, 12, 13, 14, 28,
29, 30, 31].

In this article, we show how delicate band structures can be classified
systematically using the theory of homotopy on CW complexes [32]. Similar
techniques have previously been used by Moore and Balents to rederive the
quantum spin Hall effect and to obtain the classification of topological
insulators in three dimensions [33]. Our classification procedure is defined
for a fixed number of occupied and empty bands, thereby yielding the
classification of delicate topological phases. Classifications for the stable
and fragile equivalence schemes naturally follow by taking the limit of a
large number of conduction and/or valence bands. We illustrate the
classification procedure by (re)deriving delicate classifications of spinless
band structures in two and three dimensions without crystalline symmetry, and
with ${\mathrm{C}}_{2}{\mathrm{T}}$, ${\mathrm{C}}_{2}$, ${\mathrm{C}}_{4}$,
and ${\mathrm{D}}_{4}$ symmetries, where ${\mathrm{C}}_{n}$ and
${\mathrm{D}}_{n}$ refer to $n$-fold rotation and dihedral symmetries and
${\mathrm{T}}$ is time-reversal. Band structures with
${\mathrm{C}}_{2}{\mathrm{T}}$, ${\mathrm{C}}_{2}$, and ${\mathrm{C}}_{4}$
symmetry have been used as paradigmatic examples of unstable topology
protected by crystalline symmetries [6, 34, 28, 31, 35, 36, 37, 29, 30], which
are systematized by our classification.

Stable topology is an essential prerequisite for the existence of anomalous
gapless boundary states. This follows from elementary considerations: As the
rules of fragile and delicate topology restrict the number of occupied and/or
unoccupied bands, the size of the unit cell must be fixed, which requires the
existence of a discrete translation symmetry in all spatial directions. Since
a boundary necessarily breaks the translation symmetry, it follows that
anomalous boundary states cannot exist without stable protection 111 Anomalous
boundary states also exist for the so-called weak [72] topology, whereby the
topological invariant for a $d$-dimensional system is protected by discrete
translation symmetry in less than $d$ dimensions. The anomalous boundary modes
occur only at boundaries that respect the discrete translation symmetry. . A
more subtle route to protected gapless boundary states beyond conventional
stable topology was pointed out by Song et al. [30], Alexandradinata et al.
[37], and Kobayashi and Furusaki [31] for a ${\mathrm{C}}_{4}$-symmetric
lattice model originally proposed by Fu [39]. In this case, anomalous surface
states are protected by topology that remains stable under addition of bands
that are not only constrained by a symmetry, but also by its particular
representation (i.e., the types of orbitals). We term this representation-
protected stable topology. Following our systematic classification, we
identify such phases and the associated anomalous boundary states not only for
${\mathrm{C}}_{4}$-symmetric band structures, but also for ${\mathrm{C}}_{2}$
and ${\mathrm{D}}_{4}$ symmetries. For the latter, the representation-
protected phase is intimately linked to the existence of a two-dimensional
irreducible representation of the discrete nonabelian group
${\mathrm{D}}_{4}$.

Despite the absence of anomalous boundary states, unstable topology has
physical relevance, first and foremost, because it may signal an obstruction
to the construction of a real-space basis with localized orbitals within the
available number of bands, as in the case for magic-angle twisted bilayer
graphene [15, 16, 17, 18, 19]. Unstable topology has been shown to affect bulk
physics, such as the appearance of Bloch oscillations with an extended period
[40] and connected, unbounded Hofstadter butterflies in the presence of a
magnetic field [41, 42, 43]. Various _boundary signatures_ beyond anomalous
gapless boundary states have also been proposed 222Some of these signatures
can be interpreted as manifestations of various other forms of stable
topology, viz, that of topologically localized insulators [107] for Hopf
insulators and that of obstructed atomic limits for the twisted boundary
signature and the fractional corner charges., such as a nonzero Chern number
for the surface bands in the case of Hopf insulators[45, 46], boundary states
for a pre-defined crystal termination [7], gap closing under twisted boundary
condition [29, 47], and fractional corner charges [48, 18]. In this article,
we do not venture into such manifestations of unstable topology and focus
exclusively on the classification problem and the anomalous boundary states
associated with various forms of stable topology.

The remainder of this article is organized as follows: In Sec. II, we describe
the general classification procedure and the required mathematical background.
In Sec. III we apply the classification procedure to band structures without
crystalline symmetries. Section IV contains general observations concerning
the classification of band structures with crystalline symmetries, which is
followed by specific examples of increasing complexity: Band structures with
${\mathrm{C}}_{2}{\mathrm{T}}$ symmetry in Sec. V, with ${\mathrm{C}}_{2}$
symmetry in Sec. VI, with ${\mathrm{C}}_{4}$ symmetry in Sec. VII, and with
${\mathrm{D}}_{4}$ symmetry in Sec. VIII. We conclude with a discussion in
Sec. IX. Various technical details of the classification procedure have been
relegated to the appendices.

##  II Classification strategy

In this section, we describe the homoptopic classification for band structures
with arbitrary lattice symmetries, which is based on well-established methods
of algebraic topology for CW complexes. Our classification procedure is
applicable to all ten Altland-Zirnbuaer classes [49]; however, to avoid
overburdenening the present exposition, we restrict the explicit discussion to
Bloch Hamiltonians without antisymmetry requirements, corresponding to the
Altland-Zirnbauer classes A, AI, and AII. These classes include all
topological crystalline insulators, but not topological superconductors or
lattice models with a sublattice symmetry. In the examples of Secs. III-VIII
we further restrict to symmetry classes A and AI, describing insulators with
spin-rotation symmetry with and without time-reversal symmetry.

###  II.1 Stable, fragile, delicate, and representation protected topologies

We begin with some general remarks on various notions of topological
equivalence for translation invariant systems, which are completely described
by a finite-dimensional Bloch Hamiltonian $H(\mathbf{k})$. Two Bloch
Hamiltonians $H_{1,2}$ are considered _homotopically_ equivalent,

$H_{1}\sim H_{2},$ (1)

if $H_{1}$ can be continuously deformed to $H_{2}$ without closing the gap or
violating the symmetry constraints imposed on $H_{1,2}$. A looser notion is
that of _stable_ equivalence, whereby two Bloch Hamiltonians $H_{1,2}$ are
considered topologically equivalent if there exists a gapped Bloch Hamiltonian
$H$, subject to the same symmetry constraints as $H_{1,2}$, so that

$H_{1}\oplus H\sim H_{2}\oplus H$ (2)

in the homotopic sense 333 This definition requires that $H_{1,2}$ have equal
number of conduction and valence bands. However, if the symmetries allow for
the definition of a canonical “trivial” band, then this definition can be
further relaxed to allow comparison of systems with different number of bands
by adding the trivial bands. . Finally, _fragile_ equivalence is defined by
Eq. (2) with the further constraint on $H$ that it consist only of unoccupied
bands. In mathematical terms, the problem of fragile and stable classification
can be expressed as the computation of equivalence classes of vector bundles,
with the equivalence being isomorphism [51, 52, 6] for the former case and
isomorphism _up to Whitney sums_ for the latter, which falls under the purview
of topological K-theory [5]. Given a choice of a reference band structure
$H_{\text{ref}}$, a Bloch Hamiltonian $H$ is said to have “stable topology” if
it is not equivalent to $H_{\text{ref}}$ under stable equivalence. If $H$ and
$H_{\text{ref}}$ are topologically equivalent under stable equivalence, but
not under homotopic equivalence or vector bundle isomorphism, then $H$ is said
to possess “unstable” topology. The space of stably equivalent band structures
can be endowed with a group structure, the group operation corresponding to
the direct sum of the Bloch Hamiltonians 444 Strictly speaking, defining the
trivial element for the group operation does not require the identification of
a trivial band structure, but, instead, proceeds via the Grothendieck
construction, which considers topological equivalence of pairs
$(H,H^{\prime})$ of gapped band structures. Two pairs $(H_{1},H_{1}^{\prime})$
and $(H_{2},H_{2}^{\prime})$ are then considered topologically equivalent if
$H_{1}\oplus H_{2}^{\prime}$ can be smoothly deformed into $H_{2}\oplus
H_{1}^{\prime}$, without closing the excitation gap and without violating the
symmetry constraints. . As this group operation is incompatible with the
restrictions imposed by fragile and delicate topology, they do not have a
similar group structure.

The rules of topological classification can be generalized by _a priori_
limiting the types of orbitals and Wyckoff positions allowed, i.e.,
restricting the band representations. Such _representation-protected_ phases
have been previously investigated in Refs. [39, 30, 31], whereby
${\mathrm{C}}_{4}$-symmetric band structures were shown to exhibit a
topological phase for arbitrary number of bands, provided one only allows
$p$-orbitals. As in the unrestricted case, one may have delicate, fragile, and
stable versions of representation-protected topological classifications. All
of these are sometimes referred to in the literature as “fragile” [30, 31];
however, stable representation-protected topology exhibits many of the
characteristics of conventional stable topology, such as a group structure and
the existence of anomalous states at boundaries that respect the symmetry
restrictions imposed on the bulk.

Our classification procedure starts by fixing the numbers $n_{\mathrm{o}}$ and
$n_{\mathrm{e}}$ of occupied and unoccupied bands and thus naturally
identifies all possible delicate topological phases. The fragile
classification is obtained by taking the limit $n_{\mathrm{e}}\to\infty$ for a
fixed $n_{\mathrm{o}}$, while the stable classification is obtained when we
take $n_{\mathrm{o},\mathrm{e}}\to\infty$. As our goal is to obtain a full
homotopic classification, we respect symmetry-imposed obstructions between
atomic-limit phases and do not follow the sometimes-used custom [54, 55, 56]
to refer to all band structures that are homotopically equivalent to an
atomic-limit as “trivial”. To infer whether a certain homotopic equivalence
class corresponds to an atomic-limit phase, one has to compare the topological
invariants to the topological invariants of atomic limit phases. An exhaustive
listing of atomic-limit phases for all crystalline symmetry groups has been
obtained in Ref. [54] in the framework of “topological quantum chemistry”.

###  II.2 Lattice symmetries and the Bloch Hamiltonian

We consider a $n\times n$ Bloch Hamiltonian $H(\mathbf{k})$ in the basis of
Bloch states $\ket{\mathbf{k},\alpha}$, which are defined as

$\ket{\mathbf{k},\alpha}=\sum_{\mathbf{R}}\mathrm{e}^{\mathrm{i}\mathbf{k}\cdot(\mathbf{R}+\mathbf{r}_{\alpha})}\ket{\mathbf{R},\alpha}.$
(3)

Here, $\ket{\mathbf{R},\alpha}$ denote the position-space basis states, with
$\mathbf{R}\in\mathbb{Z}^{d}$ the center of the unit cell and
$\alpha=1,\ldots,n$ an index labeling a set of $n$ orbitals at positions
$\mathbf{r}_{\alpha}\in(-1/2,1/2]^{d}$ inside the unit cell. The lattice
symmetries impose constraints on $H(\mathbf{k})$, which are encoded by a
representation $C(g)$ of the _reciprocal-space group_ $G_{\Lambda}$ [35] as

$H(g\mathbf{k})=C^{\dagger}(g)H(\mathbf{k})C(g),\ \ g\in G_{\Lambda}.$ (4)

The reciprocal-space group is generated by the point group and translations by
reciprocal lattice vectors. The latter act nontrivially on $H(\mathbf{k})$,
since our gauge choice for the Bloch states — made to ensure that the matrices
$C(g)$ representing the reciprocal-space group are independent of $\mathbf{k}$
— yields a Bloch Hamiltonian that is not periodic under reciprocal lattice
translations. Instead, it satisfies

$H(\mathbf{k}+2\pi\mathbf{e}_{a})=\Lambda_{a}^{\dagger}H(\mathbf{k})\Lambda_{a}^{\phantom{\dagger}},$
(5)

where $a\in\\{x,y,z\\}$ and we have set $\Lambda_{a}\equiv C(g_{a})$, where
$g_{a}$ denotes translation by a reciprocal unit vector along the
$a$-direction. Antiunitary symmetries, if present, are also included in
$G_{\Lambda}$; however, for an antiunitary $\bar{g}\in G_{\Lambda}$, Eq. (4)
must be replaced with

$H(\bar{g}\mathbf{k})=C^{\dagger}(\bar{g})H^{*}(\mathbf{k})C(\bar{g}),\quad\bar{g}\in
G_{\Lambda}.$ (6)

The precise form of the representation matrices $C(g)$ depends on which
orbitals are present. The relevant information is encoded in the Wyckoff
positions of the orbitals and their representations under the relevant point
group symmetries, which must be considered “input data” for the homotopic
classification.

In view of the symmetry constraints (4) and (6), the Bloch Hamiltonian
$H(\mathbf{k})$ for all $\mathbf{k}$ can be derived from its knowledge over a
subset of the full reciprocal space. Such a subset is referred to as the
“irreducible Brillouin zone”, the “effective Brillouin zone” [33], or the
“fundamental domain” corresponding to the reciprocal-space group $G_{\Lambda}$
555 Mathematically, the fundamental domain can be defined as the closure of
the largest open set in the BZ on which the group action of $G$ is free (i.e.,
no point maps onto itself under the action of any non-identity element of the
$G$). . By construction, no two momenta in the fundamental domain are related
by a symmetry operation in $G_{\Lambda}$, so that, within the fundamental
domain, all symmetry restrictions on $H(\mathbf{k})$ are local in
$\mathbf{k}$.

#### Fundamental domain as a CW complex

The fundamental domain for a $d$-dimensional system with a given group of
symmetries $G_{\Lambda}$ can be naturally constructed in terms of “$p$-cells”,
which are homotopically equivalent to the $p$-dimensional cube $(0,1)^{p}$ for
$p\geq 1$ and points for $p=0$. Starting from a disjoint union of $p$-cells
with $0\leq p\leq d$, one recovers the structure of the fundamental domain by
imposing a set of _gluing maps_. For each $p$-cell, these maps assign to its
boundary a closed loop formed by $q$-cells of lower dimensions, which may
include multiple copies of a given $q$-cell related by symmetry operations in
$G_{\Lambda}$. This construction endows the fundamental domain with the
structure of a _CW complex_ 666A similar construction, but in position space,
appears in the construction of a “topological crystal” in Ref. [108]. — a
construct well known in the field of algebraic topology [32].

For example, in $d=1$ with inversion symmetry, we can take the set $[0,\pi]$
as the fundamental domain, which can be constructed by gluing the two 0-cells
$\Gamma$ and ${\mathrm{M}}$, which consists of the points $k=0$ and $k=\pi$,
respectively, to the ends of the 1-cell $(0,\pi)$. Further examples of the
fundamental domain and its decomposition into various $p$-cells are shown in
Fig. 1 for a three-dimensional band structure without additional crystalline
symmetries and without (top) and with (bottom) time-reversal symmetry. For the
fundamental domain shown in Fig. 1 (top), the gluing condition identifies the
boundary $\partial\mathbb{D}_{3}$ of the 3-cell with the three 2-cells, the
three 1-cells, and the single 0-cell, with each of them occuring two, four and
eight times on the boundary, respectively. Similarly, in the presence of time-
reversal symmetry, the boundary $\partial\mathbb{D}_{3}$ of the 3-cell in Fig.
1 (bottom) is identified with the four 2-cells, seven 1-cells, and eight
0-cells and with their images under translation with a reciprocal wavevector
and/or time-reversal, which sends $\mathbf{k}\to-\mathbf{k}$.

Figure 1:  Fundamental domain $\mathbb{D}$ for a three-dimensional band
structure on a simple cubic lattice without (top) and with (bottom) time-
reversal symmetry. Knowledge of the Bloch Hamiltonian on the fundamental
domain is sufficient to determine it everywhere in the reciprocal space. Also
shown is the decomposition of the fundamental domain into $p$-cells with
$p=0,1,2,3$ (red dots, line segments, rectangles, and cuboid), endowing it
with the structure of a “CW complex”. Copies of 0-cells on the boundary
$\partial\mathbb{D}$ that are obtained by a reciprocal lattice transformation
are shown using open dots.

We hereafter denote the fundamental domain with this additional structure by
$\mathbb{D}$ and its $p$-dimensional subset by $\mathbb{D}_{p}$. The action of
the reciprocal-space group $G_{\Lambda}$ on $\mathbb{D}$ has two aspects:

  1. 1.

A given $p$-cell $\alpha$ may be left invariant by a subgroup
$G_{\alpha}\subset G_{\Lambda}$, which is termed its _little group_. These
must be compatible with the gluing maps, in that the little groups of the
$q$-cells $\beta\in\partial\alpha$ satisfy $G_{\beta}\supseteq G_{\alpha}$.
Cells with $G_{\alpha}\neq 1$ are usually referred to as high-symmetry
points/lines/planes.

  2. 2.

If multiple copies of a $q$-cell $\beta$ exist in the boundary
$\partial\alpha$, then these copies are related by an element of the
reciprocal-space group $G_{\Lambda}$, which is unique up to elements of the
little group $G_{\beta}$.

Both aspects are evident in Fig. 1 (bottom): All $0$-cells have a nontrivial
little group isomorphic to $\mathbb{Z}_{2}$, generated by a combination of
time-reversal and a translation by a reciprocal lattice vector, while the
1-cells have trivial little groups. Furthermore, in defining the boundary of
the 2-cells, five of the seven 1-cells occur four times, whereas two of the
1-cells occur twice, with copies related by time-reversal and/or a reciprocal
lattice translation.

A $G_{\Lambda}$-symmetric Bloch Hamiltonian $H$ can be defined abstractly as a
map $H\colon\mathbb{D}\to\mathcal{X}$, where the target space $\mathcal{X}$
consists of all gapped Hermitian $n\times n$ matrices. This map consists of a
collection of maps $H_{\alpha}\colon\alpha\to\mathcal{X}_{\alpha}$ for each
$p$-cell $\alpha$, which are constrained by the CW-complex structure of
$\mathbb{D}$. Explicitly, these maps must be compatible with the gluing map,
i.e., if $\beta\in\partial\alpha$, then the restriction of $H_{\alpha}$ to
$\beta$ is equal to $H_{\beta}$. Since $\alpha$ is an open set, we assume that
the maps $H_{\alpha}$ can be continuously extended to its boundary. For
instance, in Fig. 1 (top), the gluing condition implies that the maps
$H_{\beta}$ for the three 2-cells $\beta$ as well as their images under a
reciprocal lattice translation must coincide with $H_{\alpha}$ defined on the
3-cell $\alpha$, as one approaches the respective boundaries of $\alpha$.

The maps constituting the Bloch Hamiltonian satisfy further constraints, which
descend from the corresponding constraints on the $p$-cells:

  1. 1′.

The target space $\mathcal{X}_{\alpha}\subseteq\mathcal{X}$ consists of
matrices that are invariant under the action of the little group
$G_{\alpha}\subset G_{\Lambda}$, see Eqs. (4) and (6). (This property assumes
a representation of $G_{\Lambda}$ on $\mathcal{X}$, see Subsec. II.2.)
Consequently, the eigenstates of $H_{\alpha}$ can be labeled by the irreps of
$G_{\alpha}$. For $\beta\in\partial\alpha$, the relation between little groups
$G_{\beta}\supseteq G_{\alpha}$ implies that
$\mathcal{X}_{\beta}\subseteq\mathcal{X}_{\alpha}$.

  2. 2′.

If two copies of $\beta$ in $\partial\alpha$ are related by $g\in
G_{\Lambda}$, then the Hamiltonian on those two copies is related by the
symmetry transformation of Eq. (4) or (6).

###  II.3 Homotopic classification

We have reduced the classification problem of band structures with a given
lattice symmetry to the classification of maps from a CW complex $\mathbb{D}$
to certain target spaces. The latter can be performed using the cellular
decomposition of $\mathbb{D}$, starting at $0$-cells and inductively working
our way up to the $d$-cells. We refer to the classification of $H_{\alpha}$
defined on $p$-cells $\alpha$ as the “level-$p$ topological classification”.
As shown below, classifications at different levels are interlinked in both
directions:

  1. 1.

The precise value of the topological invariants of levels $<p$ determines the
allowed types of level-$p$ topological invariants, and

  2. 2.

The consideration of level-$p$ topology yields compatibility relations
constraining the topological invariants associated with all levels $\leq p$.

####  II.3.1 Topology at level-0: Symmetry indicators

We begin with a classification of zero-dimensional Hamiltonians, viz, the
restriction $H_{S}$ of the Bloch Hamiltonian $H$ to the 0-cell $S$. $H_{S}$
can be block diagonalized, with blocks corresponding to the irreps of the
little group $G_{S}$. Since there are no further constraints within each
block, their topological class is completely determined by
$n_{\mathrm{o}/\mathrm{e},\rho}^{S}\geq 0$, the number of occupied/empty
orbitals of irrep $\rho$. This can be generalized to any $p$-cell $\alpha$,
where the numbers of occupied/empty orbitals
$n_{\mathrm{o}/\mathrm{e},\rho}^{\alpha}\geq 0$ are well-defined for the
restriction of $H$ to $\alpha$. If $G_{\alpha}$ has a single irrep, then the
corresponding level-0 invariants are simply the total number of occupied/empty
bands $n_{\mathrm{o}/\mathrm{e}}$.

Together, the natural numbers $n_{\mathrm{o}/\mathrm{e},\rho}^{\alpha}$
constitute the level-0 invariants, which form the semigroup
$\prod_{\alpha}\mathbb{N}^{\nu_{\alpha}}$, where $\nu_{\alpha}$ is the number
of irreps of $G_{\alpha}$. For a given $p$-cell $\alpha$, if two $q$-cells
$\beta,\beta^{\prime}\in\partial\alpha$, then
$\mathcal{X}_{\beta},\mathcal{X}_{\beta^{\prime}}\subseteq\mathcal{X}_{\alpha}$
and $H_{\beta}$ and $H_{\beta^{\prime}}$ are homotopically equivalent within
$\mathcal{X}_{\alpha}$. Put differently, $H_{\beta}$ and $H_{\beta^{\prime}}$
are homotopically equivalent, when we lift the symmetry restrictions on them
to $G_{\alpha}$ for any cell $\alpha$ connecting them. This property
constrains the values level-$0$ invariants can take [59, 55, 54]. In
particular, since all $p$-cells $\alpha$ with $p<d$ lie on the boundary of
$\mathbb{D}$, the corresponding level-0 invariants satisfy

$n_{\mathrm{o}/\mathrm{e}}=\sum_{\rho=1}^{\nu_{\alpha}}\dim(\rho)\;n_{\mathrm{o}/\mathrm{e},\rho}^{\alpha},$
(7)

where $\nu_{\alpha}$ is the number of irreps of $G_{\alpha}$ and $\dim(\rho)$
is the dimension of the irrep $\rho$. Physically, this constraint expresses
that the number of occupied/empty bands $n_{\mathrm{o}/\mathrm{e}}$ must be
the same throughout reciprocal space for gapped band structures. This
constraint is an example of a _level-0 compatibility condition_. There may be
further constraints on the level-0 invariants of a $q$-cell by their inclusion
in a $p$-cells for $p<d$, which we discuss as required.

For a given band, the set of representations of the little groups $G_{\alpha}$
define a _band representation_. An arbitrary band representation can be
decomposed into a direct sum of _elementary band representations_ (EBRs),
which can be enumerated by considering all possible orbital types localized at
high-symmetry points of the lattice. A systematic study of possible band
representations has been central to the partial classifications based on
“symmetry-based indicators” [59, 55, 60, 61, 62, 63] and “topological quantum
chemistry” [54, 64, 65, 66]. The full homotopic classification is thus an
extension of these partial classifications.

Figure 2:  For two Hamiltonians $H$ and $H_{\text{ref}}$ whose topological
invariants on the boundary of a $q$-cell are identical, we can define a
boundary deformations (shown in gray) that renders the two Hamiltonians equal
at the boundary. One thereby obtains a Bloch Hamiltonian defined on a
$q$-sphere, which can be classified by a homotopy group. The figure shows this
procedure for $q=1$ (top) and $q=2$ (bottom).

####  II.3.2 Topology at levels $p>0$

We now turn to topology associated with the $p$-cells for $p>0$. We start with
two observations:

  * •

The topological class of $H$ is always determined relative to a reference
$H_{\rm ref}$. The classification thus addresses _pairs_ of Bloch Hamiltonians
$(H,H_{\text{ref}})$.

  * •

If $H$ and $H_{\rm ref}$ have different invariants on level-$q$ for any $q<p$,
then they are topologicaly inequivalent. Thus, for classification on
$p$-cells, we may assume that $H$ and $H_{\text{ref}}$ have identical
invariants up to level $p-1$.

The latter observation implies that for a $p$-cell $\alpha$, the restrictions
of $H$ and $H_{\text{ref}}$ to $\partial\alpha$ can be smoothly deformed into
each other, and, furthermore, that this deformation can be done simultaneously
for all $p$-cells. Choosing such a smooth deformation, the “difference”
between $H$ and $H_{\text{ref}}$ then defines a map from the $p$-sphere
$S^{p}$ to $\mathcal{X}_{\alpha}$ for each $p$-cell $\alpha$. This is
illustrated in Fig. 2 for the cases $p=1$ and $p=2$. Such maps are classified
by the homotopy spaces $K_{\alpha}\equiv\pi_{p}[\mathcal{X}_{\alpha}]$, which
we term the _parent classification set_ of the $p$-cell $\alpha$. The full
level-$p$ parent classification set then reads

$K_{p}=\prod_{\alpha}K_{\alpha};\quad
K_{\alpha}=\pi_{p}\left[\mathcal{X}_{\alpha}\right].$ (8)

The parent classification set $K_{p}$ is, however, not the actual classifying
set for equivalence classes of Bloch Hamiltonians $H$, because a given pair
$(H,H_{\text{ref}})$ can yield different elements in $K_{p}$, depending on the
deformation used to achieve equality of $H$ and $H_{\text{ref}}$ on
$\partial\mathbb{D}_{p}$. Such elements of $K_{p}$ must be identified to get a
true descriptor of homotopic equivalence classes. Explicitly, any deformation
used to achieve equality of $H$ and $H_{\rm ref}$ on $\partial\mathbb{D}_{p}$
can be written as a $p$-dimensional “Bloch Hamiltonian”
$H_{\partial}(\mathbf{k}_{\partial},t)$, where
$\mathbf{k}_{\partial}\in\partial\mathbb{D}_{p}$, $t\in[0,1]$, and
$H_{\partial}$ satisfies

$H_{\partial}(\mathbf{k}_{\partial},0)=H(\mathbf{k}_{\partial}),\quad
H_{\partial}(\mathbf{k}_{\partial},1)=H_{\text{ref}}(\mathbf{k}_{\partial}),$
(9)

as well as symmetry constraints for each $q$-cell in $\partial\mathbb{D}_{p}$
for any $t$. The “difference” of two boundary deformations transforming $H$ to
$H_{\rm ref}$ on $\partial\mathbb{D}_{p}$ can be viewed as a boundary
deformation that connects $H_{\rm ref}$ to itself, i.e., with

$H_{\partial}(\mathbf{k}_{\partial},0)=H_{\partial}(\mathbf{k}_{\partial},1)=H_{\rm
ref}(\mathbf{k}_{\partial}).$ (10)

These “boundary deformations” can be visualized as in Fig. 3, whereby one
imagines that $H_{\partial}(\mathbf{k}_{\partial},t)$ is continuously “grown”
onto the band structure defined by $H$ on $\alpha$ for each $p$-cell
$\alpha\in\mathbb{D}_{p}$. The level-$p$ topological invariants are then
obtained from $K_{p}$ by identification of elements related by deformations of
this type.

Figure 3:  Boundary deformation of a band structure $H(\mathbf{k})$ defined on
a 1-cell (top) and on a 2-cell (bottom). A boundary deformation
$H_{\partial}(\mathbf{k}_{\partial},t)$, $0\leq t\leq 1$, is continously added
onto $H(\mathbf{k})$ at the boundary of the cell, such that $H(\mathbf{k})$
coincides with the reference band structure $H_{\rm ref}(\mathbf{k})$ on the
cell boundary $\partial\mathbb{D}_{m}$ for $t=0$ and $t=1$ and $H(\mathbf{k})$
satisfies all symmetry constraints at each point in the transformation.

To determine the action of boundary deformations $H_{\partial}$ on the
elements of $K_{p}$, it is sufficient to consider at least one boundary
deformation from each topological equivalence class of boundary deformations.
Considering multiple boundary deformations from the same topological class is
unproblematic, because they have the same action on $K_{p}$ and therefore lead
to identification of the same elements. Since the domain
$\partial\mathbb{D}_{p}\times[0,1]$ of a boundary deformation is again a CW
complex, these topological equivalence classes can be determined in the same
manner as described above. It is, in fact, sufficient to only determine the
corresponding parent classification set

$K_{p}^{\partial}=\prod_{\alpha\in\partial\mathbb{D}_{p}}\pi_{r_{\alpha}}[\mathcal{X}_{\alpha}],$
(11)

where $r_{\alpha}=\dim\alpha+1$. This is because the parent classification set
$K_{p}^{\partial}$ is “overcomplete”, so that considering one boundary
deformation from each class in $K_{p}^{\partial}$ ensures that at least one
boundary transformation from each (true) topological class is considered. Only
having to determine $K_{p}^{\partial}$ leads to a substantial simplification
by avoiding having to deal with “boundary transformations of boundary
transformations”. For the reader that is nevertheless interested in obtaining
the exact boundary classification set, we review a recursive procedure in
Appendix C.

There are additional constraints on the (parent) topological invariants, which
arise from the fact that the individual cells are contractible. For any
$(p+1)$-cell $\alpha$ with $1\leq p\leq d-1$, the boundary $\partial\alpha$
can thus be continuously deformed to a point within $\alpha$. Consequently,
the topology of any Bloch Hamiltonian $H$ restricted to $\partial\alpha$,
evaluated with respect the (larger) target space
$\mathcal{X}_{\alpha}\supseteq\mathcal{X}_{\beta}$ for
$\beta\in\partial\alpha$, must be trivial. This implies that the “sum” of
level-$p$ invariants for $p$-cells $\beta\in\partial\alpha$ must vanish. We
term these conditions — one for each $(p+1)$-cell — the _level- $p$
compatibility conditions_. Since this sum over level-$p$ invariants may also
depend on topological invariants at levels $q<p$, the level-$p$ compatibility
conditions may affect invariants at all levels $q\leq p$. The compatibility
condition equally apply to the boundary deformations and their parent
classification $K_{p}^{\partial}$.

To summarize, the topological invariants at all levels $0\leq p\leq d$, modulo
the ambiguities associated with boundary deformations, and constrained by the
compatibility relations at all levels, constitute the complete classification
of the $d$-dimensional systems. The compatibility relations ensure that for
each set of topological invariants there is a gapped band structure with these
invariants, while the robustness to boundary deformations ensures that the
topological invariants can be uniquely associated with a gapped band
structure. The classification naturally has a hierarchical reciprocal-space
structure [6], since the classification at level-$p$ depends on the invariants
at all lower level, starting with the symmetry based indicators at level-0.

###  II.4 Target spaces

We now discuss the various target spaces $\mathcal{X}$ that arise in the
classification for symmetry classess A, AI, and AII and describe their
homotopy groups.

####  II.4.1 Grassmannians and their topology

We are interested in the space of $n\times n$ Hermitian matrices with a fixed
number of occupied and empty bands (i.e., negative and positive eigenvalues)
$n_{\mathrm{o}/\mathrm{e}}$ satisfying

$n=n_{\mathrm{o}}+n_{\mathrm{e}}.$ (12)

For a topological classification, $H(\mathbf{k})$ may be replaced by a
Hermitian matrix with a “flattened spectrum” by deforming all
positive/negative eigenvalues to $\pm 1$, so that $H(k)$ is unitarily
equivalent to
$\text{diag}\\{\openone_{n_{\mathrm{e}}},-\openone_{n_{\mathrm{o}}}\\}$. The
space of such Hamiltonians is the _complex Grassmannian_
$\mathrm{Gr}_{\mathbb{C}}(n_{\mathrm{o}},n_{\mathrm{e}})$, which is a manifold
that parametrizes $n_{\mathrm{o}}$-dimensional (or equivalently,
$n_{\mathrm{e}}$-dimensional) complex planes in $\mathbb{C}^{n}$ (see App. D
for more details). We also encounter Bloch Hamiltonians $H(\mathbf{k})$ with
real $(\mathbb{R})$ or quaternion $(\mathbb{H}$) entries, for which the target
space is the corresponding Grassmannian
$\mathrm{Gr}_{\mathbb{R}}(n_{\mathrm{o}},n_{\mathrm{e}})$ or
$\mathrm{Gr}_{\mathbb{H}}(n_{\mathrm{o}},n_{\mathrm{e}})$, respectively. These
three spaces constitute the top-level target spaces $\mathcal{X}$ for the
symmetries considered here.

For a $p$-cell $\alpha$ with a nontrivial little group $G_{\alpha}$, the
target spaces $\mathcal{X}_{\alpha}\subseteq\mathcal{X}$ can be written as a
product of Grassmannians. This is because the eigenstates of $H$ restricted to
$\alpha$ can be labeled with irreps $\rho$ of $G_{\alpha}$, so that after a
spectral flattening, we get

$\mathcal{X}_{\alpha}=\prod_{\rho}\mathrm{Gr}_{{\mathbb{F}}_{\rho}}\left(n_{\mathrm{o},\rho}^{\alpha},n_{\mathrm{e},\rho}^{\alpha}\right),$
(13)

where $\rho$ runs over the irreps of $G_{\alpha}$ and $\mathbb{F}_{\rho}$ is a
number field, which, depending on $\rho$, may be $\mathbb{C}$, $\mathbb{R}$,
or $\mathbb{H}$. The latter two occur only if $G_{\alpha}$ contains
antiunitary symmetries. Since we are interested in
$\pi_{p}[\mathcal{X}_{\alpha}]$ and the homotopy of a product of spaces
satisfies

$\pi_{k}\Big{[}\prod_{\alpha}X_{\alpha}\Big{]}=\prod_{\alpha}\pi_{k}\left[X_{\alpha}\right],$
(14)

we only need to compute
$\pi_{p}[\mathrm{Gr}_{\mathbb{F}}(n_{\mathrm{o}},n_{\mathrm{e}})]$.

Homotopy groups of Grassmannians have been extensively studied in the
mathematical literature. Their computation is briefly outlined in Appendix D.
We, however, make two general observations here:

  * •

$\pi_{p}[\mathrm{Gr}_{\mathbb{F}}(n_{\mathrm{o}},n_{\mathrm{e}})]=0$ for all
$p>0$ if $n_{\mathrm{o}}=0$ or $n_{\mathrm{e}}=0$, since in that case,
$\mathrm{Gr}_{\mathbb{F}}(n_{\mathrm{o}},n_{\mathrm{e}})$ degenerates to a
single point.

  * •

$\pi_{p}[\mathrm{Gr}_{\mathbb{F}}(n_{\mathrm{o}},n_{\mathrm{e}})]$ becomes
independent of $n_{\mathrm{o}}$ and $n_{\mathrm{e}}$ (“stabilizes”) once
$n_{\mathrm{o}}$ and $n_{\mathrm{e}}$ are large enough, which typically means
$n_{\mathrm{o}/\mathrm{e}}\gtrsim p$.

As to the former observation, while $n_{\mathrm{o},\mathrm{e}}>0$ for gapped
band structure, $\mathrm{Gr}_{\mathbb{F}}(n_{\mathrm{o}},n_{\mathrm{e}})$ with
$n_{\mathrm{o}}=0$ or $n_{\mathrm{e}}=0$ may occur in individual symmetry
sectors at high-symmetry manifolds. The latter observation forms the basis for
the classification of “stable” topology.

The homotopy invariants of Grassmannians can often be expressed in terms of
Wilson lines or loops [67, 68]. Given a path $\gamma$ in the reciprocal space,
the Wilson line ${W}^{\mathrm{o}/\mathrm{e}}(\gamma)\in\mathrm{U}(n)$ can be
associated with the occupied/empty bands and encodes the evolution of the
corresponding subspaces under parallel transport along $\gamma$. Wilson lines
are not gauge invariant, since they depend on the choice of basis at the end
points of $\gamma$. They are unambiguously defined only if there are
additional constraints on the end points of $\gamma$ that fix the basis or if
the path $\gamma$ is a closed loop. A local-in-$\mathbf{k}$ symmetry on
$H(\mathbf{k})$ implies a restriction on Wilson lines/loops; in particular, if
the Hamiltonian is restricted by symmetries to be real- or quaternion-valued,
then they are restricted to
$\mathrm{O}(n),\,\mathrm{Sp}(n)\subset\mathrm{U}(n)$, respectively.

####  II.4.2 Topological invariants for complex Grassmannians

We now discuss the nontrivial topological invariants associated with complex
Grassmannians, which are summarized in Table. 1.

_$\pi_{2}[\mathrm{Gr}_{\mathbb{C}}(n_{\mathrm{o}},n_{\mathrm{e}})]$ —_ These
stabilize to $\mathbb{Z}$ for all $n_{\mathrm{o}/\mathrm{e}}>0$, with the
corresponding invariant being the Chern number $\mathrm{C}$. For a complex
Hamiltonian, $\det{W}^{\mathrm{o}/\mathrm{e}}(\gamma)$ is a phase factor,
i.e., it is defined on the unit circle. To define the Chern number, we take a
family of Wilson loops ${W}^{\mathrm{o}/\mathrm{e}}(\gamma_{t})$ with
$t\in[0,1]$ and $\gamma_{0}=\gamma_{1}$ so that the $\gamma_{t}$ cover the
two-sphere $S^{2}$ precisely once, in a manner compatible with the orientation
of $S^{2}$. The Chern number $\mathrm{C}$ is then equal to the winding number
of $\det({W}^{\mathrm{o}}({\gamma_{t}}))$ for $0\leq t\leq 1$. As the total
Chern number of a set of bands must vanish, the Chern number of the unoccupied
bands equals $-\mathrm{C}$.

_$\pi_{3}[\mathrm{Gr}_{\mathbb{C}}(n_{\mathrm{o}},n_{\mathrm{e}})]$ —_ The
only nontrivial case occurs for $n_{\mathrm{o}}=n_{\mathrm{e}}=1$, with the
topological invariant being the $\mathbb{Z}$-valued Hopf invariant. This
arises from the fact that $\mathrm{Gr}_{\mathbb{C}}(1,1)\cong S^{2}$, so that
its third homotopy group $\pi_{3}(S^{2})\cong\mathbb{Z}$ is generated by the
Hopf map. We refer to App. E for more details.

$d$ | Bands | Topological Invariant
---|---|---
1 | — | —
2 | — | $\mathbb{Z}$ | Chern number $\mathrm{C}$
3 | $n_{\mathrm{o}},n_{\mathrm{e}}=1$ | $\mathbb{Z}$ | Hopf invariant $\mathrm{H}$
$n_{\mathrm{o}}+n_{\mathrm{e}}>2$ | —
Table 1: Topological invariants associated with complex Grassmannians
$\mathrm{Gr}_{\mathbb{C}}(n_{\mathrm{o}},n_{\mathrm{e}})\cong\mathrm{Gr}_{\mathbb{C}}(n_{\mathrm{e}},n_{\mathrm{o}})$.

####  II.4.3 Topological invariants for real Grassmannians

The nontrivial topological invariants associated with real Grassmannians are
summarized in Table. 2. These exhibit multiple low-dimensional exceptions.

_$\pi_{1}[\mathrm{Gr}_{\mathbb{R}}(n_{\mathrm{o}},n_{\mathrm{e}})]$ —_ These
groups stabilize to $\mathbb{Z}_{2}$ for $n_{\mathrm{o}/\mathrm{e}}>1$, with
the corresponding topological invariant being the _first Stiefel-Whitney
invariant_ $\mathrm{SW}_{1}$. To define this invariant, we note that the
Wilson loops are orthogonal matrices if the Hamiltonian is real, so that
$\det{W}^{\mathrm{o}/\mathrm{e}}(\gamma)\in\\{\pm 1\\}$ for any closed loop
$\gamma$. The first Stiefel-Whitney invariant is defined as [69]

$\mathrm{SW}_{1}\equiv{\rm
sgn}\,\left(\det{W}^{\mathrm{o}/\mathrm{e}}(\gamma)\right).$ (15)

As the total set of bands must be topologically trivial, $\mathrm{SW}_{1}$ may
be calculated either from the occupied or from the empty bands. Since
${W}^{\mathrm{o}/\mathrm{e}}(\gamma)\in\mathrm{O}(n_{\mathrm{o}/\mathrm{e}})$
encodes the basis transformation of the occupied/empty subspace under parallel
transport along $\gamma$, a nontrivial $\mathrm{SW}_{1}$ represents an
obstruction to defining an orientation of these subspaces continuously along
$\gamma$.

For $n_{\mathrm{o}}=n_{\mathrm{e}}=1$, this $\mathbb{Z}_{2}$ classification
can be refined to a $\mathbb{Z}$ classification. To see this, we note that
after flattening the spectrum, a gapped real $2\times 2$ Hamiltonian can be
written as

$H(\mathbf{k})=\cos\theta_{\mathbf{k}}\,\sigma_{1}+\sin\theta_{\mathbf{k}}\,\sigma_{3},$
(16)

where $\sigma_{j}$ are the Pauli matrices. Thus, for any closed path in the
reciprocal space, $\mathbf{k}\mapsto\theta_{\mathbf{k}}$ defines a map from
$S^{1}$ to $S^{1}$. Such maps are classified by a winding number
$\mathrm{Eu}_{1}\in\pi_{1}[S^{1}]\cong\mathbb{Z}$, termed the _first Euler
class_. The sign of the Euler class depends on a choice of orientation for
$\mathrm{Gr}_{\mathbb{R}}(1,1)$, since a unitary transformation can flip the
sign of $\sigma_{1}$, thereby flipping the sign of $\mathrm{Eu}_{1}$ [28].
Thus, in absence of a preferred orientation, $\mathrm{Eu}_{1}$ is defined only
up to a sign, so that the space of topological invariants is $\mathbb{N}$
(instead of $\mathbb{Z}$). The first Euler invariant is related to the first
Stiefel-Whitney invariant as

$\mathrm{SW}_{1}=\mathrm{Eu}_{1}\mod 2.$ (17)

$d$ |  | Topological invariant(s)
---|---|---
1 | $n_{\mathrm{o}},n_{\mathrm{e}}=1$ | $\mathbb{Z}$ | 1${}^{\text{st}}$ Euler invariant $\mathrm{Eu}_{1}$
$n_{\mathrm{o}},n_{\mathrm{e}}>1$ | $\mathbb{Z}_{2}$ | 1${}^{\text{st}}$ Stiefel-Whitney invariant $\mathrm{SW}_{1}$
2 | $n_{\mathrm{o}},n_{\mathrm{e}}=1$ | — | —
$n_{\mathrm{o}}=1$, | $2\mathbb{Z}$ | 2${}^{\text{nd}}$ Euler invariant $\mathrm{Eu}_{2}^{\mathrm{o}}$
$n_{\mathrm{e}}=2$
$n_{\mathrm{o}}=2$, | $2\mathbb{Z}$ | 2${}^{\text{nd}}$ Euler invariant $\mathrm{Eu}_{2}^{\mathrm{e}}$
$n_{\mathrm{e}}=1$
$n_{\mathrm{o}},n_{\mathrm{e}}=2$ | $\mathbb{Z}^{2}|_{\rm p}$ | 2${}^{\text{nd}}$ Euler invariants $(\mathrm{Eu}_{2}^{\mathrm{e}},\mathrm{Eu}_{2}^{\mathrm{e}})$
satisfying $\mathrm{Eu}_{2}^{\mathrm{o}}+\mathrm{Eu}_{2}^{\mathrm{e}}=0\mod 2$
$n_{\mathrm{o}}=2$, | $\mathbb{Z}$ | 2${}^{\text{nd}}$ Euler invariant $\mathrm{Eu}_{2}^{\mathrm{o}}$
$n_{\mathrm{e}}>2$
$n_{\mathrm{o}}>2$, | $\mathbb{Z}$ | 2${}^{\text{nd}}$ Euler invariant $\mathrm{Eu}_{2}^{\mathrm{e}}$
$n_{\mathrm{e}}=2$
$n_{\mathrm{o}},n_{\mathrm{e}}>2$ | $\mathbb{Z}_{2}$ | 2${}^{\text{nd}}$ Stiefel-Whitney invariant $\mathrm{SW}_{2}$
Table 2: Topological invariants associated with real Grassmannians
$\mathrm{Gr}_{\mathbb{R}}(n_{\mathrm{o}},n_{\mathrm{e}})\cong\mathrm{Gr}_{\mathbb{R}}(n_{\mathrm{e}},n_{\mathrm{o}})$.
The $\mathbb{Z}$-valued invariants are defined only up to a sign ambiguity if
the Grassmannian does not have a preferred orientation. The set
$\mathbb{Z}^{2}|_{\rm p}$ consists of all ordered integer pairs $(p,q)$ with
$p=q\\!\\!\mod 2$.

_$\pi_{2}[\mathrm{Gr}_{\mathbb{R}}(n_{\mathrm{o}},n_{\mathrm{e}})]$ —_ These
groups stabilize to $\mathbb{Z}_{2}$ for $n_{\mathrm{o}/\mathrm{e}}>2$, with
the corresponding topological invariant being the _second Stiefel-Whitney
invariant_ $\mathrm{SW}_{2}$. If $n_{\mathrm{o}/\mathrm{e}}=2$, the
corresponding invariants are the _second Euler invariants_
$\mathrm{Eu}_{2}^{\mathrm{o}/\mathrm{e}}$. To define these invariants, we
consider a one-parameter family of Wilson loops
${W}^{\mathrm{o}/\mathrm{e}}({\gamma_{t}})\in\mathrm{O}(n_{\mathrm{o}/\mathrm{e}})$,
where $t\in[0,1]$ such that $\gamma_{0}=\gamma_{1}$ and the $\gamma_{t}$ cover
the two-sphere $S^{2}$ precisely once, in a matter compatible with its
orientation. Such a family of Wilson loops defines a closed curve in
$\mathrm{O}(n_{\mathrm{o}/\mathrm{e}})$. For $n_{\mathrm{o}/\mathrm{e}}=2$,
one associates an integer-valued winding number
$\mathrm{Eu}_{2}^{\mathrm{o}/\mathrm{e}}$ with such a closed loop, since
$2\times 2$ orthogonal matrices are parameterized by a single phase variable.
For $n_{\mathrm{o}/\mathrm{e}}>2$ the winding number is only defined mod $2$,
with the parity being the $\mathbb{Z}_{2}$-valued second Stiefel-Whitney
number $\mathrm{SW}_{2}^{\mathrm{o}/\mathrm{e}}$ [69].

Since the combination of all bands must be topologically trivial, one has
$\mathrm{SW}_{2}^{\mathrm{o}}=\mathrm{SW}_{2}^{\mathrm{e}}$, so that one may
write $\mathrm{SW}_{2}$ instead of $\mathrm{SW}_{2}^{\mathrm{o}}$ and
$\mathrm{SW}_{2}^{\mathrm{e}}$. The second Euler and Stiefel-Whitney
invariants (if defined) further satisfy the parity constraint

$\mathrm{Eu}_{2}^{\mathrm{o}}=\mathrm{Eu}_{2}^{\mathrm{e}}=\mathrm{SW}_{2}\mod
2,$ (18)

If $n_{\mathrm{o}}=1$ or $n_{\mathrm{e}}=1$, continuity enforces that
${W}^{\mathrm{o}}(\gamma_{t})=1$ or ${W}^{\mathrm{e}}(\gamma_{t})=1$ for all
$t\in[0,1]$, respectively, so that $\mathrm{SW}_{2}$ must be trivial and
$\mathrm{Eu}_{2}^{\mathrm{e}}$ or $\mathrm{Eu}_{2}^{\mathrm{o}}$, if defined,
must be even. The sign of $\mathrm{Eu}_{2}^{\mathrm{o}}$ and
$\mathrm{Eu}_{2}^{\mathrm{e}}$ depends on the orientation of the basis of
$\mathrm{Gr}_{\mathbb{R}}(n_{\mathrm{o}},n_{\mathrm{e}})$. If this orientation
is not fixed, then $\mathrm{Eu}_{2}^{\mathrm{o}}$ and
$\mathrm{Eu}_{2}^{\mathrm{e}}$ have a sign ambiguity, which is a joint sign
ambiguity if both $\mathrm{Eu}_{2}^{\mathrm{o}}$ and
$\mathrm{Eu}_{2}^{\mathrm{e}}$ are defined. In this case the classifying space
reduces to $\mathbb{N}$ and $\mathbb{N}\times\mathbb{Z}|_{\rm p}$,
respectively, where $\mathbb{N}\times\mathbb{Z}_{\rm p}$ consists of all
ordered integer pairs $(p,q)$ with $p\geq 0$ and $p=q\mod 2$.

####  II.4.4 Topological invariants for symplectic Grassmannians

The first two homotopy groups of the symplectic Grassmannians are trivial,

$\pi_{1}[\mathrm{Gr}_{\mathbb{H}}(n_{\mathrm{o}},n_{\mathrm{e}})]=\pi_{2}[\mathrm{Gr}_{\mathbb{H}}(n_{\mathrm{o}},n_{\mathrm{e}})]=0.$
(19)

###  II.5 Topology and bulk-boundary correspondence

Band structures with nontrivial stable topology in $d$ dimensions exhibit
anomalous gapless boundary states on $(d-1)$-dimensional boundaries that
respect all symmetries — a phenomenon known as the “bulk-boudary
correspondence” [1, 2, 3, 4]. In this context, a boundary mode is called
“anomalous” if it cannot be realized by any $(d-1)$-dimensional lattice model
that respects all the symmetries. Of the three types of topological
classification — based on delicate, fragile, and stable equivalence — only the
first allows for anomalous boundary states as a signature of the bulk
topology. This, however, also includes the representation protected
topological phases (defined in Sec. II.1), provided the boundary respects the
symmetry as well as the representation constraints.

Our classification procedure can also be used to classify anomalous gapless
states at a symmetry-preserving boundary. To this end, we first enumerate the
possible gapless phases consistent with given symmetries by looking for all
possible violations of the compatibility conditions in $d-1$ dimensions. Such
violations indicate gapless phases, since a level-$p$ compatibility condition
follows from the contractibility of a $(p+1)$-cell $\alpha$, so that its a
violation implies that $H_{\alpha}:\alpha\to\mathcal{X}_{\alpha}$ is defined
only for a noncontractible subset of $\alpha$, i.e., the Bloch Hamiltonian is
gapless somewhere on $\alpha$. A gapless ($d-1$)-dimensional Bloch Hamiltonian
$H$ is anomalous if it necessarily violates a constraint imposed on the
topological invariants by a gluing condition or by symmetry. We illustrate
this classification scheme for anomalous boundary states by deriving some well
known examples in the next section.

##  III Classification without crystalline symmetries

We first apply our classification strategy to band structures without
additional crystalline symmetries. Since these classification results are well
known, this section mainly serves as an illustration of our classification
scheme following the more abstract discussion of Sec. II.

Figure 4:  Fundamental domain of a band structure on a square lattice, without
(left) or with (right) time-reversal symmetry, but without additional
crystalline symmetries. In the latter case, two of the 0-cells and all 1-cells
occur twice in $\partial\mathbb{D}$.

###  III.1 Classification without time-reversal symmetry

In the absence of time-reversal symmetry (symmetry class A), the fundamental
domain is the entire Brillouin zone, which can be decomposed into cells as
shown in the left column of Fig. 4 and Fig. 1 in two and three dimensions,
respectively. The topological invariants for the unique 0-cell are the numbers
of occupied and unoccupied bands $n_{\mathrm{o}/\mathrm{e}}$. The level-0
topological invariant is thus the pair
$(n_{\mathrm{o}},n_{\mathrm{e}})\in\mathbb{N}^{2}$. The target space
$\mathcal{X}$ for all cells is

$\mathcal{X}=\mathrm{Gr}_{\mathbb{C}}(n_{\mathrm{o}},n_{\mathrm{e}}).$ (20)

We can now read off the parent topological invariants from Tab. 1.

There are no level-1 invariants, since the first homotopy group of complex
Grassmannians is always trivial. At level-2, the parent classification set is
$K_{1}=\pi_{2}[\mathcal{X}]\cong\mathbb{Z}$ and the corresponding topological
invariant is the $\mathbb{Z}$-valued Chern number $\mathrm{C}$. To get the
true topological invariant, however, we need to check if it is robust to
deformations at the boundary of a 2-cell. As discussed in Sec. II.3, the
parent classification set of boundary deformations $K_{2}^{\partial}$ is
generated by deformations along the 0-cells and the 1-cells, whose equivalence
classes are parametrized by $\pi_{1}[\mathcal{X}]=0$ and
$\pi_{2}[\mathcal{X}]\cong K_{2}\cong\mathbb{Z}$, respectively. Thus, boundary
deformations may carry a nonzero Chern number along a 1-cell on
$\partial\mathbb{D}$. However, since each 1-cell occurs twice in
$\partial\mathbb{D}$ and the corresponding deformations carry opposite Chern
numbers (see Fig. 5), the net Chern number of any boundary deformation is
zero, rendering the parent Chern number robust. This argument applies for each
2-cell in the fundamental domain, so that the level-2 invariants are the
single Chern number $\mathrm{C}$ in 2d and three Chern numbers
$\mathrm{C}_{x,y,z}$ in 3d.

Figure 5:  Generators of the boundary deformations at the boundary of a 2-cell for a band structure without time-reversal symmetry and without additional crystalline symmetries. Periodicity of the Bloch Hamiltonian in reciprocal space forces that the net Chern number associated with the boundary deformation is zero. Hence, a boundary deformation cannot change the Chern number associated with the 2-cell. | Bands | Topological invariant
---|---|---
Level 1 | — | —
Level 2 | — |  $\mathrm{C}_{x}$, $\mathrm{C}_{y}$, $\mathrm{C}_{z}\in\mathbb{Z}$
Level-3 | $n_{\mathrm{o}}=n_{\mathrm{e}}=1$ | if $\mathrm{C}_{x}=\mathrm{C}_{y}=\mathrm{C}_{z}=0$,
then $\mathrm{H}\in\mathbb{Z}$
else: $\mathrm{H}\in\mathbb{Z}_{2\mathrm{C}}$, where
$\mathrm{C}\equiv\textrm{gcd}\,(\mathrm{C}_{x},\mathrm{C}_{y},\mathrm{C}_{z})$
$n_{\mathrm{o}}+n_{\mathrm{e}}>2$ | —
Table 3: Homotopic classification of gapped band structures with broken time-
reversal symmetry and no additional crystalline symmetries.

At level-3, the parent invariant is the $\mathbb{Z}$-valued Hopf invariant
$\mathrm{H}$ if $n_{\mathrm{o}}=n_{\mathrm{e}}=1$ [8]. The boundary
deformations in $K_{3}^{\partial}$ depend on the three level-2 invariants
$\mathrm{C}_{x,y,z}$ [10]. There are six generators of the boundary
deformations: Three carry unit Chern number associated with segments along one
of the three 1-cells and its three images under translation by a reciprocal
lattice vector and three carry unit Hopf number associated with segments along
one of the three 2-cells and its image under translation by a reciprocal
lattice vector. Because of the periodicity constraint in reciprocal space, the
latter leave the Hopf number $\mathrm{H}$ invariant, while the former change
$\mathrm{H}$ by $2\mathrm{C}_{\alpha}$, $\alpha=x$, $y$, $z$ (see App. E for
details). The minimum change to $\mathrm{H}$ one may thus obtain is
$2\,\mbox{gcd}(\mathrm{C}_{x},\mathrm{C}_{y},\mathrm{C}_{z})$, where “gcd”
denotes the greatest common divisor. Hence, $\mathrm{H}$ is well defined up to
multiples of $2\,\mbox{gcd}(\mathrm{C}_{x},\mathrm{C}_{y},\mathrm{C}_{z})$, so
that the level-3 invariant is $\mathbb{Z}$–valued if
$\mathrm{C}_{x}=\mathrm{C}_{y}=\mathrm{C}_{z}=0$ and $\mathbb{Z}_{2C}$–valued
otherwise, where
$C\equiv\mbox{gcd}(\mathrm{C}_{x},\mathrm{C}_{y},\mathrm{C}_{z})$.

These classification results are summarized in Tab. 3.

###  III.2 Classification with time-reversal symmetry

####  III.2.1 Symmetry class AI (spinless fermions)

In the presence of time-reversal and spin-rotation symmetry, the Bloch
Hamiltonian satisfies

$H(\mathbf{k})=H^{*}(-\mathbf{k}).$ (21)

The fundamental domain is half the Brillouin zone, which can be decomposed as
shown in Fig. 4 and Fig. 1 for two and three dimensions, respectively. The
level-0 invariants are given by the number of occupied/empty bands
$n_{\mathrm{o}/\mathrm{e}}^{S}$ for each 0-cell $S$. Since
$n_{\mathrm{o}/\mathrm{e}}$ must be constant throughout the Brillouin zone for
a gapped Hamiltonian, one has
$n_{\mathrm{o}/\mathrm{e}}^{S}=n_{\mathrm{o}/\mathrm{e}}$ for all $S$, leading
to a single pair of level-0 invariants
$(n_{\mathrm{o}},n_{\mathrm{e}})\in\mathbb{N}^{2}$. At the 0-cells $S$, time-
reversal symmetry constraints the Hamiltonian $H_{S}$ to be real-valued, so
that the corresponding target spaces are

$\displaystyle\mathcal{X}_{S}=$
$\displaystyle\,\mathrm{Gr}_{\mathbb{R}}(n_{\mathrm{o}},n_{\mathrm{e}}),$ (22)

For the $p$-cells with $p>0$, we have
$\mathcal{X}=\mathrm{Gr}_{\mathbb{C}}(n_{\mathrm{o}},n_{\mathrm{e}})$. Thus,
beyond level-0, the _parent topological invariant_ for individual $p$-cells
are identical to the symmtry class A, i.e., no invariants at level 1, a
$\mathbb{Z}$-valued Chern number for each 2-cell at level-2 and a
$\mathbb{Z}$-valued Hopf invariant at level-3 if
$n_{\mathrm{o}}=n_{\mathrm{e}}=1$.

At level-2, the parent classification set of boundary deformations
$K_{2}^{\partial}$ consists of a copy of $\pi_{1}[\mathcal{X}_{S}]$ for each
0-cell and $\pi_{2}[\mathcal{X}]=\mathbb{Z}$ for the 1-cells. The former
homotopy set is nontrivial for each 0-cell, and since the 0-cells $\Gamma$ and
${\mathrm{X}}$ occur only once in $\partial\mathbb{D}$, a boundary deformation
that is topologically nontrivial at one of these can change the Chern number
by an arbitrary integer. In App. F, we construct such a deformation explicitly
as well as present another argument for the existence of such a boundary
deformation. Thus, in contrast to symmetry class A, there are no level-2
topological invariants for symmetry class AI. Similarly, at level-3, there
exists a boundary deformation that changes the Hopf number $\mathrm{H}$ of the
parent classification by one, as shown in Appendix E, rendering the level-3
invariant trivial.

####  III.2.2 Symmetry class AII (spinful fermions)

In the presence of time-reversal symmetry without spin-rotation symmetry, the
Hamiltonian satisfies

$H(\mathbf{k})=\Sigma_{2}H^{\ast}(-\mathbf{k})\Sigma_{2},$ (23)

where $\Sigma_{2}=\openone_{n}\otimes\sigma_{2}$ is the time-reversal operator
and $2n$ is the total number of bands. Kramers degeneracy implies that the
number of occupied and unoccupied bands must be even, so that the level-0
invariants $(n_{\mathrm{o}}^{S},n_{\mathrm{e}}^{S})$ are _half_ the number of
occupied/empty bands. The level-0 compatibility condition is
${n}_{\mathrm{o}/\mathrm{e}}=n_{\mathrm{o}/\mathrm{e}}^{S}$, where
$2{n}_{\mathrm{o}/\mathrm{e}}$ are the total number of filled/empty bands.

To derive the target space $\mathcal{X}_{S}$, we note that at the high
symmetry points, time-reversal implies that the Hamiltonian can be written as

$H_{S}=h_{0}\otimes\sigma^{0}+h_{j}\otimes\mathrm{i}\sigma^{j},$ (24)

where $h_{0}$ and $h_{j}$ are real-valued $n\times n$ symmetric and
antisymmetric matrices, respectively. Since the set of $2\times 2$ conplex
matrices
$\\{\sigma^{0},\mathrm{i}\sigma^{1},\mathrm{i}\sigma^{2},\mathrm{i}\sigma^{3}\\}$
constitute a representation of the quaternion algebra, $H_{S}$ in Eq. (24) can
be interpreted as a quaternion-valued matrix. The target space
$\mathcal{X}_{S}$ is thus a quaternionic Grassmannian

$\mathcal{X}_{S}=\mathrm{Gr}_{\mathbb{H}}(n_{\mathrm{o}},n_{\mathrm{e}}).$
(25)

For $p$-cells with $p>0$, we have
$\mathcal{X}=\mathrm{Gr}_{\mathbb{C}}(2n_{\mathrm{o}},2n_{\mathrm{e}})$.

The topological classification proceeds as for the symmetry class AI, with one
important difference: While considering the boundary deformations to level-2
invariants, $\pi_{1}[\mathcal{X}_{S}]=0$ for the quaternionic Grassmannian, so
that there are no nontrivial deformations at the 0-cells. The boundary
deformations at 1-cells can carry a Chern number. These cancel for the two
copies of ${\mathrm{Y}}{\mathrm{M}}$, but add up for the two copies of
$\Gamma{\mathrm{Y}}$ and ${\mathrm{M}}{\mathrm{X}}$, so that boundary
deformations may add an even integer to the Chern number of the 2-cell. This
renders the level-2 invariant $\mathbb{Z}_{2}$-valued. Thus, at level-2, we
get one and four $\mathbb{Z}_{2}$-valued invariants for $d=2$ and $d=3$,
respectively [33]. These are the well-known classifications of the quantum
spin-Hall effect and of three-dimensional topological insulators [70, 71, 72,
33, 73].

###  III.3 Anomalous boundary states

In the symmetry class A, anomalous boundary states are associated with a
nontrivial value of the only stable invariant, viz, the Chern number. In two
dimensions, these are the chiral edge states of the quantum Hall effect, while
in three dimensions, these are chiral surface states of a weak Chern
insulator. We now derive these invariants from a boundary perspective.

Consider a 1d Bloch Hamiltonian in symmetry class A, whose fundamental domain
is the entire 1d Brillouin zone, as shown in Fig. 6 (left). We assume that $H$
is gapped at the 0-cells, so that the level-0 invariants
$n_{\mathrm{o}/\mathrm{e}}^{\bar{S}}$ are well defined for
$\bar{S}\in\\{{\overline{\mathrm{M}}},{\overline{\mathrm{M}}}^{\prime}\\}$.
The presence of a chiral state crossing the Fermi level implies that the
level-0 invariants $n_{\mathrm{o}/\mathrm{e}}^{{\overline{\mathrm{M}}}}\neq
n_{\mathrm{o}/\mathrm{e}}^{{\overline{\mathrm{M}}}^{\prime}}$, which violates
both the compatibility condition and the gluing condition. The former
violation signifies that a band structure with a chiral state crossing the
Fermi level is necessarilty gapless, while the latter violation signals that
such a band structure is anomalous, i.e., it can not be realized in a one-
dimensional lattice model 777 If $H$ has a zero energy mode at
${\overline{\mathrm{M}}}$, then we shift the Hamiltonian by an infinitesimal
constant to ensure that $n_{\mathrm{o}/\mathrm{e}}^{\overline{\mathrm{M}}}$ is
well-defined. . An analogous argument holds for a chiral band crossing the
Fermi energy in 2d.

Figure 6:  Fundamental domain of a one-dimensional band structure, without
(left) or with (right) time-reversal symmetry, and without additional
crystalline symmetries.

In symmetry class AI, there is no nontrivial bulk topology and, hence, there
are no associated anomalous boundary states. The same conclusion can be
obtained from a boundary perspective. To wit, the fundamental domain of a 1d
Bloch Hamiltonian in symmetry class AI is half the 1d Brillouin zone, as shown
in Fig. 6 (right). A band crossing between ${\overline{\Gamma}}$ amd
${\overline{\mathrm{M}}}$ violates the level-0 compatibility condition for the
$n_{\mathrm{o}/\mathrm{e}}^{\bar{S}}$ for
$\bar{S}\in\\{{\overline{\Gamma}},{\overline{\mathrm{M}}}\\}$ — signalling a
gapless phase — but it does not imply the violation of a gluing condition, so
that such a band crossing is not anomalous. Instead, time-reveral symmetry
imposes the presence of two opposite crossings of the Fermi energy in the full
boundary Brillouin zone, which can be realized in a 1d lattice model and which
may deformed to a gapped Hamiltonian. Thus, there are no anomalous boundary
modes in symmetry class AI.

In symmetry class AII, anomalous boundary modes are associated with the
$\mathbb{Z}_{2}$-valued stable invariant. In two dimensions, these are the
helical edge states of the quantum spin Hall effect, while in three
dimensions, these are the surface Dirac cones of a strong topological
insulator. In this case, the fundamental domain is identical to that of
symmetry class AI (Fig. 6), but the level-0 invariants
$n_{\mathrm{o}/\mathrm{e}}^{\bar{S}}$ are _half_ the number of
occupied/unoccupied bands. Thus, if a chiral mode crosses the Fermi level
between ${\overline{\Gamma}}$ and ${\overline{\mathrm{M}}}$, then the total
number of occupied/empty bands differ by an odd integer, which is incompatible
with the level-0 invariants being integers, rendering such crossings
anomalous. Application of a similar arguments to any pair of high-symmetry
points in the surface Brillouin zone indicates the existence of anomalous
surface states for 3d topological insulators [72, 33, 73].

##  IV Classification in three dimensions with crystalline symmetries: General
remarks

In the next sections, we apply the classification strategy of Sec. II to band
structures with ${\mathrm{C}}_{2}{\mathrm{T}}$, ${\mathrm{C}}_{2}$,
${\mathrm{C}}_{4}$, and ${\mathrm{D}}_{4}$ symmetries. Since these symmetries
are all two-dimensional and include a rotation, in two dimensions they do not
exhibit a boundary that is invariant under the symmetries; however, they all
exhibit invariant planes in three dimensions, which may host anomalous
boundary states. We thus consider a simple-cubic lattice with unit lattice
constant, with the rotation axis (and mirror planes for ${\mathrm{D}}_{4}$)
parallel to the $z$-axis. For the crystalline symmetries considered in this
article, this simplification does not affect our conclusions. Since $C_{2}$ is
an element of all of these groups, for the time-reversal-symmetric case, we
choose a basis such that ${\mathrm{C}}_{2}{\mathrm{T}}$ is represented by the
identity matrix, so that

$H(k_{x},k_{y},k_{z})=H^{*}(k_{x},k_{y},-k_{z}).$ (26)

For $k_{z}=0,\pi$ (and for 2d systems), this reduces to a reality condition on
$H(\mathbf{k})$.

The fundamental domain in all such cases is a prism with the two-dimensional
fundamental domain as the base and $k_{z}\in(-\pi,\pi]$ without and
$k_{z}\in[0,\pi]$ with time-reversal symmetry. (See Fig. 1 for the fundamental
domains without crystalline symmetries.) The “vertical” faces are formed by
extruding the 1-cells of the fundamental domain $\mathbb{D}_{2}$ along $z$, so
that the vertical faces and edges inherit the little groups of the 1- and
0-cells of which they are an extrusion. The “horizontal” faces at
$k_{z}=\pm\pi$ for the time-reversal broken case and $k_{z}=0,\pi$ for the
time-reversal-symmetric case are copies of $\mathbb{D}_{2}$, and thus satisfy
the corresponding constraints.

The classification on the horizontal faces depends on the details of the
symmetry in question, but that on the vertical faces can be described more
generally. Since time-reversal (if present) does not impose any additional
constraints on the side faces for a two-dimensional symmetry, the target space
for the vertical 1- and 2-cells is always the complex Grassmannian. Thus,
there are no level-1 invariants for the vertical 1-cells, while the level-2
parent invariants are given by a set of Chern numbers, one for each irrep on
the 1-cells of $\mathbb{D}_{2}$. The various Chern numbers satisfy a
compatibility constraint arising from the fact that taken together, the
2-cells form the boundary of the 3-cell.

The parent classification set on the 3-cell is identical to that obtained
without additional symmetries, i.e., the $\mathbb{Z}$-valued Hopf invariant
$\mathrm{H}$ only if $n_{\mathrm{o}}=n_{\mathrm{e}}=1$. Interestingly, it is
robust even if the vertical 2-cells possess nonzero Chern numbers, as shown in
App. E. This can alternatively be seen by extending the calculation of the
Hopf invariant to the full Brillouin zone, for which the ${\mathrm{C}}_{2}$ or
${\mathrm{C}}_{4}$ symmetry rules out a nonzero Chern number associated with
the vertical faces of the full Brillouin zone. However, if a cross section of
the fundamental domain $\mathbb{D}$ at constant $k_{z}$ has a nonzero Chern
number $\mathrm{C}$, then the Hopf invariant reduces to a
$\mathbb{Z}_{2\mathrm{C}}$-valued invariant, owing to a deformation along the
vertical faces that changes $\mathrm{H}$ by an even multiple of $\mathrm{C}$
(see App. E).

##  V ${\mathrm{C}}_{2}{\mathrm{T}}$ symmetry

As a first application with crystalline symmetries, we obtain the homotopic
classification of two- and three-dimensional band structures with only a
combined ${\mathrm{C}}_{2}{\mathrm{T}}$ symmetry. In two dimensions,
${\mathrm{C}}_{2}{\mathrm{T}}$-symmetric Bloch Hamiltonians are real and the
corresponding topological invariants are formulated in terms of the Euler
class and the Stiefel-Whitney invariants (see Sec. II.4.3).

A stable classification of ${\mathrm{C}}_{2}{\mathrm{T}}$-symmetric
topological insulators was first given by Fang and Fu [34]. Delicate and
fragile classifications for the two-dimensional case were obtained by Kennedy
and Guggenheim [6] and by Bouhon, Bzdusek, and Slager [28]. These authors
showed that level-2 classification spaces for delicate and fragile topology
depends on the values of the level-1 invariants. Our discussion below shows
that the topological classification may also depend on the number of orbitals
present at each Wyckoff position. To expose this feature, we begin with the
analogous case in one dimension, where the combination of inversion
(${\mathrm{I}}$) and time-reversal imposes a reality condition on $H(k)$.

###  V.1 ${\mathrm{I}}{\mathrm{T}}$ symmetry in one-dimension

Figure 7:  Special Wyckoff positions (left) and fundamental domain of a one-
dimensional ${\mathrm{I}}{\mathrm{T}}$ symmetric band structure.

####  V.1.1 Wyckoff positions and the fundamental domain

A lattice with ${\mathrm{I}}{\mathrm{T}}$ symmetry has two special Wyckoff
positions “a” and “b” at $x=0$ and $x=1/2$ (shown in Fig. 7) as well as one
generic Wyckoff position pair “g” at $x=\pm a$. Denoting the number of
orbitals at Wyckoff position $\alpha$ by $n_{\alpha}$, the total number of
orbitals $n$ is

$n=n_{a}+n_{b}+2n_{g}.$ (27)

With a suitable choice of the basis orbitals at the Wyckoff positions a, b,
and g, the constraint on $H(\mathbf{k})$ imposed by ${\mathrm{I}}{\mathrm{T}}$
symmetry can be cast in a form that is the same for all Wyckoff positions,

$H(k)=H^{\ast}(k).$ (28)

The symmetry relation (5) for translations by reciprocal lattice vectors,
however, bears reference to the Wyckoff positions:

$H(k+2\pi)=\Lambda_{x}^{\dagger}H(k)\Lambda_{x},$ (29)

with

$\Lambda_{x}=\text{diag}\left\\{\openone_{n_{\mathrm{a}}},-\openone_{n_{\mathrm{b}}},\openone_{n_{g}}\otimes\mathrm{e}^{2\pi\mathrm{i}a\sigma_{2}}\right\\}.$
(30)

Since a pair of orbitals at the generic Wyckoff position can always be moved
to one of the special Wyckoff positions, we may without loss of generality
assume that all orbitals are at the Wyckoff positions a and b.

In reciprocal space, the fundamental domain $\mathbb{D}$ consists of (two
copies of) one 0-cell ${\mathrm{M}}$, corresponding to $k=\pi$, and one 1-cell
${\mathrm{M}}{\mathrm{M}}^{\prime}$, corresponding to $-\pi<k<\pi$, as
depicted in Fig. 7. The Hamiltonians at ${\mathrm{M}}$ and
${\mathrm{M}}^{\prime}$ are related by Eq. (29).

| Condition(s) | Invariants
---|---|---
$n_{\mathrm{o}}=n_{\mathrm{e}}=1$ | Identical WPs | $\mathrm{Eu}_{1}\in\mathbb{Z}$
Distinct WPs | $\mathrm{Eu}_{1}\\!\mod 2\in\mathbb{Z}_{2}$
$n_{\mathrm{o}}+n_{\mathrm{e}}>2$ | — | $\mathrm{SW}_{1}\in\mathbb{Z}_{2}$
Table 4: Homotopic classification of gapped band structures with
${\mathrm{I}}{\mathrm{T}}$ symmetry in one dimension. The classification for
$n_{\mathrm{o}}=n_{\mathrm{e}}=1$ depends on the Wyckoff positions (WPs) in
the lattice model.

####  V.1.2 Homotopic classification

The level-0 invariants are given by the number of occupied/empty bands
$n_{\mathrm{o}/\mathrm{e}}$. For fixed level-0 invariants, the target space
for all $p$-cells is

$\mathcal{X}=\mathrm{Gr}_{\mathbb{R}}(n_{\mathrm{o}},n_{\mathrm{e}}).$ (31)

At level-1, the parent classification set is $K_{1}=\mathbb{Z}$ when
$n_{\mathrm{o}}=n_{\mathrm{e}}=1$ and $\mathbb{Z}_{2}$ otherwise, the
corresponding topological invariants being the first Euler number
$\mathrm{Eu}_{1}$ and the first Stiefel-Whitney invariant $\mathrm{SW}_{1}$,
respectively (see Sec. II.4 for more details).

The parent classification set for the boundary deformations $K_{1}^{\partial}$
consists of a copy of $\pi_{1}[\mathcal{X}]\cong K_{1}$ for each end point
$k=\pm\pi$. The deformations at these two points are related by Eq. (29) as

$H_{\partial}(-\pi;t)=\Lambda_{x}H_{\partial}(\pi,t)\Lambda_{x}^{\dagger}.$
(32)

Since $\Lambda_{x}$ depends on the Wyckoff positions occupied, so does the
effect of deformations on the 1d invariants. We now discuss the effect of
boundary deformations on the Euler and Stiefel-Whitney invariant separately.

_First Euler invariant —_ In this case, $n_{\mathrm{e}}=n_{\mathrm{o}}=1$ and
$\Lambda_{x}$ is proportional to $\sigma^{0}$ or $\sigma^{3}$, depending on
whether the two Wyckoff positions involved are identical or distinct. In the
former case, the windings of $H_{\partial}(\pm\pi;t)$ are equal and opposite
and thus cancel each other, rendering $\mathrm{Eu}_{1}$ a robust topological
invariant. In the latter case, the winding numbers associated with
$H_{\partial}(\pm\pi;t)$ add up, so that the boundary deformation can add an
even integer to $\mathrm{Eu}_{1}$. In this case, the true topological
invariant is $\mathbb{Z}_{2}$ valued and is given by $\mathrm{Eu}_{1}\\!\mod
2$.

Interestingly, in the latter case, the two topological classes are both
homotopic to atomic limits, i.e., Bloch Hamiltonians in either of these
classes can be continuously deformed to $k$-independent Hamiltonians. To see
how an atomic limit Hamiltonian can be assigned a nontrivial Euler invariant,
we set $\Lambda_{x}=\sigma^{3}$ and (arbitrarily) take
$H_{\text{ref}}=-\sigma^{3}$, corresponding to Wyckoff position “a” being
occupied. With respect to this reference Hamiltonian, $H(k)=\sigma^{3}$ —
corresponding to Wyckoff position “b” being occupied — carries a nontrivial
level-1 invariant. This is precisely the invariant carried by the boundary
deformation, which must rotate $\sigma^{3}$ to $-\sigma^{3}$ in opposite
senses for $k=\pm\pi$.

| Bands | Additional condition(s) | Topological invariant
---|---|---|---
Level 1 | $n_{\mathrm{o}}=n_{\mathrm{e}}=1$ | Identical Wyckoff positions | $\mathrm{Eu}_{1,x},\mathrm{Eu}_{1,y}\in\mathbb{Z}$
Distinct Wyckoff positions | $\mathrm{SW}_{1}\in\mathbb{Z}_{2}$
$n_{\mathrm{o}}\geq 1$, $n_{\mathrm{e}}>1$ | — | $\mathrm{SW}_{1,x},\mathrm{SW}_{1,y}\in\mathbb{Z}$
or $n_{\mathrm{o}}>1$, $n_{\mathrm{e}}\geq 1$
Level 2 | $n_{\mathrm{o}}=n_{\mathrm{e}}=1$ | ——
$n_{\mathrm{o}}=1$, $n_{\mathrm{e}}=2$ | if ${\cal Q}^{\mathrm{e}}$ | $\mathrm{Eu}_{2}^{\mathrm{e}}\in 2\mathbb{N}$
if not ${\cal Q}^{\mathrm{e}}$ | $\mathrm{SW}_{2}^{\mathrm{e}}\in\mathbb{Z}_{2}$
$n_{\mathrm{o}}=1$, $n_{\mathrm{e}}>2$ | ——
$n_{\mathrm{o}}=n_{\mathrm{e}}=2$ | if ${\cal Q}^{\mathrm{o}}$ and ${\cal Q}^{\mathrm{e}}$ | $(\mathrm{Eu}_{2}^{\mathrm{o}},\mathrm{Eu}_{2}^{\mathrm{e}})\in\mathbb{N}\times\mathbb{Z}|_{\rm p}$
if ${\cal Q}^{\mathrm{o}}$, but not ${\cal Q}^{\mathrm{e}}$ | $\mathrm{Eu}_{2}^{\mathrm{o}}\in\mathbb{N}$
if ${\cal Q}^{\mathrm{e}}$, but not ${\cal Q}^{\mathrm{o}}$ | $\mathrm{Eu}_{2}^{\mathrm{e}}\in\mathbb{N}$
if not ${\cal Q}^{\mathrm{o}}$ and not ${\cal Q}^{\mathrm{e}}$ | $\mathrm{SW}_{2}\in\mathbb{Z}_{2}$
$n_{\mathrm{o}}=2$, $n_{\mathrm{e}}>2$ | if ${\cal Q}^{\mathrm{o}}$ | $\mathrm{Eu}_{2}^{\mathrm{o}}\in\mathbb{N}$
if not ${\cal Q}^{\mathrm{o}}$ | $\mathrm{SW}_{2}\in\mathbb{Z}_{2}$
|  $n_{\mathrm{o}}$, $n_{\mathrm{e}}>2$ | — | $\mathrm{SW}_{2}\in\mathbb{Z}_{2}$
Table 5: Homotopic classification of gapped band structures with
${\mathrm{C}}_{2}{\mathrm{T}}$ symmetry in two dimensions. The classification
for $n_{\mathrm{o}}=n_{\mathrm{e}}=1$ depends on the Wyckoff positions of the
two sites. The symbol ${\cal Q}^{\mathrm{o}/\mathrm{e}}$ refers to the
condition ${\cal Q}^{a,\mathrm{o}/\mathrm{e}}_{2}$ of Eq. (36) being met for
$a=x$ and for $a=y$. The set $\mathbb{N}\times\mathbb{Z}_{\rm p}$ consists of
all ordered integer pairs $(p,q)$ with $p\geq 0$ and $p=q\mod 2$.

_First Stiefel-Whitney invariant —_ This invariant is given by
$\mathrm{SW}_{1}={\rm sgn}\,\large(\det{W}^{\mathrm{o}/\mathrm{e}}\large)$,
see Eq (15). The Wilson lines associated with the deformations at $\pm\pi$ are
related as

${W}^{\mathrm{o}/\mathrm{e}}_{\partial}(\pi)=\Lambda_{x}^{\mathrm{o}/\mathrm{e}}{W}_{\partial}^{\mathrm{o}/\mathrm{e}}(-\pi)\Lambda_{x}^{\mathrm{o}/\mathrm{e}\dagger}.$
(33)

where $\Lambda_{x}^{\mathrm{o}/\mathrm{e}}$ denotes the projection of
$\Lambda_{x}$ onto the occupied/empty subspaces. Thus,

$\det{W}_{\partial}^{\mathrm{o}/\mathrm{e}}(\pi)=\det{W}_{\partial}^{\mathrm{o}/\mathrm{e}}(-\pi)\in\\{\pm
1\\},$

so that $\det W^{\mathrm{o}/\mathrm{e}}$ for the entire loop remains unchanged
under boundary deformation. The first Stiefel-Whitney invariant is thus robust
under boundary deformations.

These classification results are summarized in Tab. 4.

###  V.2 ${\mathrm{C}}_{2}{\mathrm{T}}$ symmetry in two dimensions

####  V.2.1 Wyckoff positions and the fundamental domain

In two dimensions, there are four special Wyckoff positions “a”, “b”, “c”, and
“d”, as shown in Fig. 8, and one pair of generic Wyckoff positions “g” at
coordinates $(a_{x},a_{y})$. Denoting the number of orbitals at the Wyckoff
position $\alpha$ by $n_{\alpha}$, the total number of orbitals is

$n=n_{\mathrm{a}}+n_{\mathrm{b}}+n_{\mathrm{c}}+n_{\mathrm{d}}+2n_{\mathrm{g}}.$
(34)

We choose the basis vectors that are invariant under
${\mathrm{C}}_{2}{\mathrm{T}}$ (see Appendix A.2), so that the Bloch
Hamiltonian satisfies $H^{\ast}(\mathbf{k})=H(\mathbf{k})$. The reciprocal-
space group $G_{\Lambda}$ is generated by the two reciprocal lattice
translations. We denote the corresponding representation matrices (see Eq.
(4)) by $\Lambda_{x/y}$, which are explicitly given by

$\displaystyle\Lambda_{x}$
$\displaystyle=\text{diag}\left\\{\openone_{n_{\mathrm{a}}},\openone_{n_{\mathrm{b}}},-\openone_{n_{\mathrm{c}}},-\openone_{n_{\mathrm{d}}},\openone_{n_{g}}\mathrm{e}^{2\pi\mathrm{i}a_{x}\sigma_{2}}\right\\},$
$\displaystyle\Lambda_{y}$
$\displaystyle=\text{diag}\left\\{\openone_{n_{\mathrm{a}}},-\openone_{n_{\mathrm{b}}},-\openone_{n_{\mathrm{c}}},\openone_{n_{\mathrm{d}}},\openone_{n_{g}}\mathrm{e}^{2\pi\mathrm{i}a_{y}\sigma_{2}}\right\\}.$
(35)

For the purpose of classifying topological phases, we may continuously shift
any orbitals at the Wyckoff position “g” to the special Wyckoff position “a”
and use the corresponding expressions for $\Lambda_{x/y}$. The fundamental
domain is the same as symmetry class A, as depicted in Fig. 4.

####  V.2.2 Homotopic classification

The level-0 invariants are given by the number of occupied/empty bands
$n_{\mathrm{o}/\mathrm{e}}$. The target space for all cells is again the real
Grassmannian
$\mathcal{X}=\mathrm{Gr}_{\mathbb{R}}(n_{\mathrm{o}},n_{\mathrm{e}})$. The
level-1 classification is similar to the one-dimensional case discussed above:
each of the two 1-cells contribute the $\mathbb{Z}$-valued first Euler classes
$\mathrm{Eu}_{1,x/y}$ if $n_{\mathrm{o}}=n_{\mathrm{e}}=1$ and
$\mathbb{Z}_{2}$-valued Stiefel-Whitney invariants $\mathrm{SW}_{1,x/y}$
otherwise. The latter are robust to boundary deformations, while the former
are robust only if both orbitals lie at Wyckoff position with the same $x/y$
coordinate, and reduce to a $\mathbb{Z}_{2}$ invariant otherwise.

The level-1 invariants are further constrained by compatibility relations.
Since the boundary of the 2-cell consists of two copies of each 1-cell, the
$\mathbb{Z}_{2}$-valued $\mathrm{SW}_{1}$ always add up to $0$, so that there
are no additional restrictions on $\mathrm{SW}_{1,x/y}$. There are nontrivial
compatibility constraints on $\mathrm{Eu}_{1,x/y}$, however, that depend on
the reciprocal-space translation operators $\Lambda_{x/y}$. If the two
orbitals are at the same Wyckoff position, such that
$\Lambda_{x/y}\propto\sigma^{0}$, then there are no additional constraints on
$\mathrm{Eu}_{1,x/y}$, so that the level-1 classification set is
$\mathbb{Z}^{2}$. If only one of the translation operators ($\Lambda_{x}$,
say) is proportional to $\sigma^{3}$, then $\mathrm{Eu}_{1,x}$ is only defined
modulo 2, while $\mathrm{Eu}_{1,y}$ adds up for the two copies of the 1-cell
along $y$, so that the compatibility constraint requires
$2\,\mathrm{Eu}_{1,y}=0$. This leaves only one $\mathbb{Z}_{2}$-valued
invariant, _viz_ , $\mathrm{SW}_{1}\equiv\mathrm{Eu}_{1,x}\mod 2$. On the
other hand, if both $\Lambda_{x/y}\propto\sigma^{3}$ (such as when the two
orbitals are at Wyckoff positions “a” and “d”), then the compatibility
condition reads

$2\,\mathrm{Eu}_{1,x}-2\,\mathrm{Eu}_{1,y}=0\implies\mathrm{Eu}_{1,x}=\mathrm{Eu}_{1,y}\equiv\mathrm{Eu}_{1}.$

Furthermore, the boundary deformations (described for the 1d case)
simultaneously changes both $\mathrm{Eu}_{1,x}$ and $\mathrm{Eu}_{1,y}$ by an
even integer. Thus, we are again left with only one $\mathbb{Z}_{2}$-valued
invariant $\mathrm{Eu}_{1}$.

At level-2, $K_{2}=\pi_{2}[\mathcal{X}]$ depends on
$n_{\mathrm{o}/\mathrm{e}}$ (see Tab. 2), and the corresponding invariants can
be derived from the winding number $\mathrm{Eu}_{2}^{\mathrm{o}/\mathrm{e}}$
of a family of Wilson loops. These are the $\mathbb{Z}$-valued second Euler
invariant (up to a parity constraint, see Tab. 2) when
$n_{\mathrm{o}/\mathrm{e}}=2$, while for $n_{\mathrm{o}/\mathrm{e}}>2$, we get
the $\mathbb{Z}_{2}$-valued second Stiefel-Whitney invariant
$\mathrm{SW}_{2}\equiv\mathrm{Eu}_{2}^{\mathrm{o}/\mathrm{e}}\mod 2$. For the
boundary deformations, $K_{2}^{\partial}$ consists of a copy of
$\pi_{1}[\mathcal{X}]$ for the 0-cell and $\pi_{2}[\mathcal{X}]\cong K_{2}$
for each of the 1-cells. We now discuss their effect on the Euler and Stiefel-
Whitney invariant separately.

_Second Euler invariant —_ Since these invariants are defined only when
$n_{\mathrm{o}}=2$ or $n_{\mathrm{e}}=2$, the contribution of the 0-cell to
$K_{2}^{\partial}$ is $\pi_{1}[\mathcal{X}]=\mathbb{Z}_{2}$. We first consider
deformations that are trivial on the 0-cell. Nontrivial deformation along each
1-cell can carry a second Euler invariant given by the winding number of the
Wilson loops in $\mathrm{SO}(2)$ (see Sec. II.4.3). However, each 1-cell
occurs twice in $\partial\mathbb{D}$, and the corresponding windings cancel
each other, unless the orientation of the basis states is changed while going
from one copy of the 1-cell to the other. This change of orientation can
originate from either a reciprocal space translation operator or a nontrivial
first Stiefel-Whitney invariant. Explicitly, denoting the 1-cells along $x$
and $y$ by $\alpha_{x}$ and $\alpha_{y}$, respectively, the windings on two
copies of $\alpha_{x}$ cancel if either $\det\Lambda_{y}=-1$ or
$\mathrm{SW}_{1,y}=1$. Thus, $\mathrm{Eu}_{2}$ is robust to boundary
deformations only if

${\cal Q}_{2,a}^{\mathrm{o}/\mathrm{e}}:\ (-1)^{{\rm
SW}_{1,a}}\det\Lambda_{a}^{\mathrm{o}/\mathrm{e}}=1.$ (36)

for $a=x,y$. Note that while $\mathrm{SW}_{1,a}$ and
$\det\Lambda_{x/y}^{\mathrm{o}/\mathrm{e}}$ both depend on the choice of the
reference Hamiltonian $H_{\rm ref}$, this condition does not. If this
condition is obeyed, then $\mathrm{Eu}_{2}^{\mathrm{o}/\mathrm{e}}$ is robust
to boundary deformations, while if it is violated, the boundary deformations
can change it by even integers, rendering the level-2 invariant
$\mathbb{Z}_{2}$-valued.

A boundary deformation that is nontrivial at a 0-cell changes the sign of
$\mathrm{Eu}_{2}^{\mathrm{o}}$ and $\mathrm{Eu}_{2}^{\mathrm{e}}$
simultaneously, since it changes the orientation of the bases for both the
occupied and the unoccupied states [28].

_Second Stiefel-Whitney invariant —_ These are robust to boundary
deformations, since the effect of the latter always cancels out modulo 2.

The classification results are summarized in Tab. 5. They are consistent with
the conclusions of Ref. [28], which considered the case
$\Lambda_{x/y}=\openone$, corresponding to all orbitals being at the same
Wyckoff position. In the stable limit, there are two $\mathbb{Z}_{2}$-valued
level-1 invariants and one $\mathbb{Z}_{2}$-valued level-2 invariant.

###  V.3 ${\mathrm{C}}_{2}{\mathrm{T}}$ symmetry in three dimensions

In three dimensions, ${\mathrm{C}}_{2}{\mathrm{T}}$ symmetric Hamiltonians
satisfy Eq. (26). The fundamental domain is a cuboid with the top/bottom faces
(at $k_{z}=0,\pi$) formed by copies of the 2d fundamental domain, and the side
faces formed by copies of two distinct 2-cells.

####  V.3.1 Homotopic classification

The level-1 invariants are two independent copies of the invariants obtained
for the 2d classification. As discussed in Sec. IV, there are no topological
invariants at the 1-cells parallel to $k_{z}$.

The level-2 invariants for the $k_{z}=0,\pi$ planes are copies of those
obtained for the 2d classification. For the two distinct faces parallel to the
$k_{z}$-axis, the parent topological invariants are the Chern numbers
$\mathrm{C}_{x/y}$. There are two kinds of boundary deformations for these
invariants: those along the 1-cells parallel to the $k_{z}$-axis and those
perpendicular to it. The former cannot change the Chern number, since the two
vertical boundaries of each face are copies of the the same 1-cell, so that a
deformation with Chern number $\mathrm{C}$ on one copy of the 1-cell is
exactly canceled by the other. The latter deformations cannot carry a Chern
number, since the Hamiltonian is constrained to be real at the boundary. Thus,
the Chern numbers $\mathrm{C}_{x/y}$ are robust topological invariants. There
are no additional constraints on them from level-3 topology, since each of
these 2-cells occurs twice in the boundary of the 3-cell, so that the total
Chern number always vanishes.

The $\mathbb{Z}$-valued $\mathrm{C}_{x/y}$ can be combined with the
$\mathbb{Z}_{2}$ valued difference between $\mathrm{SW}_{1,x/y}$ computed at
$k_{z}=0$ and $\pi$ to get a $\mathbb{Z}$-valued invariant
$\mathrm{C}^{\text{full}}_{x/y}$, which is simply the Chern number computed on
the $k_{x}/k_{y}$ plane for the full range $k_{z}\in(-\pi,\pi)$. This is
because the symmetry condition
$H(k_{x},k_{y},k_{z})=H^{\ast}(k_{x},k_{y},-k_{z})$ implies that the Chern
numbers for $k_{z}\in(-\pi,0)$ and $(0,\pi)$ add up to give even integers.
Furthermore, a difference between $\mathrm{SW}_{1,x/y}$ at $k_{z}=0$ and $\pi$
implies that the Wilson loop along $k_{x}/k_{y}$ changes from $0$ to $\pi$
along $k_{z}$, which, combined with the symmetry of Wilson loops under time
reversal, implies that the Chern number computed for $k_{z}\in(-\pi,\pi)$ is
unity (see Appendix F.1 for a similar argument). Combining this with the even
total Chern number yields two $\mathbb{Z}$-valued invariants
$\mathrm{C}^{\text{full}}_{x/y}$, which represent weak Chern insulator phases
protected by discrete translation symmetry in the $x$ and $y$ direction,
respectively.

At level-3, we get the Hopf invariant when $n_{\mathrm{o}}=n_{\mathrm{e}}=1$.
This is not affected by deformations, since the Bloch Hamiltonian is real for
$k_{z}=0$, $\pi$ and thus cannot carry a Chern number (see Sec. IV for
details).

The classification results are summarized in Tab. 6.

| Bands | Topological invariant
---|---|---
Level 1 | — | Two copies of invariants
from Tab. 5
Level 2 | — |  $\mathrm{C}_{x}$, $\mathrm{C}_{y}\in\mathbb{Z}$ and two copies
of invariants from Tab. 5
Level-3 | $n_{\mathrm{o}}=n_{\mathrm{e}}=1$ | $\mathrm{H}\in\mathbb{Z}$
$n_{\mathrm{o}}+n_{\mathrm{e}}>2$ | —
Table 6: Homotopic classification of gapped band structures with
${\mathrm{C}}_{2}{\mathrm{T}}$ symmetry in three dimensions.

####  V.3.2 Anomalous boundary states

The presence of stable topological indices for a 3d band structure with
${\mathrm{C}}_{2}{\mathrm{T}}$ symmetry implies the existence of anomalous
boundary states on boundaries invariant under ${\mathrm{C}}_{2}$, i.e., a
surface parallel to the $xy$ plane [34]. The stable topological invariants are
the $\mathbb{Z}$-valued $\mathrm{C}_{x,y}^{\text{full}}$ associated with the
vertical 2-cells and the $\mathbb{Z}_{2}$-valued difference between
$\mathrm{SW}_{2}$ associated with the planes $k_{z}=0,\pi$ 888 A band
structure with identical invariants for $k_{z}=0,\pi$ and
$\mathrm{C}_{x}^{\text{full}}=\mathrm{C}_{y}^{\text{full}}=0$ is a weak
topological phase that is protected by the discrete translation symmetry in
the $z$ direction. As a surface parallel to the $xy$ plane breaks this
symmetry, we do not generically get surface modes. . The former lead to
anomalous chiral states at the surface that propagate along $x$ and $y$,
respectively, while the latter leads to a surface Dirac cone.

We now derive these anomalous states from the boundary perspective. Following
the argument for symmetry class A (Sec. III.3), a chiral state violates the
level-0 compatibility condition as well as the gluing condition and is thus
anomalous. To see that a surface Dirac cone is anomalous, one first observes
that a closed path $\gamma$ around a single Dirac cone has Berry phase $\pi$,
so that $\det{W}^{\mathrm{o}/\mathrm{e}}(\gamma)=-1$ [34]. On the other hand,
for any set of $\mathbb{Z}_{2}$-valued level-1 invariants
$\mathrm{SW}_{1,x/y}$, the level-1 compatibility condition arising from the
triviality of the Wilson loop
$\det{W}^{\mathrm{o}/\mathrm{e}}(\partial\mathbb{D})=1$ is trivially
satisfied, since each 1-cell occurs twice in $\partial\mathbb{D}$ and the
corresponding invariants add up to $0\mod 2$. The two copies of a 1-cell have
identical $\mathrm{SW}_{1}$, since the Hamiltonians at them are related by a
unitary transformation given by $\Lambda_{x,y}$. As a Dirac cone manifestly
violates the level-1 compatibility condition, it can be realized only if
$\mathrm{SW}_{1}$ is different for the two copies of a 1-cell, i.e., if the
the gluing condition is violated. This implies that a single surface Dirac
cone must be anomalous.

##  VI ${\mathrm{C}}_{2}$ symmetry

A classification of ${\mathrm{C}}_{2}$-symmetric topological insulators in the
stable limit was previously obtained in Refs. [22, 23, 76, 24, 25]. A partial
fragile classification, based on level-0 invariants only, was obtaied using
the method of topological quantum chemistry [54]. Delicate topological
invariants at levels 1 and 2 were constructed by Kobayashi and Furusaki [31],
who also identified representation-protected phases. The full homotopic
classification we present below rederives and completes these results from the
literature in a consistent mathematical framework.

###  VI.1 Wyckoff positions, basis orbitals, and symmetries

With twofold rotation symmetry and in two dimensions, there are four special
Wyckoff positions “a”, “b”, “c”, and “d” (shown in Fig. 8) and one pair of
generic Wyckoff positions “g”. At each special Wyckoff position, electron
orbitals are even ($+$) or odd ($-$) under twofold rotation. Combining
orbitals at the two generic Wyckoff positions, one even ($+$) and one odd
($-$) orbital may be obtained. Denoting the number of orbitals of parity $\pm$
by $n_{\pm}$, the total number of orbitals is

$\displaystyle n=$
$\displaystyle\,n_{\mathrm{a}}+n_{\mathrm{b}}+n_{\mathrm{c}}+n_{\mathrm{d}}+2n_{\mathrm{g}}=n_{+}+n_{-}.$
(37)

The reciprocal-space group $G_{\Lambda}$ is generated by twofold rotation
$g_{2}$ and two reciprocal-space translations $g_{x}$ and $g_{y}$. We denote
the corresponding representation matrices (see Eq. (4)) by $C_{2}$,
$\Lambda_{x}$, and $\Lambda_{y}$, respectively. The Bloch Hamiltonian
satisfies

$\displaystyle H(k_{x},k_{y})=C_{2}H(-k_{x},-k_{y})C_{2}^{\dagger}$ (38)

as well as Eq. (5) for the reciprocal-space translations $g_{x,y}$. Since the
matrices $C_{2}$, $\Lambda_{x}$, and $\Lambda_{y}$ generate a representation
of $G_{\Lambda}$, they satisfy

$C_{2}\Lambda_{x,y}=\Lambda_{x,y}^{\dagger}C_{2}^{\phantom{\dagger}}.$ (39)

We choose the over-all phase factor of the twofold rotation operator $C_{2}$
such that $C_{2}^{2}=\openone$. Hence, in its eigenbasis, it can be written
explicitly as

$C_{2}=\text{diag}\\{\openone_{n_{+}},-\openone_{n_{-}}\\}.$ (40)

Explicit expressions for $\Lambda_{x,y}$ are derived in Appendix B.1.

Figure 8:  (Left) The Wyckoff positions for a square lattice with
${\mathrm{C}}_{2}$ symmetry. (Right) The fundamental domain $\mathbb{D}$ for
two-dimensional band structures with ${\mathrm{C}}_{2}$ symmetry.

We take the fundamental domain $\mathbb{D}$ as the right half of the Brillouin
zone, as shown in Fig. 8. It consists of four 0-cells, three 1-cells and one
2-cell. The 0-cells all have a nontrivial little group $G_{S}=\\{0,g_{S}\\}$,
with $g_{\Gamma}=g_{2}$ a twofold rotation, whereas $g_{{\mathrm{X}}}$,
$g_{{\mathrm{Y}}}$, and $g_{{\mathrm{M}}}$ are combinations of a twofold
rotation and a reciprocal lattice translation. The corresponding
representation matrices $C_{S}\equiv C(g_{S})$ are

$\displaystyle C_{\Gamma}$ $\displaystyle=C_{2},$ $\displaystyle
C_{{\mathrm{X}},{\mathrm{Y}}}$
$\displaystyle=\Lambda_{x,y}^{\dagger}C_{2}=C_{2}\Lambda_{x,y},$
$\displaystyle C_{\mathrm{M}}$
$\displaystyle=\Lambda_{x}^{\dagger}\Lambda_{y}^{\dagger}C_{2}=C_{2}\Lambda_{x}\Lambda_{y}.$
(41)

These matrices all square to $\openone$ and thus have eigenvalues $\pm 1$. The
eigenstates of $H_{S}$ are thus labeled by the $C_{S}$-eigenvalue $\pm 1$. The
four matrices $C_{S}$ are not independent, since they satisfy

$C_{\Gamma}C_{{\mathrm{X}}}C_{{\mathrm{M}}}C_{{\mathrm{Y}}}=\openone.$ (42)

The eigenvalues of $C_{S}$ for band structures with a minimal set of orbitals
at a single Wyckoff position constitute the EBRs [54, 77, 55]. These are
derived in Appendix B.2 and listed in Tab. 15.

The three 1-cells $\Gamma{\mathrm{Y}}$, ${\mathrm{Y}}{\mathrm{M}}$, and
${\mathrm{M}}{\mathrm{X}}$ each have two copies, which are related as

$\displaystyle H(0,-k)=$
$\displaystyle\,C_{\Gamma}^{\dagger}H(0,k)C_{\Gamma},$ $\displaystyle
H(\pi,-k)=$ $\displaystyle\,C_{X}^{\dagger}H(\pi,k)C_{X},$ $\displaystyle
H(k,\pi)=$ $\displaystyle\,\Lambda_{y}^{\dagger}H(k,-\pi)\Lambda_{y}.$ (43)

The last expression is independent of $C_{2}$, since the two copies of
${\mathrm{Y}}{\mathrm{M}}$ are related by a translation $2\pi\mathbf{e}_{y}$.
There are no local-in-$\mathbf{k}$ constraints on the Hamiltonian at the
1-cells or the 2-cell. If time-reversal symmetry is present, we choose a basis
where $C_{2}T=\openone$, so that the Hamiltonian satisfies the reality
condition $H(\mathbf{k})=H^{*}(\mathbf{k})$.

###  VI.2 Homotopic classification in two dimensions

####  VI.2.1 Without time-reversal (symmetry class A)

The level-$0$ invariants are given by the number of occupied/empty bands
$n_{\mathrm{o}/\mathrm{e},\pm}^{S}$ for each 0-cell $S$ with even/odd parity
under ${\mathrm{C}}_{2}$. These satisfy a “level-0 compatibility relation”
(see Eq. (7))

$n_{\mathrm{o}/\mathrm{e}}=\sum_{x=\pm}n_{\mathrm{o}/\mathrm{e},x}^{S}.$ (44)

Additionally, there are constraints on the level-0 invariants that follow from
the restrictions imposed on the representation matrices $C_{S}$. In
particular, from Eqn. (42),it follows that $\prod_{S}\det C_{S}=1$, so that

$\displaystyle\sum_{S}\left(n_{\mathrm{o},-}^{S}+n_{\mathrm{e},-}^{S}\right)=0\mod
2.$ (45)

This is sufficient to ensure that the level-0 invariants, summed over occupied
and unoccupied bands, constitute a linear combination of the EBRs listed in
Tab. 15.

For fixed level-0 invariants, the target spaces are

$\mathcal{X}_{S}=\mathrm{Gr}_{\mathbb{C}}(n_{\mathrm{o},+}^{S},n_{\mathrm{e},+}^{S})\times\mathrm{Gr}_{\mathbb{C}}(n_{\mathrm{o},-}^{S},n_{\mathrm{e},-}^{S}).$
(46)

for the 0-cells and
$\mathcal{X}=\mathrm{Gr}_{\mathbb{C}}(n_{\mathrm{o}},n_{\mathrm{e}})$ for the
1- and the 2-cell. The corresponding parent invariants are identical to those
obtained for class A (Sec. III), i.e., trivial at level-1 and a
$\mathbb{Z}$-valued Chern invariant $\mathrm{C}$ at level-2. Proceeding as in
Sec. III, one sees that the Chern invariant is robust to boundary
deformations, because the Chern number of a boundary deformation along a
1-cell enters through both its copies at the boundary of the 2-cell, which
precisely cancel each other. Thus, we obtain the Chern number
$\mathrm{C}\in\mathbb{Z}$ as the level-2 topological invariant.

These results are summarized in Tab. 7.

####  VI.2.2 With time-reversal (symmetry class AI)

The level-$0$ topological invariants are identical to the time-reversal-broken
case, but now subject to an additional constraint

$\displaystyle\sum_{S}n_{\mathrm{o},-}^{S}=0\mod 2,$ (47)

which arises from the level-1 topology (as we show below) and thus constitutes
a level-1 compatibility relation. Following Eq. (45), an identical constraint
also applies to the unoccupied bands. The target spaces now consist of real
Grassmannians, and are explicitly given for the 0-cells by

$\displaystyle\mathcal{X}_{S}=$
$\displaystyle\,\mathrm{Gr}_{\mathbb{R}}(n_{\mathrm{o},+}^{S},n_{\mathrm{e},+}^{S})\times\mathrm{Gr}_{\mathbb{R}}(n_{\mathrm{o},-}^{S},n_{\mathrm{e},-}^{S})$
(48)

and for the 1-cells and the 2-cell by
$\mathcal{X}=\mathrm{Gr}_{\mathbb{R}}(n_{\mathrm{o}},n_{\mathrm{e}})$. Thus,
the parent classification is similar to that of ${\mathrm{C}}_{2}{\mathrm{T}}$
discussed in Sec. V. We summarize our classification results in Tab. 7 and
briefly discuss their derivation here.

#### Classification on 1-cells

At level-1, the parent invariants for the 1-cell $SS^{\prime}$ are the
$\mathbb{Z}$-valued first Euler invariant
$\mathrm{Eu}_{1}^{SS^{\prime}}=-\mathrm{Eu}_{1}^{S^{\prime}S}$ if
$n_{\mathrm{o}}=n_{\mathrm{e}}=1$ and the $\mathbb{Z}_{2}$-valued first
Stiefel-Whitney number $\mathrm{SW}_{1}^{SS^{\prime}}$ otherwise. The parent
classification set for boundary deformations $K_{1}^{\partial}$ consists of a
factor $\pi_{1}[\mathcal{X}_{S}]$ for each 0-cell $S$. The boundary
deformations at $S$ do not have an effect on the level-1 invariants if
$\pi_{1}[\mathcal{X}_{S}]=0$, which happens if
$\mathrm{Gr}_{\mathbb{R}}(n_{\mathrm{o},a}^{S},n_{\mathrm{e},a}^{S})$
degenerates to a point for both $a=\pm$. In terms of the level-0 invariants,
this happens when

${\cal Q}_{1}^{S}:\
n_{\mathrm{o},+}^{S}n_{\mathrm{e},+}^{S}=n_{\mathrm{o},-}^{S}n_{\mathrm{e},-}^{S}=0.$
(49)

We now discuss the effect of boundary deformations on the Euler and Stiefel-
Whitney invariant separately.

_First Euler invariant —_ A boundary deformation at the 0-cell $S$ may add an
integer simultaneously to $\mathrm{Eu}_{1}^{SS^{\prime}}$ for all 1-cells with
$S$ as an end point if $\pi_{1}[\mathcal{X}_{S}]\neq 0$. Thus,
$\mathrm{Eu}_{1}^{SS^{\prime}}$ is robust to boundary deformations only if the
conditions ${\cal Q}^{S}_{1}$ and ${\cal Q}^{S^{\prime}}_{1}$ are satisfied,
see Eq. (49). If ${\cal Q}^{S}_{1}$ is obeyed at all four 0-cells, then all
three first Euler numbers $\mathrm{Eu}_{1}^{{\mathrm{X}}{\mathrm{M}}}$,
$\mathrm{Eu}_{1}^{{\mathrm{M}}{\mathrm{Y}}}$, and
$\mathrm{Eu}_{1}^{{\mathrm{Y}}\Gamma}$ are well defined. If this condition is
violated, then using $n_{\mathrm{o}/\mathrm{e},\pm}^{S}\in\\{0,1\\}$ and Eq.
(47), we conclude that it must be violated at either two or all four 0-cells.
In these cases, upon taking into account the ambiguity from boundary
deformations, we are left with one and no first Euler invariant, respectively.

The number of first Euler invariants are further reduced by the level-1
compatibility conditions. If ${\cal Q}_{1}^{S}$ is obeyed at all four 0-cells,
then one has the symmetry relations
$\mathrm{Eu}_{1}^{\Gamma{\mathrm{Y}}^{\prime}}=-\mathrm{Eu}_{1}^{\Gamma{\mathrm{Y}}}$,
$\mathrm{Eu}_{1}^{{\mathrm{Y}}^{\prime}{\mathrm{M}}^{\prime}}=-\mathrm{Eu}_{1}^{{\mathrm{Y}}{\mathrm{M}}}$,
and
$\mathrm{Eu}_{1}^{{\mathrm{M}}^{\prime}{\mathrm{X}}}=-\mathrm{Eu}_{1}^{{\mathrm{X}}{\mathrm{M}}}$.
Triviality of the first Euler invariant computed for the boundary of the
2-cell then gives the constraint

$\mathrm{Eu}_{1}^{{\mathrm{X}}{\mathrm{M}}}+\mathrm{Eu}_{1}^{{\mathrm{M}}{\mathrm{Y}}}+\mathrm{Eu}_{1}^{{\mathrm{Y}}\Gamma}=0,$
(50)

which effectively removes one level-1 invariant. There is a similar constraint
if ${\cal Q}_{1}^{S}$ is obeyed only at two of the 0-cells, which removes the
only level-1 invariant present in that case. Thus, there are two independent
first Euler invariants if ${\cal Q}_{1}^{S}$ is obeyed at all 0-cells $S$,
while there are none otherwise.

_First Stiefel-Whitney invariant —_ If a boundary deformation flips the
orientation at a 0-cell $S$, then it flips the sign of
$\det{W}^{\mathrm{o}/\mathrm{e}}_{SS^{\prime}}$ for all 1-cells having $S$ as
their end point. Such a boundary deformation is allowed if ${\cal Q}_{1}^{S}$
is violated, thereby effectively “removing” one level-1 invariant.

The level-1 compatibility condition follows from triviality of the Wilson loop
around the boundary of the 2-cell

${W}_{\partial\mathbb{D}}={W}_{\Gamma{\mathrm{Y}}^{\prime}}{W}_{{\mathrm{Y}}^{\prime}{\mathrm{M}}^{\prime}}{W}_{{\mathrm{M}}^{\prime}{\mathrm{X}}}{W}_{{\mathrm{X}}{\mathrm{M}}}{W}_{{\mathrm{M}}{\mathrm{Y}}}{W}_{{\mathrm{Y}}\Gamma}=\openone,$
(51)

where we have skipped the superscript $\mathrm{o}/\mathrm{e}$ to avoid
notational clutter. The Wilson loops at the symmetry-related 1-cells satisfy

$\displaystyle{W}^{\mathrm{o}/\mathrm{e}}_{\Gamma{\mathrm{Y}}^{\prime}}$
$\displaystyle=C^{\mathrm{o}/\mathrm{e}\dagger}_{\Gamma}{W}^{\mathrm{o}/\mathrm{e}}_{\Gamma{\mathrm{Y}}}C^{\mathrm{o}/\mathrm{e}}_{{\mathrm{Y}}},$
$\displaystyle{W}^{\mathrm{o}/\mathrm{e}}_{{\mathrm{Y}}^{\prime}{\mathrm{M}}^{\prime}}$
$\displaystyle={W}^{\mathrm{o}/\mathrm{e}}_{{\mathrm{Y}}{\mathrm{M}}},$
$\displaystyle{W}^{\mathrm{o}/\mathrm{e}}_{{\mathrm{M}}^{\prime}{\mathrm{X}}}$
$\displaystyle=C^{\mathrm{o}/\mathrm{e}\dagger}_{{\mathrm{M}}}{W}^{\mathrm{o}/\mathrm{e}}_{{\mathrm{M}}{\mathrm{X}}}C^{\mathrm{o}/\mathrm{e}}_{{\mathrm{X}}}.$
(52)

Using these and $\det{W}_{SS^{\prime}}\in\\{\pm 1\\}$, we get

$\displaystyle\det{W}^{\mathrm{o}/\mathrm{e}}_{\partial\mathbb{D}}$
$\displaystyle=\det\left(C^{\mathrm{o}/\mathrm{e}\dagger}_{\Gamma}C^{\mathrm{o}/\mathrm{e}}_{{\mathrm{Y}}}C^{\mathrm{o}/\mathrm{e}\dagger}_{{\mathrm{M}}}C^{\mathrm{o}/\mathrm{e}}_{{\mathrm{X}}}\right),$
(53)

so that the compatibility condition becomes

$1=\prod_{S}\det
C^{\mathrm{o}/\mathrm{e}}_{S}=\prod_{S}(-1)^{n_{\mathrm{o}/\mathrm{e},-}^{S}},$
(54)

which is equivalent to Eq. (47). Thus, the compatibility condition does not
constrain the level-1 invariants, but imposes a constraint on the level-0
invariants.

| Bands | Additional condition(s) | Topological invariant
---|---|---|---
A, Level 1 |  $n_{\mathrm{o}}$, $n_{\mathrm{e}}\geq 1$ | ——
A, Level 2 |  $n_{\mathrm{o}}$, $n_{\mathrm{e}}\geq 1$ | — |  $\mathrm{C}\in\mathbb{Z}$ [S]
AI, Level 1 | $n_{\mathrm{o}}=n_{\mathrm{e}}=1$ | if ${\cal Q}_{1}^{\land}$ |  $\mathrm{Eu}_{1}^{\Gamma{\mathrm{Y}}}$, $\mathrm{Eu}_{1}^{{\mathrm{Y}}{\mathrm{M}}}\in\mathbb{Z}$
if not ${\cal Q}_{1}^{\land}$ | —
$n_{\mathrm{o}}\geq 1$, $n_{\mathrm{e}}>1$ | — |  $\mathrm{SW}_{1}^{\Gamma{\mathrm{Y}}}$, $\mathrm{SW}_{1}^{{\mathrm{Y}}{\mathrm{M}}}$, $\mathrm{SW}_{1}^{{\mathrm{M}}{\mathrm{X}}}\in\mathbb{Z}_{2}$
or $n_{\mathrm{o}}>1$, $n_{\mathrm{e}}\geq 1$ | one less for every violation of ${\cal Q}_{1}^{S}$
AI, Level 2 | $n_{\mathrm{o}}=n_{\mathrm{e}}=1$ | ——
$n_{\mathrm{o}}=1$, $n_{\mathrm{e}}=2$ | if ${\cal Q}_{1}^{\lor}$ and ${\cal Q}_{2}^{\mathrm{e}}$ | $\mathrm{Eu}_{2}^{\mathrm{e}}\in 2\mathbb{Z}$
if ${\cal Q}_{2}^{\mathrm{e}}$, but not ${\cal Q}_{1}^{\lor}$ | $\mathrm{Eu}_{2}^{\mathrm{e}}\in 2\mathbb{N}$
if not ${\cal Q}_{2}^{\rm e}$ | $\mathrm{Eu}_{2}^{\mathrm{e}}\in\mathbb{Z}_{2}$
$n_{\mathrm{o}}=1$, $n_{\mathrm{e}}>2$ | ——
$n_{\mathrm{o}}=n_{\mathrm{e}}=2$ | if ${\cal Q}_{1}^{\lor}$ and ${\cal Q}_{2}^{\mathrm{e}}$ and ${\cal Q}_{2}^{\mathrm{o}}$ | $(\mathrm{Eu}_{2}^{\mathrm{o}},\mathrm{Eu}_{2}^{\mathrm{e}})\in\mathbb{Z}^{2}|_{\rm p}$
if ${\cal Q}_{2}^{\mathrm{e}}$ and ${\cal Q}_{2}^{\mathrm{o}}$, but not ${\cal Q}_{1}^{\lor}$ | $(\mathrm{Eu}_{2}^{\mathrm{o}},\mathrm{Eu}_{2}^{\mathrm{e}})\in\mathbb{N}\times\mathbb{Z}|_{\rm p}$
if ${\cal Q}_{1}^{\lor}$ and ${\cal Q}_{2}^{\mathrm{o}}$, but not ${\cal Q}_{2}^{\mathrm{e}}$ | $\mathrm{Eu}_{2}^{\mathrm{o}}\in\mathbb{Z}$
if ${\cal Q}_{1}^{\lor}$ and ${\cal Q}_{2}^{\mathrm{e}}$, but not ${\cal Q}_{2}^{\mathrm{o}}$ | $\mathrm{Eu}_{2}^{\mathrm{e}}\in\mathbb{Z}$
if ${\cal Q}_{2}^{\mathrm{o}}$, but not ${\cal Q}_{1}^{\lor}$ and not ${\cal Q}_{2}^{\mathrm{e}}$ | $\mathrm{Eu}_{2}^{\mathrm{o}}\in\mathbb{N}$
if ${\cal Q}_{2}^{\mathrm{e}}$, but not ${\cal Q}_{1}^{\lor}$ and not ${\cal Q}_{2}^{\mathrm{o}}$ | $\mathrm{Eu}_{2}^{\mathrm{e}}\in\mathbb{N}$
if ${\cal Q}_{2}^{\land}$, but not ${\cal Q}_{2}^{\mathrm{e}}$ and not ${\cal Q}_{2}^{\mathrm{o}}$ | $\mathrm{SW}_{2}\in\mathbb{Z}_{2}$
if not ${\cal Q}_{2}^{\land}$ | —
$n_{\mathrm{o}}=2$, $n_{\mathrm{e}}>2$ | if ${\cal Q}_{1}^{\lor}$ and ${\cal Q}_{2}^{\mathrm{o}}$ | $\mathrm{Eu}_{2}^{\mathrm{o}}\in\mathbb{Z}$
if ${\cal Q}_{2}^{\mathrm{o}}$, but not ${\cal Q}_{1}^{\lor}$ |  $\mathrm{Eu}_{2}^{\mathrm{o}}\in\mathbb{N}$ [F]
if ${\cal Q}_{2}^{\land}$, but not ${\cal Q}_{2}^{\mathrm{o}}$ | $\mathrm{SW}_{2}\in\mathbb{Z}_{2}$
if not ${\cal Q}_{2}^{\land}$ | —
| $n_{\mathrm{o}}$, $n_{\mathrm{e}}>2$ | if ${\cal Q}_{2}^{\land}$ |  $\mathrm{SW}_{2}\in\mathbb{Z}_{2}$ [F]
| if not ${\cal Q}_{2}^{\land}$ | —
Table 7: Homotopic classification of gapped band structures with
${\mathrm{C}}_{2}$ symmetry in two dimensions for various possible
combinations of number of bands and level 0 invariants, the latter encoded in
the ${\cal Q}$’s. In particular, ${\cal Q}_{1}^{S}$, ${\cal
Q}_{2}^{\mathrm{o}/\mathrm{e}}$, ${\cal Q}_{2}^{S}$, and ${\cal
Q}_{2}^{\land}$ are defined by Eqs. (49), (55), (56), and (57), respectively.
The condition ${\cal Q}_{1}^{\land}$ holds if ${\cal Q}_{1}^{S}\,\forall S$,
while ${\cal Q}_{1}^{\lor}$ holds if $\exists S$ where ${\cal Q}_{1}^{S}$ is
true. The set $\mathbb{Z}^{2}|_{\rm p}$ consists of all ordered integer pairs
$(p,q)$ with $p=q\mod 2$, while the set $\mathbb{N}\times\mathbb{Z}_{\rm p}$
consists of all ordered integer pairs $(p,q)$ with $p\geq 0$ and $p=q\mod 2$.
The table lists the cases $n_{\mathrm{o}}\leq n_{\mathrm{e}}$ only. The
remaining classifications with $n_{\mathrm{o}}>n_{\mathrm{e}}$ can be obtained
from the entries in the Table by exchanging the labels “$\mathrm{o}$” and
“$\mathrm{e}$”. The fragile and stable topological phases are marked by [F]
and [S], respectively.

#### Classification on the 2-cell

At level-2, the the parent classification set is given by
$K_{2}=\pi_{2}[\mathcal{X}]$, with the corresponding invariants being the
$\mathbb{Z}$-valued second Euler invariant or the $\mathbb{Z}_{2}$-valued
second Stiefel-Whitney invariant, depending on $n_{\mathrm{o}/\mathrm{e}}$
(see Table 2). The parent classification set for boundary deformations
$K_{2}^{\partial}$ consists of a factor of $\pi_{2}[\mathcal{X}]\cong K_{2}$
for each 1-cell and of $\pi_{1}[\mathcal{X}_{S}]$ for each 0-cell, which
consists of two factors corresponding to the two parity sectors (see Eq.
(48)). We now discuss the effect of boundary deformations on the Euler and
Stiefel-Whitney invariant separately.

_Second Euler invariant —_ For the occupied/empty subspaces,
$\mathrm{Eu}_{2}^{\mathrm{o}/\mathrm{e}}$ is defined if
$n_{\mathrm{o}/\mathrm{e}}=2$, whereby it is $\mathbb{Z}$-valued if
$n_{\mathrm{e}/\mathrm{o}}\geq 2$ and $2\mathbb{Z}$-valued if
$n_{\mathrm{e}/\mathrm{o}}=1$. For the former case, boundary deformations that
are nontrivial along a 1-cell (i.e., in $\pi_{2}[\mathcal{X}]\cong K_{2}$) can
add an arbitrary integer to $\mathrm{Eu}_{2}^{\mathrm{o}/\mathrm{e}}$.
However, as each 1-cell occurs twice in $\partial\mathbb{D}$, following the
argument for ${\mathrm{C}}_{2}{\mathrm{T}}$ (Sec. V), these individual
contributions cancel out unless the orientation of the occupied/empty subspace
is flipped between the two copies of the 1-cell. From Eq. (52), this can
happen if $\det C_{S}^{\mathrm{o}/\mathrm{e}}=-1$ for any of the 0-cells.
Thus, the two boundary contributions cancel if $\det
C_{S}^{\mathrm{o}/\mathrm{e}}=1$ for all $S$, or, equivalently, if

${\cal Q}_{2}^{\mathrm{o}/\mathrm{e}}:\
n^{S}_{\mathrm{o}/\mathrm{e},+}n^{S}_{\mathrm{o}/\mathrm{e},-}=0\ \mbox{for
all $S$}.$ (55)

If this condition is violated, then the boundary deformations at the two
copies of the 1-cell add up. For $n_{\mathrm{o}/\mathrm{e}}=2$ and
$n_{\mathrm{e}/\mathrm{o}}\geq 2$, such a boundary deformation may add an even
integer to $\mathrm{Eu}^{\mathrm{o}/\mathrm{e}}$, rendering the classification
group $\mathbb{Z}/2\mathbb{Z}\cong\mathbb{Z}_{2}$. On the other hand, for
$n_{\mathrm{o}/\mathrm{e}}=2$ and $n_{\mathrm{e}/\mathrm{o}}=1$, such boundary
deformations may add an arbitrary multiple of four to
$\mathrm{Eu}_{2}^{\mathrm{o}/\mathrm{e}}$, so that the classification group
becomes $2\mathbb{Z}/4\mathbb{Z}=\mathbb{Z}_{2}$.

Next, we consider boundary deformations that are nontrivial at a 0-cell $S$,
which may be nontrivial in either one or both factors of
$\pi_{1}[\mathcal{X}_{S}]$. In the former case, boundary deformations can flip
the orientation of the basis of the occupied/empty subspace for one of the
parity sectors at $S$, and hence of that subspace also away from $S$. By
continuity, such a deformation must be nontrivial at each 0-cell, so that it
cannot exist if $\pi_{1}[\mathcal{X}_{S}]=0$ for at least one 0-cell, i.e., if
$\exists S$ for which ${\cal Q}_{1}^{S}$ holds. If that is not the case, then
such a boundary deformation changes the sign of both
$\mathrm{Eu}_{2}^{\mathrm{o}}$ and $\mathrm{Eu}_{2}^{\mathrm{e}}$, replacing
the level-2 classifying spaces $\mathbb{Z}$ by $\mathbb{N}$ and
$\mathbb{Z}^{2}|_{\rm p}$ by $\mathbb{N}\times\mathbb{Z}|_{\rm p}$.

Boundary deformations with nontrivial topology in both factors of
$\pi_{1}[\mathcal{X}_{S}]$ cannot change the orientation of the occupied/empty
subspaces, so that they are trivial away from $S$. Such deformations may,
however, change both $\mathrm{Eu}_{2}^{\mathrm{o}}$ and
$\mathrm{Eu}_{2}^{\mathrm{e}}$ by arbitrary integers. (See App. F for an
explicit construction of such a boundary deformation.) These deformations do
not exist if either Grassmannian in $\mathcal{X}_{S}$ degenerates to a point,
i.e., if

${\cal Q}_{2}^{S}:\
n^{S}_{\mathrm{o},+}n^{S}_{\mathrm{o},-}n^{S}_{\mathrm{e},+}n^{S}_{\mathrm{e},-}=0.$
(56)

We further define

${\cal Q}_{2}^{\land}:\ {\cal Q}_{2}^{S}\ \mbox{for all}\ S.$ (57)

Thus, if ${\cal Q}_{2}^{\land}$ is violated, i.e., if ${\cal Q}_{2}^{S}$ is
violated at any $S$, then the boundary deformation around $S$ can change both
$\mathrm{Eu}_{2}^{\mathrm{o}}$ and $\mathrm{Eu}_{2}^{\mathrm{e}}$ by arbitrary
integers, so that there are no topological invariants at level-2.

_Second Stiefel-Whitney invariant —_ The boundary deformations at 1-cells do
not affect $\mathbb{Z}_{2}$-valued $\mathrm{SW}_{2}$, since the contributions
from its two copies always cancel out. On the other hand, boundary
deformations at the 0-cells with nontrivial topology in both factors of
$\pi_{1}[\mathcal{X}_{S}]$, if they exist, can add arbitrary integers to
$\mathrm{SW}_{2}$, rendering the topology at level-2 trivial.

|  | Level 1 | Level 2 | Level 3
---|---|---|---|---
A | $n_{\mathrm{o}}=n_{\mathrm{e}}=1$ | — |  $\mathrm{C}_{z}$, $\mathrm{C}^{\Gamma{\mathrm{Y}}}$, $\mathrm{C}^{{\mathrm{Y}}{\mathrm{M}}}$, $\mathrm{C}^{{\mathrm{M}}{\mathrm{X}}}\in\mathbb{Z}$, | if $\mathrm{C}_{z}=0$: $\mathrm{H}\in\mathbb{Z}$
one less for every violation of ${\cal Q}_{1}^{S}$ | if $\mathrm{C}_{z}\neq 0$: $\mathrm{H}\in\mathbb{Z}_{2\mathrm{C}_{z}}$
$n_{\mathrm{o}}$, $n_{\mathrm{e}}\geq 1$, | — |  $\mathrm{C}\in\mathbb{Z}$ [S]; $\mathrm{C}^{\Gamma{\mathrm{Y}}}$, $\mathrm{C}^{{\mathrm{Y}}{\mathrm{M}}}$, $\mathrm{C}^{{\mathrm{M}}{\mathrm{X}}}\in\mathbb{Z}$, | —
$n_{\mathrm{o}}+n_{\mathrm{e}}>2$ | one less for every violation of ${\cal Q}_{1}^{S}$
AI | $n_{\mathrm{o}}=n_{\mathrm{e}}=1$ | Two copies of invariants |  $\mathrm{C}^{\Gamma{\mathrm{Y}}}$, $\mathrm{C}^{{\mathrm{Y}}{\mathrm{M}}}$, $\mathrm{C}^{{\mathrm{M}}{\mathrm{X}}}\in\mathbb{Z}$, | $\mathrm{H}\in\mathbb{Z}$
from Tab. 7 | one less for every violation of ${\cal Q}_{1}^{S}$
$n_{\mathrm{o}}$, $n_{\mathrm{e}}\geq 1$, | Two copies of invariants | two copies of invariants from | —
$n_{\mathrm{o}}+n_{\mathrm{e}}>2$ | from Tab. 7 |  Tab. 7 [F]; $\mathrm{C}^{\Gamma{\mathrm{Y}}}$, $\mathrm{C}^{{\mathrm{Y}}{\mathrm{M}}}$, $\mathrm{C}^{{\mathrm{M}}{\mathrm{X}}}\in\mathbb{Z}$,
one less for every violation of ${\cal Q}_{1}^{S}$
Table 8: Homotopic classification of gapped band structures with
${\mathrm{C}}_{2}$ symmetry in three dimensions. The condition ${\cal
Q}_{1}^{S}$ is given in Eq. (49). Nontrivial fragile and stable invariants are
denoted with [F] and [S], respectively.

#### Summary

Table 7 summarizes the classification for ${\mathrm{C}}_{2}$-symmetric gapped
band structures with time-reversal symmetry. To obtain the fragile and stable
classifications, besides taking the limits $n_{\mathrm{e}}\gg 1$ or
$n_{\mathrm{o}}$, $n_{\mathrm{e}}\gg 1$, one must also verify compatibility of
the various conditions for the existence of these invariants with the rules of
fragile and stable topology. In particular, the conditions ${\cal Q}_{1}^{S}$
and ${\cal Q}_{2}^{\mathrm{e}}$ are incompatible with fragile classification
rules, since the latter allows for addition of unoccupied bands with arbitrary
band representations. Furthermore, ${\cal Q}_{2}^{\land}$ and ${\cal
Q}_{2}^{\mathrm{o}}$ become equivalent, since $n_{\mathrm{e},\pm}>0$ after a
possible addition of empty bands. Finally, all of these conditions are
incompatible with the rules of stable classification, which allows for adding
occupied/empty bands with arbitrary band representations. Thus, there are no
${\mathrm{C}}_{2}$-symmetric stable topological phases. For $n_{\mathrm{e}}\gg
1$, we get the classification of fragile phases consistent with the fragile
band structures identified via topological quantum chemistry [54]. The
delicate topological invariants agree with those obtained by Kobayashi and
Furusaki [31].

We also encounter our first example of _representation-protected stable
topology_ , whereby a stable topological phase exists if we constrain the band
representations such that the condition ${\cal Q}_{2}^{\land}$ is always
satisfied, without limiting the number of occupied and unoccupied bands. This
can be ensured by demanding that all bands at each 0-cell must have the same
parity. To ensure the representation constraint in real space, we turn to the
EBRs [54, 77, 55] of a ${\mathrm{C}}_{2}$-symmetric band structure, listed in
Tab. 15. Since each EBR, labelled by a special Wyckoff position and a parity,
corresponds to a fixed set of parities at each 0-cell, we can ensure a
representation-protected stable phase only by allowing a _single_ orbital type
at a single Wyckoff position. Inspection of Tab. 15 further shows that the
presence of multiple EBRs always leads to the existence of different parity
bands at at least one of the high-symmetry points, thereby violating the
condition for representation-protected stable topology.

###  VI.3 Classification in three dimensions

Figure 9:  Fundamental domains $\mathbb{D}$ for a three-dimensional band
structure with twofold rotation symmetry without (left) and with (right) time-
reversal symmetry.

####  VI.3.1 Without time-reversal symmetry

The fundamental domain is a cuboid as shown in Fig. 9(left). There are no
level-1 invariants, as follows for horizontal and vertical 1-cells from the 2d
classification and Sec. IV, respectively. At level-2, the parent invariants
are Chern numbers for each 2-cell. For the horizontal 2-cell, $\mathrm{C}_{z}$
is robust to boundary deformations and can be computed for any fixed-$k_{z}$
slice of $\mathbb{D}$. For the vertical 2-cells, boundary deformations at the
vertical line through a 0-cell $S$ can add arbitrary integers to all
$\mathrm{C}^{SS^{\prime}}=-\mathrm{C}^{S^{\prime}S}$. Such deformations are
allowed if $\pi_{1}[\mathcal{X}_{S}]\neq 0$, i.e., if ${\cal Q}_{1}^{S}$ (Eq.
(47)) is violated. Thus, the number of level-2 invariants associated with
vertical faces is one less than the number of 0-cells satisfying ${\cal
Q}_{1}^{S}$. These invariants were previously identified by Nelson et al.
[78], who refer to them as “returning Thouless pump”. At level-3, we get the
$\mathbb{Z}$-valued Hopf invariant when $n_{\mathrm{o}}=n_{\mathrm{e}}=1$.
This reduces to a $\mathbb{Z}_{2\mathrm{C}_{z}}$-valued invariant if
$\mathrm{C}_{z}\neq 0$.

####  VI.3.2 With time-reversal symmetry

The fundamental domain is a cuboid as shown in Fig. 9(right). At level-0,
level-1, and level-2, we get two independent sets of topological invariants as
for the two-dimensional case, associated with the planes $k_{z}=0$ and
$k_{z}=\pi$. At level-2, we additionally get three Chern numbers as parent
invariants, associated with the vertical faces of $\mathbb{D}$. The conditions
for their robustness with respect to boundary deformations is the same as in
the case of broken time-reversal symmetry. At level-3, we get the Hopf
invariant if $n_{\mathrm{o}}=n_{\mathrm{e}}=1$, which is always
$\mathbb{Z}$-valued, since the Chern number associated to a cross section of
$\mathbb{D}$ at constant $k_{z}$ vanishes.

These classification results are summarized in Table 8. Fragile and stable
invariants in 3d derive entirely from those in 2d (see Sec. VI.2).

####  VI.3.3 Anomalous boundary states

In the absence of time-reversal symmetry, a nontrivial Chern number
$\mathrm{C}_{z}$ for the horizontal 2-cell indicates a weak Chern-insulator
phase, which is a stable topological phase with topologically protected chiral
surface states on surfaces parallel to the rotation axis.

In the presence of time-reversal symmetry, we get a pair of surface Dirac
cones related by ${\mathrm{C}}_{2}$ [31], which are associated with a
nontrivial representation-protected stable topology if the level-2 invariants
at $k_{z}=0,\pi$ are different. From the boundary perspective, following the
${\mathrm{C}}_{2}{\mathrm{T}}$-symmetric case (Sec. V.3.2), a single Dirac
node in the fundamental domain implies that
$\det{W}^{\mathrm{o}/\mathrm{e}}(\partial\mathbb{D})=-1$, which violates the
level-1 compatibility constraint of Eq. (47). This condition cannot be
satisfied for a lattice model with only orbitals of a single parity at a
single Wyckoff position, as evident from the EBRs of Tab. 15. Hence, a surface
band structure with a pair of Dirac cones (or an odd number of such pairs) is
anomalous under the rules of representation-protected stable topology.

##  VII ${\mathrm{C}}_{4}$ symmetry

A four-band lattice model of spinless electrons in three dimensions with
${\mathrm{C}}_{4}$ and ${\mathrm{T}}$ symmetries was introduced by Fu, as an
early example of topology protected by a crystalline symmetry [39]. Song,
Elcoro, and Bernevig discuss a model of spinless electrons with $C_{4}$
symmetry in two dimensions and show that it is topologically inequivalent to a
insulator with localized orbitals under the rules of fragile topology. Both of
these are particularly instructive, because, depending on the precise
constraints imposed, they exhibit nontrivial stable, fragile, or delicate
topology. Previous theoretical analyses of these models have identified
multiple topological invariants specific to delicate or fragile
classifications [35, 36, 37, 29, 30, 31]. We now show how these invariants
arise in the framework of a systematic homotopic classification.

Figure 10:  (Left) The Wyckoff positions for a square lattice with
${\mathrm{C}}_{4}$ symmetry. (Right) The fundamental domains $\mathbb{D}$ for
two-dimensional band structures with ${\mathrm{C}}_{4}$ symmetry.

###  VII.1 Wyckoff positions and basis orbitals

With fourfold rotation symmetry, there are three special Wyckoff positions
“a”, “b”, and “c”, as well as one pair of generic Wyckoff positions “g”, as
shown in Fig. 10. The special Wyckoff positions “a” and “b” have fourfold
rotation symmetry, whereas the special Wyckoff position “c” has twofold
rotation symmetry only. Correspondingly, one may choose a basis of orbitals at
the Wyckoff positions that transform under one of the irreducible
representations of the fourfold or twofold rotation operation, respectively.
For “a” and “b”, this means that orbitals can be of $s$, $d$, or $p\pm$ type,
corresponding to eigenvalue 1, $-1$, or $\pm i$ under fourfold rotation. With
time-reversal symmetry the $p\pm$ orbitals must be combined into a doublet of
$p$ orbitals that transforms under a two-dimensional representation of a
fourfold rotation. For the two “c” positions, orbitals are even ($+$) or odd
($-$) under twofold rotation. Combining orbitals at the two “c” positions, a
pair of even-parity orbitals may be rearranged as one $s$ orbital and one $d$
orbital, whereas the pair of odd-parity orbitals may be arranged as a pair of
$p$ orbitals. Finally, at the generic Wyckoff position orbitals appear in
quartets that can be arranged as one $s$ orbital, one $d$ orbital, and a pair
of $p$ orbitals. The parity of $s$ and $d$ orbitals under twofold rotation is
even ($+$); the parity of $p$ orbitals under twofold rotation is odd ($-$).

The reciprocal-space group $G_{\Lambda}$ is generated by fourfold rotation
$g_{4}$ and two reciprocal-space translations $g_{x}$ and $g_{y}$. The
corresponding representation matrices $C(g)$ (cf. Eq. (4)) are denoted
$C_{4}$, $\Lambda_{x}$, and $\Lambda_{y}$, respectively. The Bloch Hamiltonian
satisfies

$\displaystyle
H(k_{x},k_{y})=C_{4}^{\phantom{\dagger}}H(-k_{y},k_{x})C_{4}^{\dagger}$ (58)

for a fourfold rotation, as well as Eq. (5) under $g_{x,y}$. Since the
matrices $C_{4}$, $\Lambda_{x}$, and $\Lambda_{y}$ generate a representation
of $G_{\Lambda}$, they satisfy

$C_{4}\Lambda_{x}=\Lambda_{y}C_{4},\qquad
C_{4}\Lambda_{y}=\Lambda_{x}^{\dagger}C_{4}^{\phantom{\dagger}}.$ (59)

We choose the overall phase factor of $C_{4}$ such that $C_{4}^{4}=\openone$.
In its eigenbasis, it can be written explicitly as

$C_{4}=\text{diag}\\{\openone_{n_{s}},-\openone_{n_{\mathrm{d}}},\mathrm{i}\openone_{n_{p+}},-\mathrm{i}\openone_{n_{p-}}\\}.$
(60)

In the presence of time-reversal symmetry, the $p_{\pm}$ orbitals always come
in pairs, and changing to a basis where $C_{4}$ is manifestly real, we can
write

$\displaystyle
C_{4}=\text{diag}\\{\openone_{n_{s}},-\openone_{n_{\mathrm{d}}},\mathrm{i}\sigma_{2}\openone_{n_{p}}\\}.$
(61)

Explicit expresions for $\Lambda_{x,y}$ are derived in Appendix B.2.

We take the fundamental domain $\mathbb{D}$ as the right quadrant of the
Brillouin zone enclosed between the lines $k_{y}=\pm k_{x}$, as shown in Fig.
10. It consists of three 0-cells, two 1-cells and one 2-cell. The little
groups of the 0-cells are $G_{\Gamma}\cong
G_{\mathrm{M}}\cong{\mathrm{C}}_{4}$ and $G_{\mathrm{X}}\cong\mathbb{Z}_{2}$,
with $G_{\Gamma}$ generated by the fourfold rotation $g_{\Gamma}=g_{4}$,
$G_{\mathrm{M}}$ generated by
$g_{\mathrm{M}}=g_{x}^{-1}g_{4}^{\phantom{\dagger}}$ and $G_{\mathrm{X}}$ by
the twofold rotation followed by a reciprocal lattice translation
$g_{\mathrm{X}}=g_{x}^{-1}g_{4}^{2}$. The corresponding matrices $C_{S}\equiv
C(g_{S})$ are thus given by

$\displaystyle C_{\Gamma}$ $\displaystyle=C_{4},$ $\displaystyle
C_{\mathrm{M}}$
$\displaystyle=\Lambda_{x}^{\dagger}C_{4}^{\phantom{\dagger}}=C_{4}\Lambda_{y},$
$\displaystyle C_{\mathrm{X}}$
$\displaystyle=\Lambda_{x}^{\dagger}C_{4}^{2}=C_{4}^{2}\Lambda_{x}^{\phantom{\dagger}}$
(62)

and satisfy

$C_{\mathrm{X}}C_{\mathrm{M}}C_{\Gamma}=\openone.$ (63)

We can derive the spectrum of these matrices without knowing their explicit
form. By definition, $C_{\Gamma}^{4}=\openone$, and furthermore, using Eq.
(59), we get

$\displaystyle
C_{\mathrm{M}}^{4}=\left(\Lambda^{\dagger}_{x}C_{4}^{\phantom{\dagger}}\right)^{4}=\Lambda^{\dagger}_{x}\Lambda^{\dagger}_{y}C^{4}\Lambda_{x}\Lambda_{y}=\openone.$
(64)

Thus, the spectrum of $C_{\Gamma,{\mathrm{M}}}$ is
$\\{1,-1,\mathrm{i},-\mathrm{i}\\}$, corresponding to
$\\{s,d,p_{+},p_{-}\\}$-type orbitals. in the presence of time-reversal
symmetry, the $p_{\pm}$-type orbitals always come in pairs. Finally,

$\displaystyle
C_{\mathrm{X}}^{2}=(\Lambda_{x}^{\dagger}C_{4}^{2})(C_{4}^{2}\Lambda_{x}^{\phantom{\dagger}})=\Lambda_{x}^{\dagger}\Lambda_{x}^{\phantom{\dagger}}=\openone,$
(65)

so that the spectrum of $C_{\mathrm{X}}$ consists of $\pm 1$, i.e., an even
and an odd orbital. The EBRs for a ${\mathrm{C}}_{4}$-symmetric band structure
are listed in the appendix, see Table 17.

###  VII.2 Homotopic classification in two dimensions

The homotopic classification for this case closely follows that for
${\mathrm{C}}_{2}$-symmetric band structures described in Sec. VI.2. We
summarize these results in Tab. 9.

####  VII.2.1 Without time-reversal symmetry

The level-0 invariants are given by $n_{\mathrm{o}/\mathrm{e},\rho}^{S}$, the
number of occupied/empty bands at $S$, with $\rho\in\\{s,p_{\pm},d\\}$ for
$S=\Gamma,{\mathrm{M}}$ and $\rho\in\\{\pm\\}$ for $S={\mathrm{X}}$. The
“level-0 compatibility relation” (Eq. (7)) reads

$n_{\mathrm{o}/\mathrm{e}}=\sum_{\rho\in\\{s,p_{\pm},d\\}}n_{\mathrm{o}/\mathrm{e},x}^{S}=\sum_{\rho=\pm}n_{\mathrm{o}/\mathrm{e},\rho}^{\mathrm{X}},$
(66)

where $S=\Gamma,{\mathrm{M}}$. Additionally, there are constraints on the
level-0 invariants that follow from Eq. (63). The target spaces for the
0-cells are given by

$\displaystyle\mathcal{X}_{S}=\prod_{\rho}\mathrm{Gr}_{\mathbb{C}}\left(n_{\mathrm{o},\rho}^{S},n_{\mathrm{o},\rho}^{S}\right),$
(67)

where the direct product is over $\rho=s$, $d$, $p+$, and $p-$ for $S=\Gamma$,
${\mathrm{M}}$ and $\rho=\pm$ for $S={\mathrm{X}}$. For the 1-cells and the
2-cell, the target space is
$\mathcal{X}=\mathrm{Gr}_{\mathbb{C}}(n_{\mathrm{o}},n_{\mathrm{e}})$.

At level-1, there are no topological invariants, while at level-2, we get a
$\mathbb{Z}$-valued topological invariant, _viz_ , the Chern number,
irrespective of the level-0 invariants. The latter is robust to boundary
deformations following an argument identical to Sec. VI.

####  VII.2.2 With time-reversal symmetry

In the presence of time-reversal symmetry, the $p$ bands always come in pairs
at the high-symmetry points $S=\Gamma$, ${\mathrm{M}}$. Denoting the number of
these doublets by $n_{\mathrm{o}/\mathrm{e},p}^{S}$, the level-0 invariants
for the $\Gamma$ and ${\mathrm{M}}$ point are
$\large(n_{\mathrm{o}/\mathrm{e},s}^{S},n_{\mathrm{o}/\mathrm{e},d}^{S},n_{\mathrm{o}/\mathrm{e},p}^{S}\large)$,
while those at ${\mathrm{X}}$ are identical to the time-reversal broken case.
The level-0 compatibility relation (7) takes the form

$\displaystyle n_{\mathrm{o}/\mathrm{e}}=$
$\displaystyle\,n_{\mathrm{o}/\mathrm{e},+}^{\mathrm{X}}+n_{\mathrm{o}/\mathrm{e},-}^{\mathrm{X}}$
$\displaystyle=$
$\displaystyle\,n_{\mathrm{o}/\mathrm{e},s}^{S}+n_{\mathrm{o}/\mathrm{e},d}^{S}+2n_{\mathrm{o}/\mathrm{e},p}^{S},$
(68)

where $S=\Gamma$, $M$. The level-0 invariants also satisfy the level-1
compatibility condition

$\displaystyle
n_{\mathrm{o}/\mathrm{e},-}^{\mathrm{X}}=n_{\mathrm{o}/e,d}^{\Gamma}+n_{\mathrm{o}/\mathrm{e},d}^{\mathrm{M}}\mod
2,$ (69)

which will be derived below. The target spaces for the high-symmetry points
are given by

$\displaystyle\mathcal{X}_{S}$
$\displaystyle=\mathrm{Gr}_{\mathbb{R}}(n_{\mathrm{o},s}^{S},n_{\mathrm{e},s}^{S})\times\mathrm{Gr}_{\mathbb{R}}(n_{\mathrm{o},d}^{S},n_{\mathrm{e},d}^{S})\times\mathrm{Gr}_{\mathbb{C}}(n_{\mathrm{o},p}^{S},n_{\mathrm{e},p}^{S}),$
$\displaystyle\mathcal{X}_{\mathrm{X}}$