is identical to the that of the bulk topology, after removal of the strong
topological phases. On the other hand, no complete corner-to-bulk
correspondence or hinge-to-bulk correspondence exists for topological
crystalline phases protected by a twofold rotation symmetry or by inversion
symmetry, since these symmetries no not allow for symmetry-invariant corners
or hinges in two and three dimensions. Instead, there is a partial
correspondence, which relates the parity of the number of corner states or
hinge modes to the bulk topology.

For topological crystalline phases in which the number $d_{\parallel}$ of
inverted spatial dimensions is smaller than the spatial dimension $d$, such as
phases protected by mirror reflection for $d\geq 2$ or twofold rotation for
$d\geq 3$, there is a bulk-to-boundary correspondence, which uniquely links
the bulk topology with the boundary states on a symmetry-invariant boundary.
The corner-to-bulk correspondence or hinge-to-bulk correspondence for those
phases shows that they may have protected states at corners or hinges, too,
but it does not provide information beyond what is already known from
considering symmetry-invariant boundaries. This is different for topological
crystalline phases with $d_{\parallel}=d$, such as two-fold rotation symmetry
for $d=2$ or inversion symmetry for $d=3$, for which there are no symmetry-
invariant boundaries and, hence, no (first-order) bulk-to-boundary
correspondence. In this case the $\mathbb{Z}_{2}$ sum rule for the number of
corner states or hinge modes that we derive here provides a unique boundary
signature of a nontrivial topological crystalline phase for a case in which no
other boundary signatures are known to exist.Fang and Fu (2017); Khalaf  _et
al._ (2017) Correspondingly, the demonstration that a nontrivial topological
crystalline phase implies the existence of protected corner states or hinge
modes cannot start from a theory of gapless boundary modes, as it does for
$d_{\parallel}<d$,Langbehn  _et al._ (2017) but, instead, must start from the
gapped bulk, as is done in Ref. Khalaf  _et al._ , 2017 and Sec. VI for
specific examples, or, as a general construction, by dimensional reduction
from a hypothetical higher-dimensional topological crystalline phase for which
symmetry-invariant boundaries exist. This is the route we take in Sec. V,
using a real-space dimensional reduction scheme based on the scattering
matrix.Fulga  _et al._ (2012); Trifunovic and Brouwer (2017)

It is important to stress that, although crystalline symmetries are key to our
construction of second-order topological phases, second-order topological
phases are immune to weak perturbations that break the crystalline symmetry,
as long as the boundary gaps are preserved.Langbehn  _et al._ (2017) In our
description, this stability is reflected in the use of two classification
schemes: An extrinsic classification scheme, which classifies corner states or
hinge modes with respect to continuous transformations of the Hamiltonian that
preserve both bulk and boundary gaps, and an intrinsic classification, which
allows transformations of the Hamiltonian in which the boundary gap is closed,
as long as the bulk gap is preserved. The intrinsic classification depends on
the bulk topology only, and is independent of the lattice termination. On the
other hand, it is the extrinsic classification, with the possible inclusion of
local symmetry-breaking perturbations, that captures the robustness of the
phenomena associated with a second-order topological phase to weak symmetry-
breaking perturbations.

Not all two-dimensional materials with corner states or all three-dimensional
materials with gapless hinge modes are in a second-order topological phase —
just like not all materials with a gapped bulk and gapless boundary states are
topological. For a second-order topological phase it is necessary that the
corner states or hinge modes have a topological protection. A classification
of the type that we present here is a key prerequisite to determine whether a
true topological protection can exist, or whether the existence of corner
states or hinge modes in a given model is merely a matter of coincidence. For
example, the existence of zero-energy corner modes always requires that the
Hamiltonian satisfy an antisymmetry, ruling out a second-order phase in a two-
dimensional lattice model with symmetries only — in contrast to recent claims
in the literature.Ezawa (2018a, b); Xu  _et al._ (2017)

The phenomenology of a second-order topological phase — the existence of
protected zero-energy corner states or gapless hinge modes on an otherwise
gapped boundary — is not the only possible manifestation of a nontrivial bulk
topology if the standard bulk-to-boundary correspondence does not apply. As
pointed out in Refs. Lau  _et al._ , 2016; van Miert  _et al._ , 2017;
Benalcazar  _et al._ , 2017a, b; Ezawa, 2018a, a nontrivial bulk crystalline
topology may also manifest itself through a nontrivial quantized electric
multipole moment or through the existence of fractional end or corner charges.
(Note that a corner charge is different from a zero-energy corner state: A
zero-energy corner state implies a degeneracy of the many-body ground state,
whereas a corner charge implies the local accumulation of charge in an
otherwise non-degenerate many-particle ground state.) If the Hamiltonian
possesses an antisymmetry, as is the case for certain models considered in the
literature,Lau  _et al._ (2016); Benalcazar  _et al._ (2017a) a nontrivial
electric multipole moment and protected zero-energy corner states can exist
simultaneously, but this need not always be case. A counterexample is the
“breathing pyrochlore lattice” of Ref. Ezawa, 2018a, for which the nontrivial
bulk topology manifests itself through a quantized bulk polarization, whereas
the zero-energy corner states of Ref. Ezawa, 2018a lack topological
protection.

Only few materials have been proposed as realizations of second-order phases.
Examples are strained SnTeSchindler  _et al._ (2017) or odd-parity
superconducting order in doped nodal-loop materialsShapourian  _et al._ (2018)
both with mirror symmetry, and bismuth Schindler  _et al._ (2018) with
inversion symmetry. Simultaneously, the phenomenology of second-order phases
has been reproduced experimentally in artificial “materials”, such as
electricalImhof  _et al._ (2017) or microwavePeterson  _et al._ (2018)
circuits, or coupled mechanical oscillators.Serra-Garcia  _et al._ (2018) We
hope that the complete classification presented here will help to identify new
material candidates for the solid-state realizations of second-order
topological insulators and superconductors.

###### Acknowledgements.

We thank Andrei Bernevig, Eslam Khalaf, and Felix von Oppen for stimulating
discussions. We acknowledge support by project A03 of the CRC-TR 183 and by
the priority programme SPP 1666 of the German Science Foundation (DFG). A
recent article by Eslam Khalaf on the classification of inversion-symmetric
higher-order topological insulators and superconductors contains closely
related results. We thank Eslam Khalaf for making his manuscript available to
us prior to publication.

##  Appendix A Reflection-matrix-based dimensional reduction scheme

In this appendix we describe details of the reflection-matrix based
dimensional reduction scheme. We first review how this method works in the
absence of crystalline symmetry, following the original article by Fulga et
al.,Fulga  _et al._ (2012) and then show how to include order-two crystalline
symmetries with $d_{\parallel}<d$, generalizing the analysis of Ref.
Trifunovic and Brouwer, 2017. The reflection-matrix based dimensional
reduction scheme leaves $d_{\parallel}$ unchanged, so that the minimal
dimension it can achieve is $d=d_{\parallel}$. The main text discusses how the
reflection-matrix based dimensional reduction scheme can also be applied to
second-order topological insulators and superconductors.

###  A.1 Altland-Zirnbauer classes without crystalline symmetries

The key step in the method of Ref. Fulga  _et al._ , 2012 is the construction
of a $(d-1)$-dimensional gapped Hamiltonian $H_{d-1}$ for each $d$ dimensional
gapped Hamiltonian $H_{d}$. The Hamiltonians $H_{d}$ and $H_{d-1}$ have
different symmetries, but the same (strong) topological invariants. Fulga et
al. show how the Hamiltonian $H_{d-1}$ can be constructed from the reflection
matrix $r_{d}$ if a gapped system with Hamiltonian $H_{d}$ is attached to an
ideal lead with a $(d-1)$-dimensional cross section.

To be specific, following Ref. Fulga  _et al._ , 2012 we consider a
$d$-dimensional gapped insulator with Hamiltonian
$H_{d}(\mathbf{k})=H_{d}(\mathbf{k}_{\perp},k_{d})$, occupying the half space
$x_{d}>0$ and periodic boundary conditions in the transverse directions, see
Fig. 18. The half space $x_{d}<0$ consists of an ideal lead with transverse
modes labeled by the $d-1$ dimensional wavevector $\mathbf{k}_{\perp}$. The
amplitudes $a_{\rm out}(\mathbf{k}_{\perp})$ and $a_{\rm
in}(\mathbf{k}_{\perp})$ of outgoing and incoming modes are related by the
reflection matrix $r_{d}(\mathbf{k}_{\perp})$,

$a_{\rm out}(\mathbf{k}_{\perp})=r_{d}(\mathbf{k}_{\perp})a_{\rm
in}(\mathbf{k}_{\perp}).$ (46)

Since $H_{d}$ is gapped, $r_{d}(\mathbf{k}_{\perp})$ is unitary. Time-reversal
symmetry, particle-hole antisymmetry, or chiral antisymmetry pose additional
constraints on $r_{d}(\mathbf{k}_{\perp})$. These follow from the action of
these symmetries on the amplitudes $a_{\rm in}$ and $a_{\rm out}$,

$\displaystyle{\cal T}a_{\rm in}(\mathbf{k}_{\perp})$ $\displaystyle=Q_{{\cal
T}}\,a_{\rm out}^{*}(-\mathbf{k}_{\perp}),$ $\displaystyle{\cal T}a_{\rm
out}(\mathbf{k}_{\perp})$ $\displaystyle=\,V_{{\cal T}}a_{\rm
in}^{*}(-\mathbf{k}_{\perp}),$ (47) $\displaystyle{\cal P}a_{\rm
in}(\mathbf{k}_{\perp})$ $\displaystyle=V_{{\cal P}}\,a_{\rm
in}^{*}(-\mathbf{k}_{\perp}),$ $\displaystyle{\cal P}a_{\rm
out}(\mathbf{k}_{\perp})$ $\displaystyle=Q_{{\cal P}}\,a_{\rm
out}^{*}(-\mathbf{k}_{\perp}),$ (48) $\displaystyle{\cal C}a_{\rm
in}(\mathbf{k}_{\perp})$ $\displaystyle=Q_{{\cal C}}\,a_{\rm
out}(\mathbf{k}_{\perp}),$ $\displaystyle{\cal C}a_{\rm
out}(\mathbf{k}_{\perp})$ $\displaystyle=V_{{\cal C}}\,a_{\rm
in}(\mathbf{k}_{\perp}),$ (49)

where $V_{{\cal T}}$, $Q_{{\cal T}}$, $V_{{\cal P}}$, $Q_{{\cal P}}$,
$V_{{\cal C}}$, and $Q_{{\cal C}}$ are $\mathbf{k}_{\perp}$-independent
unitary matrices that satisfy $V_{\cal T}Q_{{\cal T}}^{*}=Q_{\cal T}V_{{\cal
T}}^{*}={\cal T}^{2}$, $V_{{\cal P}}V_{{\cal P}}^{*}=Q_{{\cal P}}Q_{{\cal
P}}^{*}={\cal P}^{2}$, and $Q_{{\cal C}}V_{{\cal C}}={\cal C}^{2}=1$. Systems
with both time-reversal symmetry and particle-hole antisymmetry also have a
chiral antisymmetry, with $Q_{{\cal C}}=V_{{\cal P}}Q_{{\cal T}}^{*}={\cal
T}^{2}{\cal P}^{2}Q_{{\cal T}}Q_{{\cal P}}^{*}$ and $V_{{\cal C}}=Q_{{\cal
P}}V_{{\cal T}}^{*}={\cal T}^{2}{\cal P}^{2}V_{{\cal T}}V_{{\cal P}}^{*}$. For
the reflection matrix $r_{d}(\mathbf{k}_{\perp})$ the presence of time-
reversal symmetry, particle-hole antisymmetry, and/or chiral antisymmetry
leads to the constraints

$\displaystyle r_{d}(\mathbf{k}_{\perp})=Q_{{\cal T}}^{\rm
T}r_{d}(-\mathbf{k}_{\perp})^{\rm T}V_{{\cal T}}^{*},$ (50) $\displaystyle
r_{d}(\mathbf{k}_{\perp})=Q_{{\cal P}}^{\rm
T}r_{d}(-\mathbf{k}_{\perp})^{*}V_{{\cal P}}^{*},$ (51) $\displaystyle
r_{d}(\mathbf{k}_{\perp})=Q_{{\cal
C}}^{\dagger}r_{d}(\mathbf{k}_{\perp})^{\dagger}V_{{\cal C}}.$ (52)

Figure 18:  Schematic picture of a $d$-dimensional gapped crystalline
insulator occupying the half space $x_{d}>0$, with periodic boundary
conditions applied along the remaining $(d-1)$-dimension, coupled to an ideal
lead with a $(d-1)$-dimensional cross section. The reflection matrix
$r_{d}(\mathbf{k}_{\perp})$ relates the amplitudes $a_{\rm
out}(\mathbf{k}_{\perp})$ and $a_{\rm in}(\mathbf{k}_{\perp})$ of outgoing and
incoming modes in the lead.

The effective Hamiltonian $H_{d-1}$ is constructed out of
$r_{d}(\mathbf{k}_{\perp})$ in different ways, depending on the presence or
absence of chiral symmetry. With chiral symmetry one sets

$H_{d-1}(\mathbf{k})\equiv Q_{{\cal C}}r_{d}(\mathbf{k}),$ (53)

using Eq. (52) to verify that $H_{d-1}$ is indeed hermitian. (Recall that
$V_{{\cal C}}=Q_{{\cal C}}^{\dagger}$ since $Q_{{\cal C}}V_{{\cal C}}={\cal
C}^{2}=1$.) Equation (53) simplifies to Eq. (9) of the main text if the basis
of scattering states is chosen such that $Q_{\cal C}=V_{\cal C}=1$. Without
chiral symmetry one defines $H_{d-1}$ as

$H_{d-1}(\mathbf{k})=\begin{pmatrix}0&r_{d}(\mathbf{k})\\\
r^{\dagger}_{d}(\mathbf{k})&0\end{pmatrix},$ (54)

which is manifestly hermitian and satisfies a chiral symmetry with $U_{\cal
C}=\sigma_{3}=\mbox{diag}\,(1,-1)$. Hence, for the complex classes the
dimensional reduction procedure $H_{d}\to H_{d-1}$ maps a Hamiltonian with
chiral symmetry to one without, and vice versa, corresponding to the period-
two sequence

$\mbox{A}\xrightarrow{d-1}\mbox{AIII}\xrightarrow{d-1}\mbox{A}.$

Bulk-boundary correspondence implies that the bulk, which is described by the
Hamiltonian $H_{d}(\mathbf{k})$, and the boundary, which determines the
reflection matrix $r_{d}(\mathbf{k}_{\perp})$, have the same topological
classification. Since $r_{d}(\mathbf{k}_{\perp})$ is in one-to-one
correspondence with the Hamiltonian $H_{d-1}(\mathbf{k}_{\perp})$, this
implies that $H_{d}$ and $H_{d-1}$ have the same topological classification.

Central point in the construction of Ref. Fulga  _et al._ , 2012 is that if
the Hamiltonian $H_{d}$ possesses an additional antiunitary symmetry and/or
antisymmetry, placing it in one of the real Altland-Zirnbauer classes labeled
$s=0,1,\ldots,7$, then $H_{d-1}$ possesses an antiunitary symmetry and/or
antisymmetry, too, such that it is in Altland Zirnbauer class $s-1$.Fulga  _et
al._ (2012); Trifunovic and Brouwer (2017) Hence, for the real Altland-
Zirnbauer classes, the reflection-matrix based dimensional reduction scheme
generates the period-eight sequence

CI
$\displaystyle\xrightarrow{d-1}\mbox{C}\xrightarrow{d-1}\mbox{CII}\xrightarrow{d-1}\mbox{AII}\xrightarrow{d-1}\mbox{DIII}$
$\displaystyle\xrightarrow{d-1}\mbox{D}\xrightarrow{d-1}\mbox{BDI}\xrightarrow{d-1}\mbox{AI}\xrightarrow{d-1}\mbox{CI},$
(55)

which is the well-known period-eight Bott periodicity known from the
classification of topological insulators and superconductors.Schnyder  _et
al._ (2008); Kitaev (2009); Stone  _et al._ (2011); Wen (2012); Abramovici and
Kalugin (2012); Kennedy and Zirnbauer (2016)

###  A.2 With order-two crystalline symmetries

Bulk-boundary correspondence continues to exist in the presence of an order-
two crystalline symmetry with $d_{\parallel}<d$, if the sample surface is left
invariant under the symmetry operation. (For $d_{\parallel}=d$ there are no
such invariant surfaces.) Correspondingly, the reflection-matrix based
dimensional reduction scheme may be used in the presence of such crystalline
symmetries, too, as was shown for the case of reflection symmetry by two of us
in Ref. Trifunovic and Brouwer, 2017.

Labeling the coordinates as in Sec. II, the coordinate $x_{d}$ is left
invariant by the crystalline symmetry operation ${\cal S}$ if
$d_{\parallel}<d$. Hence, taking the same geometry as above, the lead and the
lead-crystal interface are mapped to themselves under ${\cal S}$. We now
discuss the four cases of unitary symmetry, unitary antisymmetry, antiunitary
symmetry, and antiunitary antisymmetry separately.

Unitary symmetry.— As with the non-spatial symmetries, the action of a unitary
symmetry operation ${\cal S}$ on the amplitudes $a_{\rm in}$ and $a_{\rm out}$
of incoming and outgoing states in the leads involves multiplication with
$\mathbf{k}_{\perp}$-independent unitary matrices,

$\displaystyle{\cal S}a_{\rm in}(\mathbf{k}_{\perp})$ $\displaystyle=V_{{\cal
S}}\,a_{\rm in}(S\mathbf{k}_{\perp}),$ $\displaystyle{\cal S}a_{\rm
out}(\mathbf{k}_{\perp})$ $\displaystyle=\,Q_{{\cal S}}a_{\rm
out}(S\mathbf{k}_{\perp}),$ (56)

where
$S\mathbf{k}_{\perp}=(-k_{1},\ldots,-k_{d_{\parallel}},k_{d_{\parallel}+1},\ldots,k_{d-1})$
denotes the action of the symmetry operation on the mode vector $\mathbf{k}$
and the matrices $V_{{\cal S}}$ and $Q_{{\cal S}}$ satisfy $V_{{\cal
S}}^{2}=Q_{{\cal S}}^{2}={\cal S}^{2}=1$. The presence of the order-two
crystalline symmetry leads to a constraint on the reflection matrix,

$r_{d}(\mathbf{k}_{\perp})=Q_{{\cal
S}}^{\dagger}r_{d}(S\mathbf{k}_{\perp})V_{{\cal S}}.$ (57)

The algebraic relations involving the matrices $Q_{{\cal S}}$, $V_{{\cal S}}$
depend on whether the symmetry operation ${\cal S}$ commutes or anticommutes
with the non-spatial symmetry operations ${\cal T}$, ${\cal P}$, and ${\cal
C}$, $Q_{{\cal T}}Q_{{\cal S}}^{*}=\eta_{{\cal T}}V_{{\cal S}}Q_{{\cal T}}$,
$V_{{\cal T}}V_{{\cal S}}^{*}=\eta_{{\cal T}}Q_{{\cal S}}V_{{\cal T}}$,
$V_{{\cal P}}V_{{\cal S}}^{*}=\eta_{{\cal P}}V_{{\cal S}}V_{{\cal P}}$,
$Q_{{\cal P}}Q_{{\cal S}}^{*}=\eta_{{\cal P}}Q_{{\cal S}}Q_{{\cal P}}$,
$Q_{{\cal C}}Q_{{\cal S}}=\eta_{{\cal C}}V_{{\cal S}}Q_{{\cal C}}$, and
$V_{{\cal C}}V_{{\cal S}}=\eta_{{\cal C}}Q_{{\cal S}}V_{{\cal C}}$.

Unitary antisymmetry.— An order-two unitary antisymmetry ${\cal CS}$ also
exchanges incoming and outgoing modes, such that one has

$\displaystyle{\cal CS}a_{\rm in}(\mathbf{k}_{\perp})$ $\displaystyle=Q_{{\cal
CS}}\,a_{\rm out}(S\mathbf{k}_{\perp}),$ $\displaystyle{\cal CS}a_{\rm
out}(\mathbf{k}_{\perp})$ $\displaystyle=\,V_{{\cal CS}}a_{\rm
in}(S\mathbf{k}_{\perp}),$ (58)

with $S\mathbf{k}_{\perp}$ defined a above. For an antisymmetry operation
${\cal CS}$ the matrices $V_{{\cal CS}}$ and $Q_{{\cal CS}}$ satisfy $V_{{\cal
CS}}Q_{{\cal CS}}=Q_{{\cal SC}}V_{{\cal CS}}=({\cal CS})^{2}=1$. The presence
of the crystalline unitary antisymmetry ${\cal CS}$ implies that the
reflection matrix satisfies

$r_{d}(\mathbf{k}_{\perp})=Q_{{\cal
CS}}^{\dagger}r_{d}(S\mathbf{k}_{\perp})^{\dagger}V_{{\cal CS}},$ (59)

and the matrices $Q_{{\cal CS}}$ and $V_{{\cal CS}}$ satisfy the algebraic
relations $Q_{{\cal T}}V_{{\cal CS}}^{*}=\eta_{{\cal T}}Q_{{\cal CS}}V_{{\cal
T}}$, $V_{{\cal T}}Q_{{\cal CS}}^{*}=\eta_{{\cal T}}V_{{\cal CS}}Q_{{\cal
T}}$, $V_{{\cal P}}Q_{{\cal CS}}^{*}=\eta_{{\cal P}}Q_{{\cal CS}}Q_{{\cal
P}}$, $Q_{{\cal P}}V_{{\cal CS}}^{*}=\eta_{{\cal P}}V_{{\cal CS}}V_{{\cal
P}}$, $Q_{{\cal CS}}V_{{\cal C}}=\eta_{{\cal C}}Q_{{\cal C}}V_{{\cal CS}}$,
$V_{{\cal CS}}Q_{{\cal C}}=\eta_{{\cal C}}V_{{\cal C}}Q_{{\cal CS}}$.

Antiunitary symmetry.— The action of an order-two antiunitary symmetry
$\mathcal{T^{\pm}S}$ on the scattering amplitudes is represented by unitary
matrices $V_{\cal TS}$ and $Q_{\cal TS}$,

$\displaystyle{\cal TS}a_{\rm in}(\mathbf{k}_{\perp})$ $\displaystyle=Q_{{\cal
TS}}\,a_{\rm out}^{*}(-S\mathbf{k}_{\perp}),$ $\displaystyle{\cal TS}a_{\rm
out}(\mathbf{k}_{\perp})$ $\displaystyle=\,V_{{\cal TS}}a_{\rm
in}^{*}(-S\mathbf{k}_{\perp}),$ (60)

with $V_{{\cal TS}}Q_{{\cal TS}}^{*}=Q_{{\cal TS}}V_{{\cal TS}}^{*}=({\cal
TS})^{2}=\pm 1$. The presence of the order-two crystalline antiunitary
symmetry leads to a constraint on the reflection matrix,

$r_{d}(\mathbf{k}_{\perp})=Q_{{\cal TS}}^{\rm
T}r_{d}(-S\mathbf{k}_{\perp})^{\rm T}V_{{\cal TS}}^{*},$ (61)

and the matrices $Q_{{\cal TS}}$ and $V_{{\cal TS}}$ satisfy the algebraic
relations $Q_{{\cal T}}V_{{\cal TS}}^{*}=\eta_{{\cal T}}Q_{{\cal TS}}V_{{\cal
T}}^{*}$, $V_{{\cal T}}Q_{{\cal TS}}^{*}=\eta_{{\cal T}}V_{{\cal TS}}Q_{{\cal
T}}^{*}$, $V_{{\cal P}}Q_{{\cal TS}}^{*}=\eta_{{\cal P}}Q_{{\cal TS}}Q_{{\cal
P}}^{*}$, $Q_{{\cal P}}V_{{\cal TS}}^{*}=\eta_{{\cal P}}V_{{\cal TS}}V_{{\cal
P}}^{*}$, $Q_{{\cal TS}}V_{{\cal C}}^{*}=\eta_{{\cal C}}Q_{{\cal C}}V_{{\cal
TS}}$, $V_{{\cal TS}}Q_{{\cal C}}^{*}=\eta_{{\cal C}}V_{{\cal C}}Q_{{\cal
TS}}$.

Antiunitary antisymmetry.— Finally, for an antiunitary antisymmetry
$\mathcal{P^{\pm}S}$ one has

$\displaystyle{\cal PS}a_{\rm in}(\mathbf{k}_{\perp})$ $\displaystyle=V_{{\cal
PS}}\,a_{\rm in}^{*}(-S\mathbf{k}_{\perp}),$ $\displaystyle{\cal PS}a_{\rm
out}(\mathbf{k}_{\perp})$ $\displaystyle=\,Q_{{\cal PS}}a_{\rm
out}^{*}(-S\mathbf{k}_{\perp}),$ (62)

with $V_{{\cal PS}}V_{{\cal PS}}^{*}=Q_{{\cal PS}}Q_{{\cal PS}}^{*}=({\cal
PS})^{2}=\pm 1$. The reflection matrix satisfies

$r_{d}(\mathbf{k}_{\perp})=Q_{{\cal PS}}^{\rm
T}r_{d}(-S\mathbf{k}_{\perp})^{*}V_{{\cal PS}}^{*}$ (63)

and the algebraic relations involving the matrices $Q_{{\cal PS}}$ and
$V_{{\cal PS}}$ are $Q_{{\cal T}}Q_{{\cal PS}}^{*}=\eta_{{\cal T}}V_{{\cal
PS}}Q_{{\cal T}}^{*}$, $V_{{\cal T}}V_{{\cal PS}}^{*}=\eta_{{\cal T}}Q_{{\cal
PS}}V_{{\cal T}}^{*}$, $V_{{\cal P}}V_{{\cal PS}}^{*}=\eta_{{\cal P}}V_{{\cal
PS}}V_{{\cal P}}^{*}$, $Q_{{\cal P}}Q_{{\cal PS}}^{*}=\eta_{{\cal P}}Q_{{\cal
PS}}Q_{{\cal P}}^{*}$, $Q_{{\cal C}}Q_{{\cal PS}}=\eta_{{\cal C}}V_{{\cal
PS}}Q_{{\cal C}}^{*}$, and $V_{{\cal C}}V_{{\cal PS}}=\eta_{{\cal C}}Q_{{\cal
PS}}V_{{\cal C}}^{*}$.

To see how the presence of an order-two crystalline symmetry or antisymmetry
affects the dimensional reduction scheme we first consider the complex classes
A and AIII. We start from a Hamiltonian in Shiozaki-Sato symmetry class AS,
$(s,t)=(0,0)$, which is represented by a Hamiltonian $H_{d}$ in symmetry class
A with a unitary symmetry ${\cal S}$. Constructing a Hamiltonian $H_{d-1}$ in
class AIII as described above, we find that the unitary symmetry ${\cal S}$
imposes the additional constraint

$H_{d-1}(\mathbf{k}_{\perp})=U_{\cal
S}^{\dagger}H_{d-1}(S\mathbf{k}_{\perp})U_{\cal S},$ (64)

on $H_{d-1}$, with

$U_{\cal S}=\begin{pmatrix}Q_{{\cal S}}&0\\\ 0&V_{{\cal S}}\end{pmatrix}.$
(65)

Since $U_{\cal S}$ commutes with $\sigma_{3}$ and $U_{\cal S}^{2}=1$, we
conclude that dimensional reduction maps the class AS to class AIII${}^{{\cal
S}_{+}}$. In the classification of Shiozaki and Sato this class is labeled
$(s,t)=(1,0)$. Similarly, if $H_{d}$ is a Hamiltonian in Shiozaki class
$(s,t)=(0,1)$, which is represented by a unitary antisymmetry ${\cal CS}$, the
mapped Hamiltonian $H_{d-1}$ satisfies the additional symmetry

$U_{\cal CS}^{\dagger}H_{d-1}(S\mathbf{k}_{\perp})U_{\cal CS}$ (66)

with

$U_{\cal CS}=\begin{pmatrix}0&V_{{\cal CS}}\\\ Q_{{\cal CS}}&0\end{pmatrix}.$
(67)

This is a unitary symmetry operation that anticommutes with the chiral
operation $\sigma_{3}$, so that the mapped Hamiltonian is in Shiozaki-Sato
class AIII${}^{{\cal S}_{-}}$, $(s,t)=(1,1)$. Finally, starting with a
Hamiltonian with symmetry of type $(s,t)=(1,t)$, represented by a class AIII
Hamiltonian with an order-two crystalline symmetry ${\cal S}$ commuting
($\eta_{\cal C}=1)$ or anticommuting ($\eta_{\cal C}=-1$) with ${\cal C}$ for
$t=0$, $1$, respectively, the mapped Hamiltonian satisfies the constraint

$H_{d-1}(\mathbf{k}_{\perp})=\eta_{\cal C}V_{\cal
S}^{\dagger}H_{d-1}(S\mathbf{k}_{\perp})V_{\cal S},$ (68)

which is a symmetry of Shiozaki-Sato type $(s,t)=(0,t)$, $t=0,1$. (It is a
unitary symmetry for $\eta_{\cal C}=1$ and an antisymmetry for $\eta_{\cal
C}=-1$.)

A similar procedure can be applied to the remaining complex Shiozaki-Sato
classes, which are labeled by a single integer $s=0,1,\ldots,7$. Starting with
a Hamiltonian of Shiozaki classes $s=0$ and $s=4$ (classes
A${}^{\mathcal{T^{+}S}}$ and A${}^{\mathcal{T^{-}S}}$, antiunitary symmetry
${\cal TS}$ squaring to $1$ and $-1$, respectively), we find that the mapped
Hamiltonian $H_{d-1}$ satisfies the constraint

$H_{d-1}(\mathbf{k}_{\perp})=U_{\cal
TS}^{\dagger}H_{d-1}(-S\mathbf{k}_{\perp})^{*}U_{\cal TS}$ (69)

with

$U_{\cal TS}=\begin{pmatrix}0&V_{{\cal TS}}^{*}\\\ Q_{{\cal
TS}}^{*}&0\end{pmatrix}.$ (70)

Hence, $H_{d-1}$ satisfies an antiunitary symmetry that anticommutes with the
chiral operation ${\cal C}$ and squares to $1$ or $-1$, so that the mapped
Hamiltonian is in Shiozaki classes $s=7$ and $s=3$, respectively. Similarly,
for symmetry classes $s=2$ and $s=6$, corresponding to an antiunitary
antisymmetry squaring to $1$ or $-1$, respectively, we find that $H_{d-1}$
satisfies the constraint

$H_{d-1}(\mathbf{k}_{\perp})=U_{\cal
PS}^{\dagger}H_{d-1}(-S\mathbf{k}_{\perp})^{*}U_{\cal PS}$ (71)

with

$U_{\cal PS}=\begin{pmatrix}Q_{{\cal PS}}^{*}&0\\\ 0&V_{{\cal
PS}}^{*}\end{pmatrix}.$ (72)

This is an antiunitary symmetry that commutes with the chiral operation and
squares to $1$ or $-1$, corresponding to symmetry classes $s=1$ and $s=5$,
respectively. Finally, for the remaining symmetry classes we may start from an
antiunitary symmetry squaring to $\pm 1$ and find that the mapped Hamiltonian
satisfies

$H_{d-1}(\mathbf{k}_{\perp})=\eta_{\cal C}(Q_{\cal C}V_{\cal
TS})^{\dagger}H_{d-1}(-S\mathbf{k}_{\perp})^{*}(Q_{\cal C}V_{\cal TS}),$ (73)

which is an antiunitary symmetry (for $\eta_{\cal C}=1$) or antisymmetry (for
$\eta_{\cal C}=-1$) that squares to $\pm\eta_{\cal C}$, so that under
dimensional reduction the class $s=1$, $3$, $5$, $7$ is mapped to $s=0$, $2$,
$4$, and $6$, respectively.

For the real classes we may proceed in the same way. One finds that under
dimensional reduction the Shiozaki-Sato symmetry class $(s,t)$ is mapped to
$(s-1,t)$, with $s$ modulo $8$. The derivation is identical to that given in
Ref. Trifunovic and Brouwer, 2017 for the case of mirror reflection symmetry.

##  Appendix B Classification of mirror-symmetric corners of two-dimensional
crystals

In this Appendix we explain the origin of the entries in Tables 8–10.
Throughout we will use the convention that the chiral operation ${\cal C}$
squares to one.

###  B.1 Complex classes with antiunitary symmetries and antisymmetries

Class A${}^{\mathcal{T^{+}M}}$, $s=0$.— The topological crystalline phases
coincide with the strong topological phases of Altland-Zirnbauer class A. No
protected zero-energy corner states can persist in the trivial strong phase.

Class AIII${}^{\mathcal{T^{+}M_{+}}}$, $s=1$.— Since the antiunitary mirror
reflection operation ${\cal TM}$ commutes with the chiral operation ${\cal
C}$, corner state have a well-defined parity $\sigma_{\cal C}$ under ${\cal
C}$ and can be chosen to be mapped to themselves under the antiunitary mirror
reflection operation ${\cal TM}$. Two corner states with opposite
$\sigma_{\cal C}$ can be gapped out by a reflection-symmetric mass term, so
that we may use the (extrinsic) integer topological index $N=N_{+}-N_{-}$ to
characterize the zero-energy states at a corner.

A decoration of the edges by a topologically nontrivial one-dimensional chain
leads to the addition of two zero-energy states ${|\rm L\rangle}$ and ${|\rm
R\rangle}={\cal TM}{|\rm L\rangle}$ placed symmetrically around the corner as
in Fig. 8. Since ${\cal TM}$ commutes with ${\cal C}$, these corner states
have the same value of $\sigma_{\cal C}$. Moreover, the linear combinations
${|\rm L\rangle}+{|\rm R\rangle}$ and $i({|\rm L\rangle}-{|\rm R\rangle})$ map
to themselves under ${\cal TM}$, so that they meet the classification criteria
for corner states formulated above. Hence, by changing the lattice termination
we may change $N_{+}$ or $N_{-}$ and, hence, $N$, by two. The parity of $N$
remains unchanged under such a change of termination, which corresponds to an
intrinsic $\mathbb{Z}_{2}$ topological index.

If the antiunitary mirror reflection symmetry is broken locally near the
corner these conclusions do not change. We may still define $N=N_{+}-N_{-}$ as
a topological invariant, which can not change without closing a boundary gap
or the bulk gap, and by changing the lattice termination one may still change
add pairs of zero modes to the corner, so that $N\mod 2$ is the appropriate
invariant if topological equivalence is defined with respect to
transformations that possibly close boundary gaps.

Class A${}^{\mathcal{P^{+}M}}$, $s=2$.— In this symmetry class the antiunitary
reflection antisymmetry ${\cal PM}$ may protect a single zero-energy state at
a mirror-symmetric corner. A pair of zero-energy states can, however, be
gapped out by a mirror-antisymmetric perturbation. To see this, consider two
zero modes ${|1\rangle}$ and ${|2\rangle}$, for which we may assume that they
are both invariant under ${\cal PM}$. Then $i({|1\rangle}{\langle
2|}-{|2\rangle}{\langle 1|})$ is a local perturbation that obeys the mirror
reflection antisymmetry and gaps out the states ${|1\rangle}$ and
${|2\rangle}$. We conclude that a mirror-symmetric corner is described by a
$\mathbb{Z}_{2}$ index.

Class AIII${}^{\mathcal{T^{-}M_{-}}}$, $s=3$.— The bulk phase is always
topologically trivial.Shiozaki and Sato (2014) However, a single pair of
corner states can be obtained by symmetrically decorating mirror-related edges
with topologically nontrivial one-dimensional chains, as in Fig. 8. To see
this, denote states ${|\rm L\rangle}$ and ${|\rm R\rangle}={\cal TM}{|\rm
L\rangle}$, as in Fig. 8. Since $({\cal TM})^{2}=-1$ the states ${|\rm
L\rangle}$ and ${|\rm R\rangle}$ form a Kramers pair under the antiunitary
mirror reflection operation, ${|\rm L\rangle}=-{\cal TM}{|\rm R\rangle}$. A
single such pair of zero-energy states can not be gapped out by a perturbation
that respects the antiunitary mirror reflection symmetry.

Class A${}^{\mathcal{T^{-}M}}$, $s=4$.— The nontrivial topological crystalline
insulator phases in this symmetry class are also strong topological phases,
i.e., they have protected edge modes on all edges, not only on mirror-
symmetric edges. A second-order topological insulator phase with gapped edges
and protected corner states does not exist for this symmetry class.

Class AIII${}^{\mathcal{T^{-}M_{+}}}$, $s=5$.— The bulk phase is topologically
trivial.Shiozaki and Sato (2014) However, (multiple) pairs of corner states
can be obtained by symmetrically decorating mirror-related edges with
topologically nontrivial one-dimensional chains, as in Fig. 8. To see this,
denote states ${|\rm L\rangle}$ and ${|\rm R\rangle}={\cal TM}{|\rm
L\rangle}$, as in Fig. 8. The states ${|\rm L\rangle}$ and ${|\rm R\rangle}$
have the same parity under the chiral operation ${\cal C}$, since the
antiunitary mirror reflection operation ${\cal TM}$ commutes with ${\cal C}$.
Antisymmetry of the Hamiltonian under ${\cal C}$ protects corner states of
equal parity, corresponding to a $2\mathbb{Z}$ topological index.

Class A${}^{\mathcal{P^{-}M}}$, $s=6$.— In this symmetry class the bulk phase
is topologically trivial. Alternatively, one can see that no protected zero-
energy corner states can be consistent with the existence of an antiunitary
mirror reflection antisymmetry ${\cal PM}$ with $({\cal PM})^{2}=-1$: Such
corner states would have to appear in pairs ${|0\rangle}$, ${\cal
PM}{|0\rangle}$, which can be gapped out by the mass term ${|0\rangle}{\langle
0|}{\cal PM}+{\cal PM}{|0\rangle}{\langle 0|}$, which obeys the required
antisymmetry under ${\cal PM}$.

Class AIII${}^{\mathcal{T^{+}M_{-}}}$, $s=7$.— This symmetry class is
topologically trivial as a bulk phase and no corner states can be obtained by
symmetrically decorating mirror-related edges with topologically nontrivial
one-dimensional chains. To see, we again denote these end states ${|\rm
L\rangle}$ and ${|\rm R\rangle}={\cal TM}{|\rm L\rangle}$, as in Fig. 8. The
states ${|\rm L\rangle}$ and ${|\rm R\rangle}$ have opposite parity under the
chiral operation ${\cal C}$, since the antiunitary mirror reflection operation
${\cal TM}$ anticommutes with ${\cal C}$. The Hamiltonian ${|\rm
R\rangle}{\langle\rm L|}+{|\rm L\rangle}{\langle\rm R|}$ anticommutes with
${\cal C}$, commutes with ${\cal TM}$, and gaps out the zero modes ${|\rm
L\rangle}$ and ${|\rm R\rangle}$.

###  B.2 Real classes

We represent the Shiozaki-Sato classes using unitary mirror reflection
symmetries ${\cal M}$ or antisymmetries ${\cal CM}$ squaring to one.

Class AI${}^{\mathcal{M_{+}}}$, $(s,t)=(0,0)$.— This class has a topologically
trivial bulk phase and does not allow for protected corner modes.

Class BDI${}^{\mathcal{M_{++}}}$, $(s,t)=(1,0)$.— In a mirror-symmetric
corner, corner states can be chosen to have well-defined parities
$\sigma_{\cal C}$ and $\sigma_{\cal M}$ with respect to the chiral operation
${\cal C}$ and mirror reflection ${\cal M}$. We use $N_{\sigma_{\cal
C}\sigma_{\cal M}}$ to denote the number of corner states with the
corresponding parities. No mass terms can be added that gap out states with
the same parity $\sigma_{\cal C}$. Local mass terms may gap out pairs of
corner states with different $\sigma_{\cal C}$, but only if they have the same
value of $\sigma_{\cal M}$; corner states with different $\sigma_{\cal C}$ and
different $\sigma_{\cal M}$ are protected. As a result, $N_{++}-N_{-+}$ and
$N_{+-}-N_{--}$ are two independent topological invariants describing a
mirror-symmetric corner. This gives rise to an extrinsic $\mathbb{Z}^{2}$
topological index.

Upon changing the lattice termination while preserving the global mirror
reflection symmetry, e.g., by “glueing” a topologically nontrivial one-
dimensional chain to the crystal edges as in Fig. 8, a pair of corner states
with the same parity $\sigma_{\cal C}$ and opposite parities $\sigma_{\cal M}$
can be added to a corner. Such changes of the lattice termination change the
invariants $N_{++}-N_{-+}$ and $N_{+-}-N_{--}$, but leaves their difference
$N_{++}-N_{-+}-N_{+-}+N_{--}$ unaffected. Hence, if crystals that differ only
by lattice termination are considered equivalent, the relevant topological
invariant is $N_{++}-N_{-+}-N_{+-}+N_{--}$, corresponding to an intrinsic
$\mathbb{Z}$ topological index.

If mirror reflection symmetry is broken locally at the corner, corner states
can be characterized by their parity $\sigma_{\cal C}$ only. Using
$N_{\sigma_{\cal C}}$ to denote the number of corner states with parity
$\sigma_{\cal C}$, $N_{+}-N_{-}$ is a topological invariant — corresponding to
an extrinsic $\mathbb{Z}$ classification —, which is defined modulo $2$ only
if crystals that differ only lattice termination are considered equivalent.

Class D${}^{\mathcal{M_{+}}}$, $(s,t)=(2,0)$.— In this class zero-energy
corner states can be chosen to be particle-hole symmetric (i.e., they are
Majorana states) and with well-defined parity $\sigma_{\cal M}$ under mirror
reflection ${\cal M}$. We use $N_{\sigma_{\cal M}}$ to denote the number of
corner states at parity $\sigma_{\cal M}$. The numbers $N_{+}$ and $N_{-}$ are
defined modulo two only, since two zero modes of the same parity can be gapped
out by a mirror-symmetric mass term. This gives an extrinsic
$\mathbb{Z}_{2}^{2}$ topological classification of mirror-symmetric corners.

A change of lattice termination, e.g., by the addition of topologically
nontrivial one-dimensional superconductors on mirror-related edges, adds two
zero modes of opposite mirror parity to the corner. This reduces the extrinsic
$\mathbb{Z}_{2}^{2}$ classification to an intrinsic $\mathbb{Z}_{2}$
classification in case that crystals differ only by their lattice termination
are considered equivalent. Without local mirror reflection symmetry at the
corner, any pair of Majorana zero modes can gap out, corresponding to a
$\mathbb{Z}_{2}$ classification.

Class DIII${}^{\mathcal{M_{++}}}$, $(s,t)=(3,0)$.— Since particle-hole
conjugation ${\cal P}$ and time-reversal ${\cal T}$ anticommute with the
chiral operation ${\cal C}$ — recall that we require that ${\cal C}^{2}=1$ —
zero-energy corner states always appear in Kramers pairs ${|0\rangle}$ and
${\cal T}{|0\rangle}$, which have opposite parities under ${\cal C}$. Since
both states of such a Kramers pair have the same mirror parity $\sigma_{\cal
M}$, we may characterize the corner states with the help of the number
$N_{\sigma_{\cal M}}$ of Kramers pairs of corner states of mirror parity
$\sigma_{\cal M}$. Mirror reflection symmetry forbids the gapping out of
Kramers pairs at opposite mirror parity $\sigma_{\cal M}$, but allows two
Kramers pairs at same $\sigma_{\cal M}$ to annihilate. As a result, both
$N_{+}$ and $N_{-}$ are defined modulo two only, giving rise to a
$\mathbb{Z}_{2}^{2}$ topological classification.

A change of lattice termination, e.g., by the addition of topologically
nontrivial one-dimensional superconductors on mirror-related edges, adds two
Kramers pairs of zero-energy states of opposite mirror parity to the corner,
thus reducing the extrinsic $\mathbb{Z}_{2}^{2}$ classification to an
intrinsic $\mathbb{Z}_{2}$ classification. Without local mirror reflection
symmetry at the corner, any two Kramers pairs of Majorana zero modes can gap
out, corresponding to a $\mathbb{Z}_{2}$ classification.

Class AII${}^{\mathcal{M_{+}}}$, $(s,t)=(4,0)$.— This class has a
topologically trivial bulk phase and does not allow for protected corner
modes.

Class CII${}^{\mathcal{M_{++}}}$, $(s,t)=(5,0)$.— For Altland-Zirnbauer class
CII the chiral operation ${\cal C}$ commutes with particle-conjugation ${\cal
P}$ and time reversal ${\cal T}$, so that a corner hosts Kramers pairs of zero
modes at the same parity $\sigma_{\cal C}$ under the chiral operation ${\cal
C}$. Both states in such a Kramers pair have the same mirror parity
$\sigma_{\cal M}$, which allows us to count the number $N_{\sigma_{\cal
C},\sigma_{\cal M}}$ of Kramers pairs with the corresponding parities
$\sigma_{\cal C}$ and $\sigma_{\cal M}$. Two Kramers pairs with opposite
$\sigma_{\cal C}$ but the same $\sigma_{\cal M}$ may be gapped by a local
mirror reflection-symmetric perturbation to the Hamiltonian, giving rise to
integer topological invariants $N_{++}-N_{-+}$ and $N_{+-}-N_{--}$. This
corresponds to a $2\mathbb{Z}^{2}$ extrinsic topological classification.

A change of lattice termination, e.g., by the addition of topologically
nontrivial one-dimensional chains on mirror-related edges, adds two Kramers
pairs of zero modes of opposite parity $\sigma_{\cal M}$ to the corner. This
leaves $N_{++}-N_{-+}-N_{+-}+N_{--}$ as the only integer invariant,
corresponding to a $2\mathbb{Z}$ classification.

Without local mirror reflection symmetry at the corner, Kramers pairs corner
states are characterized by their parity $\sigma_{\cal C}$ only. The
difference $N_{+}-N_{-}$ of the number of zero-energy Kramers doublets with
the corresponding parities $\sigma_{\cal C}$ is an integer topological
invariant. If crystals that differ only lattice termination are considered
equivalent, this difference is defined modulo two only, leading to a
$\mathbb{Z}_{2}$ topological invariant.

Classes C${}^{\mathcal{M_{+}}}$, $(s,t)=(6,0)$, and CI${}^{\mathcal{M_{++}}}$,
$(s,t)=(7,0)$.— These classes have a topologically trivial bulk phase and do
not allow for protected corner modes.

Class AI${}^{\mathcal{CM_{-}}}$, $(s,t)=(0,1)$.— This class has a
topologically trivial bulk phase and does not allow for protected corner
modes. This conclusion holds despite the presence of a mirror reflection
antisymmetry ${\cal CM}$. Since ${\cal CM}$ anticommutes with the time-
reversal operation, corner modes can not be simultaneously eigenstates of
${\cal CM}$ and of the time-reversal operation ${\cal T}$. Hence, corner modes
appear as pairs, which can be chosen such that the two states ${|\pm\rangle}$
in the pair are invariant under ${\cal T}$ and ${\cal CM}{|\pm\rangle}=\pm
i{|\mp\rangle}$. Then the local perturbation
${|+\rangle}{\langle-|}+{|-\rangle}{\langle+|}$ satisfies time-reversal
symmetry and mirror reflection antisymmetry and gaps out these two corner
states.

Class BDI${}^{\mathcal{M_{+-}}}$, $(s,t)=(1,1)$.— This class has a
topologically trivial bulk phase. To see whether stable corner states may be
induced by a suitably chosen lattice termination, we consider adding two
topologically nontrivial one-dimensional chains in a symmetric fashion to two
symmetry-related crystal edges, as in Fig. 8. The chains have zero energy end
states ${|\rm L\rangle}$ and ${|\rm R\rangle}={\cal M}{|\rm L\rangle}$, which
may be chosen to be invariant under time reversal. Since the mirror reflection
operation ${\cal M}$ anticommutes with ${\cal C}$, the states ${|\rm
L\rangle}$ and ${|\rm R\rangle}$ have opposite parity under ${\cal C}$. The
Hamiltonian ${|\rm L\rangle}{\langle\rm R|}+{|\rm R\rangle}{\langle\rm L|}$ is
mirror reflection symmetric, satisfies the symmetry constraints of class BDI
and gaps out the states ${|\rm L\rangle}$ and ${|\rm R\rangle}$. We conclude
that no stable corner states may be induced on top of an otherwise trivial
bulk by suitably adapting the lattice termination.

Class D${}^{\mathcal{CM_{+}}}$, $(s,t)=(2,1)$.— Particle-hole symmetric (i.e.,
Majorana) corner states can be counted according to their parity $\sigma_{\cal
CM}$ under the mirror reflection antisymmetry. Since a pair of corner states
${|\pm\rangle}$ with opposite parity $\sigma_{\cal CM}$ can be gapped by the
local perturbation $i({|+\rangle}{\langle-|}-{|-\rangle}{\langle+|})$, whereas
corner states with equal parity $\sigma_{\cal CM}$ are protected by the mirror
reflection antisymmetry, the difference $N_{+}-N_{-}$ of the numbers
$N_{\sigma_{\cal C}}$ of corner states with parity $\sigma_{\cal C}$ is a
well-defined topological invariant. This number remains unchanged if one-
dimensional topological superconductors are “glued” to mirror-related edges,
since this procedure adds a pair of zero-energy states with opposite
$\sigma_{\cal CM}$. We conclude that there is a $\mathbb{Z}$ topological
classification.

If the mirror reflection symmetry is broken locally at the corner, any pair of
Majorana states can be gapped out by a local perturbation, and one arrives at
a $\mathbb{Z}_{2}$ topological classification.

Class DIII${}^{\mathcal{M_{-+}}}$, $(s,t)=(3,1)$.— Since time-reversal ${\cal
T}$ anticommutes with ${\cal C}$, zero-energy corner states appear as Kramers
pairs with opposite parity with respect to the chiral operation ${\cal C}$. We
denote such a Kramers pair as ${|+\rangle}$ and ${|-\rangle}={\cal
T}{|+\rangle}$, where the sign $\pm$ refers to the ${\cal C}$-eigenvalue.
Since mirror reflection ${\cal M}$ anticommutes with ${\cal C}$, these states
can not be chosen to simultaneously be ${\cal M}$-eigenstates. However,
multiple Kramers pairs of zero modes can always be organized in such a way
that ${\cal M}$ acts within a single Kramers pair. Since ${\cal M}$
anticommutes with ${\cal C}$ and ${\cal M}^{2}=1$, one has ${\cal
M}{|\pm\rangle}=e^{\pm i\phi}{|\mp\rangle}$, where we may fix the phases of
the basis kets ${|\pm\rangle}$ such that $\phi=0$. Whereas a single such
Kramers pair is topologically protected, two Kramers pairs ${|\pm,1\rangle}$
and ${|\pm,2\rangle}$ can be gapped out by the local perturbation
$i({|+,1\rangle}{\langle-,2|}-{|-,2\rangle}{\langle+,1|}-{|+,2\rangle}{\langle-,1|}+{|-,1\rangle}{\langle+,2|})$,
which obeys all relevant symmetries. We conclude that the only invariant is
the parity of the number of zero-energy Kramers pairs, which gives a
$\mathbb{Z}_{2}$ topological classification.

Class AII${}^{\mathcal{CM_{-}}}$, $(s,t)=(4,1)$.— A corner may host Kramers
pairs of zero modes, which may also be chosen to have a well-defined parity
under the mirror reflection antisymmetry ${\cal CM}$. Since ${\cal CM}$
anticommutes with time-reversal ${\cal T}$, the two states in the Kramers pair
have opposite ${\cal CM}$-parity. Denoting the two members of a Kramers pair
by ${|\pm\rangle}$, time-reversal symmetry forbids perturbations that have a
nonzero matrix element between the states ${|+\rangle}$ and ${|-\rangle}$,
whereas mirror reflection antisymmetry forbids perturbations that have nonzero
matrix elements between ${|+\rangle}$ and ${|+\rangle}$ and between
${|-\rangle}$ and ${|-\rangle}$. We conclude that a single such Kramers pair
is protected by the combination of time-reversal symmetry and mirror
reflection antisymmetry. A pair of such Kramers pairs can, however, be gapped
out: Denoting the states in the two Kramers pairs by ${|\pm,1\rangle}$ and
${|\pm,2\rangle}$, such a pair of Kramers pairs is gapped out by the local
perturbation
$i({|+,1\rangle}{\langle-,2|}-{|-,2\rangle}{\langle+,1|}-{|+,2\rangle}{\langle-,1|}+{|-,1\rangle}{\langle+,2|})$.
As a result, we find that this class has a $\mathbb{Z}_{2}$ topological index.
If mirror reflection (anti)symmetry is locally broken at the crystal corner, a
Kramers pair can obtain a finite energy and no protected zero-energy corner
states exist.

Class CII${}^{\mathcal{M_{+-}}}$, $(s,t)=(5,1)$.— This class has a
topologically trivial bulk phase and does not admit corner states. To see
this, note that a mirror reflection operator with ${\cal M}^{2}=1$ represents
a hermitian operator which satisfies all symmetry requirements for this class:
It commutes with time-reversal ${\cal T}$ and itself, and anticommutes with
particle-hole conjugation ${\cal P}$. Hence ${\cal M}$ is a valid term in the
Hamiltonian, which gaps out any mirror-symmetric configuration of corner
states.

Class C${}^{\mathcal{CM_{+}}}$, $(s,t)=(6,1)$.— Corner states appear as pairs
related by particle-hole conjugation ${\cal P}$. Since the mirror reflection
antisymmetry ${\cal CM}$ commutes with ${\cal P}$, the two states in the
Kramers pair have the same mirror eigenvalue $\sigma_{\cal CM}$. Multiple
Kramers pairs with the same $\sigma_{\cal CM}$ can not be gapped out by a
mirror reflection-antisymmetric Hamiltonian, but Kramers pairs of opposite
parity $\sigma_{\cal CM}$ may be mutually gapped out by a local mirror
reflection-antisymmetric Hamiltonian. (For example, in a representation in
which ${\cal P}=\sigma_{2}K$ and ${\cal CM}=\tau_{3}$, $\tau_{2}$ gaps out two
pairs of Kramers pairs at opposite ${\cal CM}$-parity.) We conclude that the
difference $N=N_{+}-N_{-}$ between the numbers of Kramers pairs with ${\cal
CM}$-parity $\sigma_{\cal CM}$ is a well-defined topological invariant, giving
a $\mathbb{Z}$ topological classification. Since Altland-Zirnbauer class C
does not allow one-dimensional chains with protected zero-energy end states,
this conclusion does not depend on whether freedom of lattice termination
plays a role in the topological classification. No corner states can be
stabilized in the absence of mirror reflection antisymmetry.

Classes CI${}^{\mathcal{M_{-+}}}$, $(s,t)=(7,1)$, and
AI${}^{\mathcal{M_{-}}}$, $(s,t)=(0,2)$.— These classes have a topologically
trivial bulk phase and do not allow for protected corner modes.

Class BDI${}^{\mathcal{M_{--}}}$, $(s,t)=(1,2)$.— This class has a
topologically trivial bulk phase. To see whether stable corner states may be
induced by a suitably chosen lattice termination, we consider adding two
topologically nontrivial one-dimensional chains in a symmetric fashion to two
symmetry-related crystal edges, as in Fig. 8. The chains have zero energy end
states ${|\rm L\rangle}$ and ${|\rm R\rangle}={\cal M}{|\rm L\rangle}$, which
may be chosen to be invariant under time reversal and particle-hole
conjugation. Since the mirror reflection operation ${\cal M}$ commutes with
${\cal C}$, the states ${|\rm L\rangle}$ and ${|\rm R\rangle}$ have equal
parity $\sigma_{\cal C}$ under ${\cal C}$. Taking symmetric and antisymmetric
linear combinations of the states ${|\rm L\rangle}$ and ${|\rm R\rangle}$, one
obtains a corner state doublet with opposite parity under ${\cal M}$, but
equal $\sigma_{\cal C}$. Multiple doublets of this type with the same
$\sigma_{\cal C}$ cannot be gapped out by a local perturbation, whereas two
corner state doublets with opposite $\sigma_{\cal C}$ can. Hence,
$N_{+}-N_{-}$ is a valid integer topological invariant, where $N_{\sigma_{\cal
C}}$ is the number of zero-energy doublets of ${\cal C}$-parity $\sigma_{\cal
C}$.

Class D${}^{\mathcal{M_{-}}}$, $(s,t)=(2,2)$.— This class has a topologically
trivial bulk phase. No zero-energy corner states can be induced by a suitably
chosen lattice termination. To see this, we consider a mirror reflection
symmetry ${\cal M}$ that squares to one, so that ${\cal M}$ is represented by
a hermitian operator. Since ${\cal M}$ anticommutes with particle-hole
conjugation ${\cal P}$, ${\cal M}$ is itself a valid perturbation to the
Hamiltonian which gaps out any set of corner states.

Class DIII${}^{\mathcal{M_{--}}}$, $(s,t)=(3,2)$.— Since ${\cal T}^{2}=-1$ and
${\cal T}$ anticommutes with ${\cal C}$, corner modes consist of Kramers
Majorana pairs of opposite parity under the chiral operation ${\cal C}$. Since
the product ${\cal MC}$ commutes with the time-reversal operation ${\cal T}$
and with the chiral operation ${\cal C}$, both states in a Kramers pair have
the same “mixed parity” $\sigma_{\cal MC}$ under ${\cal MC}$. Two Kramers
pairs of equal mixed parity $\sigma_{\cal MC}$ can not be gapped out by a
mirror-symmetric perturbation, since ${\cal MC}$ anticommutes with the
Hamiltonian. Two Kramers pairs of opposite mixed parity $\sigma_{\cal MC}$ can
be gapped out by a local perturbation satisfying ${\cal T}$ and ${\cal M}$
symmetries and ${\cal C}$ antisymmetry. (For example, the two Kramers pairs
${|\sigma_{\cal MC},\sigma_{\cal C}\rangle}$ with $\sigma_{\cal MC}$ and
$\sigma_{\cal C}=\pm$ and ${|\sigma_{\cal MC},-\rangle}={\cal T}{|\sigma_{\cal
MC},+\rangle}$, are gapped out by the perturbation
${|+,+\rangle}{\langle-,-|}+{|-,-\rangle}{\langle+,+|}-{|+,-\rangle}{\langle-,+|}-{|-,+\rangle}{|+,-\rangle}$.)
Denoting the number of zero-energy Kramers pairs with mixed parity
$\sigma_{\cal MC}$ by $N_{\sigma_{\cal MC}}$, we thus find that $N_{+}-N_{-}$
is a valid integer topological invariant. This invariant can not be changed by
changing the lattice termination, since addition of two one-dimensional
topological superconductors on mirror-related crystal edges as in Fig. 8 leads
to the addition of two Kramers doublets with opposite mixed parities
$\sigma_{\cal MC}$. If the mirror symmetry is broken locally at the corner,
any pair of Majorana Kramers doublets can gap out, and the $\mathbb{Z}$
topological classification is reduced to a $\mathbb{Z}_{2}$ classification.

Class AII${}^{\mathcal{M_{-}}}$, $(s,t)=(4,2)$.— The bulk crystalline phase is
a strong topological phase, and no stable zero-energy states can be induced by
a suitably chosen lattice termination in the trivial bulk phase.

Class CII${}^{\mathcal{M_{--}}}$, $(s,t)=(5,2)$.— Corners allow Kramers
doublets with equal ${\cal C}$ parity $\sigma_{\cal C}$ but opposite ${\cal
M}$ parity $\sigma_{\cal M}$. Two doublets at the same ${\cal C}$-parity
$\sigma_{\cal C}$ can not be gapped out, but two doublets with opposite ${\cal
C}$ can. (For example, the two Kramers doublets ${|\sigma_{\cal
C},\sigma_{\cal M}\rangle}$ with $\sigma_{\cal C}$ and $\sigma_{\cal M}=\pm$
and ${|\sigma_{\cal C},-\rangle}={\cal T}{|\sigma_{\cal C,+}\rangle}$, are
gapped out by the perturbation
${|+,+\rangle}{\langle-,+|}+{|-,+\rangle}{\langle+,+|}+{|-,-\rangle}{\langle+,-|}+{|+,-\rangle}{|-,-\rangle}$.)
Denoting the number of Kramers pairs with ${\cal C}$-parity $\sigma_{\cal C}$
by $N_{\sigma_{\cal C}}$, we thus find that $N_{+}-N_{-}$ is a well-defined
integer topological invariant.

A change of lattice termination, e.g., by the addition of topologically
nontrivial one-dimensional chains on mirror-related edges, adds two Kramers
pairs of zero modes of the same parity $\sigma_{\cal C}$ to the corner. Taking
symmetric and antisymmetric linear combinations these can be reorganized into
two Kramers pairs ${|\sigma_{\cal C},\pm\rangle}$ of the type discussed above.
Since changing the lattice termination allows one to change $N_{+}-N_{-}$ by
an even number, it is only the modulo two part if this invariant which is
determined by the bulk band structure. The above analysis does not change if
the mirror reflection symmetry is broken locally at the corner.

Class C${}^{\mathcal{M_{-}}}$, $(s,t)=(6,2)$.— This class has a topologically
trivial bulk phase and does not allow for protected corner modes.

Class CI${}^{\mathcal{M_{--}}}$, $(s,t)=(7,2)$.— Corner states appear in
doublets related by particle-hole conjugation ${\cal P}$. Such doublets have
opposite parity under the chiral operation, since ${\cal P}$ and the chiral
operation ${\cal C}$ anticommute for this class. The product ${\cal MC}$
commutes with ${\cal P}$ and ${\cal C}$, so that both states in a doublet have
the same mixed parity $\sigma_{\cal MC}$ under ${\cal MC}$. Two doublets of
equal mixed parity $\sigma_{\cal MC}$ can not be gapped out by a mirror-
symmetric perturbation, since ${\cal MC}$ anticommutes with the Hamiltonian.
Two doublets of opposite mixed parity $\sigma_{\cal MC}$ can be gapped out by
a local perturbation satisfying ${\cal M}$ symmetry and ${\cal P}$ and ${\cal
C}$ antisymmetries. (For example, the two doublets ${|\sigma_{\cal
MC},\sigma_{\cal C}\rangle}$ with $\sigma_{\cal MC}$ and $\sigma_{\cal C}=\pm$
and ${|\sigma_{\cal MC},-\rangle}={\cal P}{|\sigma_{\cal MC},+\rangle}$, are
gapped out by the perturbation
${|+,+\rangle}{\langle-,-|}+{|-,-\rangle}{\langle+,+|}+{|+,-\rangle}{\langle-,+|}+{|-,+\rangle}{|+,-\rangle}$.)
Denoting the number of zero-energy Kramers pairs with mixed parity
$\sigma_{\cal MC}$ by $N_{\sigma_{\cal MC}}$, we thus find that $N_{+}-N_{-}$
is a valid integer topological invariant. This invariant can not by changed by
changing the lattice termination, since the Altland-Zirnbauer class CI does
not allow a nontrivial one-dimensional phase with protected end states.

Class AI${}^{\mathcal{CM_{+}}}$, $(s,t)=(0,3)$.— The mirror reflection
antisymmetry ${\cal CM}$ allows for the protection of corner states at a
mirror-symmetric corner. Corner states can be chosen to be real and with well-
defined parity $\sigma_{\cal CM}$ under mirror reflection. Corner states of
equal parity can not be gapped out because of the mirror reflection
antisymmetry; corner states with opposite parity can be gapped out. Hence
$N=N_{+}-N_{-}$ is an appropriate topological invariant, with $N_{\sigma_{\cal
CM}}$ the number of corner states with ${\cal CM}$-parity $\sigma_{\cal CM}$.

Class BDI${}^{\mathcal{M_{-+}}}$, $(s,t)=(1,3)$.— This class has a trivial
bulk phase. To see whether stable corner states may be induced by a suitably
chosen lattice termination, we consider adding two topologically nontrivial
one-dimensional chains in a symmetric fashion to two symmetry-related crystal
edges, as in Fig. 8. The chains have zero energy end states ${|\rm L\rangle}$
and ${|\rm R\rangle}={\cal M}{|\rm L\rangle}$, which may be chosen to be
invariant under particle-hole conjugation since the mirror reflection
operation ${\cal M}$ commutes with particle-hole conjugation ${\cal P}$. A
pair of zero-energy states ${|\rm L,R\rangle}$ is protected by the combination
of ${\cal P}$ antisymmetry and ${\cal M}$ symmetry. Two such doublets ${|{\rm
L,R},1\rangle}$ and ${|{\rm L,R},2\rangle}$, however, can be gapped out by the
local perturbation $i({|{\rm L},1\rangle}{\langle{\rm R},2|}-{|{\rm
R},2\rangle}{\langle{\rm L},1|}-{|{\rm L},2\rangle}{\langle{\rm R},1|}+{|{\rm
R},1\rangle}{\langle{\rm L},2|})$, which obeys ${\cal P}$ and ${\cal C}$
antisymmetries and ${\cal M}$ symmetry. We conclude that the only invariant is
the parity of the number of such zero-energy doublets, which gives a
$\mathbb{Z}_{2}$ topological classification. If mirror reflection symmetry is
broken locally at the corner, the ${\cal M}$-induced protection of a single
doublet disappears, and even a single doublet of zero-energy corner states can
be gapped out.

Class D${}^{\mathcal{CM_{-}}}$, $(s,t)=(2,3)$.— This class is a strong
topological phase, which has doublets of chiral Majorana modes at edges. A
single chiral Majorana mode is not compatible with the symmetries, since such
mode would have to be invariant under ${\cal P}$ and ${\cal CM}$, which is not
possible since ${\cal P}$ and ${\cal CM}$ anticommute. Nevertheless, by a
suitable choice of lattice termination, a protected pair of Majorana zero
modes can be localized at a mirror-symmetric corner in the topologically
trivial bulk phase. To see this we consider adding two one-dimensional
superconductors with Majorana end states ${|\rm L\rangle}$ and ${|\rm
R\rangle}$ to mirror-related crystal edges of an otherwise topologically
trivial bulk crystal, as in Fig. 8. The end states ${|\rm L,R\rangle}$ are
chosen invariant under particle-hole conjugation ${\cal P}$. Since ${\cal CM}$
anticommutes with ${\cal P}$ we have ${|\rm R\rangle}=i{\cal CM}{|\rm
L\rangle}$. A zero-energy doublet ${|\rm L,R\rangle}$ is then protected by the
combination of ${\cal CM}$ and ${\cal P}$ antisymmetries. Two such doublets,
however, can be gapped out by a local perturbation, which results in a
$\mathbb{Z}_{2}$ topological classification.

Class DIII${}^{\mathcal{M_{+-}}}$, $(s,t)=(3,3)$.— This class has a trivial
bulk phase and cannot host protected corner states. (In a representation in
which ${\cal M}^{2}=1$ the mirror reflection operation ${\cal M}$ is
represented by a hermitian operator, which satisfies ${\cal P}$ antisymmetry
and ${\cal T}$ symmetry and gaps out any set of zero-energy states localized
at the corner.)

Class AII${}^{\mathcal{CM_{+}}}$, $(s,t)=(4,3)$.— This class allows mirror-
protected zero-energy Kramers pairs at corners. Since the mirror reflection
antisymmetry ${\cal CM}$ commutes with the time-reversal operator ${\cal T}$,
such Kramers pairs have the same parity $\sigma_{\cal CM}$ under mirror
reflection. Reflection antisymmetry protects zero-energy Kramers pairs with
equal parity $\sigma_{\cal CM}$, but allows the mutual gapping out of Kramers
pairs with opposite $\sigma_{\cal CM}$. Hence, $N=N_{+}-N_{-}$ is a valid
integer topological index for this class, with $N_{\sigma_{\cal CM}}$ the
number of zero-energy corner states with ${\cal CM}$ parity $\sigma_{\cal
CM}$.

Class CII${}^{\mathcal{M_{-+}}}$, $(s,t)=(5,3)$.— This class has a
topologically trivial bulk phase, and does not allow protected zero-energy
states at corners. To see this, we consider the addition of two topologically
nontrivial one-dimensional chains in a symmetric fashion to two symmetry-
related crystal edges, as in Fig. 8. We denote the doublets at the two chains
by ${|\rm L\rangle}$, ${|\rm L^{\prime}\rangle}={\cal P}{|\rm L\rangle}$,
${|\rm R\rangle}={\cal M}{|\rm L\rangle}$ and ${|\rm R^{\prime}\rangle}={\cal
P}{|\rm R\rangle}={\cal P}{\cal M}{|\rm L\rangle}$. Since ${\cal M}$
anticommutes with ${\cal C}$, doublets at the ends of the left and right
chains have opposite parity under the chiral operation ${\cal C}$. These four
states can be gapped out by the perturbation $i({|\rm L\rangle}{\langle\rm
R^{\prime}|}-{|\rm R^{\prime}\rangle}{\langle\rm L|}+{|\rm
R\rangle}{\langle\rm L^{\prime}|}-{|\rm L^{\prime}\rangle}{\langle\rm R|})$.

Class C${}^{\mathcal{CM_{-}}}$, $(s,t)=(6,3)$.— This class is a strong
topological phase, which has chiral modes at edges. No corner modes can be
constructed in the trivial bulk phase, because the Altland-Zirnbauer class C
is topologically trivial in one dimension.

Class CI${}^{\mathcal{M_{+-}}}$, $(s,t)=(7,3)$.— This class is topologically
trivial and does not allow for protected zero-energy corner states.

##  Appendix C Edge-to-corner correspondence for two-dimensional mirror-
symmetric crystals

A nontrivial mirror-symmetric topological crystalline bulk phase implies the
existence of protected gapless states on mirror-symmetric edges. If the
topological crystalline insulator or superconductor is not in a strong
topological phase, these edge states can be gapped out for edges that are not
invariant under the mirror operation. In that case protected zero-energy
states remain at mirror-symmetric corners. The main text discusses this
scenario in detail for the complex Altland-Zirnbauer classes with unitary
mirror symmetries and antisymmetries. In this appendix we give details for the
complex classes with antiunitary mirror symmetries and antisymmetries and for
the real Altland-Zirnbauer classes. For completeness, we repeat the discussion
of those mirror-symmetric topological phases that were already contained in
Ref. Langbehn  _et al._ , 2017.

Throughout this appendix we will use $x$ as a coordinate along a mirror-
symmetric edge, see Fig. 9a, or along an edge that is symmetrically deformed
from a mirror-symmetric edge, with a mirror-symmetric corner located at $x=0$,
see Fig. 9b. Further, $v$ is a constant with the dimension of velocity, and we
use $\sigma_{j}$, $\tau_{j}$, $j=0,1,2,3$ to refer to Pauli matrices acting on
different spinor degrees of freedom, and $\openone_{N}$ to denote the $N\times
N$ unit matrix. Edge Hamiltonians are always given in the simplest possible
form, after a suitable basis transformation and after rescaling of energies
and coordinates.

###  C.1 Complex Altland-Zirnbauer classes with antiunitary symmetries and
antisymmetries

Class A${}^{\mathcal{T^{+}M}}$, $s=0$.— Representing ${\cal TM}$ by complex
conjugation $K$, this phase allows chiral edge modes with Hamiltonian $H_{\rm
edge}=-iv\openone_{N}\partial_{x}$. This is a strong topological phase, which
does not allow localized zero-energy states at corners.

Class AIII${}^{\mathcal{T^{+}M_{+}}}$, $s=1$.— We represent the chiral
operation ${\cal C}$ using $U_{\cal C}=\sigma_{3}$ and the antiunitary mirror
reflection operation using $U_{\cal TM}=1$, so that the bulk Hamiltonian
$H(k_{x},k_{y})$ satisfies the constraints
$H(k_{x},k_{y})=-\sigma_{3}H(k_{x},k_{y})\sigma_{3}=H^{*}(k_{x},k_{y})$. A
nontrivial mirror-symmetric edge is described by the edge Hamiltonian

$H_{\rm edge}=-iv\sigma_{1}\partial_{x}.$ (74)

This edge allows a unique mass term $m\sigma_{2}$, which is odd under ${\cal
TM}$. The intersection of two mirror-related edges represents a domain wall
with respect to such a mass term and hosts a protected zero-energy mode. The
chiral parity $\sigma_{\cal C}$ of such a corner state depends on the sign of
$m$ far away from the corner, such that $\sigma_{\cal C}$ is negative if
$m(x)$ is positive for $x\to\infty$. The $\mathbb{Z}$ (extrinsic)
classification of corner states follows from the observation that corner
states at equal parity $\sigma_{\cal C}$ can not mutually gap out.

Class A${}^{\mathcal{P^{+}M}}$, $s=2$.— We represent the antiunitary mirror
antisymmetry ${\cal PM}$ by complex conjugation $K$, so that
$H(k_{x},k_{y})=-H^{*}(k_{x},-k_{y})$. The edge Hamiltonian at a mirror-
symmetric edge is

$H_{\rm edge}=-iv\sigma_{2}\partial_{x}.$ (75)

Upon deforming the edge away symmetrically around a corner at $x=0$, two mass
terms $m_{1}(x)\sigma_{1}+m_{2}(x)\sigma_{3}$ are allowed, with
$m_{1,2}(x)=-m_{1,2}(-x)$. Such a Hamiltonian hosts a zero mode symmetrically
located around the corner at $x=0$. A mirror-symmetry-breaking perturbation
near $x=0$ can however push this state away from zero energy.

Class A${}^{\mathcal{T^{-}M}}$, $s=2$.— We represent ${\cal TM}$ by
$\sigma_{2}K$, where $K$ is complex conjugation. A mirror-symmetric edge can
host multiple Kramers pairs of chiral modes, described by the edge Hamiltonian
$H_{\rm edge}=-iv\sigma_{0}\openone_{N}\partial_{x}$. This is a strong
topological phase which does not allow for localized states at corners.

###  C.2 Real classes

Class BDI${}^{\mathcal{M_{++}}}$, $(s,t)=(1,0)$.— We use $U_{\mathcal{T}}=1$,
$U_{\mathcal{P}}=\sigma_{3}$, $U_{\cal C}=\sigma_{3}$, and
$U_{\mathcal{M}}=\sigma_{3}\tau_{3}$ to represent time-reversal, particle-hole
conjugation, chiral operation, and mirror reflection. The integer topological
invariant $N$ for class BDI${}^{\mathcal{M_{++}}}$ counts the difference of
the number of helical edge states with positive and negative mixed parity
$\sigma_{\cal MC}$ at zero energy. For a minimal mirror-invariant edge with
$N\geq 0$, all edge states have the same (positive) mixed parity, so that
effectively we may set $U_{\cal M}=\sigma_{3}$. With a suitable choice of
basis and after rescaling the edge Hamiltonian takes the form

$H_{\rm edge}=-iv\sigma_{2}\openone_{N}\partial_{x}.$ (76)

The unique mass term $m\sigma_{1}$, with $m$ a $N\times N$ hermitian matrix,
is odd under reflection. The intersection of two mirror-related edges
represents a domain wall with respect to such a mass term and hosts $N$
protected zero-energy modes. The parity $\sigma_{\cal C}$ of these modes
depends on the sign of the eigenvalues of the matrix $m(x)$ away from the
corner at $x=0$, such that a positive eigenvalue for $x\to\infty$ corresponds
to a corner state with positive $\sigma_{\cal C}$. This reproduces the
$\mathbb{Z}^{2}$ (extrinsic) classification of corner states derived in App.
B.

Class D${}^{\mathcal{M_{+}}}$, $(s,t)=(2,0)$.— We choose the unitary matrices
$U_{\cal P}=1$ and $U_{\cal M}=\sigma_{1}$ to represent particle-hole
conjugation and mirror reflection. The bulk topological crystalline phase has
a $\mathbb{Z}_{2}$ classification, for which the nontrivial phase has
counterpropagating edge modes at a mirror-symmetric edge described by the edge
Hamiltonian $H_{\rm edge}=-iv\sigma_{3}\partial_{x}$. This edge Hamiltonian
has a unique mass term $m\sigma_{2}$. The intersection between two mirror-
related edges represents a domain wall and hosts a localized zero-energy
state. The mirror parity $\sigma_{\cal M}$ depends on the sign of $m$ far away
from the corner at $x=0$, such that a positive value of $m$ for $x\to\infty$
corresponds to a positive mirror parity $\sigma_{\cal M}$. Since no matrix
elements may exist between two corner state with opposite mirror parity
$\sigma_{\cal M}$, this reproduces the $\mathbb{Z}_{2}^{2}$ (extrinsic)
classification of corner states derived in App. B.

Class DIII${}^{\mathcal{M_{++}}}$, $(s,t)=(3,0)$.— We set $U_{\cal
T}=\sigma_{2}$, $U_{\cal P}=1$, and $U_{\cal M}=\sigma_{2}\tau_{2}$. With a
suitable choice of basis, a mirror-invariant edge in the nontrivial
topological crystalline phase has a pair of counter-propagating Majorana modes
with Hamiltonian

$H_{\rm edge}=-iv\sigma_{3}\tau_{0}\partial_{x}.$ (77)

The unique mass term $m\sigma_{1}\tau_{2}$ is odd under ${\cal M}$. As a
result, intersection between two mirror-related edges represents a domain wall
and hosts a Kramers pair localized zero-energy states. The mirror parity
$\sigma_{\cal M}$ depends on the sign of $m$ far away from the corner at
$x=0$. The $\mathbb{Z}_{2}^{2}$ (extrinsic) classification of corner states
derived in App. B follows upon noting that no matrix elements may exist
between two corner state with opposite mirror parity $\sigma_{\cal M}$.

Class CII${}^{\mathcal{M_{++}}}$, $(s,t)=(5,0)$.— We set $U_{\cal
T}=\sigma_{2}$, ${\cal U}_{\rm P}=\sigma_{2}\tau_{3}$, so that $U_{\cal
C}=\tau_{3}$. The $2\mathbb{Z}$ bulk classification of this symmetry class
counts the difference $N$ of “edge quartets” with positive and negative “mixed
parity” $\sigma_{\cal CM}$. For a minimal edge all edge modes have the same
mixed parity, so that effectively ${\cal M}$ may be represented by $U_{\cal
M}=\tau_{3}$. A minimal edge has Hamiltonian

$H_{\rm edge}=-iv\partial_{x}\sigma_{0}\tau_{2}\openone_{N}.$ (78)

The unique mass term gapping out such edge modes is $m\tau_{1}$, with $m$ a
real symmetric $N\times N$ matrix. This mass term is odd under mirror
reflection, ensuring the existence of $N$ Kramers pairs of corner states at
the intersection between two mirror-related edges. Both states in such a
Kramers pair have the same parity $\sigma_{\cal C}$, which is determined by
the sign of the eigenvalues of $m$ far away from the corner at $x=0$. This
corresponds to the $2\mathbb{Z}^{2}$ (extrinsic) classification of corner
states derived in Sec. B.

Class D${}^{\mathcal{CM_{+}}}$, $(s,t)=(2,1)$.— We represent particle-hole
conjugation ${\cal P}$ by complex conjugation and the mirror antisymmetry
${\cal CM}$ by $U_{\cal CM}=\sigma_{3}$. We use $N_{{\rm L}\sigma_{\cal CM}}$
and $N_{{\rm R}\sigma_{\cal CM}}$ to denote the numbers of left-moving and
right-moving edge modes with mirror parity $\sigma_{\cal M}$ at zero energy,
respectively. Since edge modes moving in opposite directions and with opposite
mirror parity can mutually gap out, the differences $N_{{\rm R}+}-N_{{\rm
L}-}$ and $N_{{\rm R}-}-N_{{\rm L}+}$ are topological invariants, giving a
$\mathbb{Z}^{2}$ classification of edge states. The sum $N_{{\rm R}+}-N_{{\rm
L}-}+N_{{\rm R}-}-N_{{\rm L}+}$ is a strong topological invariant. For a
second-order topological superconductor phase, we are interested in the case
$N_{{\rm R}+}-N_{{\rm L}-}+N_{{\rm R}-}-N_{{\rm R}+}=0$, a minimal realization
of which has $N_{{\rm L}-}=N_{{\rm R}-}=0$ and $N=N_{{\rm R}+}=N_{{\rm L}+}$.
With a suitable choice of basis and after rescaling, the corresponding edge
Hamiltonian reads

$H_{\rm edge}=-iv\tau_{3}\openone_{N}\partial_{x},$ (79)

where $\tau_{3}$ is a Pauli matrix in the left mover–right mover basis. The
unique mass term $m\tau_{2}$ is odd under the mirror antisymmetry, so that the
intersection between two mirror-related edges hosts $N$ Majorana corner
states. All $N$ corner states have the same mirror parity, so that no further
classification is possible. This is consistent with the $\mathbb{Z}$
(extrinsic) classification derived in App. B.

Class DIII${}^{\mathcal{M_{-+}}}$, $(s,t)=(3,1)$.— Here we choose the
representations $U_{\cal T}=\sigma_{2}$ and $U_{\cal P}=\sigma_{1}$, so that
$U_{\cal C}=\sigma_{3}$. Although in the most general case the representation
of ${\cal M}$ requires the introduction of additional spinor degrees of
freedom, the generators for the nontrivial topological phases can be
constructed using the simpler representation $U_{\cal M}=\sigma_{1}$. The two
generators of the $\mathbb{Z}_{2}^{2}$ topological crystalline classification
have edge Hamiltonians $H_{{\rm edge},1}=-iv\sigma_{2}\partial_{x}$ and
$H_{{\rm edge},2}=-iv\sigma_{2}\tau_{3}\partial_{x}$. The former edge
Hamiltonian represents a strong topological phase and is not compatible with a
second-order topological phase. The latter edge Hamiltonian has a unique mass
term $m\sigma_{2}\tau_{2}$, which is odd under mirror reflection. As a result,
the intersection of two mirror-related edges hosts a Kramers pair of Majorana
zero modes.

Class AII${}^{\mathcal{CM_{-}}}$, $(s,t)=(4,1)$.— We represent ${\cal T}$ by
$\sigma_{2}K$ and ${\cal CM}$ by $\sigma_{3}$. The two generators of the
$\mathbb{Z}_{2}^{2}$ topological crystalline classification have edge
Hamiltonians $H_{{\rm edge},1}=-iv\sigma_{3}\partial_{x}$ and $H_{{\rm
edge},2}=-iv\tau_{2}\sigma_{0}\partial_{x}$, where $x$ is the coordinate along
the mirror-symmetric edge and the Pauli matrix $\tau_{2}$ acts on a separate
spinor degree of freedom. The former edge Hamiltonian $H_{{\rm edge},1}$
describes a strong phase in which the edge state is protected by time-reversal
symmetry alone and can not be gapped out. The latter Hamiltonian $H_{{\rm
edge},2}$ has two mass terms
$m_{1}\tau_{1}\sigma_{0}+m_{2}\tau_{3}\sigma_{0}$, which are both odd under
mirror reflection. Such a Hamiltonian hosts a zero mode symmetrically located
around a mirror-symmetric corner at $x=0$. A local perturbation near the
corner at $x=0$ that breaks the mirror symmetry can move this state away from
zero energy.

Class C${}^{\mathcal{CM_{+}}}$, $(s,t)=(6,1)$.— We set $U_{\cal
P}=\sigma_{2}$. This phase allows a strong topological phase with doublets of
particle-hole conjugated co-propagating chiral edge modes. Pairs of
counterpropagating doublets are prevented from mutually gapping out if they
have the same parity under ${\cal CM}$. Hence, within the relevant subspace we
may represent ${\cal CM}$ by the identity, $U_{\cal CM}=1$. The edge
Hamiltonian for $N$ such pairs of counterpropagating doublets reads

$H_{\rm edge}=-iv\sigma_{1}\tau_{2}\partial_{x}\openone_{N},$ (80)

where $\tau_{2}$ is a Pauli matrix acting on a different spinor degree of
freedom than the $\sigma$ matrices. Upon deforming the edge away symmetrically
around a corner at $x=0$, four mass terms
$m_{1}(x)\rho_{0}\sigma_{2}+m_{2}(x)\rho_{0}\sigma_{3}+m_{3}(x)\sigma_{1}\rho_{1}+m_{4}(x)\sigma_{1}\rho_{3}$
are allowed under a global reflection symmetry, with $m_{j}(x)=-m_{j}(-x)$,
$j=1,2,3,4$. Such a Hamiltonian hosts $N$ doublets of zero modes symmetrically
located around the corner at $x=0$.

Class DIII${}^{\mathcal{M_{--}}}$, $(s,t)=(3,2)$.— We choose the
representations $U_{\cal T}=\sigma_{2}$ and $U_{\cal P}=\sigma_{1}$, so that
$U_{\cal C}=\sigma_{3}$. The $\mathbb{Z}$ bulk topological invariant $N$ is
the difference of the numbers of helical edge doublets with positive and
negative “mixed parity” $\sigma_{\cal MC}$. For a “minimal” edge with $N\geq
0$ all modes have the same (positive) mixed parity and we may represent
$U_{\cal M}=\sigma_{3}$. Only topological crystalline phases with an even
number $N$ of pairs of helical modes can be used for the construction of a
second-order topological insulator, since a single helical Majorana mode
corresponds to a strong topological phase. With a suitable rescaling and basis
choice, an edge with $N$ pairs of helical modes is described by the edge
Hamiltonian

$H_{\rm edge}=-iv\sigma_{1}\tau_{3}\openone_{N/2}\partial_{x}.$ (81)

This edge Hamiltonian has the unique mass term $m\tau_{2}\sigma_{1}$, where
$m$ is a real symmetric $N/2\times N/2$ matrix. The mass term is odd under
mirror reflection, ensuring the existence of $N/2$ Majorana Kramers pairs at a
mirror-symmetric corner.

Class AII${}^{\mathcal{M_{-}}}$, $(s,t)=(4,2)$.— We represent time-reversal
and mirror symmetry using $U_{\cal T}=\sigma_{2}$ and $U_{\cal M}=\sigma_{3}$,
respectively. The bulk has a $\mathbb{Z}_{2}$ topological crystalline
classification, the generator of which has edge Hamiltonian $H_{\rm
edge}=-iv\sigma_{3}\partial_{x}$, with $x$ a coordinate along a mirror
symmetric edge. This is a strong topological phase.

Class CII${}^{\mathcal{M_{--}}}$, $(s,t)=(5,2)$.— We choose $U_{\cal
T}=\sigma_{2}$, ${\cal U}_{\rm P}=\sigma_{2}\tau_{3}$, and $U_{\cal
M}=\sigma_{3}$. With a suitable choice of basis, the nontrivial topological
crystalline phase has edge Hamiltonian

$H_{\rm edge}=-iv\sigma_{1}\tau_{1}\partial_{x}.$ (82)

The unique mass term $m\sigma_{1}\tau_{2}$ for this Hamiltonian is odd under
mirror reflection, ensuring the existence of a Kramers pair of zero-energy
states at a mirror-symmetric corner. A pair of corner states has a well
defined parity $\sigma_{\cal C}$ with respect to the chiral operation ${\cal
C}$, which depends on the asymptotic sign of the mass $m$ far away from the
corner. Multiple corner doublets with the same $\sigma_{\cal C}$ cannot gap
out, consistent with the $2\mathbb{Z}$ (extrinsic) classification of corner
states derived in App. B.

Class CI${}^{\mathcal{M_{--}}}$, $(s,t)=(7,2)$.— We represent ${\cal T}$ by
$\sigma_{1}K$, ${\cal P}$ by $\sigma_{2}K$, so that $U_{\cal C}=\sigma_{3}$.
An edge allows multiple pairs of counterpropagating states with support on
orbitals with the same parity under the product ${\cal MC}$, so one may
represent ${\cal M}$ by $U_{\cal M}=\sigma_{3}$ on a minimal edge. With a
suitable basis transformation and after rescaling, an edge with $N$ such pairs
of counterpropagating modes is described by the edge Hamiltonian

$H_{\rm edge}=-iv\sigma_{1}\tau_{2}\partial_{x}\openone_{N},$ (83)

where $\tau_{2}$ is a Pauli matrix acting on an additional spinor degree of
freedom. Upon deforming the edge away symmetrically around a corner at $x=0$,
three mass terms
$m_{1}(x)\sigma_{1}\tau_{1}+m_{2}(x)\sigma_{2}+m_{3}(x)\sigma_{1}\tau_{3}$ are
allowed under a global reflection symmetry, with $m_{i}$ real symmetric
matrices satisfying $m_{i}(x)=-m_{i}(-x)$, $i=1,2,3$. Such a Hamiltonian hosts
$2N$ zero-energy Majorana states symmetrically located around the corner at
$x=0$.

Class AI${}^{\mathcal{CM_{+}}}$, $(s,t)=(0,3)$.— We represent ${\cal T}$ by
complex conjugation $K$. An edge allows multiple pairs of counterpropagating
states with support on orbitals with the same mirror parity, so that we may
represent the mirror antisymmetry ${\cal CM}$ using $U_{\cal CM}=1$ for a
minimal edge. The corresponding edge Hamiltonian reads

$H_{\rm edge}=-iv\tau_{2}\partial_{x}\openone_{N},$ (84)

where $\tau_{2}$ is a Pauli matrix acting on an additional spinor degree of
freedom. Upon deforming the edge away symmetrically around a corner at $x=0$,
two mass terms $m_{1}(x)\tau_{1}+m_{2}(x)\tau_{3}$ are allowed under a global
reflection symmetry, with $m_{1}$ and $m_{2}$ real symmetric matrices
satisfying $m_{1}(x)=-m_{1}(-x)$ and $m_{2}(x)=-m_{2}(-x)$. Such a Hamiltonian
hosts $N$ zero modes symmetrically located around the corner at $x=0$.

Class D${}^{\mathcal{CM_{-}}}$, $(s,t)=(2,3)$.— We choose $U_{\cal P}=1$ and
$U_{\cal CM}=\sigma_{2}$. These symmetries allow a chiral edge Hamiltonian
$H_{\rm edge}=-iv\sigma_{0}\openone_{N}\partial_{x}$, with $x$ a coordinate
along the edge and $\openone_{N}$ the $N\times N$ identity matrix. Such an
edge represents a strong topological phase.

Class AII${}^{\mathcal{CM_{+}}}$, $(s,t)=(4,3)$.— We represent ${\cal T}$ by
$\sigma_{2}K$. An edge allows multiple pairs of helical modes with support on
orbitals with the same ${\cal CM}$ parity, so that we may represent ${\cal
CM}$ using $U_{\cal CM}=1$ for a minimal model. An insulator with an odd
number of such helical edge modes is a strong topological insulator. With a
suitable choice of basis, a “minimal” edge with an even number $N$ of helical
modes is described by the edge Hamiltonian

$H_{\rm edge}=-iv\sigma_{1}\tau_{0}\partial_{x}\openone_{N}/2,$ (85)

where $\tau_{0}$ the $2\times 2$ identity matrix acting an additional spinor
degrees of freedom. Upon deforming the edge away symmetrically around a corner
at $x=0$, two mass terms
$m_{1}(x)\tau_{2}\sigma_{2}+m_{2}(x)\tau_{2}\sigma_{3}$ are allowed under a
global reflection symmetry, with $m_{1}$ and $m_{2}$ real symmetric $N/2\times
N/2$ matrices satisfying $m_{1}(x)=-m_{1}(-x)$ and $m_{2}(x)=-m_{2}(-x)$. Such
a Hamiltonian hosts $N/2$ Kramers pairs of zero modes symmetrically located
around the corner at $x=0$.

Class C${}^{\mathcal{CM_{-}}}$, $(s,t)=(6,3)$.— We set $U_{\cal P}=\sigma_{2}$
and $U_{\cal CM}=\sigma_{3}$. An edge allows multiple pairs of chiral modes,
described by the edge Hamiltonian $H_{\rm
edge}=-iv\sigma_{1}\openone_{N}\partial_{x}$, where $x$ is a coordinate along
the edge and $\openone_{N}$ the $N\times N$ identity matrix. This is a strong
topological phase.

##  Appendix D Surface-to-hinge correspondence with twofold rotation symmetry

In this appendix we give details for the surface-to-hinge correspondence for
topological crystalline insulators and superconductors with twofold rotation
symmetry or antisymmetry and with mirror symmetry or antisymmetry, starting
from a symmetry characterization of the gapless surface states on symmetry-
invariant surfaces. The general idea underlying the surface-to-hinge
correspondence is the same as for edge-to-corner correspondence with mirror-
symmetric edges and corners, see Sec. IV.2 and App. C. The low-energy theory
of the surface states is given in terms of one or multiple Dirac cones that
are compatible with the non-spatial and spatial symmetries of the
corresponding Shiozaki-Sato class.Shiozaki and Sato (2014) Tilting the surface
away from the invariant direction, as in Fig. 11, allows for mass terms which
must be odd under twofold rotation or mirror reflection — because otherwise
they would be allowed for the symmetry-invariant surface orientation. If the
mass term is unique, the intersection of surfaces with opposite sign of the
mass constitutes a domain wall, hosting a gapless hinge state. If the mass
term is not unique, a mirror-symmetric hinge will still host a gapless mode,
but there is no protection for gapless hinge modes in the rotation-symmetric
case.

Throughout this appendix, $x$ and $y$ are coordinates on a (eventually tilted)
rotation-invariant or mirror-symmetric surface, where the mirror reflections
sends $x\to-x$, $\openone_{N}$ is the $N\times N$ unit matrix, and
$\sigma_{i}$, $\tau_{i}$, $\rho_{i}$, and $\mu_{i}$, $i=0,1,2,3$, are Pauli
matrices acting on different spinor degrees of freedom. We will restrict our
discussion to symmetry classes with a nontrivial bulk topological crystalline
phase, see Ref. Shiozaki and Sato, 2014 and Tables 6 and 7.

###  D.1 Complex classes with antiunitary symmetries and antisymmetries

Classes A${}^{\mathcal{T^{+}R}}$, $s=0$, and A${}^{\mathcal{P^{+}M}}$, $s=2$.—
We choose $U_{\mathcal{TR}}=\sigma_{1}$ and $U_{\mathcal{PM}}=\sigma_{3}$ to
represent the magnetic point group symmetry $\mathcal{TR}$ and mirror
antisymmetry $\mathcal{PM}$, respectively. These symmetries can protect a
single gapless surface state with a Dirac-like dispersion $H_{\rm
surface}=-iv(\sigma_{1}\partial_{x}+\sigma_{2}\partial_{y})$ (with a suitable
choice of basis). The unique mass term $m\sigma_{3}$ is odd under ${\cal TR}$
and ${\cal PM}$. A hinge at the intersection of crystal surfaces with opposite
signs of $m$ host a gapless hinge mode.

Classes AIII${}^{\mathcal{T^{+}R_{+}}}$, $s=1$, and
AIII${}^{\mathcal{T^{-}M_{-}}}$, $s=3$.— We choose $U_{\cal C}=\sigma_{3}$,
$U_{\cal TR}=1$, $U_{\cal TM}=\sigma_{2}\tau_{3}$. The nontrivial phase hosts
a pair of Dirac cones with dispersion $H_{\rm
surface}=-iv\sigma_{1}(\tau_{1}\partial_{x}+\tau_{3}\partial_{y})$ (with a
suitable choice of basis). There are two mass terms that may gap the Dirac
cones if the surface is tilted away from the invariant direction,
$m_{1}\sigma_{2}\tau_{0}+m_{2}\sigma_{1}\tau_{2}$, where both $m_{1}$ and
$m_{2}$ must change sign upon shifting to the rotated/mirror-reflected tilt
direction. With two mass terms, there is a protected hinge mode at a mirror-
symmetric hinge, but not generically in the presence of the twofold rotation
symmetry ${\cal TR}$.

Classes AIII${}^{\mathcal{T^{-}R_{-}}}$, $s=3$ and
AIII${}^{\mathcal{T^{-}M_{+}}}$, $s=5$.— We use $U_{\cal C}=\sigma_{3}$,
$U_{\cal TR}=\sigma_{2}$, $U_{\cal TM}=\sigma_{3}\tau_{2}$ to represent the
operations ${\cal C}$, ${\cal TR}$, and ${\cal TM}$, respectively. The twofold
rotation symmetry is compatible with pairs of Dirac cones with dispersion
$\propto-iv\tau_{2}(\sigma_{1}\partial_{x}\pm\sigma_{2}\partial_{y})$, which
defines the chirality $\pm$. The $2\mathbb{Z}$ bulk topological crystalline
index for this symmetry class counts the difference $N=N_{+}-N_{-}$ of such
Dirac cones with positive and negative chirality. For a “minimal” surface all
surface Dirac cones have the same chirality, so that after rescaling and with
suitable choice of basis the surface Hamiltonian reads $H_{\rm
surface}=-iv\tau_{2}(\sigma_{1}\partial_{x}+\sigma_{2}\partial_{y})\openone_{N}$.
Since such surface states are protected by chiral antisymmetry alone, this
represents a strong topological phase.

Classes AIII${}^{\mathcal{{T^{+}R}_{-}}}$, $s=7$, and
AIII${}^{\mathcal{T^{+}M_{+}}}$, $s=1$.— Like the previous case, this is a
strong phase, with gapless surface states on all surfaces. We choose $U_{\cal
C}=\sigma_{3}$, $U_{\cal TR}=\sigma_{1}$ and $U_{\cal M}=1$. The integer bulk
topological crystalline index counts the difference $N=N_{+}-N_{-}$ of surface
Dirac cones with dispersion $\propto-
iv(\partial_{x}\sigma_{1}\pm\partial_{y}\sigma_{2})$. For a “minimal” surface
all surface Dirac cones have the same chirality, so that after rescaling and
with suitable choice of basis the surface Hamiltonian reads $H_{\rm
surface}=-iv(\partial_{x}\sigma_{1}+\partial_{y}\sigma_{2})\openone_{N}$. Such
surface states are protected by chiral antisymmetry alone.

###  D.2 Real classes

Classes BDI${}^{\mathcal{R}_{++}}$, $(s,t)=(1,0)$, and
BDI${}^{\mathcal{M_{-+}}}$, $(s,t)=(1,3)$.— We represent time-reversal and
particle-hole conjugation using $U_{\mathcal{T}}=\sigma_{0}$,
$U_{\mathcal{P}}=\sigma_{3}$, $U_{\cal C}=\sigma_{3}$, $U_{\cal
R}=\sigma_{3}\rho_{3}$, and $U_{\cal M}=\sigma_{2}$. A symmetry-invariant
surface may host multiple pairs of Dirac cones with dispersion $\propto-
iv(\sigma_{1}\tau_{2}\partial_{x}\pm\sigma_{2}\tau_{0}\partial_{y})$, which
defines the “mirror chirality” $\pm$ for class BDI${}^{\mathcal{M_{-+}}}$. The
integer invariant $N$ counts the number of such pairs of Dirac cones, weighted
by the parity under ${\cal RC}$ (for class BDI${}^{\mathcal{R}_{++}}$) or by
mirror chirality (for class BDI${}^{\mathcal{M_{-+}}}$). A minimal surface
with $N\geq 0$ has pairs of Dirac cones of the same mirror chirality or the
same ${\cal RC}$-parity, so that effectively we may use $U_{\cal
R}=\sigma_{3}$ to represent ${\cal R}$. The corresponding surface Hamiltonian
is

$H_{\rm
surface}=-iv(\sigma_{1}\tau_{2}\partial_{x}+\sigma_{2}\tau_{0}\partial_{y})\openone_{N}.$
(86)

Two mass terms $m_{1}\sigma_{1}\tau_{1}+m_{2}\sigma_{1}\tau_{3}$, with $m_{1}$
and $m_{2}$ $N\times N$ real symmetric matrices, are allowed upon tilting the
surface away from the symmetry-invariant orientation. These mass terms are odd
under ${\cal R}$ and ${\cal M}$. Since there are two such mass terms, there
are no protected hinge modes for the rotation-symmetric case. However, there
are protected hinge modes at mirror-symmetric hinges in the mirror-symmetric
case.

Classes DIII${}^{\mathcal{R}_{++}}$, $(s,t)=(3,0)$, and
DIII${}^{\mathcal{M}_{+-}}$, $(s,t)=(3,3)$.— We set
$U_{\mathcal{T}}=\sigma_{2}$, $U_{\mathcal{P}}=\sigma_{1}$, $U_{\cal
M}=\sigma_{1}\tau_{2}$, and $U_{\cal R}=\tau_{3}$. Without rotation or mirror
symmetry, there are protected surface states with dispersion
$-iv(\sigma_{1}\partial_{x}\pm\sigma_{2}\partial_{y})$, which defines the
chirality $\pm$. The integer topological invariant $N$ counts the number of
such surface Dirac cones, weighted by the chirality. One such Dirac cone is
not compatible with ${\cal R}$ or ${\cal M}$ symmetry on a symmetry-invariant
surface, but two Dirac cones with the same chirality are, the dispersion for a
pair of Dirac cones being
$-iv\tau_{1}(\sigma_{1}\partial_{x}\pm\sigma_{2}\partial_{y})$. Since they
have the same chirality, such a pair of Dirac cones is protected by ${\cal T}$
and ${\cal P}$ alone. A phase with multiple such pairs of Dirac cones is a
strong topological phase with gapless surface states for surfaces of arbitrary
orientation.

Classes CII${}^{\mathcal{R}_{++}}$, $(s,t)=(5,0)$, and
CII${}^{\mathcal{M}_{-+}}$, $(s,t)=(5,3)$.— We let time-reversal be
represented by $U_{\mathcal{T}}=i\sigma_{2}$ and particle-hole by
$U_{\mathcal{P}}=i\sigma_{2}\tau_{3}$, so that $U_{\cal C}=\tau_{3}$. We
further set $U_{\cal M}=\tau_{3}$ and $U_{\cal R}=\tau_{3}\rho_{3}$. A
symmetry-invariant surface admits pairs of gapless surface states with Dirac-
like dispersion
$\propto-i(\sigma_{0}\tau_{2}\partial_{x}\pm\sigma_{1}\tau_{1}\partial_{y})$,
which defines the mirror chirality $\pm$ for class CII${}^{\mathcal{M}_{-+}}$.
The integer invariant $N$ counts the number of such pairs of Dirac cones,
weighted by the parity under ${\cal RC}$ (for class
CII${}^{\mathcal{R}_{++}}$) or by mirror chirality (for class
CII${}^{\mathcal{M_{-+}}}$). A minimal surface with $N\geq 0$ has pairs of
Dirac cones of the same mirror chirality or the same ${\cal RC}$-parity, so
that effectively we may use $U_{\cal R}=\tau_{3}$ to represent ${\cal R}$. A
single pair of Dirac cones is protected by ${\cal T}$ and ${\cal P}$ symmetry
alone, corresponding to a strong topological phase with gapless surface states
on all surfaces. A purely crystalline phase requires an even number $N$ of
pairs of surface Dirac cones, so that the corresponding surface Hamiltonian is

$H_{\rm
surface}=-iv(\tau_{1}\sigma_{1}\partial_{x}+\tau_{2}\sigma_{0}\partial_{y})\mu_{0}\openone_{N/2}.$
(87)

Such a Hamiltonian admits two mass terms
$m_{1}\tau_{1}\sigma_{2}\mu_{2}+m_{2}\tau_{1}\sigma_{3}\mu_{2}$, where $m_{1}$
and $m_{2}$ are $N/2\times N/2$ real symmetric matrices, which change sign
under mirror reflection and twofold rotation. Since there are two mass terms,
the rotation-symmetric class CII${}^{\mathcal{R}_{++}}$ does not have
protected hinge modes, whereas the mirror-symmetric class
CII${}^{\mathcal{M}_{-+}}$ has protected gapless modes at mirror-symmetric
hinges.

Classes CI${}^{\mathcal{R}_{++}}$, $(s,t)=(7,0)$, and
CI${}^{\mathcal{M}_{+-}}$, $(s,t)=(7,3)$.— We set $U_{\mathcal{T}}=\sigma_{1}$
and $U_{\mathcal{P}}=\sigma_{2}$, so that $U_{\cal C}=\sigma_{3}$. We further
set $U_{\cal R}=\tau_{3}$ and $U_{\cal M}=\sigma_{1}\tau_{3}$. A symmetry-
invariant surface admits surface states with dispersion
$-i\tau_{2}(\sigma_{1}\partial_{x}\pm\sigma_{2}\partial_{y})$, where $\pm$
defines the chirality. The integer invariant counts the number of such pairs
of surface Dirac cones, weighted by chirality. Since such pairs of surface
Dirac cones do not rely on crystalline symmetries for their protection this is
a strong phase, which has gapless surface states on surfaces of arbitrary
orientation.

Classes D${}^{\mathcal{CR}_{+}}$, $(s,t)=(2,2)$, and D${}^{\mathcal{M}_{+}}$,
$(s,t)=(2,1)$.— We set $U_{\cal P}=1$, $U_{\cal CR}=\tau_{3}$, and $U_{\cal
M}=\sigma_{3}$. A symmetry-invariant surface admits surface states with
dispersion $\propto-i(\sigma_{1}\partial_{x}\pm\sigma_{3}\partial_{y})$, which
defines the mirror chirality $\pm$ (for class D${}^{\mathcal{M}_{+}}$). This
class has an integer topological invariant $N$, which counts the differences
of the number of Dirac cones with positive and negative ${\cal CR}$-parity at
zero energy or mirror chirality, as appropriate. On a minimal surface with
$N\geq 0$ all surface Dirac cones have the same mirror chirality or ${\cal
CR}$-parity, so that we may effectively represent ${\cal CR}$ by $U_{\cal
CR}=1$. The corresponding surface Hamiltonian reads

$H_{\rm
surface}=-iv(\sigma_{1}\partial_{x}+\sigma_{3}\partial_{y})\openone_{N}.$ (88)

Such a surface admits a unique mass term $\sigma_{2}m$, with $m$ an $N\times
N$ real symmetric matrix, which changes sign under mirror reflection or under
the rotation antisymmetry. Correspondingly, a mirror-symmetric hinge admits
gapless modes. With twofold rotation antisymmetry, gapless hinge modes are
guaranteed to exist if $N$ is odd.

Classes DIII${}^{\mathcal{R}_{-+}}$, $(s,t)=(3,1)$, and
$DIII^{\mathcal{M}_{++}}$, $(s,t)=(3,0)$.—  We set $U_{\cal T}=\sigma_{2}$ and
$U_{\cal P}=\sigma_{1}$, so that $U_{\cal C}=\sigma_{3}$, $U_{\cal
R}=\sigma_{1}\tau_{3}$, $U_{\cal M}=\tau_{3}$. The crystalline bulk phase has
a $\mathbb{Z}_{2}$ topological classification, for which the nontrivial phase
has a surface state with Hamiltonian

$H_{\rm
surface}=-iv(\sigma_{1}\tau_{1}\partial_{x}+\sigma_{2}\tau_{0}\partial_{y})$
(89)

at a symmetry-invariant surface. There is a unique mass term
$m\sigma_{1}\tau_{2}$, which is odd under twofold rotation and under mirror
reflection. We conclude that the conditions for the existence of gapless hinge
modes are met.

Classes AII${}^{\mathcal{CR}_{-}}$, $(s,t)=(4,1)$ and
AII${}^{\mathcal{M}_{+}}$, $(s,t)=(4,0)$.— For a minimal model we choose
$U_{\cal T}=\sigma_{2}$, $U_{\cal CR}=\sigma_{1}$, and $U_{\cal M}=\tau_{3}$.
The crystalline bulk phase has a $\mathbb{Z}_{2}$ topological classification,
for which the nontrivial phase has a surface state with Hamiltonian

$H_{\rm surface}=-iv\sigma_{1}(\tau_{1}\partial_{x}+\tau_{3}\partial_{y})$
(90)

The model admits a unique mass term $m\sigma_{1}\tau_{2}$, which changes sign
under the twofold rotation antisymmetry operation and under mirror reflection.
Correspondingly, this model admits a helical gapless hinge mode.

Classes C${}^{\mathcal{CR}_{+}}$, $(s,t)=(6,1)$ and C${}^{\mathcal{M}_{+}}$,
$(s,t)=(6,0)$.— We set $U_{\cal P}=\sigma_{2}$, $U_{\cal CR}=\rho_{3}$, and
$U_{\cal M}=\tau_{2}\sigma_{3}$. The surface admits pairs of surface states
with a dispersion
$-iv\tau_{2}(\sigma_{1}\partial_{x}\pm\sigma_{3}\partial_{y})$, which defines
the mirror chirality (for class C${}^{\mathcal{M}_{+}}$). The integer
topological invariant $N$ for class C${}^{\mathcal{CR}_{+}}$ counts the
difference of the number of surface Dirac cones with $\mathcal{CR}$-eigenvalue
$1$ and $-1$ on a symmetry-invariant surface, and we may use $U_{\cal CR}=1$
to represent ${\cal CR}$ on a minimal surface with $N\geq 0$. With mirror
symmetry, $N$ counts the number of pairs of surface Dirac cones weighted by
mirror chirality. In both cases a minimal surface with $N\geq 0$ has
Hamiltonian

$H_{\rm
surface}=-iv\tau_{2}(\sigma_{1}\partial_{x}+\sigma_{3}\partial_{y})\openone_{N}.$
(91)

Such a surface has a unique mass term $m\tau_{2}\sigma_{2}$, with $m$ a real
symmetric $N\times N$ matrix which is odd under ${\cal CR}$ and ${\cal M}$.
Correspondingly, this mirror-symmetric model admits helical gapless hinge
modes at a mirror-symmetric hinge for all $N$, whereas the rotation-
antisymmetric model has gapless hinge modes if $N$ is odd.

Classes DIII${}^{\mathcal{R}_{--}}$, $(s,t)=(3,2)$ and
DIII${}^{\mathcal{M}_{-+}}$, $(s,t)=(3,1)$.— We set $U_{\cal T}=\sigma_{2}$,
$U_{\cal P}=\sigma_{1}$, $U_{\cal C}=\sigma_{3}$, $U_{\cal
R}=\sigma_{3}\tau_{3}$, and $U_{\cal M}=\sigma_{1}$. These classes admit
surface states with Dirac dispersion
$-iv(\sigma_{2}\partial_{x}\pm\sigma_{1}\partial_{y})$, which defines the
chirality $\pm$. Such a surface state is compatible with ${\cal R}$ and ${\cal
M}$ symmetries, but protected by chiral antisymmetry ${\cal C}$ alone. The
corresponding strong integer index counts their number, weighted by chirality.
A pair of surface states of opposite chirality, with dispersion
$-iv(\sigma_{2}\rho_{3}\partial_{x}\pm\sigma_{1}\rho_{0}\partial_{y})$, where
the sign $\pm$ defines the mirror chirality for class
DIII${}^{\mathcal{M}_{-+}}$, is protected by rotation or mirror symmetry. The
associated integer topological index $N$ counts the number of such pairs of
surface Dirac cones, weighted by ${\cal RC}$-parity (for class
DIII${}^{\mathcal{R}_{--}}$) or by mirror chirality (for class
DIII${}^{\mathcal{M}_{-+}}$). This allows one to effectively set $U_{\cal
R}=\sigma_{3}$ for a minimal surface with $N\geq 0$. The corresponding surface
Hamiltonian reads

$H_{\rm
surface}=-iv(\sigma_{2}\rho_{3}\partial_{x}+\sigma_{1}\rho_{0}\partial_{y})\openone_{N}.$
(92)

The surface Hamiltonian admits a unique mass term $m\sigma_{2}\rho_{2}$, with
$m$ a real symmetric $N\times N$ matrix which changes sign under the twofold
rotation antisymmetry operation and under mirror reflection. Correspondingly,
this mirror-symmetric model admits helical gapless hinge modes at a mirror-
symmetric hinge for all $N$, whereas the rotation-antisymmetric model has
gapless hinge modes if $N$ is odd.

Classes AII${}^{\mathcal{R}_{-}}$, $(s,t)=(4,2)$, and
AII${}^{\mathcal{CM}_{-}}$, $(s,t)=(4,1)$ We set $U_{\cal T}=\sigma_{2}$,
$U_{\cal R}=\sigma_{3}$, and $U_{\cal CM}=\sigma_{1}$. These classes have a
$\mathbb{Z}_{2}^{2}$ classification, with purely crystalline part
$\mathbb{Z}_{2}$. A generator for the strong phase has a surface state with
Dirac dispersion $-iv(\sigma_{1}\partial_{x}+\sigma_{2}\partial_{y})$, which
is protected by time-reversal symmetry alone. The generator for the purely
crystalline topological phase has a pair of surface Dirac cones with surface
Hamiltonian

$H_{\rm
surface}=-iv(\sigma_{1}\tau_{0}\partial_{x}+\sigma_{2}\tau_{3}\partial_{y}).$
(93)

This surface Hamiltonian has a unique mass term $m\sigma_{2}\tau_{2}$, which
is odd under ${\cal R}$ or ${\cal M}$. We conclude that these classes admits a
protected hinge mode.

Classes CII${}^{\mathcal{R}_{--}}$, $(s,t)=(5,2)$ and
CII${}^{\mathcal{M}_{+-}}$, $(s,t)=(5,1)$.— We set $U_{\cal T}=\sigma_{2}$,
$U_{\cal P}=\sigma_{2}\tau_{3}$, $U_{\cal C}=\tau_{3}$, $U_{\cal
R}=\sigma_{3}$ and $U_{\cal M}=\sigma_{2}\tau_{2}$. These classes have a
$\mathbb{Z}_{2}^{2}$ classification, with purely crystalline part
$\mathbb{Z}_{2}$. On a symmetry-invariant surface, the generator for the
strong phase has a pair of surface Dirac cones with dispersion
$-iv\tau_{1}(\sigma_{2}\partial_{x}+\sigma_{1}\partial_{y})$, which is
compatible with ${\cal R}$ and ${\cal M}$ symmetries, but does not rely on
those symmetry for its protection. The nontrivial purely crystalline phase has
two pairs of surface Dirac cones with Hamiltonian

$H_{\rm
surface}=-iv\tau_{1}(\sigma_{1}\rho_{0}\partial_{x}+\sigma_{2}\rho_{3}\partial_{y}).$
(94)

This surface Hamiltonian admits two mass terms
$m_{1}\sigma_{2}\tau_{1}\rho_{2}+m_{2}\sigma_{1}\tau_{2}\rho_{1}$, which is
odd under ${\cal R}$ or ${\cal M}$. We conclude that class
CII${}^{\mathcal{M}_{+-}}$ admits a protected hinge mode along mirror-
symmetric hinges, whereas class CII${}^{\mathcal{R}_{--}}$ does not allow
protected hinge modes.

Classes CI${}^{\mathcal{R}_{--}}$, $(s,t)=(7,2)$, and
CI${}^{\mathcal{M}_{-+}}$, $(s,t)=(7,1)$.— We choose $U_{\cal T}=\sigma_{1}$,
$U_{\cal P}=\sigma_{2}$, $U_{\cal C}=\sigma_{3}$, $U_{\cal
R}=\sigma_{3}\rho_{3}$, $U_{\cal M}=\sigma_{2}\tau_{2}$. These classes admit
pairs of surface states with dispersion
$-iv\tau_{2}(\sigma_{1}\partial_{x}\pm\sigma_{2}\partial_{y})$, which defines
the chirality $\pm$. Such a surface state is compatible with ${\cal R}$ and
${\cal M}$ symmetries, but protected by chiral antisymmetry ${\cal C}$ alone.
The corresponding strong integer index counts their number, weighted by
chirality. Two pairs of surface states of opposite chirality, with dispersion
$-iv\tau_{2}(\sigma_{1}\mu_{3}\partial_{x}\pm\sigma_{2}\mu_{0}\partial_{y})$,
where the sign $\pm$ defines the mirror chirality for class
CI${}^{\mathcal{M}_{-+}}$, are protected by rotation or mirror symmetry. The
associated integer topological index $N$ counts the number of such pairs of
surface Dirac cones, weighted by ${\cal RC}$-parity (for class
CI${}^{\mathcal{R}_{--}}$) or by mirror chirality (for class
CI${}^{\mathcal{M}_{-+}}$). This allows one to effectively set $U_{\cal
R}=\sigma_{3}$ for a minimal surface with $N\geq 0$. The corresponding surface
Hamiltonian reads

$H_{\rm
surface}=-iv\tau_{2}(\sigma_{1}\mu_{3}\partial_{x}+\sigma_{2}\mu_{0}\partial_{y})\openone_{N}.$
(95)

The surface Hamiltonian admits four mass terms
$m_{1}\sigma_{1}\tau_{2}\mu_{2}+m_{2}\sigma_{2}\tau_{1}\mu_{1}+m_{3}\sigma_{2}\tau_{3}\mu_{1}+m_{4}\sigma_{1}\tau_{0}\mu_{1}$,
with $m_{1}$, $m_{2}$, $m_{3}$, and $m_{4}$ real symmetric $N\times N$
matrices which change sign under the twofold rotation antisymmetry operation
and under mirror reflection. Correspondingly, this mirror-symmetric model
admits helical gapless hinge modes at a mirror-symmetric hinge for all $N$,
but the rotation-symmetric model has no protected hinge states.

Classes AI${}^{\mathcal{CR}_{+}}$, $(s,t)=(0,3)$, and
AI${}^{\mathcal{M}_{-}}$, $(s,t)=(0,2)$.— We choose $U_{\cal T}=1$, $U_{\cal
CR}=\rho_{3}$, and $U_{\cal M}=\sigma_{2}\tau_{3}$ to represent time reversal,
twofold rotation antisymmetry, and mirror reflection symmetry, respectively. A
symmetry-invariant surface admits pairs of surface states with a dispersion
$-iv\sigma_{2}(\tau_{1}\partial_{x}\pm\tau_{3}\partial_{y})$, which defines
the mirror chirality (for class AI${}^{\mathcal{M}_{-}}$). The integer
topological invariant $N$ counts the number of such pairs of surface Dirac
cones, weighted by $\mathcal{CR}$-parity or by mirror chirality, as
appropriate. On a minimal surface with $N\geq 0$ we may use $U_{\cal CR}=1$ to
represent ${\cal CR}$. The corresponding surface Hamiltonian reads

$H_{\rm
surface}=-iv\sigma_{2}(\tau_{1}\partial_{x}+\tau_{3}\partial_{y})\openone_{N}.$
(96)

Such a surface has three mass terms
$m_{1}\sigma_{1}\tau_{1}+m_{2}\sigma_{2}\tau_{2}+m_{3}\sigma_{3}\tau_{0}$,
with $m_{1}$, $m_{2}$, and $m_{3}$ real symmetric $N\times N$ matrices.
Correspondingly, this mirror-symmetric model admits helical gapless hinge
modes at a mirror-symmetric hinge for all $N$, but the rotation-antisymmetric
model has no protected hinge states.

Classes AII${}^{\mathcal{CR}_{+}}$, $(s,t)=(4,3)$ and
AII${}^{\mathcal{M}_{-}}$, $(s,t)=(4,2)$.— We set $U_{\cal T}=\sigma_{2}$,
$U_{\cal CR}=\tau_{3}$, and $U_{\cal M}=\sigma_{2}$. This phase allows surface
Dirac cones on symmetry-invariant surfaces with dispersion
$-iv(\sigma_{1}\partial_{x}\pm\sigma_{2}\partial_{y})$, which defines the
mirror chirality for class AII${}^{\mathcal{M}_{-}}$. The integer invariant
$N$ counts the number of such surface Dirac cones, weighted by ${\cal
CR}$-parity or mirror chirality, as appropriate. Odd values of $N$ correspond
to strong phases, which have gapless surface states irrespective of the
surface orientation. For even $N$ one has a purely crystalline phase. For a
minimal model with $N\geq 0$ one may effectively use $U_{\cal R}=1$ to
represent twofold rotation. The corresponding surface Hamiltonian is

$H_{\rm
surface}=-iv\rho_{0}(\sigma_{1}\partial_{x}+\sigma_{2}\partial_{y})\openone_{N/2}.$
(97)

There is a unique mass term $m\sigma_{3}\rho_{2}$, with $m$ an $N/2\times N/2$
matrix, which is odd under ${\cal CR}$ or ${\cal M}$. Correspondingly, a
mirror-symmetric hinge has $N/2$ protected helical modes, whereas there are
protected hinge modes in the presence of twofold rotation antisymmetry if
$N/2$ is odd.

Classes CII${}^{\mathcal{R}_{-+}}$, $(s,t)=(5,3)$, and
CII${}^{\mathcal{M}_{--}}$, $(s,t)=(5,2)$.— We set
$U_{\mathcal{T}}=\sigma_{2}$, $U_{\mathcal{P}}=\sigma_{2}\tau_{3}$, $U_{\cal
C}=\tau_{3}$, $U_{\cal R}=\sigma_{3}\tau_{1}$, and $U_{\cal M}=\sigma_{1}$.
These classes have a $\mathbb{Z}_{2}$ classification, for which the nontrivial
phase has a pair of Dirac cones with dispersion
$-i\tau_{1}(\sigma_{1}\partial_{x}+\sigma_{2}\partial_{y})$ on a symmetry-
invariant surface. Such a pair of Dirac cones is protected by time-reversal
symmetry and particle-hole antisymmetry alone, so that this is a strong
topological phase, which has gapless modes on all surfaces.

Classes C${}^{\mathcal{CR}_{-}}$, $(s,t)=(6,3)$, and C${}^{\mathcal{M}_{-}}$,
$(s,t)=(6,2)$.— We choose $U_{\cal P}=\tau_{2}$, $U_{\cal CR}=\tau_{3}$, and
$U_{\cal M}=\tau_{3}\sigma_{3}$. These classes have a $\mathbb{Z}_{2}$
classification, for which the nontrivial phase has a pair of Dirac cones with
dispersion $-i\tau_{0}(\sigma_{1}\partial_{x}+\sigma_{3}\partial_{y})$ on a
symmetry-invariant surface. Such a surface admits a unique mass term
$m\sigma_{2}\tau_{0}$, which is odd under ${\cal CR}$ and ${\cal M}$. We
conclude that the conditions for gapless hinge modes on a mirror-symmetric
hinge or with rotation-symmetric crystal termination at surfaces are met.

## References

  * Hasan and Kane (2010) M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
  * Bernevig and Hughes (2013) B. A. Bernevig and T. L. Hughes,  _Topological Insulators and Topological Superconductors_ (Princeton University Press, 2013).
  * Qi and Zhang (2011) X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).
  * Schindler  _et al._ (2017) F. Schindler, A. M. Cook, M. G. Vergniory, Z. Wang, S. S. P. Parkin, B. A. Bernevig, and T. Neupert, arXiv:1708.03636 (2017).
  * Benalcazar  _et al._ (2014) W. A. Benalcazar, J. C. Y. Teo, and T. L. Hughes, Phys. Rev. B 89, 224503 (2014).
  * Benalcazar  _et al._ (2017a) W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Science 357, 61 (2017a), http://science.sciencemag.org/content/357/6346/61.full.pdf .
  * Benalcazar  _et al._ (2017b) W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Phys. Rev. B 96, 245115 (2017b).
  * Peng  _et al._ (2017) Y. Peng, Y. Bao, and F. von Oppen, Phys. Rev. B 95, 235143 (2017).
  * Langbehn  _et al._ (2017) J. Langbehn, Y. Peng, L. Trifunovic, F. von Oppen, and P. W. Brouwer, Phys. Rev. Lett. 119, 246401 (2017).
  * Sitte  _et al._ (2012) M. Sitte, A. Rosch, E. Altman, and L. Fritz, Phys. Rev. Lett. 108, 126807 (2012).
  * Zhang  _et al._ (2013) F. Zhang, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 110, 046404 (2013).
  * Volovik (2010) G. E. Volovik, JETP Letters 91, 201 (2010).
  * Song  _et al._ (2017) Z. Song, Z. Fang, and C. Fang, Phys. Rev. Lett. 119, 246402 (2017).
  * Fang and Fu (2017) C. Fang and L. Fu, arXiv:1709.01929 (2017).
  * Schnyder  _et al._ (2008) A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, Phys. Rev. B 78, 195125 (2008).
  * Schnyder  _et al._ (2009) A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, AIP Conference Proceedings 1134, 10 (2009).
  * Kitaev (2009) A. Kitaev, AIP Conference Proceedings 1134, 22 (2009).
  * Altland and Zirnbauer (1997) A. Altland and M. R. Zirnbauer, Phys. Rev. B 55, 1142 (1997).
  * Shiozaki and Sato (2014) K. Shiozaki and M. Sato, Phys. Rev. B 90, 165114 (2014).
  * Chiu  _et al._ (2013) C.-K. Chiu, H. Yao, and S. Ryu, Phys. Rev. B 88, 075142 (2013).
  * Morimoto and Furusaki (2013) T. Morimoto and A. Furusaki, Phys. Rev. B 88, 125129 (2013).
  * Trifunovic and Brouwer (2017) L. Trifunovic and P. W. Brouwer, Phys. Rev. B 96, 195109 (2017).
  * Lu and Lee (2014) Y.-M. Lu and D.-H. Lee, arXiv:1403.5558v1 (2014).
  * Shiozaki  _et al._ (2016) K. Shiozaki, M. Sato, and K. Gomi, Phys. Rev. B 93, 195413 (2016).
  * Fang  _et al._ (2012) C. Fang, M. J. Gilbert, and B. A. Bernevig, Phys. Rev. B 86, 115112 (2012).
  * Fang  _et al._ (2013) C. Fang, M. J. Gilbert, and B. A. Bernevig, Phys. Rev. B 87, 035119 (2013).
  * Slager  _et al._ (2013) R.-J. Slager, A. Mesaros, V. Juricic, and J. Zaanen, Nature Phys. 9, 98 (2013).
  * Jadaun  _et al._ (2013) P. Jadaun, D. Xiao, Q. Niu, and S. K. Banerjee, Phys. Rev. B 88, 085110 (2013).
  * Liu  _et al._ (2014) X.-J. Liu, J. J. He, and K. T. Law, Phys. Rev. B 90, 235141 (2014).
  * Alexandradinata  _et al._ (2014) A. Alexandradinata, C. Fang, M. J. Gilbert, and B. A. Bernevig, Phys. Rev. Lett. 113, 116403 (2014).
  * Dong and Liu (2016) X.-Y. Dong and C.-X. Liu, Phys. Rev. B 93, 045429 (2016).
  * Kruthoff  _et al._ (2017) J. Kruthoff, J. de Boer, J. van Wezel, C. L. Kane, and R.-J. Slager, Phys. Rev. X 7, 041069 (2017).
  * Po  _et al._ (2017) H. C. Po, A. Vishwanath, and H. Watanabe, Nature Comm. 8, 50 (2017).
  * Bradlyn  _et al._ (2017) B. Bradlyn, L. Elcoro, J. Cano, M. G. Vergniory, Z. Wang, C. Felser, M. I. Aroyo, and B. A. Bernevig, Nature 547, 298 (2017).
  * Shiozaki  _et al._ (2017) K. Shiozaki, M. Sato, and K. Gomi, Phys. Rev. B 95, 235425 (2017).
  * Khalaf  _et al._ (2017) E. Khalaf, H. C. Po, A. Vishwanath, and H. Watanabe, arXiv:1711.11589 (2017).
  * Teo and Kane (2010) J. C. Y. Teo and C. L. Kane, Phys. Rev. B 82, 115120 (2010).
  * Note (1) Although ${\cal P}$ and and ${\cal C}$ are commonly referred to as “particle-hole symmetry” and “chiral symmetry”, we will refer to these as antisymmetries, because they connect $H$ to $-H$, see Eq. (1).
  * Chiu  _et al._ (2016) C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Rev. Mod. Phys. 88, 035005 (2016).
  * Note (2) For spin $1/2$ electrons often spatial symmetries squaring to $-1$ are used. Multiplication by $i$ then gives a symmetry operation squaring to $1$. Note, however, that multiplication with $i$ turns a symmetry that commutes with ${\cal T}$ or ${\cal P}$ into a symmetry that anticommutes with ${\cal T}$ or ${\cal P}$ and vice versa.
  * Stone  _et al._ (2011) M. Stone, C.-K. Chiu, and A. Roy, J. Math. Phys. A: Math. Theor. 44, 045001 (2011).
  * Wen (2012) X.-G. Wen, Phys. Rev. B 85, 085103 (2012).
  * Abramovici and Kalugin (2012) G. Abramovici and P. Kalugin, Int. J. Geom. Methods Mod. Phys. 9, 125003 (2012).
  * Kennedy and Zirnbauer (2016) R. Kennedy and M. R. Zirnbauer, Commun. Math. Phys. 342, 909 (2016).
  * Fulga  _et al._ (2012) I. C. Fulga, F. Hassler, and A. R. Akhmerov, Phys. Rev. B 85, 165409 (2012).
  * Groth  _et al._ (2014) C. W. Groth, M. Wimmer, A. R. Akhmerov, and X. Waintal, New J. Phys. 16, 063065 (2014).
  * Teo  _et al._ (2008) J. C. Y. Teo, L. Fu, and C. L. Kane, Phys. Rev. B 78, 045426 (2008).
  * Turner  _et al._ (2012) A. M. Turner, Y. Zhang, R. S. K. Mong, and A. Vishwanath, Phys. Rev. B 85, 165120 (2012).
  * Ezawa (2018a) M. Ezawa, Phys. Rev. Lett. 120, 026801 (2018a).
  * Ezawa (2018b) M. Ezawa, arXiv:1801.00437 (2018b).
  * Xu  _et al._ (2017) Y. Xu, R. Xue, and S. Wan, arXiv:1711.09202 (2017).
  * Lau  _et al._ (2016) A. Lau, J. van den Brink, and C. Ortix, Phys. Rev. B 94, 165164 (2016).
  * van Miert  _et al._ (2017) G. van Miert, C. Ortix, and C. Morais Smith, 2D Materials 4, 015023 (2017).
  * Shapourian  _et al._ (2018) H. Shapourian, Y. Wang, and S. Ryu, Phys. Rev. B 97, 094508 (2018).
  * Schindler  _et al._ (2018) F. Schindler, Z. Wang, M. G. Vergniory, A. M. Cook, A. Murani, S. Sengupta, A. Y. Kasumov, R. Deblock, S. Jeon, I. Drozdov, H. Bouchiat, S. Guéron, A. Yazdani, B. A. Bernevig, and T. Neupert, ArXiv e-prints (2018), arXiv:1802.02585 [cond-mat.mtrl-sci] .
  * Imhof  _et al._ (2017) S. Imhof, C. Berger, F. Bayer, J. Brehm, L. Molenkamp, T. Kiessling, F. Schindler, C. H. Lee, M. Greiter, T. Neupert, and R. Thomale, ArXiv e-prints (2017), arXiv:1708.03647 [cond-mat.mes-hall] .
  * Peterson  _et al._ (2018) C. W. Peterson, W. A. Benalcazar, T. L. Hughes, and G. Bahl, Nature 555, 346 (2018).
  * Serra-Garcia  _et al._ (2018) M. Serra-Garcia, V. Peri, R. Süsstrunk, O. R. Bilal, T. Larsen, L. G. Villanueva, and S. D. Huber, Nature 555, 342 (2018).