# Symmetry-based indicators for topological Bogoliubov-de Gennes Hamiltonians

Max Geier  Dahlem Center for Complex Quantum Systems and Physics Department,
Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany  Piet W.
Brouwer  Dahlem Center for Complex Quantum Systems and Physics Department,
Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany  Luka
Trifunovic  Dahlem Center for Complex Quantum Systems and Physics Department,
Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany  RIKEN Center
for Emergent Matter Science, Wako, Saitama 351-0198, Japan  Department of
Physics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich,
Switzerland

###### Abstract

We develop a systematic approach for constructing symmetry-based indicators of
a topological classification for superconducting systems. The topological
invariants constructed in this work form a complete set of symmetry-based
indicators that can be computed from knowledge of the Bogoliubov-de Gennes
Hamiltonian on high-symmetry points in Brillouin zone. After excluding
topological invariants corresponding to the phases without boundary
signatures, we arrive at natural generalization of symmetry-based indicators
[H. C. Po, A. Vishwanath, and H. Watanabe, Nature Comm. 8, 50 (2017)] to
Hamiltonians of Bogoliubov-de Gennes type.

##  I Introduction

Although topological phases of matter have been known for four decades now,
starting with the discovery of the quantized Hall effect, Klitzing  _et al._
(1980) the study of topological phases and phase transitions became central to
quantum condensed matter physics only in the early 2000s, after the
theoretical proposals for topological superconducting phases Kitaev (2001);
Ivanov (2001) and the quantum spin Hall effect Kane and Mele (2005a, b) and
the subsequent experimental observation of these phases of matter. Mourik  _et
al._ (2012); Brüne  _et al._ (2011) These theoretical and experimental
developments paved the way for the complete classification of all possible
topological phases of single-particle systems, protected by either time-
reversal, particle-hole or sublattice symmetries — the so-called “tenfold way
classification”Altland and Zirnbauer (1997) —, which, reflecting its
periodicity as a function of dimensionality and honoring its fundamental
importance to the field, was coined “periodic table of topological phases”.
Kitaev (2009); Schnyder  _et al._ (2009)

In addition to the presence or absence of the fundamental non-spatial
symmetries that define the tenfold-way classes, real materials have
crystalline symmetries. The combination of topology and crystalline symmetries
leads to an exceedingly rich set of “topological crystalline phases”. Fu
(2011) Unlike non-crystalline tenfold-way topological phases, for which a
nontrivial topology of the bulk is always associated with a unique anomalous
boundary signature, topological crystalline phases may come with a variety of
possible boundary signatures. These include the protected existence of
anomalous boundary states on all boundaries — a “first-order” topological
phase, for which the crystalline symmetry is not essential for the protection
of the nontrivial topology —, the appearance of higher-order boundary states
on hinges or corners of a crystal, Parameswaran and Wan (2017); Schindler  _et
al._ (2018); Peng  _et al._ (2017); Langbehn  _et al._ (2017); Song  _et al._
(2017); Fang and Fu (2017); Ezawa (2018); Shapourian  _et al._ (2018); Zhu
(2018); Yan  _et al._ (2018); Wang  _et al._ (2018a, b); Khalaf  _et al._
(2018); Khalaf (2018); Okuma  _et al._ (2019) or even the complete absence of
protected boundary states. The latter scenario applies to “atomic-limit
phases”, in which the electronic states can be continuously deformed to a
collection of localized orbitals, while preserving all relevant symmetries as
discussed by Po et al. Po  _et al._ (2017) and Bradlyn et al. Bradlyn  _et
al._ (2017). In such cases, the presence of the crystalline symmetry may form
an “obstruction” that prevents from different arrangements of localized
orbitals to be continuously connected to each other, 111Although atomic-limit
phases have no protected boundary states, they may have protected fractional
charges at ends or corners, see, e.g., Refs. Lau  _et al._ , 2016; Benalcazar
_et al._ , 2017a, b. allowing for the existence of multiple topologically-
distinct atomic-limit phases. Turner  _et al._ (2012); Lau  _et al._ (2015,
2016); Benalcazar  _et al._ (2017a, b); van Miert and Ortix (2017); Rhim  _et
al._ (2017)

Although the classification of topological crystalline phases can be
considered largely under control Shiozaki and Sato (2014); Shiozaki  _et al._
(2016); Chiu  _et al._ (2016); Trifunovic and Brouwer (2019); Cornfeld and
Chapman (2019); Shiozaki (2019a); Huang  _et al._ (2017); Thorngren and Else
(2018) (for initial partial classifications results, see Refs. Fang  _et al._
, 2012, 2013; Chiu  _et al._ , 2013; Jadaun  _et al._ , 2013; Morimoto and
Furusaki, 2013; Teo and Hughes, 2013; Benalcazar  _et al._ , 2014; Liu  _et
al._ , 2014; Alexandradinata  _et al._ , 2014; Zhang  _et al._ , 2013; Slager
_et al._ , 2013; Dong and Liu, 2016; Kruthoff  _et al._ , 2017) the explicit
computation of topological invariants for a given band structure is often
computationally expensive. Explicit expressions for the invariants need not
always be readily available, since a full classification does not always come
with explicit expressions for topological invariants. A practical — but
partial — solution to this problem is the use of a set of easy-to-compute
“symmetry-based indicators”, which, when nonzero, are a sufficient indicator
of a nontrivial topology of the bulk band structure. Po  _et al._ (2017) The
Fu-Kane criterion, which links the existence of a strong 222In this work, the
term “strong topological phase” is reserved for a phase that remains
topologically non-trivial after the translational symmetry is broken.
Topological crystalline phases that do not rely on the crystalline symmetry
for their protection will be referred to as “first-order”, corresponding to
the dimension of their boundary signature. topological insulator phase in an
inversion-symmetric crystal to the parity of the number of occupied bands with
odd-inversion-parity at the high-symmetry momenta, is an example of such a
symmetry-based indicator. Fu and Kane (2007) For normal-state insulating
phases, symmetry-based indicators were constructed for the complete set of
point group symmetries in two and three dimensions, Po  _et al._ (2017);
Bradlyn  _et al._ (2017); Song  _et al._ (2018a, b) taking into account the
order of the boundary states. Khalaf  _et al._ (2018); Zhang  _et al._ (2019)
Like the classifying groups of topological phases, symmetry-based indicators
of topological phases have a group structure, the group operation being the
direct sum “$\oplus$” of representative Hamiltonians. 333Although Refs. Po
_et al._ , 2017; Bradlyn  _et al._ , 2017 have equivalent definitions of
“atomic-limit phases”, Bradlyn et al. Bradlyn  _et al._ (2017) consider band
labels that lack the full group structure — one can perform addition of the
band labels, but not their subtraction. Subsequent careful comparison revealed
that some phases without protected boundary states are topologically
equivalent to a “difference” of atomic-limit phases, i.e., they can only be
deformed to an atomic-limit phase after addition of topologically trivial
bands, see H. C. Po, H. Watanabe, and A. Vishwanath, Phys. Rev. Lett. 121,
126402 (2018). Such phases were called “fragile topological insulators”. They
are trivial in the classifying scheme of Po et al. Po  _et al._ (2017), while
Bradlyn et al. Bradlyn  _et al._ (2017) label them as non-trivial. The
extension of the definition of atomic-limit insulators to superconductors
discussed in the present work allows one to define “fragile topological
superconductors” in analogous way, although this analogy is not being pursued
here.

The general strategy underlying the construction of symmetry-based indicators
for a Hamiltonian $h(\mathbf{k})$ is to replace the topological classification
of the matrix-valued function $h(\mathbf{k})$ by the simpler problem of the
topological characterization of the hermitian matrices $h(\mathbf{k}_{\rm s})$
at a selected set of “high-symmetry” points $\\{\mathbf{k}_{\rm s}\\}$ in the
Brillouin zone. Po  _et al._ (2017) The topological characterization of the
hermitian matrices $h(\mathbf{k}_{\rm s})$ may be considered a set of
“topological band labels”, the calculation of which is considerably easier to
obtain than the topological classification of the full functions
$h(\mathbf{k})$. The group SI of symmetry-based indicators then follows by
“dividing out” all combinations of topological band labels that correspond to
atomic phases and imposing a set of compatibility constraints derived from the
topological classification of the matrices $h(\mathbf{k})$ at lower-symmetry
points in the Brillouin zone.Po  _et al._ (2017); Bradlyn  _et al._ (2017)
This procedure ensures that only topological phases with a nontrivial boundary
signature have nontrivial symmetry-based indicators.

Despite their enormous computational advantage,Bradlyn  _et al._ (2017)
symmetry-based indicators are not guaranteed to give complete classification
information: There exist topologically nontrivial phases with protected
anomalous boundary signatures, but trivial symmetry-based indicators. An
example is the quantized Hall effect in the absence of any crystalline
symmetries, for which no symmetry-based indicators exist (i.e., the group SI
is trivial), in spite of the existence of topological phases with nonzero
Chern number.

In the present article, we extend the construction of symmetry-based
indicators to superconductors, which, on the mean-field level, are described
by Hamiltonians $H(\mathbf{k})$ of Bogoliubov-de Gennes (BdG) type. Gorkov
(1958); de Gennes (1966) We explicitly construct the topological band labels
and the symmetry-based indicators for selected point groups using the complete
topological classification of the BdG Hamiltonian $H(\mathbf{k}_{\rm s})$ at
high-symmetry momenta $\mathbf{k}_{\rm s}$ and compare the symmetry-based
indicators with a full classification of topological phases with nontrivial
boundary signature. Such a comparison gives information to what extent the
symmetry-based indicators can be used as a proxy for a complete
classification. Depending on the crystalline symmetries considered, we find
that certain aspects of the wealth of boundary signatures available to
superconducting phases — Majorana modes at surfaces or hinges, zero-energy
Majorana bound states at corners — are reflected in the symmetry-based
indicators, but not all.

Recently, a number of articles appeared in the literature that also consider
the construction of symmetry-based indicators for superconducting phases.
Based on a general analysis of the principles underlying the construction of
symmetry-based indicators, Ono and Watanabe Ono and Watanabe (2018) arrive at
the conclusion that the sets of symmetry-based indicators that describe
superconducting phases and the underlying normal-state phases are essentially
the same. Our construction of symmetry-based indicators, which is based on the
full classification of “zero-dimensional” Hamiltonians at high-symmetry
momenta,Shiozaki  _et al._ (2018) shows that this statement needs to be
corrected to the extent that we successfully put to use a topological band
label based on the Pfaffian of a BdG-type Hamiltonian $H(\mathbf{k}_{\rm s})$
at high-symmetry momenta $\mathbf{k}_{\rm s}$, which has no counterpart in the
normal state. A different approach to the problem of symmetry-based indicators
in superconductors was taken in a later article by these authors, together
with Yanase, Ono  _et al._ (2019) as well as by Skurativska, Neupert, and
Fischer, Skurativska  _et al._ (2019) who consider the “weak-pairing limit”
(superconducting order parameter $\Delta$ much smaller than energy scales
typical for the normal-state band structure) and derive a classification of
superconducting phases that is based on the topological classification of the
normal state and the symmetry of the superconducting order parameter. An
approach that uses the normal-state Hamiltonian as its sole input has the
practical advantage that the symmetry-based indicators can be calculated from
the vast body of band-structure knowledge available for normal phases.
Furthermore, Skurativska et al. argue that there is no alternative to such an
approach, because an atomic limit can not be defined on the level of the BdG
Hamiltonian. Skurativska  _et al._ (2019) We arrive at a different conclusion,
showing that there exists a consistent definition of an “atomic-limit
superconductor” as an “array” of zero-dimensional superconductors. Trifunovic
and Brouwer (2019) In the weak-pairing limit, the symmetry-based indicators
derived here can be expressed in terms of the normal part of the Hamiltonian
only, so that in that limit our approach offers the same computational
advantages as the approaches that rely on the weak-pairing limit at the
outset. Very recently, an article by Ono, Po, and Watanabe appeared,Ono  _et
al._ (2019) which bases its symmetry indicators on the full BdG Hamiltonian,
be it without the Pfaffian band labels, and has a definition of an atomic
limit that is consistent with ours. A recent article by Shiozaki also reports
the construction of symmetry-based indicators for Hamiltonians of BdG-type and
which has results very similar to ours. Shiozaki (2019b)

The remaining part of this article is organized as follows: In Sec. II we
discuss the symmetry of the superconducting order parameter and the set of
data that needs to be specified in order to define a “topological class” for a
crystalline superconductor. In Sec. III we discuss the classification and
topological invariants of zero-dimensional Hamiltonians. This discussion is
the cornerstone for defining topological invariants at high-symmetry momenta
in Brillouin zone. We present our main result, a method to construct and
calculate symmetry-based indicators for superconductors, in Sec. IV. Sections
V–VII contain examples for various crystalline symmetries compatible with a
square or cubic lattice structure, for which we present a detailed calculation
of symmetry-based indicators and relate this to higher-order boundary
phenomenology. We conclude in Sec. VIII. The appendices contain additional
examples as well as fully worked-out classifications of anomalous boundary
states for crystalline symmetry groups not readily available in the
literature.

##  II Superconductors with crystalline symmetries

The mean-field theory of superconductors uses an effective non-interacting
description with a Hamiltonian of the Bogoliubov-de Gennes (BdG) form Gorkov
(1958); de Gennes (1966)

$H(\mathbf{k})=\begin{pmatrix}h(\mathbf{k})&\Delta(\mathbf{k})\\\
\Delta^{\dagger}(\mathbf{k})&-h^{*}(-\mathbf{k})\end{pmatrix},$ (1)

where the normal-state Hamiltonian $h(\mathbf{k})$ is hermitian and the
superconducting order parameter $\Delta(\mathbf{k})=-\Delta(-\mathbf{k})^{\rm
T}$ is antisymmetric. The $2\times 2$ block structure describes particle and
hole degrees of freedom. The special choice of the blocks in Eq. (1) is
equivalent to imposing that $H(\mathbf{k})$ is antisymmetric under particle-
hole conjugation ${\cal P}=\tau_{1}K$,

$H(\mathbf{k})=-\tau_{1}H(-\mathbf{k})^{*}\tau_{1},$ (2)

where $\tau_{1}$ is a Pauli matrix acting within particle-hole space and $K$
is complex conjugation. (Note that $\mathbf{k}\to-\mathbf{k}$ under complex
conjugation.) In addition to the BdG structure, we assume that the system
obeys translation invariance — which is what allows us to use the Fourier
language in Eqs. (1) and (2) in the first place — and that it has additional
symmetries described by the point group $G$. We restrict ourselves to
symmorphic symmetries, for which the unit cell can be chosen in such a way
that it is left invariant under $G$. For the initial discussion we focus on
systems without time-reversal symmetry or other antiunitary symmetries.

Requiring that the normal part $h(\mathbf{k})$ be symmetric under $G$ implies
that there exists a projective representation $u(g)$ for $g\in G$ such that

$h(\mathbf{k})=u(g)h(g\mathbf{k})u^{\dagger}(g).$ (3)

Note that the unitary matrix $u(g)$ does not depend on $\mathbf{k}$, as the
symmetry group $G$ acts within the unit cell. The representation $u(g)$ is
projective, because the transformation rule (3) determines $u(g)$ up to a
phase factor only. In general, a consistent choice of these phase factors is
possible up to a sign only, the sign ambiguities being captured by the “factor
system”

$\\{z_{g,h}=u(gh)^{-1}u(g)u(h)=\pm 1\,\mbox{for}\,g,\,h\in G\\}$ (4)

of the projective representation. Two realizations $u_{1}(g)$ and $u_{2}(g)$
that have the same factor system may still differ by a one-dimensional
representation $\Theta(g)$ of $G$ with trivial factor system. Although the
mathematical structure of the theory allows many non-equivalent factor groups
for the representation $u(g)$, for physical systems only two factor groups are
relevant: The trivial one, which applies to spinless particles, and the
nontrivial factor system associated with the spinful particles. The
representation $\Theta(g)$, which describes the difference between two
representations with the same factor system, always has the trivial factor
group.

The canonical form for the representation $u(g)$ is

$u(g)=\mbox{diag}\,[r_{1}(g)\otimes\openone_{N_{1}},\ldots,r_{n}(g)\otimes\openone_{N_{n}}],$
(5)

where the $r_{\alpha}(g)$ are irreducible representations (”irreps”) of $G$
(with the appropriate factor system) and $\openone_{N_{\alpha}}$ the
$N_{\alpha}\times N_{\alpha}$ unit matrix. The dimension of $h(\mathbf{k})$,
corresponding to the total number of orbitals in the unit cell, is
$\sum_{\alpha}d_{\alpha}N_{\alpha}$, where $d_{\alpha}$ is the dimension of
the irreducible presentation $\alpha$. For notational simplicity, we choose to
set $N_{1}=N_{2}=\ldots=N_{n}\equiv N$.

The general symmetry constraint for the full BdG Hamiltonian $H(\mathbf{k})$
is obtained by allowing different realizations $u_{1,2}(g)$ of the
transformation matrices for the particle and hole degrees of freedom. Such a
symmetry constraint is compatible with the group operation of $G$ if
$u_{1}(g)$ and $u_{2}(g)$ have the same factor system, i.e., if there exists a
one-dimensional representation $\Theta(g)$ of the symmetry group $G$ with
trivial factor system such that

$u_{1}(g)\equiv u(g)=u_{2}(g)\Theta(g).$ (6)

For the BdG Hamiltonian $H(\mathbf{k})$ one then finds the general symmetry
constraint

$H(\mathbf{k})=U(g)H(g\mathbf{k})U(g)^{\dagger},$ (7)

with

$U(g)=\begin{pmatrix}u(g)&0\\\ 0&u(g)^{*}\Theta(g)\end{pmatrix}.$ (8)

For the superconducting order parameter $\Delta(\mathbf{k})$, Eqs. (7) and (8)
imply that

$\Delta(\mathbf{k})=u(g)\Delta(g\mathbf{k})u(g)^{\rm T}\Theta(g)^{*},$ (9)

i.e., $\Delta(\mathbf{k})$ transforms under a one-dimensional representation
of the group $G$.Sigrist and Ueda (1991) Whereas the projective representation
$u(g)$ is a property of the normal phase, the additional phase factor
$\Theta(g)$ exists by virtue of the superconducting order only. Together, the
representation $u$ and the one-dimensional representation $\Theta$ fully
determine the symmetry class of the Bogoliubov-de Gennes Hamiltonian (1).

Alternatively, Eqs. (7) and (8) may be recast in the form of an algebraic
relation between the elements $g$ of the symmetry group $G$ and the particle-
hole conjugation operation ${\cal P}$,

$g{\cal P}=\Theta(g){\cal P}g.$ (10)

This is the formulation used in Refs. Chiu  _et al._ , 2013; Morimoto and
Furusaki, 2013; Shiozaki and Sato, 2014; Chiu  _et al._ , 2016; Trifunovic and
Brouwer, 2019; Geier  _et al._ , 2018, which considered the full
classification of BdG Hamiltonians in the presence of a single order-two
spatial symmetry ${\cal S}$ and discriminated between the cases in which
${\cal S}$ commutes or anticommutes with the Fermi constraint ${\cal P}$.

The presence of time-reversal symmetry and/or spin-rotation symmetry does not
change the above considerations, provided these commute with the crystalline
symmetries. Time-reversal symmetry imposes the additional constraintsAltland
and Zirnbauer (1997)

$h(\mathbf{k})=\sigma_{2}h(-\mathbf{k})^{*}\sigma_{2},\ \
\Delta(\mathbf{k})=\sigma_{2}\Delta(-\mathbf{k})^{*}\sigma_{2},$ (11)

where $\sigma_{2}$ is a Pauli matrix that acts in spin space. With time-
reversal symmetry, the one-dimensional representation $\Theta(g)$ of $G$ that
characterizes the superconducting state must be real. In the presence of spin
rotation symmetry, one arrives at an effective description in terms of a BdG-
type Hamiltonian of the form (1), but with a symmetric order parameter
$\Delta(\mathbf{k})=\Delta(-\mathbf{k})^{\rm T}$, so that $H(\mathbf{k})$ is
antisymmetric under the antiunitary operation $\tau_{2}K$, which plays the
role of an effective particle-hole symmetry. If time-reversal symmetry is
present additionally, $H(\mathbf{k})$ is symmetric under complex conjugation
$K$, which plays the role of an effective time-reversal operation. Note that
while ${\cal P}$ and ${\cal T}$ square to $1$ and $-1$, respectively, in the
spinful case, the effective particle-hole conjugation and time-reversal
operations square to $-1$ and $1$, respectively, if spin-rotation symmetry is
present. Similarly, in the presence of spin-rotation symmetry, the effective
representation of crystalline symmetry operations is changed into the
corresponding “spinless” type.

| $\Gamma$ | 1 | ${\cal I}_{\pi}$
---|---|---|---
$r_{+}$ | $A_{g}$ | 1 | 1
$r_{-}$ | $A_{u}$ | 1 | -1
Table 1: Irreducible representations of the point group $C_{i}$. The second
column lists the standard crystallographic notation for the representations,
the first column gives the notation used in the text.

The general framework described here is best illustrated using examples. As a
first example, we consider a system with inversion symmetry. In this case the
group $G=C_{i}=\\{\textbf{1},{\cal I}\\}$, where 1 is the identity and ${\cal
I}$ is the inversion operation. Choosing a basis that has a well-defined
parity under inversion, one finds that one may represent $G$ by

$u(g)=\mbox{diag}\,[r_{+}(g),r_{-}(g)],$ (12)

where $r_{\pm}(g)$ are the two irreducible representations of $G$, see Table
1, which are both one-dimensional. Specifically, Eq. (12) reads

$u(\textbf{1})=\rho_{0},\ \ u({\cal I})=\rho_{3},$ (13)

where $\rho_{3}$ is a Pauli matrix that acts in the space of even/odd-parity
states and $\rho_{0}$ is the $2\times 2$ unit matrix. From the two one-
dimensional representations of $G$ we also find two possibilities for the
representation $U(g)$ for the BdG Hamiltonian $H(\mathbf{k})$. Choosing the
trivial one-dimensional representation $\Theta(g)=r_{+}(g)$ in Eq. (7) (the
$A_{g}$ representation, see Table 1), we find that the BdG Hamiltonian
$H(\mathbf{k})$ obeys the symmetry constraint

$H(\mathbf{k})=(\rho_{3}\tau_{0})H(-\mathbf{k})(\rho_{3}\tau_{0}),$ (14)

where $\tau_{0}$ is the identity matrix in particle-hole space and $\rho_{3}$
is a Pauli matrix that acts in the space of even/odd-parity states. For the
blocks $h(\mathbf{k})$ and $\Delta(\mathbf{k})$ this implies

$h(\mathbf{k})=\rho_{3}h(-\mathbf{k})\rho_{3},\ \
\Delta(\mathbf{k})=\rho_{3}\Delta(-\mathbf{k})\rho_{3},$ (15)

i.e., the order parameter is even under inversion. Alternatively, choosing the
nontrivial one-dimensional representation $\Theta(g)=u_{-}(g)$ in Eq. (7) (the
$A_{u}$ representation), we find that the BdG Hamiltonian $H(\mathbf{k})$
satisfies

$H(\mathbf{k})=(\rho_{3}\tau_{3})H(-\mathbf{k})(\rho_{3}\tau_{3}).$ (16)

In this case the blocks $h(\mathbf{k})$ and $\Delta(\mathbf{k})$ satisfy the
constraint

$h(\mathbf{k})=\rho_{3}h(-\mathbf{k})\rho_{3},\ \
\Delta(\mathbf{k})=-\rho_{3}\Delta(-\mathbf{k})\rho_{3},$ (17)

so that the order parameter is odd under inversion. The two different
transformation rules of the order parameter $\Delta(\mathbf{k})$ under
inversion imply two different algebraic relations between the inversion
operation ${\cal I}$ and the Fermi constraint ${\cal P}$: If
$\Delta(\mathbf{k})$ is even under ${\cal I}$, ${\cal I}$ and ${\cal P}$
commute, whereas ${\cal I}$ and ${\cal P}$ anticommute if $\Delta(\mathbf{k})$
is odd under inversion.

| $\Gamma$ | 1 | ${\cal R}_{\pi/2}$ | ${\cal R}^{2}_{\pi/2}$ | ${\cal R}^{3}_{\pi/2}$
---|---|---|---|---|---
$r_{0}$ | $A$ | $1$ | $1$ | $1$ | $1$
$r_{\pi/2}$ | ${}^{2}E$ | $1$ | $i$ | $-1$ | $-i$
$r_{\pi}$ | $B$ | $1$ | $-1$ | $1$ | $-1$
$r_{3\pi/2}$ | ${}^{1}E$ | $1$ | $-i$ | $-1$ | $i$
$r_{\pi/4}$ | ${}^{1}\overline{E}_{1}$ | $1$ | $e^{i\pi/4}$ | $i$ | $e^{3\pi i/4}$
$r_{3\pi/4}$ | ${}^{2}\overline{E}_{2}$ | $1$ | $e^{3\pi i/4}$ | $-i$ | $e^{i\pi/4}$
$r_{5\pi/4}$ | ${}^{1}\overline{E}_{2}$ | $1$ | $e^{5\pi i/4}$ | $i$ | $e^{7\pi i/4}$
$r_{7\pi/4}$ | ${}^{2}\overline{E}_{1}$ | $1$ | $e^{7\pi i/4}$ | $-i$ | $e^{5\pi i/4}$
Table 2: Spinless representations (top four rows) and one-dimensional spinful
representations (bottom four rows) of the point group $C_{4}$. The second
column lists the standard crystallographic notation for the representations,
the first column gives the notation used in the text.

As a second example, we consider a fourfold rotation around a fixed axis for a
system of spinful particles. We write $G=C_{4}=\\{\textbf{1},{\cal
R}_{\pi/2}^{\vphantom{1}},{\cal R}_{\pi/2}^{2},{\cal R}_{\pi/2}^{3}\\}$, where
the generator ${\cal R}_{\pi/2}\equiv{\cal R}$ denotes a clockwise rotation by
$\pi/2$. In this case one has ${\cal R}^{4}=-\textbf{1}$, which corresponds to
the factor system

$z_{{\cal R}^{k},{\cal R}^{l}}=\left\\{\begin{array}[]{ll}1&\mbox{if $0\leq
k+l<4$},\\\ -1&\mbox{if $4\leq k+l<8$}.\end{array}\right.$ (18)

Choosing basis states with well-defined angular momentum $j=\frac{1}{2}$,
$\frac{3}{2}$, $\frac{5}{2}$, and $\frac{7}{2}$ (defined modulo $4$), we find
the spinful representation

$u(g)=\mbox{diag}\,[r_{\pi/4}(g),r_{3\pi/4}(g),r_{5\pi/4}(g),r_{7\pi/4}(g)],$
(19)

see Table 2.

The transformation rule for the BdG Hamiltonian $H(\mathbf{k})$ requires the
choice of a a one-dimensional spinless representation $\Theta$ of $G$. There
are four of those, and we denote these as $r_{\theta}$ with $\theta=0$,
$\pi/2$, $\pi$, and $3\pi/2$, see Table 2. Proceeding as before, we find four
possible transformation rules for the BdG Hamiltonian $H(\mathbf{k})$ under a
$\pi/2$ rotation ${\cal R}$,

$H(\mathbf{k})=U_{\theta}({\cal R})H({\cal R}\mathbf{k})U_{\theta}({\cal
R})^{\dagger},$ (20)

with

$U_{\theta}({\cal R})=\begin{pmatrix}u({\cal R})&0\\\ 0&u({\cal
R})^{*}e^{i\theta}\end{pmatrix}.$ (21)

Alternatively, the transformation rules for the blocks $h(\mathbf{k})$ and
$\Delta(\mathbf{k})$ are

$\displaystyle h(\mathbf{k})=$ $\displaystyle\,u({\cal R})h({\cal
R}\mathbf{k})u({\cal R})^{\dagger},$ $\displaystyle\Delta(\mathbf{k})=$
$\displaystyle\,e^{i\theta}u({\cal R})\Delta({\cal R}\mathbf{k})u({\cal
R})^{\rm T},$ (22)

which corresponds to a superconducting order parameter with finite angular
momentum.

The time-reversal symmetry operation is represented by $U({\cal T})K$, where
the definition of $U({\cal T})$ in the angular momentum basis is

$U({\cal T})=\begin{pmatrix}0&0&0&-i\\\ 0&0&-i&0\\\ 0&i&0&0\\\
i&0&0&0\end{pmatrix}.$ (23)

Time-reversal commutes with ${\cal P}$ and satisfies the property ${\cal
T}^{2}=-1$. Time-reversal symmetry imposes the additional conditions (11) on
the blocks $h(\mathbf{k})$ and $\Delta(\mathbf{k})$. One easily verifies that
these additional conditions allow for a nonzero superconducting order
parameter $\Delta(\mathbf{k})$ for the real one-dimensional representations
corresponding to $\theta=0$ or $\theta=\pi$ only.

As a third example, we consider spinful fermions in a system with two
perpendicular mirror symmetries, $G=C_{2v}$. We write the point group elements
as $G=\\{\textbf{1},{\cal M}_{x},{\cal M}_{y},{\cal R}_{\pi}\\}$, where ${\cal
M}_{x}$ and ${\cal M}_{x}$ are mirror reflections in planes perpendicular to
the $x$ and $y$ axis, respectively, and ${\cal R}_{\pi}={\cal M}_{y}{\cal
M}_{x}$ is a rotation around the $z$ axis by $\pi$. For spinful particles the
factor system is nontrivial with ${\cal M}_{x}^{2}={\cal M}_{y}^{2}={\cal
R}_{\pi}^{2}=-1$. In that case there is only one (projective) representation
$u$ of $G$ (up to unitary transformations), which is two dimensional, see
Table 3. Superconducting phases are characterized by one-dimensional
representations of $G$ with a trivial factor system. There are four of these,
denoted $r_{\sigma_{x},\sigma_{y}}(g)$, with $\sigma_{x,y}=\pm 1$, see Table
3. Each of the one-dimensional representations of $G$ corresponds to a
distinct symmetry of the superconducting order parameter $\Delta$. For
example, for the one-dimensional representation $r_{--}$, the order parameter
is odd under mirror reflections in the $yz$ and $xz$ planes and even under a
twofold rotation around the $z$ axis.

| $\Gamma$ | 1 | ${\cal R}_{\pi}$ | ${\cal M}_{x}$ | ${\cal M}_{y}$
---|---|---|---|---|---
$r_{++}$ | $A_{1}$ | 1 | 1 | 1 | 1
$r_{--}$ | $A_{2}$ | 1 | 1 | -1 | -1
$r_{-+}$ | $B_{1}$ | 1 | -1 | 1 | -1
$r_{+-}$ | $B_{2}$ | 1 | -1 | -1 | 1
$u$ | $E$ | $\sigma_{0}$ | $i\sigma_{3}$ | $i\sigma_{1}$ | $i\sigma_{2}$
Table 3: Spinful representation (bottom row) and one-dimensional spinless
representations (top four rows) of the point group $C_{2v}$. The
representation $u$ is unique up to unitary transformations. The second column
lists the standard crystallographic notation for the representations, the
first column gives the notation used in the text.

##  III Classification of zero-dimensional Hamiltonians

The cornerstone of the construction of symmetry-based indicators is the
classification of zero-dimensional Hamiltonians with additional unitary
symmetries. Concretely, the classification problem is that of a Hamiltonian
$H$ with or without time-reversal symmetry, particle-hole antisymmetry, or
chiral antisymmetry, and with additional unitary symmetries specified by the
group $G$. The presence or absence of time-reversal symmetry, particle-hole
antisymmetry, and chiral antisymmetry determines the tenfold-way (Altland-
Zirnbauer) class and is indicated by the tripleAltland and Zirnbauer (1997)

$\eta=(\eta_{\cal T},\eta_{\cal P},\eta_{\cal C}).$ (24)

Here $\eta_{\cal T}={\cal T}^{2}=\pm 1$ if time-reversal symmetry is present
and $\eta_{\cal T}=0$ if time-reversal symmetry is absent. Similarly,
$\eta_{\cal P}={\cal P}^{2}=\pm 1$ or $0$ if particle-hole antisymmetry is
present or absent and $\eta_{\cal C}={\cal C}^{2}=1$ or $0$ in the presence or
absence of chiral antisymmetry, respectively. Table 4 summarizes the tenfold
way classification and lists both the notation using the triple $(\eta_{\cal
T},\eta_{\cal P},\eta_{\cal C})$ and the Cartan labeling. If the context
allows, we will use the Cartan label instead of the triple $\eta=(\eta_{\cal
T},\eta_{\cal P},\eta_{\cal C})$.

$\eta=(\eta_{\cal T},\eta_{\cal P},\eta_{\cal C})$ | Cartan | $\mathfrak{K}[0]$ | $\nu$
---|---|---|---
$(0,0,0)$ | A | $\mathbb{Z}$ | $\mathfrak{N}$
$(0,0,1)$ | AIII | $0$ | -
$(1,0,0)$ | AI | $\mathbb{Z}$ | $\mathfrak{N}$
$(1,1,1)$ | BDI | $\mathbb{Z}_{2}$ | $\mathfrak{p}$
$(0,1,0)$ | D | $\mathbb{Z}_{2}$ | $\mathfrak{p}$
$(-1,1,1)$ | DIII | $0$ | -
$(-1,0,0)$ | AII | $\mathbb{Z}$ | $\mathfrak{N}$
$(-1,-1,1)$ | CII | $0$ | -
$(0,-1,0)$ | C | $0$ | -
$(1,-1,1)$ | CI | $0$ | -
Table 4:  The tenfold-way symmetry classes are defined by the absence or
presence time-reversal symmetry, particle-hole antisymmetry, or chiral
antisymmetry. The triple $\eta=(\eta_{\cal T},\eta_{\cal P},\eta_{\cal C})$
(left column) indicates the presence or absence of these symmetries as well as
the square of the symmetry operation if it is present. The second column gives
the Cartan label for the symmetry class; the third column lists the
corresponding classifying group $\mathfrak{K}_{\eta}[0]$ for zero-dimensional
Hamiltonians. The rightmost column lists the topological invariant $\nu$,
where $\mathfrak{N}$ is the number of eigenstates (Kramers pairs in class AII)
with negative energy and $\mathfrak{p}$ is the Pfaffian invariant as defined
in Eq. (40).

For physically relevant systems, the crystalline symmetry group $G$ commutes
with the time-reversal operation ${\cal T}$, which is what we assume
throughout. The algebraic relation between $G$ and the particle-hole
conjugation operation ${\cal P}$ or the chiral operation ${\cal C}={\cal PT}$
is characterized by a one-dimensional representation $\Theta(g)$ of $G$, as
discussed in Sec. II.

With a suitable choice of basis, the (projective) representation $U(g)$ for a
Bogoliubov-de Gennes Hamiltonian $H$ can be brought into a canonical form
analogous to Eq. (5),

$U(g)=\mbox{diag}\,[r_{1}(g)\otimes\openone_{2N_{1}},\ldots,r_{n}(g)\otimes\openone_{2N_{n}}],$
(25)

where the $r_{\alpha}(g)$ are irreducible representations of $G$. To keep the
notation simple, we make the choice $N_{1}=N_{2}=\ldots=N_{n}=N$. The
Bogoliubov-de Gennes Hamiltonian $H$ is then brought into a block-diagonal
form,

$H=\mbox{diag}\,(\openone_{d_{1}}\otimes H_{1},\ldots,\openone_{d_{n}}\otimes
H_{n}),$ (26)

where $d_{\alpha}$ is the dimension of the irreducible representation
$r_{\alpha}$. The diagonal blocks $H_{\alpha}$ have dimension $2N$ and are no
longer subject to additional unitary symmetries. However, because of the basis
change involved in representing $U$ in the canonical form (25), the action of
the fundamental symmetry operations ${\cal T}$, ${\cal P}$, and/or ${\cal C}$
on these blocks need not be the same as their action on the original
Hamiltonian $H$ and different blocks may be related to each other by ${\cal
T}$, ${\cal P}$ and/or ${\cal C}$. Hence, additional considerations are needed
to determine which blocks are independent and to what tenfold-way class they
belong, which we now discuss.

Time-reversal operation ${\cal T}$.— If the irreducible representations
$r_{\alpha}$ and $r_{\alpha}^{*}$ are the same (up to a unitary
transformation), the time-reversal operation ${\cal T}$ acts within the
diagonal block corresponding to $r_{\alpha}$ and takes the form of an
effective time-reversal operation for $H_{\alpha}$, but with a square $W_{\cal
T}(\alpha)$ that may differ from $\eta_{\cal T}$. If the irreducible
representations $r_{\alpha}$ and $r_{\alpha}^{*}$ are different, ${\cal T}$
interchanges the diagonal blocks corresponding to $r_{\alpha}$ and
$r_{\alpha}^{*}$. In most examples, which of these cases applies can easily be
found by inspection, although one may also obtain the answer from the “Wigner
test” by calculating the quantityBradley (1972); Po  _et al._ (2017); Shiozaki
_et al._ (2018)

$W_{\cal T}(\alpha)=\eta_{\cal T}\left\langle
z_{g,g}\chi_{\alpha}(g^{2})\right\rangle_{G},$ (27)

where $\langle\ldots\rangle_{G}$ denotes an average over all $g\in G$,
$z_{g,g}$ is an element of the factor system, see Eq. (4), and
$\chi_{\alpha}(g)=\mbox{tr}\,r_{\alpha}(g)$ is the character of the
irreducible representation. The quantity $W_{\cal T}(\alpha)$ can take the
values $W_{\cal T}(\alpha)=\pm 1$ or $0$, corresponding to the two cases
discussed above.

Particle-hole conjugation ${\cal P}$.— Similarly, if the irreducible
representations $r_{\alpha}$ and $\Theta r_{\alpha}^{*}$ are the same, ${\cal
P}$ acts as an effective particle-hole conjugation operator for $H_{\alpha}$
with square $W_{\cal P}(\alpha)=\pm 1$, whereas if $r_{\alpha}$ and $\Theta
r_{\alpha}^{*}$ are different irreducible representations of $G$, ${\cal P}$
interchanges the corresponding diagonal blocks. Again, one may find out which
of the three cases applies by calculating

$W_{\cal P}{(\alpha)}=\eta_{\cal
P}\left\langle\Theta(g)^{*}z_{g,g}\chi_{\alpha}(g^{2})\right\rangle_{G},$ (28)

which takes the values $\pm 1$ or $0$, respectively, for the two cases
described above.

Chiral operation ${\cal C}$.— Finally, for the chiral operation ${\cal
C}={\cal PT}$ one only needs to distinguish two cases: If $r_{\alpha}$ and
$\Theta r_{\alpha}$ are the same irreducible representation of $G$, ${\cal C}$
acts as an effective antisymmetry of $H_{\alpha}$, whereas ${\cal C}$
interchanges the diagonal blocks corresponding to $r_{\alpha}$ and $\Theta
r_{\alpha}$ if they are different. This defines the quantity $W_{\cal
C}{(\alpha)}$, which takes the values $0$ or $1$.

We denote the full classifying group for zero-dimensional Hamiltonians with
the additional symmetry group $G$ as $\mathfrak{K}_{\eta}[G,\Theta]$, where
the multiindex $\eta$ indicates the tenfold-way class, see Table 4, and
$\Theta$ is the one-dimensional representation of $G$ that characterizes the
superconducting order parameter. The classifying group also depends on the
factor system of $G$, but we do not write this dependence explicitly. The
argument $\Theta$ is omitted for non-superconducting phases. The above
considerations then lead to the result

$\mathfrak{K}_{\eta}[G,\Theta]=\prod_{r_{\alpha}\,\text{irrep
of}\,G}\mathfrak{K}_{W(\alpha)}[0],$ (29)

where the multiindex $W(\alpha)=(W_{\cal T}(\alpha),W_{\cal P}(\alpha),W_{\cal
C}(\alpha))$ and the product is over the irreducible representations
$r_{\alpha}$ of $G$, where only one representative appears in the product if
multiple irreps are linked by the fundamental symmetries ${\cal T}$, ${\cal
P}$, or ${\cal C}$. The classifying groups $\mathfrak{K}_{\eta}[0]$, which
apply to tenfold-way classes without additional crystalline symmetries,Ryu
_et al._ (2010); Kitaev (2009) can be found in Table 4. We remark that the
classification approach described here also works for $d$-dimensional
Hamiltonians $H(\mathbf{k})$, as long as $G$ contains only onsite (i.e. non-
spatial, $\mathbf{k}$-independent) crystalline symmetries.

Examples.— We illustrate this general procedure using the three examples
already discussed in the previous Section. The classification results for
these three examples as well as for other relevant point groups are summarized
in Tables 5–8.

$G$ | $\Theta$ |  | $(r_{\alpha})_{W_{{\cal P}(\alpha)}}$ | $\mathfrak{K}_{{\rm D}}[G,\Theta]$
---|---|---|---|---
$C_{1}$ | $A$ | $1$ | $(A)_{\eta_{{\cal P}}}$ | $\mathbb{Z}_{2}$
$C_{i}$ | $A_{g}$ | $r_{+}$ | $(A_{g})_{\eta_{{\cal P}}},(A_{u})_{\eta_{{\cal P}}}$ | $\mathbb{Z}_{2}^{2}$
| $A_{u}$ | $r_{-}$ | $(A_{g},A_{u})_{0}$ | $\mathbb{Z}$
$C_{s}$ | $A^{\prime}$ | $r_{+}$ | $(^{1}\bar{E},^{2}\bar{E})_{0}$ | $\mathbb{Z}$
| $A^{\prime\prime}$ | $r_{-}$ | $(^{1}\bar{E})_{\eta_{{\cal P}}},(^{2}\bar{E})_{\eta_{{\cal P}}}$ | $\mathbb{Z}_{2}^{2}$
$C_{2}$ | $A$ | $r_{+}$ | $(^{1}\bar{E},^{2}\bar{E})_{0}$ | $\mathbb{Z}$
| $B$ | $r_{-}$ | $(^{1}\bar{E})_{\eta_{{\cal P}}},(^{2}\bar{E})_{\eta_{{\cal P}}}$ | $\mathbb{Z}_{2}^{2}$
$C_{3}$ | $A_{1}$ | $r_{0}$ | $\bar{E}_{\eta_{{\cal P}}},(^{1}\bar{E},^{2}\bar{E})_{0}$ | $\mathbb{Z}\times\mathbb{Z}_{2}$
| ${}^{2}E$ | $r_{2\pi/3}$ | ${}^{2}\bar{E}_{\eta_{{\cal P}}},(\bar{E},^{1}\bar{E})_{0}$ | $\mathbb{Z}\times\mathbb{Z}_{2}$
| ${}^{1}E$ | $r_{4\pi/3}$ | ${}^{1}\bar{E}_{\eta_{{\cal P}}},(\bar{E},^{2}\bar{E})_{0}$ | $\mathbb{Z}\times\mathbb{Z}_{2}$
$C_{4}$ | $A$ | $r_{0}$ | $(^{1}\bar{E}_{1},^{2}\bar{E}_{1})_{0},(^{1}\bar{E}_{2},^{2}\bar{E}_{2})_{0}$ | $\mathbb{Z}^{2}$
| $B$ | $r_{\pi}$ | $(^{1}\bar{E}_{1},^{2}\bar{E}_{2})_{0},(^{1}\bar{E}_{2},^{2}\bar{E}_{1})_{0}$ | $\mathbb{Z}^{2}$
| ${}^{2}E$ | $r_{\pi/2}$ | $(^{1}\bar{E}_{1})_{\eta_{{\cal P}}},(^{1}\bar{E}_{2})_{\eta_{{\cal P}}},(^{2}\bar{E}_{1},^{2}\bar{E}_{2})_{0}$ | $\mathbb{Z}\times\mathbb{Z}_{2}^{2}$
| ${}^{1}E$ | $r_{3\pi/2}$ | $(^{2}\bar{E}_{1})_{\eta_{{\cal P}}},(^{2}\bar{E}_{2})_{\eta_{{\cal P}}},(^{1}\bar{E}_{1},^{1}\bar{E}_{2})_{0}$ | $\mathbb{Z}\times\mathbb{Z}_{2}^{2}$
$C_{6}$ | $A$ | $r_{0}$ | $(^{1}\bar{E}_{1},^{2}\bar{E}_{1})_{0},(^{1}\bar{E}_{2},^{2}\bar{E}_{2})_{0},(^{1}\bar{E}_{3},^{2}\bar{E}_{3})_{0}$ | $\mathbb{Z}^{3}$
| $B$ | $r_{\pi}$ | $(^{1}\bar{E}_{1})_{\eta_{{\cal P}}},({}^{2}\bar{E}_{1})_{\eta_{{\cal P}}},(^{1}\bar{E}_{3},^{2}\bar{E}_{2})_{0},(^{2}\bar{E}_{3},^{1}\bar{E}_{2})_{0}$ | $\mathbb{Z}^{2}\times\mathbb{Z}_{2}^{2}$
| ${}^{2}E_{1}$ | $r_{4\pi/3}$ | $(^{1}\bar{E}_{1},^{2}\bar{E}_{3})_{0},(^{2}\bar{E}_{1},^{1}\bar{E}_{2})_{0},(^{2}\bar{E}_{2},^{1}\bar{E}_{3})_{0}$ | $\mathbb{Z}^{3}$
| ${}^{2}E_{2}$ | $r_{\pi/3}$ | $(^{1}\bar{E}_{3})_{\eta_{{\cal P}}},({}^{2}\bar{E}_{2})_{\eta_{{\cal P}}},(^{2}\bar{E}_{1},^{2}\bar{E}_{3})_{0},(^{1}\bar{E}_{1},^{1}\bar{E}_{2})_{0}$ | $\mathbb{Z}^{2}\times\mathbb{Z}_{2}^{2}$
| ${}^{1}E_{1}$ | $r_{2\pi/3}$ | $(^{1}\bar{E}_{1},^{2}\bar{E}_{2})_{0},(^{2}\bar{E}_{1},^{1}\bar{E}_{3})_{0},(^{2}\bar{E}_{3},^{1}\bar{E}_{2})_{0}$ | $\mathbb{Z}^{3}$
| ${}^{1}E_{2}$ | $r_{-\pi/3}$ | $(^{1}\bar{E}_{2})_{\eta_{{\cal P}}},({}^{2}\bar{E}_{3})_{\eta_{{\cal P}}},(^{1}\bar{E}_{1},^{1}\bar{E}_{3})_{0},(^{2}\bar{E}_{1},^{2}\bar{E}_{2})_{0}$ | $\mathbb{Z}^{2}\times\mathbb{Z}_{2}^{2}$
$C_{2v}$ | $A_{1}$ | $r_{++}$ | $(\bar{E})_{-\eta_{{\cal P}}}$ | 0
| $A_{2}$ | $r_{--}$ | $(\bar{E})_{\eta_{{\cal P}}}$ | $\mathbb{Z}_{2}$
| $B_{1}$ | $r_{-+}$ | $(\bar{E})_{\eta_{{\cal P}}}$ | $\mathbb{Z}_{2}$
| $B_{2}$ | $r_{+-}$ | $(\bar{E})_{\eta_{{\cal P}}}$ | $\mathbb{Z}_{2}$
$C_{3v}$ | $A_{1}$ |  | $(^{1}\bar{E},^{2}\bar{E})_{0},(\bar{E}_{1})_{-\eta_{{\cal P}}}$ | $\mathbb{Z}$
| $A_{2}$ |  | $(^{1}\bar{E})_{\eta_{{\cal P}}},(^{2}\bar{E})_{\eta_{{\cal P}}},(\bar{E}_{1})_{\eta_{{\cal P}}}$ | $\mathbb{Z}_{2}^{3}$
$C_{4v}$ | $A_{1}$ |  | $(\bar{E}_{1})_{-\eta_{{\cal P}}},(\bar{E}_{2})_{-\eta_{{\cal P}}}$ | 0
| $A_{2}$ |  | $(\bar{E}_{1})_{\eta_{{\cal P}}},(\bar{E}_{2})_{\eta_{{\cal P}}}$ | $\mathbb{Z}_{2}^{2}$
| $B_{1}$ |  | $(\bar{E}_{1},\bar{E}_{2})_{0}$ | $\mathbb{Z}$
| $B_{2}$ |  | $(\bar{E}_{1},\bar{E}_{2})_{0}$ | $\mathbb{Z}$
$C_{6v}$ | $A_{1}$ |  | $(\bar{E}_{1})_{-\eta_{{\cal P}}},(\bar{E}_{2})_{-\eta_{{\cal P}}},(\bar{E}_{3})_{-\eta_{{\cal P}}}$ | $0$
| $A_{2}$ |  | $(\bar{E}_{1})_{\eta_{{\cal P}}},(\bar{E}_{2})_{\eta_{{\cal P}}},(\bar{E}_{3})_{\eta_{{\cal P}}}$ | $\mathbb{Z}_{2}^{3}$
| $B_{1}$ |  | $(\bar{E}_{3})_{1},(\bar{E}_{1},\bar{E}_{2})_{0}$ | $\mathbb{Z}\times\mathbb{Z}_{2}$
| $B_{2}$ |  | $(\bar{E}_{3})_{1},(\bar{E}_{1},\bar{E}_{2})_{0}$ | $\mathbb{Z}\times\mathbb{Z}_{2}$
Table 5:  Classification of zero-dimensional BdG Hamiltonians with point group $G$ and one-dimensional representation $\Theta$ describing the symmetry of the superconducting order parameter in the absence of time-reversal symmetry and spin-rotation symmetry, corresponding to tenfold-way class D, $\eta_{\cal P}=1$. The second and third columns list the one-dimensional representation $\Theta$ using the standard crystallographic notation and the notation used in the examples in the main text, respectively. The fourth column lists the set $(r_{\alpha})_{W_{\mathcal{P}}(\alpha)}$ of irreps $r_{\alpha}$ together with the result of the Wigner test $W_{\mathcal{P}}(\alpha)$. In case irreps $r_{\alpha}$ and $\Theta r_{\alpha}^{*}$ are paired by particle-hole antisymmetry ($W_{\mathcal{P}}(\alpha)=0$), the paired representations are appear in brackets. From this information one can read off the classification, invariants, generators and representations using Table 10. For convenience we list the result for the classifying group $\mathfrak{K}_{\rm D}[G,\Theta]$ in the last column. $G$ | $\Theta$ |  | $(r_{\alpha})_{W_{{\cal P}(\alpha)}}$ | $\mathfrak{K}_{{\rm C}}[G,\Theta]$
---|---|---|---|---
$C_{1}$ | $A$ | $1$ | $(A)_{\eta_{{\cal P}}}$ | 0
$C_{i}$ | $A_{g}$ | $r_{+}$ | $(A_{g})_{\eta_{{\cal P}}},(A_{u})_{\eta_{{\cal P}}}$ | 0
| $A_{u}$ | $r_{-}$ | $(A_{g},A_{u})_{0}$ | $\mathbb{Z}$
$C_{s}$ | $A^{\prime}$ | $r_{+}$ | $(A^{\prime})_{\eta_{{\cal P}}},(A^{\prime\prime})_{\eta_{{\cal P}}}$ | 0
| $A^{\prime\prime}$ | $r_{-}$ | $(A^{\prime},A^{\prime\prime})_{0}$ | $\mathbb{Z}$
$C_{2}$ | $A$ | $r_{+}$ | $(A)_{\eta_{{\cal P}}},(B)_{\eta_{{\cal P}}}$ | 0
| $B$ | $r_{-}$ | $(A,B)_{0}$ | $\mathbb{Z}$
$C_{3}$ | $A_{1}$ | $r_{0}$ | $(A_{1})_{\eta_{{\cal P}}},(^{1}E,^{2}E)_{0}$ | $\mathbb{Z}$
| ${}^{2}E$ | $r_{2\pi/3}$ | $(^{1}E)_{\eta_{{\cal P}}},(A_{1},^{1}E)_{0}$ | $\mathbb{Z}$
| ${}^{1}E$ | $r_{4\pi/3}$ | $(^{2}E)_{\eta_{{\cal P}}},(A_{1},^{2}E)_{0}$ | $\mathbb{Z}$
$C_{4}$ | $A$ | $r_{0}$ | $A_{\eta_{\mathcal{P}}},B_{\eta_{\mathcal{P}}},(^{1}E,^{2}E)_{0}$ | $\mathbb{Z}$
| $B$ | $r_{\pi}$ | $(^{1}E)_{\eta_{{\cal P}}},(^{2}E)_{\eta_{{\cal P}}},(A,B)_{0}$ | $\mathbb{Z}$
| ${}^{2}E$ | $r_{\pi/2}$ | $(A,^{2}E)_{0},(B,^{1}E)_{0}$ | $\mathbb{Z}^{2}$
| ${}^{1}E$ | $r_{3\pi/2}$ | $(A,^{1}E)_{0},(B,^{2}E)_{0}$ | $\mathbb{Z}^{2}$
$C_{6}$ | $A$ | $r_{0}$ | $A_{\eta_{\mathcal{P}}},B_{\eta_{\mathcal{P}}},(^{1}E_{1},^{2}E_{1})_{0},(^{1}E_{2},^{2}E_{2})_{0}$ | $\mathbb{Z}^{2}$
| $B$ | $r_{\pi}$ | $(A,B)_{0},(^{1}E_{1},^{2}E_{2})_{0},(^{1}E_{2},^{2}E_{1})_{0}$ | $\mathbb{Z}^{3}$
| ${}^{2}E_{1}$ | $r_{4\pi/3}$ | $(^{1}E_{1})_{\eta_{\mathcal{P}}},(^{1}E_{2})_{\eta_{\mathcal{P}}},(A,^{2}E_{1})_{0},(B,^{2}E_{2})_{0}$ | $\mathbb{Z}^{2}$
| ${}^{2}E_{2}$ | $r_{\pi/3}$ | $(A,^{2}E_{2})_{0},(B,^{2}E_{1})_{0},({}^{1}E_{1},{}^{1}E_{2})_{0}$ | $\mathbb{Z}^{3}$
| ${}^{1}E_{1}$ | $r_{2\pi/3}$ | $(^{2}E_{1})_{\eta_{\mathcal{P}}},(^{2}E_{2})_{\eta_{\mathcal{P}}},(A,^{1}E_{1})_{0},(B,^{1}E_{2})_{0}$ | $\mathbb{Z}^{2}$
| ${}^{1}E_{2}$ | $r_{-\pi/3}$ | $(A,^{1}E_{2})_{0},(B,^{1}E_{1})_{0},({}^{2}E_{1},{}^{2}E_{2})_{0}$ | $\mathbb{Z}^{3}$
$C_{2v}$ | $A_{1}$ | $r_{++}$ | $(A_{1})_{\eta_{{\cal P}}},(A_{2})_{\eta_{{\cal P}}},(B_{1})_{\eta_{{\cal P}}},(A_{2})_{\eta_{{\cal P}}}$ | 0
| $A_{2}$ | $r_{--}$ | $(A_{1},A_{2})_{0},(B_{1},B_{2})_{0}$ | $\mathbb{Z}^{2}$
| $B_{1}$ | $r_{-+}$ | $(A_{1},B_{1})_{0},(A_{2},B_{2})_{0}$ | $\mathbb{Z}^{2}$
| $B_{2}$ | $r_{+-}$ | $(A_{1},B_{2})_{0},(B_{1},A_{2})_{0}$ | $\mathbb{Z}^{2}$
$C_{3v}$ | $A_{1}$ |  | $(A_{1})_{\eta_{\mathcal{P}}},(A_{2})_{\eta_{\mathcal{P}}},(E)_{\eta_{\mathcal{P}}}$ | $\mathbb{Z}_{2}^{3}$
| $A_{2}$ |  | $(A_{1},A_{2})_{0},(E)_{-\eta_{\mathcal{P}}}$ | $\mathbb{Z}$
$C_{4v}$ | $A_{1}$ |  | $(A_{1})_{\eta_{{\cal P}}},(A_{2})_{\eta_{{\cal P}}},(B_{1})_{\eta_{{\cal P}}},(B_{2})_{\eta_{{\cal P}}},E_{\eta_{{\cal P}}}$ | 0
| $A_{2}$ |  | $(A_{1},A_{2})_{0},(B_{1},B_{2})_{0},E_{-\eta_{{\cal P}}}$ | $\mathbb{Z}^{2}\times\mathbb{Z}_{2}$
| $B_{1}$ |  | $(A_{1},B_{1})_{0},(A_{2},B_{1})_{0},E_{\eta_{{\cal P}}}$ | $\mathbb{Z}^{2}$
| $B_{2}$ |  | $(A_{1},B_{2})_{0},(B_{1},A_{2})_{0},E_{\eta_{{\cal P}}}$ | $\mathbb{Z}^{2}$
$C_{6v}$ | $A_{1}$ |  | $(A_{1})_{\eta_{{\cal P}}},(A_{2})_{\eta_{{\cal P}}},(B_{1})_{\eta_{{\cal P}}},(B_{2})_{\eta_{{\cal P}}},(E_{1})_{\eta_{{\cal P}}},(E_{2})_{\eta_{{\cal P}}}$ | 0
| $A_{2}$ |  | $(A_{1},A_{2})_{0},(B_{1},B_{2})_{0},(E_{1})_{-\eta_{{\cal P}}},(E_{2})_{-\eta_{{\cal P}}}$ | $\mathbb{Z}^{2}\times\mathbb{Z}_{2}^{2}$
| $B_{1}$ |  | $(A_{1},B_{1})_{0},(A_{2},B_{1})_{0},(E_{1},E_{2})_{0}$ | $\mathbb{Z}^{3}$
| $B_{2}$ |  | $(A_{1},B_{1})_{0},(A_{2},B_{1})_{0},(E_{1},E_{2})_{0}$ | $\mathbb{Z}^{3}$
Table 6:  Same as Table 5, but in the presence of spin-rotation symmetry, i.e. for spinless representations of the crystalline symmetry group. BdG Hamiltonians with spin-rotation symmetry and without time-reversal symmetry correspond to tenfold-way class C, $\eta_{\cal P}=-1$. $G$ | $\Theta$ |  | $(r_{\alpha})_{W_{{\cal P}(\alpha)}}$ | $\mathfrak{K}_{{\rm DIII}}[G,\Theta]$ | $\mathfrak{K}_{{\rm BDI}}[G,\Theta]$
---|---|---|---|---|---
$C_{1}$ | $A$ | $1$ | $(A)_{(\eta_{{\cal T}},\eta_{{\cal P}},1)}$ | 0 | $\mathbb{Z}_{2}$
$C_{i}$ | $A_{g}$ | $r_{+}$ | $(A_{g})_{(\eta_{{\cal T}},\eta_{{\cal P}},1)},(A_{u})_{(\eta_{{\cal T}},\eta_{{\cal P}},1)}$ | 0 | $\mathbb{Z}_{2}^{2}$
| $A_{u}$ | $r_{-}$ | $(A_{g},A_{u})_{(\eta_{{\cal T}},0,0)}$ | $\mathbb{Z}$ | $\mathbb{Z}$
$C_{s}$ | $A^{\prime}$ | $r_{+}$ | $(^{1}\bar{E},^{2}\bar{E})_{(0,0,1)}$ | 0 | 0
| $A^{\prime\prime}$ | $r_{-}$ | $(^{1}\bar{E},^{2}\bar{E})_{(0,\eta_{{\cal P}},0)}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$
$C_{2}$ | $A$ | $r_{+}$ | $(^{1}\bar{E},^{2}\bar{E})_{(0,0,1)}$ | 0 | 0
| $B$ | $r_{-}$ | $(^{1}\bar{E},^{2}\bar{E})_{(0,\eta_{{\cal P}},0)}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$
$C_{3}$ | $A_{1}$ | $r_{0}$ | $\bar{E}_{(\eta_{{\cal T}},\eta_{{\cal P}},1)},(^{1}\bar{E},^{2}\bar{E})_{(0,0,1)}$ | 0 | $\mathbb{Z}_{2}$
$C_{4}$ | $A$ | $r_{0}$ | $(^{1}\bar{E}_{1},^{2}\bar{E}_{1})_{(0,0,1)},(^{1}\bar{E}_{2},^{2}\bar{E}_{2})_{(0,0,1)}$ | 0 | 0
| $B$ | $r_{\pi}$ | $(^{1}\bar{E}_{1},{}^{2}\bar{E}_{1},^{2}\bar{E}_{2},^{1}\bar{E}_{2})_{(0,0,0)}$ | $\mathbb{Z}$ | $\mathbb{Z}$
$C_{6}$ | $A$ | $r_{0}$ | $(^{1}\bar{E}_{1},^{2}\bar{E}_{1})_{(0,0,1)},(^{1}\bar{E}_{2},^{2}\bar{E}_{2})_{(0,0,1)},(^{1}\bar{E}_{3},^{2}\bar{E}_{3})_{(0,0,1)}$ | 0 | 0
| $B$ | $r_{\pi}$ | $(^{1}\bar{E}_{1},{}^{2}\bar{E}_{1})_{(0,\eta_{{\cal P}},0)},(^{1}\bar{E}_{2},^{2}\bar{E}_{2},{}^{2}\bar{E}_{3},^{1}\bar{E}_{3})_{(0,0,0)}$ | $\mathbb{Z}\times\mathbb{Z}_{2}$ | $\mathbb{Z}\times\mathbb{Z}_{2}$
$C_{2v}$ | $A_{1}$ | $r_{++}$ | $(\bar{E})_{(-\eta_{{\cal T}},-\eta_{{\cal P}},1)}$ | 0 | 0
| $A_{2}$ | $r_{--}$ | $(\bar{E})_{(-\eta_{{\cal T}},\eta_{{\cal P}},1)}$ | $\mathbb{Z}_{2}$ | 0
| $B_{1}$ | $r_{-+}$ | $(\bar{E})_{(-\eta_{{\cal T}},\eta_{{\cal P}},1)}$ | $\mathbb{Z}_{2}$ | 0
| $B_{2}$ | $r_{+-}$ | $(\bar{E})_{(-\eta_{{\cal T}},\eta_{{\cal P}},1)}$ | $\mathbb{Z}_{2}$ | 0
$C_{3v}$ | $A_{1}$ |  | $(^{1}\bar{E},^{2}\bar{E})_{(0,0,1)},(\bar{E}_{1})_{(-\eta_{{\cal T}},-\eta_{{\cal P}},1)}$ | 0 | 0
| $A_{2}$ |  | $(^{1}\bar{E},{}^{2}\bar{E})_{(0,\eta_{{\cal P}},0)},(\bar{E}_{1})_{(-\eta_{{\cal T}},\eta_{{\cal P}},1)}$ | $\mathbb{Z}_{2}^{2}$ | $\mathbb{Z}_{2}$
$C_{4v}$ | $A_{1}$ |  | $(\bar{E}_{1})_{(-\eta_{{\cal T}},-\eta_{{\cal P}},1)},(\bar{E}_{2})_{(-\eta_{{\cal T}},-\eta_{{\cal P}},1)}$ | 0 | 0
| $A_{2}$ |  | $(\bar{E}_{1})_{(-\eta_{{\cal T}},\eta_{{\cal P}},1)},(\bar{E}_{2})_{(-\eta_{{\cal T}},\eta_{{\cal P}},1)}$ | $\mathbb{Z}_{2}^{2}$ | 0
| $B_{1}$ |  | $(\bar{E}_{1},\bar{E}_{2})_{(-\eta_{{\cal T}},0,0)}$ | $\mathbb{Z}$ | $\mathbb{Z}$
| $B_{2}$ |  | $(\bar{E}_{1},\bar{E}_{2})_{(-\eta_{{\cal T}},0,0)}$ | $\mathbb{Z}$ | $\mathbb{Z}$
$C_{6v}$ | $A_{1}$ |  | $(\bar{E}_{1})_{(-\eta_{{\cal T}},-\eta_{{\cal P}},1)},(\bar{E}_{2})_{(-\eta_{{\cal T}},-\eta_{{\cal P}},1)},(\bar{E}_{3})_{(-\eta_{{\cal T}},-\eta_{{\cal P}},1)}$ | 0 | 0
| $A_{2}$ |  | $(\bar{E}_{1})_{(-\eta_{{\cal T}},\eta_{{\cal P}},1)},(\bar{E}_{2})_{(-\eta_{{\cal T}},\eta_{{\cal P}},1)},(\bar{E}_{3})_{(-\eta_{{\cal T}},\eta_{{\cal P}},1)}$ | $\mathbb{Z}_{2}^{3}$ | 0
| $B_{1}$ |  | $(\bar{E}_{3})_{(-\eta_{{\cal T}},\eta_{{\cal P}},1)},(\bar{E}_{1},\bar{E}_{2})_{(-\eta_{{\cal T}},0,0)}$ | $\mathbb{Z}\times\mathbb{Z}_{2}$ | $\mathbb{Z}$
| $B_{2}$ |  | $(\bar{E}_{3})_{(-\eta_{{\cal T}},\eta_{{\cal P}},1)},(\bar{E}_{1},\bar{E}_{2})_{(-\eta_{{\cal T}},0,0)}$ | $\mathbb{Z}\times\mathbb{Z}_{2}$ | $\mathbb{Z}$
Table 7:  Classification of zero-dimensional BdG Hamiltonians with point group $G$ and one-dimensional representation $\Theta$ describing the symmetry of the superconducting order parameter with time-reversal symmetry without spin-rotation symmetry, corresponding to tenfold-way classes DIII, $\eta=(-1,1,1)$. We also include systems with emergent time-reversal symmetry, corresponding to class BDI, $\eta=(1,1,1)$. The second and third columns list the one-dimensional representation using the standard crystallographic notation and in the notation used in the examples in the main text, respectively. The fourth column lists the set $(r_{\alpha})_{W_{\mathcal{P}}(\alpha)}$ of irreps $r_{\alpha}$, together with the result of the Wigner tests $W(\alpha)$. Irreps that are connected by application of the fundamental symmetry operations ${\cal T}$, ${\cal P}$, or ${\cal C}$ are shown together, using brackets. If two of the three Wigner tests $W_{\mathcal{T}}(\alpha)$, $W_{\mathcal{C}}(\alpha)$, and $W_{\mathcal{C}}(\alpha)$ are zero, the irreps form a pair and the single symmetry operation that leave the irrep invariant is the one with the nonzero Wigner label $W(\alpha)$. If all three Wigner tests are zero, $W_{\mathcal{T}}(\alpha)=W_{\mathcal{P}}(\alpha)=W_{\mathcal{C}}(\alpha)=0$, the irreps form a quartet. In that case the order of the four irreps between brackets is such, that the first two and last two irreps are interchanged by ${\cal T}$ and the first and third, and second and fourth irreps are interchanged by ${\cal P}$. From this information one can read off the classification, invariants, generators and representations using Table 9. For convenience we list the result for the classifying group $\mathfrak{K}_{\eta}[G,\Theta]$ in the last two columns. $G$ | $\Theta$ |  | $(r_{\alpha})_{W_{{\cal P}(\alpha)}}$ | $\mathfrak{K}_{{\rm CI}}[G,\Theta]$ | $\mathfrak{K}_{{\rm CII}}[G,\Theta]$
---|---|---|---|---|---
$C_{1}$ | $A$ | $1$ | $(A)_{(\eta_{{\cal T}},\eta_{{\cal P}},1)}$ | 0 | 0
$C_{i}$ | $A_{g}$ | $r_{+}$ | $(A_{g})_{(\eta_{{\cal T}},\eta_{{\cal P}},1)},(A_{u})_{(\eta_{{\cal T}},\eta_{{\cal P}},1)}$ | 0 | 0
| $A_{u}$ | $r_{-}$ | $(A_{g},A_{u})_{(\eta_{{\cal T}},0,0)}$ | $\mathbb{Z}$ | $\mathbb{Z}$
$C_{s}$ | $A^{\prime}$ | $r_{+}$ | $(A^{\prime})_{(\eta_{{\cal T}},\eta_{{\cal P}},1)},(A^{\prime\prime})_{(\eta_{{\cal T}},\eta_{{\cal P}},1)}$ | 0 | 0
| $A^{\prime\prime}$ | $r_{-}$ | $(A^{\prime},A^{\prime\prime})_{(\eta_{{\cal T}},0,0)}$ | $\mathbb{Z}$ | $\mathbb{Z}$
$C_{2}$ | $A$ | $r_{+}$ | $(A)_{(\eta_{{\cal T}},\eta_{{\cal P}},1)},(B)_{(\eta_{{\cal T}},\eta_{{\cal P}},1)}$ | 0 | 0
| $B$ | $r_{-}$ | $(A,B)_{(\eta_{{\cal T}},0,0)}$ | $\mathbb{Z}$ | $\mathbb{Z}$
$C_{3}$ | $A_{1}$ | $r_{0}$ | $(A_{1})_{(\eta_{{\cal T}},\eta_{{\cal P}},1)},(^{1}E,^{2}E)_{(0,0,1)}$ | 0 | 0
$C_{4}$ | $A$ | $r_{0}$ | $A_{(\eta_{{\cal T}},\eta_{{\cal P}},1)},B_{(\eta_{{\cal T}},\eta_{{\cal P}},1)},(^{1}E,^{2}E)_{(0,0,1)}$ | 0 | 0
| $B$ | $r_{\pi}$ | $(^{1}E,^{2}E)_{(0,\eta_{{\cal P}},0)},(A,B)_{(\eta_{{\cal T}},0,0)}$ | $\mathbb{Z}$ | $\mathbb{Z}$
$C_{6}$ | $A$ | $r_{0}$ | $A_{(\eta_{{\cal T}},\eta_{{\cal P}},1)},B_{(\eta_{{\cal T}},\eta_{{\cal P}},1)},(^{1}E_{1},^{2}E_{1})_{(0,0,1)},(^{1}E_{2},^{2}E_{2})_{(0,0,1)}$ | 0 | 0
| $B$ | $r_{\pi}$ | $(A,B)_{(\eta_{{\cal T}},0,0)},(^{1}E_{1},^{2}E_{1},{}^{2}E_{2},^{1}E_{2})_{(0,0,0)}$ | $\mathbb{Z}^{2}$ | $\mathbb{Z}^{2}$
$C_{2v}$ | $A_{1}$ | $r_{++}$ | $(A_{1})_{(\eta_{{\cal T}},\eta_{{\cal P}},1)},(A_{2})_{(\eta_{{\cal T}},\eta_{{\cal P}},1)},(B_{1})_{(\eta_{{\cal T}},\eta_{{\cal P}},1)},(A_{2})_{(\eta_{{\cal T}},\eta_{{\cal P}},1)}$ | 0 | 0
| $A_{2}$ | $r_{--}$ | $(A_{1},A_{2})_{(\eta_{{\cal T}},0,0)},(B_{1},B_{2})_{(\eta_{{\cal T}},0,0)}$ | $\mathbb{Z}^{2}$ | $\mathbb{Z}^{2}$
| $B_{1}$ | $r_{-+}$ | $(A_{1},B_{1})_{(\eta_{{\cal T}},0,0)},(A_{2},B_{2})_{(\eta_{{\cal T}},0,0)}$ | $\mathbb{Z}^{2}$ | $\mathbb{Z}^{2}$
| $B_{2}$ | $r_{+-}$ | $(A_{1},B_{2})_{(\eta_{{\cal T}},0,0)},(B_{1},A_{2})_{(\eta_{{\cal T}},0,0)}$ | $\mathbb{Z}^{2}$ | $\mathbb{Z}^{2}$
$C_{3v}$ | $A_{1}$ |  | $(A_{1})_{(\eta_{{\cal T}},\eta_{{\cal P}},1)},(A_{2})_{(\eta_{{\cal T}},\eta_{{\cal P}},1)},(E)_{(\eta_{{\cal T}},\eta_{{\cal P}},1)}$ | 0 | 0
| $A_{2}$ |  | $(A_{1},A_{2})_{(\eta_{{\cal T}},0,0)},(E)_{(\eta_{{\cal T}},-\eta_{{\cal P}},1)}$ | $\mathbb{Z}\times\mathbb{Z}_{2}$ | $\mathbb{Z}$
$C_{4v}$ | $A_{1}$ |  | $(A_{1})_{(\eta_{{\cal T}},\eta_{{\cal P}},1)},(A_{2})_{(\eta_{{\cal T}},\eta_{{\cal P}},1)},(B_{1})_{(\eta_{{\cal T}},\eta_{{\cal P}},1)},(A_{2})_{(\eta_{{\cal T}},\eta_{{\cal P}},1)},E_{(\eta_{{\cal T}},\eta_{{\cal P}},1)}$ | 0 | 0
| $A_{2}$ |  | $(A_{1},A_{2})_{(\eta_{{\cal T}},0,0)},(B_{1},B_{2})_{(\eta_{{\cal T}},0,0)},E_{(\eta_{{\cal T}},-\eta_{{\cal P}},1)}$ | $\mathbb{Z}^{2}\times\mathbb{Z}_{2}$ | $\mathbb{Z}^{2}$
| $B_{1}$ |  | $(A_{1},B_{1})_{(\eta_{{\cal T}},0,0)},(A_{2},B_{1})_{(\eta_{{\cal T}},0,0)},E_{(\eta_{{\cal T}},\eta_{{\cal P}},1)}$ | $\mathbb{Z}^{2}$ | $\mathbb{Z}^{2}$
| $B_{2}$ |  | $(A_{1},B_{2})_{(\eta_{{\cal T}},0,0)},(B_{1},A_{2})_{(\eta_{{\cal T}},0,0)},E_{(\eta_{{\cal T}},\eta_{{\cal P}},1)}$ | $\mathbb{Z}^{2}$ | $\mathbb{Z}^{2}$
$C_{6v}$ | $A_{1}$ |  | $(A_{1})_{(\eta_{{\cal T}},\eta_{{\cal P}},1)},(A_{2})_{(\eta_{{\cal T}},\eta_{{\cal P}},1)},(B_{1})_{(\eta_{{\cal T}},\eta_{{\cal P}},1)},(A_{2})_{(\eta_{{\cal T}},\eta_{{\cal P}},1)},(E_{1})_{(\eta_{{\cal T}},\eta_{{\cal P}},1)},(E_{2})_{(\eta_{{\cal T}},\eta_{{\cal P}},1)}$ | 0 | 0
| $A_{2}$ |  | $(A_{1},A_{2})_{(\eta_{{\cal T}},0,0)},(B_{1},B_{2})_{(\eta_{{\cal T}},0,0)},(E_{1})_{(\eta_{{\cal T}},-\eta_{{\cal P}},1)},(E_{2})_{(\eta_{{\cal T}},-\eta_{{\cal P}},1)}$ | $\mathbb{Z}^{2}\times\mathbb{Z}_{2}^{2}$ | $\mathbb{Z}^{2}$
| $B_{1}$ |  | $(A_{1},B_{1})_{(\eta_{{\cal T}},0,0)},(A_{2},B_{1})_{(\eta_{{\cal T}},0,0)},(E_{1},E_{2})_{(\eta_{{\cal T}},0,0)}$ | $\mathbb{Z}^{3}$ | $\mathbb{Z}^{3}$
| $B_{2}$ |  | $(A_{1},B_{1})_{(\eta_{{\cal T}},0,0)},(A_{2},B_{1})_{(\eta_{{\cal T}},0,0)},(E_{1},E_{2})_{(\eta_{{\cal T}},0,0)}$ | $\mathbb{Z}^{3}$ | $\mathbb{Z}^{3}$
Table 8:  Same as Table 7, but in the presence of spin-rotation symmetry, i.e.
for spinless representations of the crystalline symmetry group. Time-reversal
symmetric BdG Hamiltonians with spin-rotation symmetry correspond to tenfold-
way class CI, $\eta=(1,-1,1)$. We also include systems with emergent time-
reversal symmetry, corresponding to class CII, $\eta=(-1,-1,1)$.

We first consider a zero-dimensional superconductor with inversion symmetry.
The symmetry group $G=C_{i}=\\{\textbf{1},{\cal I}\\}$ and the two one-
dimensional representations $\Theta$ of $G$ that characterize the
superconducting phase are the two irreducible representations $r_{\pm}$ of
$G$, see Table 1. If the superconducting order is even under inversion,
corresponding to $\Theta=r_{+}$ (the “$A_{g}$” representation), Eqs. (5) and
(8) give

$U(\textbf{1})=\rho_{0}\tau_{0},\ \ U({\cal I})=\rho_{3}\tau_{0},$

where $\rho_{3}$ is the Pauli matrix in parity space. Correspondingly, the BdG
Hamiltonian $H=\mbox{diag}\,(H_{+},H_{-})$ is the diagonal sum of blocks with
even and odd parity under inversion. Since $U$ is real and does not involve
the particle-hole degree of freedom, the particle-hole conjugation operation
${\cal P}$ commutes with $G$, so that it acts within each parity subblock. Its
square is $W_{\cal P}=\eta_{\cal P}$, i.e., the same as for the full
Hamiltonian $H$. We conclude that

$\mathfrak{K}_{\eta}[C_{i},A_{g}]=\mathfrak{K}_{\eta}[0]^{2}$ (30)

for all tenfold-way symmetry classes $\eta$. The groups
$\mathfrak{K}_{\eta}[0]$ are listed in Table 4. This conclusion is consistent
with the Wigner test, which gives $W_{\cal P}(r_{\pm})=1$. If, on the other
hand, $\Delta$ is odd under inversion, which corresponds to the one-
dimensional representation $\Theta=r_{-}$ for the superconducting phase (the
“$A_{u}$ representation, see Table 1), the representation of $G$ for the full
BdG Hamiltonian is

$U(\textbf{1})=\rho_{0}\tau_{0},\ \ U({\cal I})=\rho_{3}\tau_{3},$

see Eq. (14). The block-diagonal structure of $H$ now involves one block with
even-parity particle-like states and odd-parity hole-like states and one block
with odd-parity particle-like states and even-parity hole-like states. The two
blocks are interchanged by particle-hole conjugation. One arrives at the same
conclusion by observing that $r_{+}$ and $\Theta r_{+}=r_{-}$ are different
irreducible representations of $G$. For the classifying group
$\mathfrak{K}_{\eta}[G,A_{u}]$ we thus find

$\mathfrak{K}_{(\eta_{\cal T},\eta_{\cal P},\eta_{\cal
C})}[C_{i},A_{u}]=\mathfrak{K}_{(\eta_{\cal T},0,0)}[0].$ (31)

Again, this conclusion is compatible with the Wigner test, which gives
$W_{\cal P}(r_{\pm})=0$.

The second example deals with a spinful system with a fourfold rotation
symmetry, $G=C_{4}$. The one-dimensional representation $\Theta\equiv
r_{\theta}$ of $G$ that characterizes the superconducting phase is labeled by
an angle $\theta$ which can take the values $0$, $\pi/2$, $\pi$, or $3\pi/2$,
see Table 2. The representation (21) of the rotation generator ${\cal
R}_{\pi/2}$ implies that $H$ has a block-diagonal structure in which particle-
like states with angular momentum $j=\frac{1}{2}$, $\frac{3}{2}$,
$\frac{5}{2}$, or $\frac{7}{2}$ are combined with hole-like states of angular
momentum $-j+2\theta/\pi\mod 4$. In the following discussion, we use the
angular momentum $j$ of the particle-like states to label the blocks. If
$\theta=0$ (the “$A$” representation), particle-hole conjugation ${\cal P}$
interchanges the blocks with $j=\frac{1}{2}$ and $\frac{7}{2}$, as well as the
blocks with $j=\frac{3}{2}$ and $\frac{5}{2}$. If $\theta=\pi/2$ (the
“${}^{2}\\!E$” representation), ${\cal P}$ interchanges the blocks with
$j=\frac{3}{2}$ and $\frac{7}{2}$, but acts within the blocks with
$j=\frac{1}{2}$ or $\frac{5}{2}$, squaring to one. If $\theta=\pi$ (the “$A$”
representation), ${\cal P}$ interchanges the blocks with $j=\frac{1}{2}$ and
$\frac{3}{2}$, as well as the blocks with $j=\frac{5}{2}$ and $j=\frac{7}{2}$.
Finally, if $\theta=3\pi/2$ (the “${}^{1}\\!E$” representation), ${\cal P}$
interchanges the blocks with $j=\frac{1}{2}$ and $\frac{5}{2}$, but acts
within the blocks with $j=\frac{3}{2}$ and $\frac{7}{2}$, again squaring to
one. We conclude that

$\displaystyle\mathfrak{K}_{{\rm D}}[C_{4},A]$
$\displaystyle=\,\mathfrak{K}_{{\rm D}}[C_{4},B]$
$\displaystyle=\,\mathfrak{K}_{{\rm A}}[0]^{2},$
$\displaystyle=\,\mathbb{Z}^{2}$ (32)

and

$\displaystyle\mathfrak{K}_{{\rm D}}[C_{4},^{1,2}\\!E]$
$\displaystyle=\,\mathfrak{K}_{{\rm A}}[0]\times\mathfrak{K}_{{\rm D}}[0]^{2}$
$\displaystyle=\,\mathbb{Z}\times\mathbb{Z}_{2}^{2}.$ (33)

The same conclusions can be obtained by performing the Wigner test.

$W_{\mathcal{P}}(\alpha)$ | Cartan | $\mathfrak{K}_{(0,W_{P}(\alpha),0)}[0]$ | $\nu$ | $H_{\text{gen}}$ | $H_{\text{ref}}$ | $\mathcal{P}$ | $U$
---|---|---|---|---|---|---|---
1 | D | $\mathbb{Z}_{2}$ | $\mathfrak{p}_{\alpha}$ | $-\tau_{3}\otimes 1_{d}$ | $\tau_{3}\otimes 1_{d}$ | $\tau_{1}K$ | $r_{\alpha}\otimes\tau_{0}$
-1 | C | $0$ | $-$ | - | $\tau_{3}\otimes 1_{d}$ | $\tau_{2}K$ | $r_{\alpha}\otimes\tau_{0}$
0 | A | $\mathbb{Z}$ | $\begin{cases}\mathfrak{N}_{\alpha}\\\ -\mathfrak{N}_{\Theta\alpha^{*}}\end{cases}$ | $-\tau_{3}\otimes 1_{d}$ | $\tau_{3}\otimes 1_{d}$ | $\tau_{1,2}K$ | $\left(\begin{array}[]{cc}r_{\alpha}&0\\\ 0&\Theta r_{\alpha}^{*}\end{array}\right)_{\tau}$
Table 9: Classifying groups $\mathfrak{K}_{W_{\cal P}(\alpha)}[0]$ and
topological invariants $\nu$ for the tenfold-way classes D and C, $\eta=(0,\pm
1,0)$ (third and fourth columns). These depend on the irrep $r_{\alpha}$
through the outcome of the Wigner test $W_{\cal P}(\alpha)$ only, which is
listed in the first column. The second column lists the effective tenfold-way
class corresponding to $W_{\cal P}(\alpha)$. The fifth and sixth columns list
a generating Hamiltonian $H_{\rm gen}$ and a reference Hamiltonian
$H_{\text{ref}}$, respectively. The seventh and eighth columns give the
representations of particle-hole conjugation $\mathcal{P}$ and the
representation $U$ of the elements of the crystalline symmetry group $G$. The
dimension of the irreducible representation $r_{\alpha}$ is denoted
$d_{\alpha}$.

In the presence of time-reversal symmetry only the real one-dimensional
representations $A$ and $B$ are relevant (corresponding to $\theta=0$, $\pi$,
respectively). In both cases the time-reversal operation ${\cal T}$
interchanges the blocks with angular momentum $j$ and angular momentum $-j\mod
4$. For the $A$ representation, ${\cal P}$ and ${\cal T}$ give the same
pairing of diagonal blocks of $H$ and the combined operation ${\cal C}={\cal
PT}$ leaves the diagonal blocks invariant. It follows that

$\displaystyle\mathfrak{K}_{{\rm DIII}}[C_{4},A]$
$\displaystyle=\,\mathfrak{K}_{{\rm A}}[0]^{2}$ $\displaystyle=\,0.$ (34)

On the other hand, for the $B$ representation, ${\cal P}$, ${\cal T}$, and
their product ${\cal C}={\cal PT}$ map all four diagonal blocks of $H$ to each
other and one has

$\displaystyle\mathfrak{K}_{{\rm DIII}}[C_{4},B]$
$\displaystyle=\,\mathfrak{K}_{{\rm A}}[0]$ $\displaystyle=\,\mathbb{Z}.$ (35)

The third example is that of spinful particles with symmetry group $G=C_{2v}$.
We refer to Table 3 for the irreducible representations with nontrivial and
trivial factor system. Since there is only one (two-dimensional) irreducible
representation $u$ — the “$\overline{E}$” representation, see Table 3 —, it
follows automatically that $u^{*}$ and $r_{\sigma_{x}\sigma_{y}}u^{*}$ are the
same representations of $G$. Indeed, one easily verifies that $u^{*}$ and
$r_{\sigma_{x},\sigma_{y}}u^{*}$ are equal to $u$ up to a unitary
transformation. To find the square of the effective time-reversal and
particle-hole conjugation operations, we consider the case of the one-
dimensional irreducible representation $\Theta=r_{++}$ (the $A_{1}$
representation) in detail and summarize results for the remaining three one-
dimensional representations $r_{+-}$, $r_{-+}$, and $r_{--}$ (the $B_{2}$,
$B_{1}$, and $A_{2}$ representations, respectively).

For $\Theta=r_{++}$ (“$A_{1}$” representation, see Table 3), and choosing the
representation of Table 3 for the representation of the generators ${\cal
M}_{x}$ and ${\cal M}_{y}$ for the normal-state block $h$, we find that their
representation for the full BdG Hamiltonian $H$ are $U({\cal
M}_{x})=i\sigma_{1}\tau_{3}$ and $U({\cal M}_{y})=i\sigma_{2}\tau_{0}$. To
bring the representation matrices to the canonical form (25) we have to
perform the basis transformation

$H\to\begin{pmatrix}1&0\\\ 0&i\sigma_{2}\end{pmatrix}H\begin{pmatrix}1&0\\\
0&-i\sigma_{2}\end{pmatrix}.$

After this basis transformation, ${\cal M}_{x}$ and ${\cal M}_{y}$ are
represented as $i\sigma_{1}\tau_{0}$ and $i\sigma_{2}\tau_{0}$, respectively,
consistent with Eq. (25). The particle-hole conjugation operation ${\cal P}$,
which was represented by $\sigma_{0}\tau_{1}K$ before the basis change, now
reads $\sigma_{2}\tau_{2}K$. Since the $C_{2v}$ symmetry enforces that $H$ is
of the form $\sigma_{0}H_{\overline{E}}$, compare with Eq. (26), the factor
$\sigma_{2}$ must be dropped from the representation of ${\cal P}$, since it
does not affect $H_{\overline{E}}$, and one arrives at the effective particle-
hole conjugation operator $\tau_{2}K$: The effective particle-hole conjugation
operator squares to minus one. The time-reversal operation ${\cal T}$, which
was given by $\sigma_{2}\tau_{0}K$ in the original basis, is unchanged by the
basis transformation. Again omitting the factor $\sigma_{2}$, one finds that
the effective time-reversal operation is $\tau_{0}K$. It follows that

$\mathfrak{K}_{(\eta_{\mathcal{T}},1,\eta_{\mathcal{C}})}[C_{2v},A_{1}]=\mathfrak{K}_{(-\eta_{\mathcal{T}},-1,\eta_{\mathcal{C}})}[0].$
(36)

The same result is found if one performs the Wigner test, see Eqs. (27) and
(28).

For $\Theta=r_{+-}$ (the “$B_{2}$” representation) one finds that the
representation $u$ of Table 3 for the normal-state block $h$ gives the
representations $U({\cal M}_{x})=i\sigma_{1}\tau_{0}$ and $U({\cal
M}_{y})=i\sigma_{2}\tau_{0}$ for the full BdG Hamiltonian $H$, which is
already in the canonical form (25). It then immediately follows that the
effective particle-hole conjugation operation is $\tau_{1}K$ and the effective
time-reversal operation is $\tau_{0}K$. For the remaining two cases $r_{-+}$,
and $r_{--}$ (the “$B_{1}$” and “$A_{2}$” representations), one may proceed as
described above for the $A_{1}$ representation, by performing a suitable basis
transformation, or use an alternative normal-state representation with
$u({\cal M}_{x})=i\sigma_{2}$, $u({\cal M}_{y})=i\sigma_{3}$ for $B_{1}$ and
$u({\cal M}_{x})=i\sigma_{3}$, $u({\cal M}_{y})=i\sigma_{1}$ for $A_{2}$, to
which the canonical form (25) applies without the need for a basis
transformation. Either way, we find that both the effective time-reversal
operation and the effective particle-hole conjugation operation square to
unity, $W_{\cal T}=-\eta_{\cal T}=1$ and $W_{\cal P}=\eta_{\cal P}=1$. As a
result, we have

$\displaystyle\mathfrak{K}_{(\eta_{\mathcal{T}},1,\eta_{\mathcal{C}})}[C_{2v},B_{1,2}]$
$\displaystyle=\mathfrak{K}_{(\eta_{\mathcal{T}},1,\eta_{\mathcal{C}})}[C_{2v},A_{2}]$
$\displaystyle=\mathfrak{K}_{(-\eta_{\mathcal{T}},1,\eta_{\mathcal{C}})}[0].$
(37)

$W(\alpha)$ | Cartan | $\mathfrak{K}_{W(\alpha)}[0]$ | $\nu$ | $H_{\text{gen}}$ | $H_{\text{ref}}$ | $\mathcal{T}$ | $\mathcal{P}$ | $U$
---|---|---|---|---|---|---|---|---
(0,0,0) | A | $\mathbb{Z}$ | $\begin{cases}\mathfrak{N}_{\alpha}\\\ \mathfrak{N}_{\alpha^{*}}\\\ -\mathfrak{N}_{\Theta\alpha^{*}}\\\ -\mathfrak{N}_{\Theta\alpha}\end{cases}$ | $-\sigma_{0}\tau_{3}\otimes 1_{d_{\alpha}}$ | $\sigma_{0}\tau_{3}\otimes 1_{d_{\alpha}}$ | $\sigma_{1,2}K$ | $\tau_{1,2}K$ | $\left(\begin{array}[]{cc}\left(\begin{array}[]{cc}r&0\\\ 0&r_{\alpha}^{*}\end{array}\right)_{\sigma}&0\\\ 0&\Theta\left(\begin{array}[]{cc}r_{\alpha}&0\\\ 0&r_{\alpha}^{*}\end{array}\right)_{\sigma}^{*}\end{array}\right)_{\tau}$
(0,0,1) | AIII | $0$ | - | - | $\sigma_{0}\tau_{3}\otimes 1_{d_{\alpha}}$ | $\sigma_{1,2}K$ | $\sigma_{1,2}\tau_{1}K$ | $\left(\begin{array}[]{cc}r_{\alpha}&0\\\ 0&r_{\alpha}^{*}\end{array}\right)_{\sigma}\otimes\tau_{0}$
(1,0,0) | AI | $\mathbb{Z}$ | $\begin{cases}\mathfrak{N}_{\alpha}\\\ -\mathfrak{N}_{\Theta\alpha^{*}}\end{cases}$ | $-\tau_{3}\otimes 1_{d_{\alpha}}$ | $\tau_{3}\otimes 1_{d_{\alpha}}$ | $K$ | $\tau_{1,2}K$ | $\left(\begin{array}[]{cc}r_{\alpha}&0\\\ 0&\Theta r_{\alpha}^{*}\end{array}\right)_{\tau}$
(1,1,1) | BDI | $\mathbb{Z}_{2}$ | $\mathfrak{p}_{\alpha}$ | $-\tau_{3}\otimes 1_{d_{\alpha}}$ | $\tau_{3}\otimes 1_{d_{\alpha}}$ | $K$ | $\tau_{1}K$ | $r_{\alpha}\otimes\tau_{0}$
(0,1,0) | D | $\mathbb{Z}_{2}$ | $\begin{cases}\mathfrak{p}_{\alpha}\\\ \mathfrak{p}_{\alpha^{*}}\end{cases}$ | $-\sigma_{0}\tau_{3}\otimes 1_{d_{\alpha}}$ | $\sigma_{0}\tau_{3}\otimes 1_{d_{\alpha}}$ | $\sigma_{1,2}K$ | $\tau_{1}K$ | $\left(\begin{array}[]{cc}r_{\alpha}&0\\\ 0&r_{\alpha}^{*}\end{array}\right)_{\sigma}\otimes\tau_{0}$
(-1,1,1) | DIII | $0$ | - | - | $\sigma_{0}\tau_{3}\otimes 1_{d_{\alpha}}$ | $\sigma_{2}K$ | $\tau_{1}K$ | $r_{\alpha}\otimes\sigma_{0}\tau_{0}$
(-1,0,0) | AII | $\mathbb{Z}$ | $\begin{cases}\mathfrak{N}_{\alpha}\\\ -\mathfrak{N}_{\Theta\alpha^{*}}\end{cases}$ | $-\sigma_{0}\tau_{3}\otimes 1_{d_{\alpha}}$ | $\sigma_{0}\tau_{3}\otimes 1_{d_{\alpha}}$ | $\sigma_{2}K$ | $\tau_{1,2}K$ | $\left(\begin{array}[]{cc}r_{\alpha}&0\\\ 0&\Theta r_{\alpha}^{*}\end{array}\right)_{\tau}\otimes\sigma_{0}$
(-1,-1,1) | CII | $0$ | - | - | $\sigma_{0}\tau_{3}\otimes 1_{d_{\alpha}}$ | $\sigma_{2}K$ | $\tau_{2}K$ | $r_{\alpha}\otimes\sigma_{0}\tau_{0}$
(0,-1,0) | C | $0$ | - | - | $\sigma_{0}\tau_{3}\otimes 1_{d_{\alpha}}$ | $\sigma_{1,2}K$ | $\tau_{2}K$ | $\left(\begin{array}[]{cc}r_{\alpha}&0\\\ 0&r_{\alpha}^{*}\end{array}\right)_{\sigma}\otimes\tau_{0}$
(1,-1,1) | CI | $0$ | - | - | $\tau_{3}\otimes 1_{d_{\alpha}}$ | $K$ | $\tau_{2}K$ | $r_{\alpha}\otimes\tau_{0}$
Table 10: Classifying groups $\mathfrak{K}_{W_{\cal P}(\alpha)}[0]$ and
topological invariants $\nu$ for the tenfold-way classes with time-reversal
symmetry (third and fourth columns). These depend on the irrep $r_{\alpha}$
through the outcome of the Wigner test $W_{\cal P}(\alpha)$ only, which is
listed in the first column. The second column lists the effective tenfold-way
class corresponding to $W_{\cal P}(\alpha)$. The fifth and sixth columns list
a generating Hamiltonian $H_{\rm gen}$ and a reference Hamiltonian
$H_{\text{ref}}$, respectively. The seventh, eighth, and ninth columns give
the representations of time-reversal ${\cal T}$, particle-hole conjugation
$\mathcal{P}$, and the representation $U$ of the elements of the crystalline
symmetry group $G$. The dimension of the irreducible representation
$r_{\alpha}$ is denoted $d_{\alpha}$.

A remark on classifying groups.— We close this section by making a few remarks
regarding the group structure of topological classification within $K$-theory.
Formally, the group structure within $K$-theory is given by the Grothendieck
construction,Nakahara (2003); Turner  _et al._ (2012) where the group elements
are represented by ordered pairs $(H_{1},H_{2})$ of hermitian matrix-valued
functions $H_{1,2}(\mathbf{k})$ of equal dimension. The two pairs
$(H_{1},H_{2})$ and $(H_{1}^{\prime},H_{2}^{\prime})$ are topologically
equivalent if $H_{1}\oplus H_{2}^{\prime}$ is continuously deformable to
$H_{1}^{\prime}\oplus H_{2}$. Loosely speaking the ordered pair
$(H_{1},H_{2})$ represents the “difference” of the two Hamiltonians $H_{1}$
and $H_{2}$.

For gapped phases with dimension $d>0$, a meaningful concept of “topologically
nontrivial” Hamiltonians can be obtained by defining a reference atomic-limit
Hamitonian $H_{\rm ref}(\mathbf{k})$ as a “topologically trivial phase”. (A
precise definition of “atomic-limit Hamiltonians” will be given in the next
Section.) Such a strategy results in a unique and well-defined topological
classification that is independent of the choice of the precise reference
Hamiltonian $H_{\rm ref}(\mathbf{k})$ if one considers Hamiltonians that
differ by an atomic-limit Hamiltonian to be in the same topological class. It
is this classification principle that underlies the classifying groups ${\cal
K}_{\eta}[G,\Theta]$ used throughout the remainder of this article for
Hamiltonians of dimension $d\geq 1$. On other hand, for zero-dimensional
Hamiltonians, there is no natural choice for the trivial phase and it is
important to adhere to the notion that a topological classification classifies
pairs of Hamiltonians only. It is this stricter notion of topological
classification that is used for the definition of the classifying groups
$\mathfrak{K}_{\eta}[G,\Theta]$ of zero-dimensional Hamiltonians, which play a
key role in the construction of symmetry-based indicators in the next Section.

Generators and invariants for the classifying groups
$\mathfrak{K}_{\eta}[G,\Theta]$.— The classifying groups
$\mathfrak{K}_{\eta}[G,\Theta]$ of zero-dimensional Hamiltonians with
additional point group $G$ and with one-dimensional representation $\Theta$
governing the pairing term $\Theta$ are determined by the effective Cartan
class of diagonal blocks corresponding to the irreducible representation
$\alpha$ or by pairs or quadruples of such blocks, as discussed above. We
tabulate the classification $\mathfrak{K}_{\eta}[0]$, the invariants $\nu$,
generators $H$ and representations $U(g)$ of all symmetry elements $g$,
$\mathcal{T}$ and $\mathcal{P}$ for all cases in Tables 9, 10. The zero-
dimensional invariants can be given as the number $\mathfrak{N}_{\alpha}$ of
negative energy eigenstates (Kramers pairs in case the block $H_{\alpha}$ is
invariant under an antiunitary symmetry $UK$ with $(UK)^{2}=-1$, i. e. in
Cartan class AII) with representation $\alpha$ or the Pfaffian invariant
$\mathfrak{p}_{\alpha}$ of the block $H_{\alpha}$ belonging to the irreducible
representation $r_{\alpha}$. In case there are multiple blocks related by
$\mathcal{T}$ or $\mathcal{P}$ we present all equivalent invariants.

For the calculation the zero-dimensional invariants
$\nu_{\alpha}=\mathfrak{N}_{\alpha}$ or $\nu_{\alpha}=\mathfrak{p}_{\alpha}$
it may be helpful to use the projector Dresselhaus  _et al._ (2007) onto a
subspace spanned by irreducible representation $\alpha$

$P_{\alpha}=d_{\alpha}\langle\chi_{\alpha}^{*}(g)U(g)\rangle_{G},$ (38)

where $\langle\ldots\rangle_{G}$ denotes the average over all elements $g\in
G$ and $\chi_{\alpha}(g)$ is the character. Choosing a basis in which the
projector $P_{\alpha}$ is block diagonal — which can be done for all irreps
$r_{\alpha}$ simultaneously, although this is not necessary for the
calculation to succeed —, the projected Hamiltonian $P_{\alpha}HP_{\alpha}$
takes the form $\text{diag}(H_{\alpha},0_{N-N_{\alpha}})$, and the topological
invariant can be computed as

$\nu_{\alpha}(H)\equiv\nu(H_{\alpha}).$ (39)

The Pfaffian invariant $\mathfrak{p}_{\alpha}$ of a subblock $H_{\alpha}$
invariant under an (effective) antiunitary antisymmetry with representation
$U$ with $(UK)^{2}=1$ is defined as

$(-1)^{\mathfrak{p}_{\alpha}}={\rm sign}\,\mbox{Pf}\,(H_{\alpha}U).$ (40)

##  IV Symmetry-based indicators of band topology

Whereas a full topological classification of a gapped Hamiltonians in $d$
dimensions — as described by the classifying group ${\cal K}_{\eta}[G,\Theta]$
— requires the analysis of matrix-valued functions $H(\mathbf{k})$ with the
momentum $\mathbf{k}$ taken on the full Brillouin zone, partial information on
the topological phase can be already obtained by inspection of the topological
class of matrices $H(\mathbf{k}_{\rm s})$ at a discrete set of high-symmetry
momenta $\mathbf{k}_{\rm s}$. Such an approach has been developed by Po et al.
for non-superconducting insulatorsPo  _et al._ (2017) (see also Refs. Bradlyn
_et al._ , 2017; Song  _et al._ , 2018b.) We here present this approach in
such a way that it can immediately be generalized to Hamiltonians of
Bogoliubov-de Gennes type.

We consider a band structure defined by the Hamiltonian $H(\mathbf{k})$, with
$\mathbf{k}$ an element of the Brillouin zone of a $d$-dimensional crystal
with (discrete) translation invariance and with symmorphic crystalline
symmetry described by the point group $G$. In addition, there may be
nonspatial symmetries such as time-reversal symmetry, particle-hole
antisymmetry, or chiral antisymmetry, which determine the tenfold-way symmetry
class. These nonspatial symmetries are characterized by the triple
$\eta=(\eta_{\cal T},\eta_{\cal P},\eta_{\cal C})$, as explained in the
previous Section. For superconductors, i.e. for Hamiltonians of BdG type, one
further needs to specify a one-dimensional representation $\Theta$ of $G$,
which characterizes the superconducting phase, see the discussion in Sec. II.

###  IV.1 Construction of a reference set of high-symmetry momenta

To define a representative set of high-symmetry momenta, we consider the group
$\tilde{G}$ of automorphisms of the Brillouin zone that are induced by
elements of $G$ and by the operations ${\cal T}$ and ${\cal P}$, if present.
(The element $g\in G$ induces an automorphism of the Brillouin zone by sending
$\mathbf{k}\to g\mathbf{k}$, whereas ${\cal T}$ and ${\cal P}$ send
$\mathbf{k}$ to $-\mathbf{k}$, in both cases identifying wavevectors that
differ by a reciprocal lattice vector.) For each momentum $\mathbf{k}$ we
define the “little group” $\tilde{G}_{\mathbf{k}}$ as the subgroup of elements
$\tilde{g}\in\tilde{G}$ such that $\tilde{g}\mathbf{k}=\mathbf{k}$. Two
momenta $\mathbf{k}_{1}$ and $\mathbf{k}_{2}$ are considered equivalent for
classification purposes if there exists an element $\tilde{g}\in\tilde{G}$ and
a continuous path between $\mathbf{k}_{1}$ and $\tilde{g}\mathbf{k}_{2}$ along
which the little group $\tilde{G}_{\mathbf{k}}$ does not change. (In
particular, this implies that $\mathbf{k}_{1}$ and $\mathbf{k}_{2}$ are
equivalent if $\mathbf{k}_{1}=\tilde{g}\mathbf{k}_{2}$ and that equivalent
momenta have the same little group.) A set of equivalent momenta is called of
“high-symmetry” if it does not border to another set of equivalent momenta
with a strictly larger little group. The representative set
$\\{\mathbf{k}_{\rm s}\\}$ of high-symmetry momenta is then constructed by
arbitrarily selecting one momentum from each set of equivalent high-symmetry
momenta.

###  IV.2 Definition of the groups “BS” and “SI”

To each high-symmetry momentum $\mathbf{k}_{\rm s}$ we may associate a
subgroup $G_{\mathbf{k}_{\rm s}}\subset G$ of crystalline symmetry operations
that leave $\mathbf{k}_{\rm s}$ invariant and a triple $\eta_{\mathbf{k}_{\rm
s}}=(\eta_{{\cal T},\mathbf{k}_{\rm s}},\eta_{{\cal P},\mathbf{k}_{\rm
s}},\eta_{\cal C})$ that indicate whether $\mathbf{k}_{\rm s}$ is invariant
under the operations ${\cal T}$ and ${\cal P}$, if present. The group
$G_{\mathbf{k}_{\rm s}}$, the symmetry indices $\eta_{\mathbf{k}_{\rm s}}$,
and the one-dimensional representation $\Theta$ of $G$ (suitably restricted to
$G_{\mathbf{k}_{\rm s}}$) determine the symmetry of the Hamiltonian
$H(\mathbf{k}_{\rm s})$ at the high-symmetry momentum $\mathbf{k}_{\rm s}$.
The group of “band labels” BL is defined as the combined set of topological
invariants that can be obtained from the topological information at these
high-symmetry momenta alone, Shiozaki  _et al._ (2018)

$\displaystyle\text{BL}_{\eta}[G,\Theta]=\prod_{\mathbf{k}_{\rm
s}}\mathfrak{K}_{\eta_{\mathbf{k}_{\rm s}}}[G_{\mathbf{k}_{\rm s}},\Theta].$
(41)

The classifying group
$\mathfrak{K}_{\eta_{\mathbf{k}}}[G_{\mathbf{k}},\Theta]$ describes zero-
dimensional topological phases protected by the onsite symmetry group
$G_{\mathbf{k}}$ and can be calculated from Eq. (29).

Although this procedure associates a well-defined topological invariant with
each Hamiltonian $H(\mathbf{k})$, there are three reasons why
$\text{BL}_{\eta}[G,\Theta]$ is different from the classifying group ${\cal
K}_{\eta}[G,\Theta]$ of topological phases with tenfold-way symmetries $\eta$,
point group $G$, and a superconducting phase with symmetry described by
$\Theta$: (i) Not every element in BL represents the band labels of a gapped
Hamiltonian $H(\mathbf{k})$, (ii) there may exist $d$-dimensional Hamiltonians
$H(\mathbf{k})$ without topologically protected anomalous boundary states for
which the band labels are nontrivial nevertheless, and (iii) there may be
Hamiltonians $H(\mathbf{k})$ with topologically protected anomalous boundary
states for which the band labels are trivial.

Problem (i) is addressed by the introduction of “compatibility relations”,
constraints on the band labels, which follow from the fact that
$H(\mathbf{k})$ is not only gapped at the high-symmetry momenta
$\mathbf{k}_{\rm s}$ — which is what allows the band labels to be defined in
the first place —, but also in the remainder of the Brillouin zone.Po  _et
al._ (2017); Bradlyn  _et al._ (2017); Kruthoff  _et al._ (2017) These
constraints appear, in the first place, because for a gapped Hamiltonian
$H(\mathbf{k})$ any band labels that can be defined for lower-symmetry momenta
$\mathbf{k}\notin\\{\mathbf{k}_{\rm s}\\}$ remain well-defined and continuous
if $\mathbf{k}$ approaches a high-symmetry momentum $\mathbf{k}_{\rm s}$.
Since band labels are essentially zero-dimensional topological invariants, see
the discussion above, we call these compatibility relations of “zero-
dimensional” (“0d”) type. Po et al. use these “0d compatibility relations” to
define the subgroup $\text{BS}\subset\text{BL}$ of “topological band labels”,

$\text{BS}=\left.\text{BL}\right|_{\text{$0d$ compatibility relations}}.$ (42)

We use $B[H(\mathbf{k})]$ to denote the element of BS associated with the
gapped Hamiltonian $H(\mathbf{k})$.

Whereas the group BL of band labels in principle depends on the choice of the
set of high-symmetry momenta $\\{\mathbf{k}_{\rm s}\\}$, the group BS is
independent of this choice, as long as sufficiently many momenta included. To
see this, we note that the inclusion of additional momenta beyond those
appearing in the set $\\{\mathbf{k}_{\rm s}\\}$ of high-symmetry momenta of
Sec. IV.1 adds band labels to BL, but not to BS, as band labels at lower-
symmetry-momenta are fully determined by the $0d$ compatibility relations.

Additional constraints on topological band labels of gapped Hamiltonians —
compatibility relations of “one-dimensional” (“$1d$”) or “two-dimensional”
(“$2d$”) type — may also appear because of the existence of topological
invariants on families of higher-dimensional subspaces of the Brillouin
zone,Turner  _et al._ (2012); Song  _et al._ (2018a) as we discuss below in
more detail. Elements in BL or BS that violate the compatibility constraints
describe “representation-enforced” gapless phases. Po  _et al._ (2017); Song
_et al._ (2018a); Ono and Watanabe (2018) Depending on the dimension of the
lower-symmetry region in reciprocal space and the type of constraint imposed
on it, these representation-enforced gapless phases may have nodal points,
nodal lines, or nodal planes.

Problem (ii) can be remedied by passing to the quotient groupPo  _et al._
(2017)

$\text{SI}=\text{BS}/\text{AI},$ (43)

the group of “symmetry-based indicators”. Here $\text{AI}\subset\text{BS}$ is
the subgroup generated by the image under $B$ of all Hamiltonians
$H(\mathbf{k})$ without boundary signature. A generating set of Hamiltonians
without boundary states consists of the “atomic-limit” Hamiltonians,
$d$-dimensional Hamiltonians that correspond to the arrangement of zero-
dimensional Hamiltonians on a suitably defined lattice. After dividing out AI,
a gapped Hamiltonian $H(\mathbf{k})$ is associated with a nontrivial element
of SI only if it has topologically protected anomalous boundary states. Po et
al. use the notation $X_{\rm BS}$ to refer to the group of symmetry-based
indicators.

There is no general solution to address problem (iii), however. This is why
the group SI is said to contain “indicators” of the topology, not a full
classification. Po  _et al._ (2017)

Although the topological invariants of zero-dimensional Hamiltonians are only
defined for pairs of Hamiltonians, the symmetry-based indicators are expressed
in terms of the band labels of a general Hamiltonian $H(\mathbf{k})$ with
$\mathbf{k}$-independent symmetry representation, without comparing to a
reference Hamiltonian.

###  IV.3 Construction of the subgroup $\text{AI}\subset\text{BS}$

Following Po et al. Po  _et al._ (2017) and Bradlyn et al. Bradlyn  _et al._
(2017), the construction of the subgroup AI proceeds in three steps: (1) One
first selects a representative collection $W$ of high-symmetry Wyckoff
positions in the unit cell. Hereto, one defines the site symmetry group
$G_{\mathbf{x}}$ for a position $\mathbf{x}$ in the unit cell as the subgroup
of $G$ that leaves $\mathbf{x}$ invariant, possibly up to lattice
translations, and arranges lattice positions with the same site symmetry group
which are related by a continuous path and/or by an element of $G$ into
equivalence classes (“Wyckoff positions”). A representative collection of
positions $W$ in the unit cell is then obtained by choosing a position from
each equivalence class that does not border on an equivalence class with a
larger site symmetry group. (2) For each position $\mathbf{x}\in W$ the
“orbit” of $\mathbf{x}$ is defined as the set $\\{g\mathbf{x}+\bm{t}|\,g\in
G\\}$ for $\bm{t}$ in Bravais lattice. The orbit defines a $G$-symmetric
Bravais lattice. We label the lattice sites within the unit cell by
$O_{\mathbf{x}}=\\{\mathbf{x}_{\sigma}\\}_{\sigma=1,2,\ldots}$, with the
convention $\mathbf{x}_{1}\equiv\mathbf{x}$. (3) For each Wyckoff position
$\mathbf{x}$ and each pair $(\alpha,\Theta)$ of an irreducible representation
of the site symmetry group $G_{\mathbf{x}}$ and the associated one-dimensional
representation $\Theta$ describing the symmetry of the superconducting order
parameter, we construct a pair of atomic-limit Hamiltonians
$H_{\mathbf{x},{\rm ref}}^{(\alpha,\Theta)}$ and $H_{\mathbf{x},{\rm
gen}}^{(\alpha,\Theta)}$ by placing the zero-dimensional reference and
generating Hamiltonians $H_{\rm ref}$ and $H_{\rm gen}$ of
$\mathfrak{K}_{\eta}[G_{\mathbf{x}},\Theta]$ on the positions
$\mathbf{x}_{\sigma}$ of the orbit of $\mathbf{x}$. One verifies that
invariance of the zero-dimensional Hamiltonians $H_{\mathbf{x},{\rm
ref}}^{(\alpha,\Theta)}$ and $H_{\mathbf{x},{\rm gen}}^{(\alpha,\Theta)}$
under the site symmetry group $G_{\mathbf{x}}$ ensures that such a procedure
yields well-defined $G$-symmetric, translationally invariant, Hamiltonians
$H_{\mathbf{x},{\rm ref}}^{(\alpha,\Theta)}(\mathbf{k})$ and
$H_{\mathbf{x},{\rm gen}}^{(\alpha,\Theta)}(\mathbf{k})$ on a $d$-dimensional
lattice. The subgroup $\text{AI}\subset\text{BS}$ is generated by the
differences $B[H_{\mathbf{x},{\rm
gen}}^{(\alpha,\Theta)}(\mathbf{k})]-B[H_{\mathbf{x},{\rm
ref}}^{(\alpha,\Theta)}(\mathbf{k})]$ of the images of the Hamiltonians
obtained in this manner.

The procedure of taking differences of images under $B$ is necessary for the
construction of AI, because, although the reference Hamiltonians $H_{\rm ref}$
in Tables 9 and 10 have been chosen such that they map to the trivial element
under $B$, the band labels need not be trivial for atomic-limit Hamiltonian
obtained by placing copies of $H_{\rm ref}$ on nontrivial Wyckoff positions.
The nontrivial band labels for the reference Hamiltonian have their origin in
the $\mathbf{k}$-dependence of the representation of the point group $G$,
which is unavoidable if minimal $0d$ superconductors are placed at nontrivial
Wyckoff positions. Such a $\mathbf{k}$ dependence of the representation is not
compatible with our construction of the topological band labels
$B[H(\mathbf{k})]$, which assume a $\mathbf{k}$-independent representation of
$G$. (The simplest example where this is the case is a one-dimensional
superconductor with inversion symmetry and odd-parity superconducting order,
see Sec. V.2).

One can resolve this issue either by adding multiple copies of the “trivial”
$0d$ Hamiltonian $H_{\rm ref}$ at the same Wyckoff position, which eventually
allows one to construct a $\mathbf{k}$-independent representation of $G$. We
here prefer to take an equivalent, but computationally more efficiently
approach, in which we remedy this situation by comparing atomic-limit
Hamiltonians with the same number of orbitals (per representation pair
$(\alpha,\Theta)$) at each Wyckoff position only. We construct a “trivial”
reference atomic-limit Hamiltonian for this case, and define the band labels
of each other atomic-limit Hamiltonian as the difference of the band labels
with the reference atomic-limit Hamiltonian. The trivial reference atomic-
limit Hamiltonian is chosen such that all Wyckoff positions and all
representation pairs $(\alpha,\Theta)$ are represented: It is the direct sum
of $0d$ reference Hamiltonians $H_{\rm ref}$ from Tables 9 and 10 for all
pairs $(\alpha,\Theta)$, placed at all Wyckoff positions. Generator atomic-
limit Hamiltonians $H^{(\alpha,\Theta)}_{\mathbf{x},{\rm gen}}(\mathbf{k})$
are obtained by replacing a $0d$ reference Hamiltonian in this direct sum by a
generator from Tables 9 and 10 for the orbit $O_{\mathbf{x}}$ and
representation pair $(\alpha,\Theta)$ only. Since the map $B$ is additive
under the taking of direct sums, taking the difference of the band labels of
this generator Hamiltonian and the reference atomic-limit Hamiltonian is the
same as taking the difference $B[H_{\mathbf{x},{\rm
gen}}^{(\alpha,\Theta)}(\mathbf{k})]-B[H_{\mathbf{x},{\rm
ref}}^{(\alpha,\Theta)}(\mathbf{k})]$ for atomic-limit Hamiltonians involving
a single Wyckoff position $\mathbf{x}$ and representation pair
$(\alpha,\Theta)$ only, which justifies the procedure outlined step (iii)
above.

Each reference Hamiltonian or generator Hamiltonian is compactly expressed
asBradley (1972); Po  _et al._ (2017); Bradlyn  _et al._ (2017)

$\displaystyle H_{\mathbf{x}}^{(\alpha,\Theta)}(\mathbf{k})$
$\displaystyle=\mathbb{1}_{|O_{\mathbf{x}}|\times|O_{\mathbf{x}}|}\otimes
H_{\mathbf{x}}^{(\alpha,\Theta)},$ (44)

where $|O_{\mathbf{x}}|$ is the number of sites in the orbit of $\mathbf{x}$.
The representation of $g\in G$ is

$\displaystyle[U^{\mathbf{x},\alpha}(g,\mathbf{k})]_{\sigma^{\prime}j,\sigma
i}$
$\displaystyle=\delta^{\prime}_{\mathbf{x}_{\sigma^{\prime}},g\mathbf{x}_{\sigma}}e^{-i\mathbf{k}\cdot(g\mathbf{x}_{\sigma}-\mathbf{x}_{\sigma^{\prime}})}$
(45)
$\displaystyle\times\frac{z_{g,g_{\sigma}}}{z_{g_{\sigma^{\prime}},g_{\sigma^{\prime}}^{-1}gg_{\sigma}}}[u_{\mathbf{x}}^{\alpha}(g_{\sigma^{\prime}}^{-1}t_{\mathbf{x}_{\sigma^{\prime}}-\mathbf{x}_{\sigma}}gg_{\sigma})]_{ji},$

where the indices $\sigma$ and $\sigma^{\prime}$ label the positions
$\mathbf{x}_{\sigma}$ in the orbit of $\mathbf{x}$ and the indices $i$ and $j$
the degree of freedom of $H_{\mathbf{x}}^{(\alpha,\Theta)}$. Further, a choice
$\\{g_{\sigma}\\}_{\sigma=1,\ldots,|O_{\mathbf{x}}|}$ from $G$ was made in
such a way that $g_{\sigma=1}$ is the identity and
$g_{\sigma}\mathbf{x}=\mathbf{x}_{\sigma}$ for
$\sigma=2,\ldots,|O_{\mathbf{x}}|$. Each group element is decomposed as
$g=t_{\mathbf{R}}g_{\sigma}h$ with $h\in G_{\mathbf{x}}$, $t_{\mathbf{R}}\in
T$ and $\mathbf{R}=g\mathbf{x}_{\sigma}-\mathbf{x}_{\sigma^{\prime}}$. The
Kronecker symbol $\delta^{\prime}_{\mathbf{x}_{1},\mathbf{x}_{2}}$ is non-zero
and equal to one only when $\mathbf{x}_{1}=\mathbf{x}_{2}+\mathbf{R}$ for some
$\mathbf{R}$ in Bravais lattice. The representation $u_{\alpha}$ is given in
Tables 9 and 10.

In this construction it is important that time-reversal ${\cal T}$ and
particle-hole conjugation ${\cal P}$ are local operations. If ${\cal P}$ is
not a local operation, an arrangement of disconnected onsite Hamiltonians can
acquire a spurious non-locality, in spite of the absence of inter-site matrix
elements. The suggestion of Ref. Skurativska  _et al._ , 2019 that no atomic
limit can be defined for Hamiltonians of Bogoliubov-de Gennes type can be
traced back to the use of a non-local particle-hole conjugation in that
reference.

###  IV.4 Compatibility relations

In the examples that we consider in the next Sections, we find that it is
sufficient to use compatibility relations for gapped hermitian matrices
$H(\mathbf{k})$ based on local-in-$\mathbf{k}$ symmetries only. These
compatibility relations may be applied to the entire matrix $H(\mathbf{k})$ or
to a diagonal block, if $H(\mathbf{k})$ has a block structure that is
preserved throughout a part of the Brillouin zone. The compatibility relations
involve topological invariants of $H(\mathbf{k})$ defined using information
about $H(\mathbf{k})$ away from the high-symmetry points $\mathbf{k}_{\rm s}$.
Depending on the dimensionality of the subspace required to define these
topological invariants, we distinguish between compatibility relations of $0d$
type (which are used to define the group BS of topological band labels) and of
$1d$ or $2d$ type.Shiozaki  _et al._ (2018) We now discuss the construction of
$0d$, $1d$, and $2d$ compatibility relations in detail.

0a.— The number of negative eigenvalues of a hermitian matrix $H(\mathbf{k})$
with a gapped spectrum does not change if $\mathbf{k}$ is changed
continuously. This gives a compatibility constraint if $\mathbf{k}$ can be
changed continuously between high-symmetry points $\mathbf{k}_{{\rm s},1}$ and
$\mathbf{k}_{{\rm s},2}$, while preserving Hermiticity of (a subblock of) the
Hamiltonian, and if the number of negative eigenvalues can be related to the
band labels at $\mathbf{k}_{{\rm s},1}$ and $\mathbf{k}_{{\rm s},2}$.

0b.— The sign of the Pfaffian of a gapped, antisymmetric matrix
$UH(\mathbf{k})$, with $U$ an appropriately chosen unitary operator, does not
change if $\mathbf{k}$ is changed continuously. This results in a
compatibility constraint for topological band labels if $\mathbf{k}$ can be
changed continuously between high-symmetry points $\mathbf{k}_{{\rm s},1}$ and
$\mathbf{k}_{{\rm s},2}$, while preserving the antisymmetry of (a subblock of)
the matrix $UH(\mathbf{k})$, and if the sign of the Pfaffian can be related to
the band labels at $\mathbf{k}_{{\rm s},1}$ and $\mathbf{k}_{{\rm s},2}$.

1.— The topological invariant of a one-dimensional Hamiltonian $H(k)$,
obtained by restricting $H(\mathbf{k})$ to a one-dimensional closed contour in
reciprocal space, does not change if this contour is continuously deformed.
Such a topological invariant can be a winding number, if $H(k)$ has a chiral
antisymmetry, but it may also be a $\mathbb{Z}_{2}$ invariant (the first
Stiefel-Whitney number Fang  _et al._ (2015); Ahn  _et al._ (2018)), if $H(k)$
or a diagonal block of it satisfy local-in-$\mathbf{k}$ symmetries that place
it effectively in tenfold-way classes AI or BDI. This gives an additional
compatibility relation if the topological invariant can be related to the
topological band labels of $H(\mathbf{k}_{\rm s})$ at high-symmetry points
$\mathbf{k}_{\rm s}$ on the contour and if the contour can be deformed so that
it can be made to pass through different sets of high-symmetry momenta, while
preserving local-in-$\mathbf{k}$ symmetries of $H(\mathbf{k})$. An example of
such a $1d$ compatibility relation is given in the example discussed in Sec.
VII.4 (tenfold-way class CI with inversion symmetry).

2.— The topological invariant of a gapped hermitian matrix $H(k_{1},k_{2})$,
which is defined on a two-dimensional plane cutting through the Brillouin
zone, does not change if the position of the plane is shifted continuously.
The topological invariant is a Chern number if the local-in-$\mathbf{k}$
crystalline symmetries are such that $H(k_{1},k_{2})$ or a diagonal block of
it are effectively in tenfold-way classes A, D, or C, or it may be a
$\mathbb{Z}_{2}$ invariant (the second Stiefel-Whitney number Fang  _et al._
(2015); Ahn  _et al._ (2018)), if $H(k_{1},k_{2})$ is effectively in tenfold-
way classes CI or AI. This yields an additional compatibility relation if the
topological invariant can be related to the band labels of $H(\mathbf{k}_{\rm
s})$ at high-symmetry points $\mathbf{k}_{\rm s}$ on the plane and if the
plane can be shifted continuously, such that it can be made to pass through
different sets of high-symmetry momenta while preserving the hermiticity of
(the relevant subblock of) $H(\mathbf{k})$. This “2$d$ compatibility relation”
is used in Ref. Po  _et al._ , 2017 to identify representation-enforced nodal
semimetals (see also Ref. Turner  _et al._ , 2012; Song  _et al._ , 2018a). It
also appears, e.g., in the example discussed in Sec. VII.1 (tenfold-way class
D with inversion symmetry).

Unlike the construction of the groups BS and AI, we are not aware of a method
that allows one to implement the compatibility relations based on higher-
dimensional topological invariants in an algorithmic way. The “bottleneck” is
the relation between the $n$-dimensional topological invariant used to
construct the compatibility relation and the topological band labels of
$H(\mathbf{k})$ at high-symmetry momenta $\mathbf{k}_{\rm s}$, which requires
knowledge of the full classification of $n$-dimensional crystalline phases.
This is not a problem if the symmetry-based indicators are used to determine
the topological phase of a given BdG Hamiltonian with a gapped spectrum, but
it does affect the use of SI as a proxy for the full boundary classification
group ${\cal K}$.

Generalizing the definition of BS, see Eq. (42), we may define the subgroup
series

$\text{BS}^{(2)}\subset\text{BS}^{(1)}\subset\text{BS}^{(0)}\equiv\text{BS},$
(46)

where $\text{BS}^{(n)}$ is the subgroup of BL obtained by imposing all
compatibility constraints involving topological invariants of dimension $\leq
n$. Correspondingly, we may define the group $\text{SI}^{(n)}\subset\text{SI}$
of symmetry-based indicators as

$\text{SI}^{(2)}\subset\text{SI}^{(1)}\subset\text{SI}^{(0)}\equiv\text{SI},$
(47)

with

$\text{SI}^{(n)}=\text{BS}^{(n)}/\text{AI}$ (48)

and $\text{SI}^{(0)}=\text{SI}$. For the examples we consider, we find that
$\text{SI}^{(d-1)}$ contains no gapless phases, implying that the
compatibility relations based on local-in-$\mathbf{k}$ symmetries are
sufficient for these cases.

###  IV.5 Weak-pairing limit

In the weak-pairing limit — superconducting order parameter $\Delta$ much
smaller than energy scales typical for the normal-state band structure — the
band labels $\mathfrak{N}_{\alpha}^{\mathbf{k}_{\rm s}}$ and
$\mathfrak{p}_{\alpha}^{\mathbf{k}_{\rm s}}$ of the BdG Hamiltonian
$H({\mathbf{k}_{\rm s}})$ at the high-symmetry momentum $\mathbf{k}_{\rm s}$
can be expressed in terms of zero-dimensional topological invariants of the
normal-state Hamiltonians $h({\mathbf{k}_{\rm s}})$ and $h(-{\mathbf{k}_{\rm
s}})$. For $\mathfrak{N}_{\alpha}^{\mathbf{k}_{\rm s}}$ one has Ono  _et al._
(2019); Shiozaki (2019b); Ono  _et al._ (2019)

$\displaystyle\mathfrak{N}_{\alpha}^{\mathbf{k}_{\rm s}}$
$\displaystyle=n_{\alpha}^{\mathbf{k}_{\rm
s}}|_{\text{occ}}+n_{\Theta\alpha^{*}}^{-{\mathbf{k}_{\rm
s}}}|_{\text{unocc}}$ (49) $\displaystyle=n_{\alpha}^{\mathbf{k}_{\rm
s}}|_{\text{occ}}-n_{\Theta\alpha^{*}}^{-{\mathbf{k}_{\rm
s}}}|_{\text{occ}}+n_{\Theta\alpha^{*}}^{-{\mathbf{k}_{\rm s}}}$

where $n^{\mathbf{k}_{\rm s}}_{\alpha}$, $n^{\mathbf{k}_{\rm
s}}_{\alpha}|_{\text{occ}}$ and $n^{\mathbf{k}_{\rm
s}}_{\alpha}|_{\text{unocc}}$ are the total number, the number of occupied,
and the number of unoccupied bands of the subblock
$h_{\alpha}({\mathbf{k}_{\rm s}})$ of the normal-state Hamiltonian,
respectively. Similarly, the Pfaffian invariant
$\mathfrak{p}_{\alpha}^{\mathbf{k}_{\rm s}}$ can be expressed as Shiozaki
(2019b)

$\mathfrak{p}_{\alpha}^{\mathbf{k}_{\rm s}}=n_{\alpha}^{\mathbf{k}_{\rm
s}}|_{\text{occ}}\mod 2.$ (50)

##  V Examples: One dimension

We now discuss symmetry-based indicators for Bogoliubov-de Gennes-type
Hamiltonians for a selected set of point groups $G$ and tenfold-way classes.

###  V.1 Trivial point group $G=C_{1}$, class D

Without crystalline symmetries, the Hamiltonian $H(k)$ is particle-hole
antisymmetric at both high-symmetry momenta $k=0$ and $k=\pi$. In the absence
of time-reversal symmetry and spin-rotation symmetry (class D), this gives a
$\mathbb{Z}_{2}$-classification for $H(0)$ and $H(\pi)$, so that
$\text{BL}=\mathbb{Z}_{2}^{2}$. There are no compatibility relations, hence

$\text{BS}=\mathbb{Z}_{2}^{2}.$ (51)

The corresponding topological invariants are $(-1)^{\mathfrak{p}^{(0)}}={\rm
sign}\,\text{Pf}\,[H(0)\tau_{1}]$ and $(-1)^{\mathfrak{p}^{(\pi)}}={\rm
sign}\,\text{Pf}\,[H(\pi)\tau_{1}]$, giving

$B[H(k)]=\\{\mathfrak{p}^{(0)},\mathfrak{p}^{(\pi)}\\}.$ (52)

Without crystalline symmetries there is only one Wyckoff position in a one-
dimensional crystal. Placing zero-dimensional Hamiltonians on the generic
Wyckoff position, one obtains two topologically different classes of atomic-
limit Hamiltonians,

$\text{AI}=\mathbb{Z}_{2}.$

As a subgroup of BS, AI corresponds to the pairs
$(\mathfrak{p}^{(0)},\mathfrak{p}^{(\pi)})$ with
$\mathfrak{p}^{(0)}=\mathfrak{p}^{(\pi)}$. The group
$\text{SI}=\text{BS}/\text{AI}=\mathbb{Z}_{2}$ is identical to the full
classification group

$\text{SI}_{{\rm D}}[C_{1}]={\cal K}_{{\rm D}}[C_{1}]=\mathbb{Z}_{2},$ (53)

the nontrivial element of which describes one-dimensional topological
superconductors (the “Kitaev chain”). The symmetry-based indicator for the
topological superconductor phase is

$z_{1}=\sum_{k_{\rm s}}\mathfrak{p}^{(k_{\rm s})}\mod 2.$ (54)

In the weak pairing limit, the indicator (54) gives the parity of the number
of Fermi level crossings between $k=0$ and $k=\pi$. Sato (2010); Fu and Berg
(2010) The same expression for the full BdG Hamiltonian was obtained in Ref.
Shiozaki, 2019b.

###  V.2 Inversion symmetry $C_{i}$, class D

Representation $\Theta=A_{g}$.— For a superconducting order parameter that
transforms according to the representation $A_{g}$, inversion may be
represented as $\rho_{3}\tau_{0}$, the Pauli matrices $\rho_{3}$ and
$\tau_{0}$ acting in parity and particle-hole space, respectively, see Sec.
II. At the high symmetry momenta $k_{\rm s}=(0),(\pi)$, the Hamiltonian
$H(k_{\rm s})=\mbox{diag}\,(H_{+}(k_{\rm s}),H_{-}(k_{\rm s}))$ is the
diagonal sum of blocks acting within the even and odd parity subspaces, where
the blocks correspond to the irreducible representations $A_{g}$ and $A_{u}$
of $C_{i}$. The preceding discussion of a one-dimensional Hamiltonian $H(k)$
without crystalline symmetries applies to the two blocks separately. In
particular, it follows that $\text{BL}=\mathbb{Z}_{2}^{4}$.

To find the compatibility constraints for gapped Hamiltonians, we note that
the combination of inversion and particle-hole conjugation gives an
antiunitary antisymmetry local in momentum space,

$H(k)=-\rho_{3}\tau_{1}H(k)^{*}\rho_{3}\tau_{1}.$

As a result, $H(k)\rho_{3}\tau_{1}$ is antisymmetric for all $k$, so that
${\rm sign}\,\mbox{Pf}\,[H(k)\rho_{3}\tau_{1}]$ is well-defined and
$k$-independent if $H(k)$ is gapped. Considering that at the high-symmetry
momenta $k=0$ and $k=\pi$ one has ${\rm
sign}\,\mbox{Pf}\,[H(k)\rho_{3}\tau_{1}]=(-1)^{\mathfrak{p}_{+}^{(k)}+\mathfrak{p}_{-}^{(k)}}$,
one finds the compatibility constraint

$\mathfrak{p}_{+}^{(0)}+\mathfrak{p}_{-}^{(0)}=\mathfrak{p}_{+}^{(\pi)}+\mathfrak{p}_{-}^{(\pi)}\mod
2$ (55)

for gapped Hamiltonians, where we use the subscripts $+$ and $-$ for the
representations $A_{g}$ and $A_{u}$, respectively. Using the compatibility
relation (55) to eliminate $\mathfrak{p}_{-}^{(\pi)}$ as an independent band
label, we find

$\text{BS}=\mathbb{Z}_{2}^{3},\ \
B[H(k)]=\\{\mathfrak{p}_{+}^{(0)},\mathfrak{p}_{+}^{(\pi)},\mathfrak{p}_{-}^{(0)}\\},$
(56)

With inversion symmetry there are two inequivalent Wyckoff positions, labeled
$x=0$ and $x=\frac{1}{2}$. A generating set of atomic-limit Hamiltonians is
obtained by placing zero-dimensional inversion-symmetric Hamiltonians at each
of the Wyckoff positions. Placing a generator of $\mathfrak{K}_{D}[0]$ with
topological invariant $\mathfrak{p}=1$ and irreducible representation $A_{g}$
or $A_{u}$ at position $x=0$ we obtain a $k$-independent Hamiltonian for which
inversion is represented as by $\tau_{0}$ or $-\tau_{0}$, respectively, and
particle-hole conjugation by $\tau_{1}K$. Under the map $B$, these
Hamiltonians are mapped to $\\{1,0,1\\}$ and $\\{0,1,0\\}$ for the $A_{g}$ and
$A_{u}$ representations, respectively. Placing a generator of
$\mathfrak{K}_{D}[0]$ with topological invariant $\mathfrak{p}=1$ and
irreducible representation $A_{g}$ or $A_{u}$ at the Wyckoff position
$x=\frac{1}{2}$, one obtains a $k$-independent Hamiltonian for which inversion
is represented by $\tau_{0}e^{ik}$ or $-\tau_{0}e^{ik}$ for $A_{g}$ and
$A_{u}$, respectively. At $k=\pi$ the irreducible representations are
interchanged, so that under $B$ such a Hamiltonian is mapped to $\\{1,0,0\\}$
and $\\{0,1,1\\}$ for the irreducible representations $A_{g}$ and $A_{u}$,
respectively. The images of these four generating atomic-limit Hamiltonians
under $B$ span the whole group $\text{AI}=\text{BS}$. The quotient group

$\text{SI}_{{\rm D}}[C_{i},A_{g}]=0.$ (57)

The conclusion that SI contains no nontrivial gapped phases is consistent with
the triviality of the classifying group

${\cal K}_{{\rm D}}[C_{i},A_{g}]=0$ (58)

for this symmetry class. Trifunovic and Brouwer (2019)

Representation $A_{u}$.— For the representation $A_{u}$ the inversion
operation for the BdG Hamiltonian is represented as $\rho_{3}\tau_{3}$, see
Sec. II. The Hamiltonian $H(k)=\mbox{diag}\,(H_{+}(k),H_{-}(k))$ is block
diagonal with respect to the eigenvalues $\pm$ of $\rho_{3}\tau_{3}$,
corresponding to the fundamental representations $A_{g,u}$ of $C_{i}$. At
$k=0$ and $k=\pi$ the two blocks are minus the particle-hole conjugate of each
other. The topological invariants of $H(0)$ and $H(\pi)$ are
$\mathfrak{N}_{+}^{(0)}$ and $\mathfrak{N}_{+}^{(\pi)}$, where
$\mathfrak{N}_{\pm}^{(k)}$ is the number of positive eigenvalues of
$H_{\pm}(k)$. (Note that the topological invariants involve the numbers
$\mathfrak{N}_{+}^{(k)}$ for $k=0,\pi$ only; Particle-hole antisymmetry
implies that $\mathfrak{N}_{+}^{(k)}=-\mathfrak{N}_{-}^{(k)}$ at $k=0,\pi$.)
Accordingly, $\text{BL}=\mathbb{Z}^{2}$.

The combination of inversion and particle-hole conjugation gives an
antiunitary antisymmetry of $H(k)$ that is local in momentum space and squares
to $-1$, thus $H(k)$ belongs to class C with trivial classification in zero-
dimension. We conclude that there are no compatibility constraints for
$\mathfrak{N}_{+}^{(0)}$ and $\mathfrak{N}_{+}^{(\pi)}$, so that

$\text{BS}=\mathbb{Z}^{2},\ \
B[H(k)]=\\{\mathfrak{N}_{+}^{(0)},\mathfrak{N}_{+}^{(\pi)}\\}.$ (59)

Figure 1: Topological band labels associated with an atomic-limit
superconductor are calculated as the difference of band labels of arrays with
the $0d$ generator Hamiltonian $H_{\rm gen}$ and with the $0d$ reference
Hamiltonian $H_{\rm ref}$ at Wyckoff position $x$. The figure shows this
schematically for the Wyckoff position $x=\frac{1}{2}$.

To construct the subgroup $\text{AI}\subset\text{BS}$ we place zero-
dimensional generator Hamiltonians at the Wyckoff positions $x=0$ or
$x=\frac{1}{2}$, which gives the $k$-independent Hamiltonian
$H=\mbox{diag}\,(-1,1)=-\tau_{3}$, with representation $\tau_{1}K$ for
particle-hole conjugation. For the Wyckoff position $x=0$ inversion is
represented by $\tau_{3}$, so that the corresponding atomic-limit Hamiltonian
maps to the element $\\{1,1\\}$ of BS. For the Wyckoff position
$x=\frac{1}{2}$ inversion is represented by $\tau_{3}e^{ik}$. The $A_{g}$ and
$A_{u}$ blocks are interchanged at $k=\pi$ and the corresponding atomic-limit
Hamiltonian maps to $\\{1,-1\\}$, where the $-1$ originates from the
difference with an atomic-limit superconductor obtained by placing the
reference Hamiltonian at the same Wyckoff position, see Fig. 1 for a schematic
picture. The subgroup $\text{AI}=\mathbb{Z}\times 2\mathbb{Z}$ thus consists
of all elements $\\{\mathfrak{N}_{+}^{(0)},\mathfrak{N}_{+}^{(\pi)}\\}$ for
which $\mathfrak{N}_{+}^{(0)}-\mathfrak{N}_{+}^{(\pi)}$ is even. The quotient
$\text{SI}=\text{BS}/\text{AI}=\mathbb{Z}_{2}$ contains the symmetry-based
indicator

$z_{1}=\sum_{k_{\rm s}}\mathfrak{N}_{+}^{k_{\rm s}}\mod 2.$ (60)

The conclusion that

$\text{SI}_{{\rm D}}[C_{i},A_{u}]=\mathbb{Z}_{2}$ (61)

is consistent with the classifying group

${\cal K}_{{\rm D}}[C_{i},A_{u}]=\mathbb{Z}_{2}$ (62)

for this symmetry class, Trifunovic and Brouwer (2019) which describes a
topological superconductor phase with a single zero-energy Majorana bound
state at each end. An example of a nontrivial Hamiltonian in this symmetry
class is

$H(k)=\mbox{diag}\,\left(\tau_{3}(1-m-\cos k)+\tau_{1}\sin k,\tau_{3}\right)$

with $0<m<2$.

The weak-pairing limit (49) of the symmetry-based indicator $z_{1}$ agrees
Refs. Skurativska  _et al._ , 2019; Shiozaki, 2019b; Ono  _et al._ , 2019. The
symmetry-based indicator $z_{1}$ for the full BdG Hamiltonian agrees with the
forms defined in Refs. Shiozaki, 2019b; Ono  _et al._ , 2019.

##  VI Examples: Two dimensions

###  VI.1 Two mirror symmetries $C_{2v}$, class D

The four high-symmetry momenta are $\mathbf{k}_{\rm s}=(0,0)$, $(0,\pi)$,
$(\pi,0)$, and $(\pi,\pi)$. Each of these four momenta is invariant under the
full group $C_{2v}$ and under particle-hole conjugation, so that the
Hamiltonian $H(\mathbf{k})$ has the symmetries corresponding to a zero-
dimensional Hamiltonian of tenfold-way class D and with symmetry group
$C_{2v}$. As discussed in Sec. III, such Hamiltonians are topologically
trivial if the superconducting phase transforms according to the $A_{1}$
representation of $C_{2v}$, whereas there is a $\mathbb{Z}_{2}$ invariant
$\mathfrak{p}$ for the representations $A_{2}$, $B_{1}$, and $B_{2}$. Hence,
we find that

$\text{BL}=\left\\{\begin{array}[]{ll}0&\mbox{for $\Theta=A_{1}$},\\\
\mathbb{Z}_{2}^{4}&\mbox{for $\Theta=A_{2}$, $B_{1}$, or
$B_{2}$.}\end{array}\right.$

Representation $\Theta=A_{1}$.— For the case $\Theta=A_{1}$ the group of
symmetry-based indicators is

$\text{SI}_{{\rm D}}[C_{2v},A_{1}]=0,$ (63)

consistent with the triviality

${\cal K}_{{\rm D}}[C_{2v},A_{1}]=0$ (64)

of the full classifying group, see App. B.4.

Representation $\Theta=B_{2}$.— For this representation we have ${\cal
M}_{x}{\cal P}=-{\cal P}{\cal M}_{x}$, ${\cal M}_{y}{\cal P}={\cal P}{\cal
M}_{y}$ and ${\cal M}_{x}^{2}={\cal M}_{y}^{2}=-1$. For $k_{y}=0$ one has the
effective “particle-hole antisymmetry” ${\cal P}{\cal M}_{x}$ with $({\cal
P}{\cal M}_{x})^{2}=1$, which is local in momentum space. The mirror operation
$i{\cal M}_{y}$ is also local in momentum space if $k_{y}=0$, squares to one,
and commutes with the effective particle-hole symmetry, $({\cal P}{\cal
M}_{x})(i{\cal M}_{y})=(i{\cal M}_{y})({\cal P}{\cal M}_{x})$. We conclude
that for each $0<k_{x}<\pi$ the Hamiltonian $H(k_{x},0)$ satisfies an onsite
“inversion symmetry” with $\Theta=A_{g}$, so that it has a nontrivial
topological classification given by the classifying group
$\mathfrak{K}_{\mathrm{D}}(C_{i},A_{g})=\mathbb{Z}_{2}^{2}$. Upon taking the
limits $k_{x}\to 0$, $\pi$, the classification of $H(k_{x},0)$ for generic
$k_{x}$ can be related to the classification at the high-symmetry momenta
$k_{x}=0$, $\pi$, which is given by the topological band labels
$\mathfrak{p}^{(0,0)}$ and $\mathfrak{p}^{(\pi,0)}$. Explicit calculation
gives that only the diagonal elements $(\mathfrak{p},\mathfrak{p})$ of
$\mathfrak{K}_{\mathrm{D}}(C_{i},A_{g})=\mathbb{Z}_{2}^{2}$ are allowed, with
$\mathfrak{p}=\mathfrak{p}^{(0,0)}=\mathfrak{p}^{(\pi,0)}$. It follows that
one has the compatibility constraint

$\mathfrak{p}^{(0,0)}=\mathfrak{p}^{(\pi,0)}.$ (65)

In the same way one finds

$\mathfrak{p}^{(0,\pi)}=\mathfrak{p}^{(\pi,\pi)}.$ (66)

One may obtain the same compatibility relations using the explicit
representations ${\cal P}=\rho_{0}\tau_{1}K$, $U({\cal
M}_{x})=i\rho_{1}\tau_{0}$, $U({\cal M}_{y})=i\rho_{2}\tau_{0}$ used in Sec.
III, so that $H(\mathbf{k})$ satisfies

$\displaystyle H(k_{x},k_{y})$
$\displaystyle=\,\rho_{1}\tau_{0}H(-k_{x},k_{y})\rho_{1}\tau_{0}$
$\displaystyle=\,\rho_{2}\tau_{0}H(k_{x},-k_{y})\rho_{2}\tau_{0}$
$\displaystyle=\,-\rho_{0}\tau_{1}H(-k_{x},-k_{y})^{*}\rho_{0}\tau_{1}.$

For $k_{y}=0$ these constraints imply that
$H(k_{x},0)=-\rho_{1}\tau_{1}H(k_{x},0)^{*}\rho_{1}\tau_{1}=\rho_{2}\tau_{0}H(k_{x},0)\rho_{2}\tau_{0}$,
so that we may write $H(k_{x},0)=\rho_{0}h_{0}+\rho_{2}h_{2}$ with
$h_{0}\tau_{1}$ and $h_{2}\tau_{1}$ antisymmetric. Further, $h_{0}\pm h_{2}$
is gapped for all $0\leq k_{x}\leq\pi$, so that the Pfaffian of
$\tau_{1}(h_{0}\pm h_{2})$ is nonzero and cannot change sign for $0\leq
k_{x}\leq\pi$. Since $h_{2}=0$ for $k_{x}=0$, $\pi$, the compatibility
relations (65) and (66) follow immediately. It follows that

$\text{BS}=\mathbb{Z}_{2}^{2},\ \
B[H(\mathbf{k})]=\\{\mathfrak{p}(\pi,0),\mathfrak{p}(\pi,\pi)\\}.$ (67)

To construct AI, we place generators at one of the four Wyckoff positions. The
Wyckoff positions are $(x,y)=(0,0)$, $(0,\frac{1}{2})$, $(\frac{1}{2},0)$, and
$(\frac{1}{2},\frac{1}{2})$. From the onsite representations $U({\cal
M}_{x})=i\rho_{1}\tau_{0}$, $U({\cal M}_{y})=i\rho_{2}\tau_{0}$ we derive the
$\mathbf{k}$-dependent representations

$\displaystyle U^{\mathbf{x}}({\cal M}_{x},\mathbf{k})$
$\displaystyle=\,i\rho_{1}\tau_{0}e^{2ik_{x}x},$ $\displaystyle
U^{\mathbf{x}}({\cal M}_{y},\mathbf{k})$
$\displaystyle=\,i\rho_{2}\tau_{0}e^{2ik_{y}y}$

for a zero-dimensional Hamiltonian placed at Wyckoff position
$\mathbf{x}=(x,y)$. The $\mathbf{k}$-dependent factors appear, because for the
Wyckoff positions other than $(0,0)$ the operations ${\cal M}_{x}$ and/or
${\cal M}_{y}$ correspond to an onsite operation followed by a translation by
a lattice vector. As these induced representations differ from the onsite
representations by a sign for half of the elements of the symmetry group,
which is a change that can be accommodated by a basis transformation, the
induced representation is the same at all high-symmetry momenta. After
verifying that placing the trivial reference Hamiltonian $H_{\rm ref}$ at the
same Wyckoff positions produces the trivial element in BS, we conclude that
the subgroup $\text{AI}\subset\text{BS}$ consists of the elements with
$\mathfrak{p}^{(0,0)}=\mathfrak{p}^{(0,\pi)}$. We conclude that

$\text{SI}_{{\rm D}}[C_{2v},B_{2}]=\mathbb{Z}_{2},$ (68)

corresponding to the symmetry-based indicator

$z_{1;x}=\sum_{\mathbf{k}_{\rm s}|k_{{\rm
s},x}=\pi}\mathfrak{p}^{\mathbf{k}_{\rm s}}\mod 2.$ (69)

This symmetry-based indicator describes a weak phase of one-dimensional
$C_{2v}$-symmetric topological superconductors in the $y$ direction, stacked
in the $x$ direction,

${\cal K}_{{\rm D}}[C_{2v},B_{2}]=\mathbb{Z}_{2},$ (70)

see App. B.4. An example of a generator Hamiltonian is

$H_{(1;x)}=\rho_{0}\tau_{3}(1-m-\cos k_{y})+\rho_{1}\tau_{2}\sin k_{y},$

with $0<m<2$.

Representation $\Theta=B_{1}$.— The discussion for the $B_{1}$ representation
is similar to the discussion above. One finds

$\displaystyle\text{SI}_{{\rm D}}[C_{2v},B_{1}]=$
$\displaystyle\,\mathbb{Z}_{2},$ (71) $\displaystyle{\cal K}_{{\rm
D}}[C_{2v},B_{1}]=$ $\displaystyle\,\mathbb{Z}_{2}.$ (72)

The only topologically nontrivial gapped phase is a weak phase of one-
dimensional $C_{2v}$-symmetric topological superconductors in the $x$
direction, stacked in the $y$ direction, for which
$z_{1;y}=\mathfrak{p}^{(0,0)}+\mathfrak{p}^{(\pi,0)}\mod 2$ is the associated
symmetry-based indicator.

Representation $\Theta=A_{2}$.— The discussion of this representation is
easiest if we choose the representation $U({\cal M}_{x})=i\rho_{3}\tau_{0}$,
$U({\cal M}_{y})=i\rho_{1}\tau_{0}$, see the discussion in Sec. III. With this
choice of representation, one finds that $H(k_{x},k_{y})=h_{0}\rho_{0}$ at the
high-symmetry momenta $(k_{x},k_{y})=(0,0)$, $(0,\pi)$, $(\pi,0)$, and
$(\pi,\pi)$. The matrix $\tau_{1}h_{0}$ is antisymmetric and the sign
$(-1)^{\mathfrak{p}}$ of its Pfaffian is used as the topological invariant of
$H(k_{x},k_{y})$. In contrast to the $B_{1}$ and $B_{2}$ representations,
there are no compatibility constraints for the $A_{2}$ representation, hence

$\displaystyle\text{BS}=\mathbb{Z}_{2}^{4},\ $ $\displaystyle
B[H(\mathbf{k})]=\\{\mathfrak{p}(0,0),\mathfrak{p}(\pi,0),\mathfrak{p}(0,\pi),\mathfrak{p}(\pi,\pi)\\}.$
(73)

The subgroup AI for the $A_{2}$ representation consists of the elements with
$\mathfrak{p}^{(0,0)}=\mathfrak{p}^{(0,\pi)}=\mathfrak{p}^{(\pi,0)}=\mathfrak{p}^{(\pi,\pi)}$,
so that

$\text{SI}_{{\rm D}}[C_{2v},A_{2}]=\mathbb{Z}_{2}^{3}.$ (74)

The classifying group ${\cal K}$ for the $A_{2}$ representation is

${\cal K}_{{\rm D}}[C_{2v},A_{2}]=\mathbb{Z}_{2}^{4},$ (75)

see App. B.4. Two factors $\mathbb{Z}_{2}$ correspond to weak phases stacked
in the $x$ and $y$ directions, with labels “$1;x$” and “$1;y$”, symmetry-based
indicators

$\displaystyle z_{1;j}=$ $\displaystyle\,\sum_{\mathbf{k}_{\rm s}|k_{{\rm
s},j}=\pi}\mathfrak{p}^{\mathbf{k}_{\rm s}}\mod 2,\ \ j=x,y,$ (76)

The corresponding generating Hamiltonians are

$H_{(1;x)}=\rho_{0}\tau_{3}(1-m-\cos k_{y})+\rho_{3}\tau_{1}\sin k_{y},$

and

$H_{(1;y)}=\rho_{0}\tau_{3}(1-m-\cos k_{x})+\rho_{1}\tau_{1}\sin k_{x},$

with $0<m<2$, respectively. The remaining two factors $\mathbb{Z}_{2}$ in the
classifying group ${\cal K}$ and the single remaining factor $\mathbb{Z}_{2}$
in SI correspond to strong second-order phases. These have generator
Hamiltonians

$\displaystyle H_{(2,\pm)}^{\prime}=$ $\displaystyle\rho_{0}\tau_{3}(2-m-\cos
k_{x}-\cos k_{y})$ $\displaystyle\pm\rho_{1}\tau_{1}\sin
k_{x}+\rho_{3}\tau_{1}\sin k_{y},$

with $0<m<2$, but only a single associated symmetry-based indicator

$z_{2}=\sum_{\mathbf{k}_{\rm s}}\mathfrak{p}^{\mathbf{k}_{\rm s}}\mod 2.$ (77)

As discussed in App. B.4, the generator Hamiltonians $H_{(2;\pm)}$ describe a
second-order phase with a single Majorana corner state at each mirror-
symmetric corner, whereas the direct sum $H_{(2;+)}\oplus H_{(2;-)}$, which is
mapped to the trivial element in SI, has pairs of even and odd parity Majorana
corner states at the corners bisected by one mirror axis, but no corner states
at the corners bisected by the other mirror axis.

The results are summarized in Table 11. Following the notation of Ref.
Trifunovic and Brouwer, 2019, we list a full boundary-signature-resolved
subgroup series for each singly-generated group ${\cal K}_{i}$ contributing to
the full classification group ${\cal K}=\prod_{i}{\cal K}_{i}$ that admits
topological phases with a higher-order boundary signature. The subgroup ${\cal
K}_{i}^{\prime}\subset{\cal K}_{i}$ is the subset of all topological phases
with boundary signature of order larger than one. Weak topological phases in
two dimensions, which are essentially stacks of one-dimensional phases, do not
have a higher-order boundary signature, so that we do not give a subgroup
sequence for factors ${\cal K}_{i}$ representing a weak phase.

Table 11 also lists the topological band labels for the four Hamiltonians
generating the classifying group ${\cal K}$, as well as their image in the
quotient group SI. Here we use the symbol $\mathrm{e}_{j}\\!^{(n)}$ to denote
the $j$th generator of SI. The superscript $n$ indicates its order, i.e.,
$n\mathrm{e}_{j}\\!^{(n)}=0$.

|  | BS | SI
---|---|---|---
$\mathcal{K}_{i}^{\prime}\subseteq\mathcal{K}_{i}$ | Phase | $\mathfrak{p}^{(0,0)}$ | $\mathfrak{p}^{(\pi,0)}$ | $\mathfrak{p}^{(0,\pi)}$ | $\mathfrak{p}^{(\pi,\pi)}$ | $\mathbb{Z}_{2}^{3}$
| $\mathbf{x}=(0,0)$ | 1 | 1 | 1 | 1 | $0$
| $\mathbf{x}=(\frac{1}{2},0)$ | 1 | 1 | 1 | 1 | $0$
| $\mathbf{x}=(0,\frac{1}{2})$ | 1 | 1 | 1 | 1 | $0$
| $\mathbf{x}=(\frac{1}{2},\frac{1}{2})$ | 1 | 1 | 1 | 1 | $0$
$\mathbb{Z}_{2}$ | $(1;x)$ | 1 | 1 | 0 | 0 | $\mathrm{e}_{1;x}^{(2)}$
$\mathbb{Z}_{2}$ | $(1;y)$ | 1 | 0 | 1 | 0 | $\mathrm{e}_{1;y}^{(2)}$
$\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}$ | $(2;+)^{\prime}$ | 1 | 0 | 0 | 0 | $\mathrm{e}_{2}^{(2)}$
$\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}$ | $(2;-)^{\prime}$ | 1 | 0 | 0 | 0 | $\mathrm{e}_{2}^{(2)}$

|  |  |
---|---|---|---
$(1;x)$ | $(1;y)$ | $(2;+)^{\prime}$ | $(2;-)^{\prime}$
$z_{1;x}=1$ | $z_{1;y}=1$ | $z_{2}=1$ | $z_{2}=1$

Table 11: Band labels and symmetry-based indicators for atomic-limit
Hamiltonians obtained by placing 0d generators at Wyckoff position
$\mathbf{x}$ (upper four rows) and for the generators $H_{(1;x)}$,
$H_{(1;y)}$, $H_{(2,+)}^{\prime}$, $H_{(2,-)}^{\prime}$ of the weak and
second-order phases for the symmetry group $C_{2v}$ and representation
$\Theta=A_{2}$ in two dimensions, class D. The weak phases $(1;x)$ and $(1;y)$
can be constructed from one dimensional topological superconductors with
Majorana bound states, as schematically indicated below (red dots with
indicated mirror parity $\pm$), stacked in the $x$ or $y$ direction,
respectively. A possible boundary signature of the generators of second-order
phases consists of Majorana bound states at mirror symmetric corners with
indicated mirror parity. The first column contains the boundary-signature-
resolved subgroup sequence for each of the factor groups contributing to the
full classification group ${\cal K}=\prod_{i}{\cal K}_{i}$.

###  VI.2 Fourfold rotation symmetry $C_{4}$, class D

General considerations.— There are three non-equivalent high-symmetry momenta,
$\\{\mathbf{k}_{\rm s}\\}=\\{(0,0),(0,\pi),(\pi,\pi)\\}$. All three non-
equivalent high symmetry momenta are invariant under particle-hole
conjugation. In addition, the momenta $(0,0)$ and $(\pi,\pi)$ are invariant
under fourfold rotation, while $(0,\pi)$ is invariant under twofold rotation
only.

On a square lattice we can choose the set $W$ of representative Wyckoff
positions as $W=\\{(0,0),(\frac{1}{2},\frac{1}{2}),(\frac{1}{2},0)\\}$. The
two Wyckoff positions $\mathbf{x}=(0,0)$ and
$\mathbf{x}=(\frac{1}{2},\frac{1}{2})$ are invariant under fourfold rotations;
the Wyckoff position $\mathbf{x}=(\frac{1}{2},0)$ is invariant under twofold
rotations only. This Wyckoff position has a nontrivial orbit consisting of the
positions $(\frac{1}{2},0)$ and $(0,\frac{1}{2})$, generated by fourfold
rotation. The induced representations $U^{\mathbf{x},\alpha}(g,\mathbf{k})$ of
orbitals located at $\mathbf{x}$ can be written in terms of the onsite (i.e.,
zero-dimensional) representations $U^{\alpha}(g)$ using

$\displaystyle U^{(0,0),\alpha}({\cal R}_{\pi/2},\mathbf{k})$
$\displaystyle=U^{\alpha}({\cal R}_{\pi/2}),$ $\displaystyle
U^{(\frac{1}{2},\frac{1}{2}),\alpha}({\cal R}_{\pi/2},\mathbf{k})$
$\displaystyle=U^{\alpha}({\cal R}_{\pi/2})e^{ik_{x}},$ (78) $\displaystyle
U^{(\frac{1}{2},0),\alpha}({\cal R}_{\pi/2},\mathbf{k})$
$\displaystyle=U^{\alpha}({\cal R}_{\pi/2})\otimes\begin{pmatrix}0&1\\\
e^{ik_{y}}&0\end{pmatrix},$

where ${\cal R}_{\pi/2}$ is a rotation by $\pi/2$ and the matrices for the
Wyckoff position $(\frac{1}{2},0)$ act in the space of orbitals contained in
its orbit. As in Sec. II we use the half-integer angular momentum $j$ to
denote the irreps $\alpha$ of the rotation symmetry. The angular momentum $j$
is defined modulo $4$ for fourfold rotation and modulo $2$ for twofold
rotation.

The computation of the band labels can be performed with the help of the
projector $P_{j}(\mathbf{k}_{\rm s})$ of Eq. (38), which projects the
Hamiltonian $H(\mathbf{k}_{\rm s})$ at the high-symmetry momentum
$\mathbf{k}_{\rm s}$ onto its diagonal block with irreducible representation
$\alpha=j$. In addition to using Eq. (38) to project on the angular momentum
$j$ subspace, a unitary basis transformation $V_{j}$ must be implemented to
ensure that $V_{j}P_{j}(\mathbf{k}_{\rm s})V_{j}^{\dagger}$ is block diagonal.
(The unitary matrix $V_{j}$ depends on $\mathbf{k}_{\rm s}$, but we suppress
this dependence to keep the notation simple.) In the present case, all irreps
are one-dimensional and the characters $\chi_{j}$ of Eq. (38) are given by

$\chi_{j}({\cal R}_{\pi/4}^{n})=r_{j}^{n}({\cal R}_{\pi/4})=e^{inj\pi/2}.$

We may then choose the unitary matrix $V\equiv V_{j}$ independent of $j$ by
requiring that $VU(g,\mathbf{k}_{\rm s})V^{\dagger}$ be diagonal as in Eq.
(19), with $U(g)=\mbox{diag}_{\alpha}[U^{\alpha}(g,\mathbf{k}_{\rm s})]$ and
$U^{\alpha}(g,\mathbf{k}_{\rm s})$ given in Eq. (VI.2). Indeed, one verifies
that with this choice $VP_{j}(\mathbf{k}_{\rm s})V^{\dagger}$ is a diagonal
matrix with unit entries for each band $n$ with $(VU({\cal
R}_{\pi/4})V^{\dagger})_{nn}=r_{j}({\cal R}_{\pi/4})$ and zeroes otherwise.
The transformed Hamiltonian $VH(\mathbf{k}_{\rm s})V^{\dagger}$ is block
diagonal and all zero-dimensional invariants can be computed from the block
$VP_{j}(\mathbf{k}_{\rm s})H(\mathbf{k}_{\rm s})P_{j}(\mathbf{k}_{\rm
s})V^{\dagger}$. In case the block is characterized by a Pfaffian invariant
$\mathfrak{p}_{j}(\mathbf{k}_{\rm s})$, the transformed representation of the
corresponding antiunitary antisymmetry $VU({\cal P})V^{T}$ acts within the
block such that $(-1)^{\mathfrak{p}}_{j}(\mathbf{k}_{\rm s})={\rm
sign}\,\text{Pf}(VP_{j}(\mathbf{k}_{\rm s})H(\mathbf{k}_{\rm s})U({\cal
P})V^{T}VP_{j}(\mathbf{k}_{\rm s})V^{\dagger})$ is well defined.

Representations $\Theta=A$ and $\Theta=B$.— These two representations have the
same algebraic structure, so that it is sufficient to discuss the case
$\Theta=A$ only. (To see this, one considers $i{\cal R}_{\pi/2}$ as the
generator of $C_{4}$ for the $B$ representation and verifies that $i{\cal
R}_{\pi/2}$ commutes with particle-hole conjugation ${\cal P}$.)

The Hamiltonian has fourfold rotation symmetry at the high-symmetry momenta
$(0,0)$ and $(\pi,\pi)$. As discussed in Sec. III, for zero-dimensional BdG
Hamiltonians with additional symmetry group $C_{4}$ and representation
$\Theta=A$, particle-hole conjugation pairs the eigensectors corresponding to
the irreducible representations $j=\frac{1}{2}$ and $j=\frac{7}{2}$ (mod $4$),
as well as $j=\frac{3}{2}$ and $j=\frac{5}{2}$ (mod $4$). The classifying