# Higher-order topological semimetals and nodal superconductors with an order-
two crystalline symmetry

Sophia Simon  Dahlem Center for Complex Quantum Systems and Physics
Department, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany
Department of Physics, University of Toronto, Toronto, Ontario M5S1A7, Canada
Max Geier  Dahlem Center for Complex Quantum Systems and Physics Department,
Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany  Center for
Quantum Devices, Niels Bohr Institute, University of Copenhagen, DK-2100
Copenhagen, Denmark  Piet W. Brouwer  Dahlem Center for Complex Quantum
Systems and Physics Department, Freie Universität Berlin, Arnimallee 14, 14195
Berlin, Germany

###### Abstract

We identify four types of higher-order topological semimetals or nodal
superconductors (HOTS), hosting (i) flat zero-energy “Fermi arcs” at crystal
hinges, (ii) flat zero-energy hinge arcs coexisting with surface Dirac cones,
(iii) chiral or helical hinge modes, or (iv) flat zero-energy hinge arcs
connecting nodes only at finite momentum. Bulk-boundary correspondence relates
the hinge states to the bulk topology protecting the nodal point or loop. We
classify all HOTS for all tenfold-way classes with an order-two crystalline
(anti-)symmetry, such as mirror, twofold rotation, or inversion.

Topological semimetals have a band structure with a touching of valence and
conduction bands that is robust to small parameter changes that preserve the
crystalline symmetries [1, 2, 3, 4, 5, 6, 7, 8]. The prime example is a Weyl
semimetal, which has twofold degenerate band crossing points around which the
band structure is effectively described in terms of massless Weyl fermions [9,
10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. Similarly, topological nodal
superconductors have protected gapless modes in the excitation spectrum of the
Bogoliubov quasiparticles [1, 20, 21, 22]. A topological classification of
nodal points or lines, indicating which type of node may be protected for
which combination of symmetries, has been obtained for the “tenfold-way”
symmetry classes [23]. These are defined by the presence or absence of time-
reversal and spin-rotation symmetry, as well as the particle-hole symmetry
associated with the mean-field description of superconductors [24]. The
results have been subsequently extended to incorporate crystalline symmetries
[25, 26, 27, 28, 29, 30, 31].

Topological nodal semimetals and superconductors may have gapless surface
excitations, with a surface band structure that connects to the projections of
the nodes of the bulk band structure onto the surface Brillouin zone. These
surface excitations are anomalous, in the sense that they cannot be removed by
a local, symmetry-allowed perturbation [32]. For example, a Weyl semimetal has
a “Fermi-arc” of surface states, whereas nodal-loop semimetals have “drumhead”
surface states, an approximately flat “Fermi disc” of surface states inside
the projection of the nodal loop on the surface [15, 33, 34, 35, 36, 37, 38,
39, 40].

There is an insightful theoretical argument that links the protected surface
states and nodal points or lines in gapless band structures to properties of
gapped topological band structures in lower dimensions. For example, the
Hamiltonian $H(k_{x},k_{y},k_{z})$ of a Weyl semimetal can be considered as a
$k_{z}$-dependent “one-parameter family” of two-dimensional Hamiltonians
$H_{k_{z}}(k_{x},k_{y})$. A topological phase transition of
$H_{k_{z}}(k_{x},k_{y})$ as a function of $k_{z}$ — a “band inversion” — then
corresponds to a protected gap closing in three dimensions [41, 14], whereas
the chiral edge states in the topological phase of $H_{k_{z}}(k_{x},k_{y})$
naturally map to the Fermi-arc surface states [11, 42, 19]. In the same way, a
gapless band structure with a nodal loop can be obtained by considering
$H(k_{x},k_{y},k_{z})$ as a “two-parameter family” of one-dimensional
Hamiltonians [43, 38, 40]. The flat disc of “drumhead” surface states is then
associated with the protected end-states of topological band structures in one
dimension [32].

With crystalline symmetries, anomalous boundary states may also exist at edges
or corners of a crystal, instead of being extended across the entire surface
[44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57]. Band structures that
impose such boundary signatures are said to possess “higher-order” topology
[44, 58]. For a large class of crystalline symmetries, classification
information of gapped topological band structures that includes the order of
the topological phase is available [49, 51, 50, 52, 59, 53, 60].

Figure 1:  (a) Rhombic pillar geometry and local density of states at energy
$E=0$ of model (1). (b) Quasi one-dimensional band structures in the rhombic
pillar geometry for the models of Eqs. (1), (3), (5), and (6) (from left to
right). The full and dashed black curves indicate the bulk and surface
excitation gaps, respectively. Symmetry-preserving perturbations have been
added to turn accidentally degenerate nodal points into nodal loops and to gap
out non-anomalous surface states, as desribed in App. S.1. In models (1), (3),
and (5) we set $t=1$, $k_{0}=0.6\pi$. In model (6) we used $m=0$, $t=1$,
$t^{\prime}=2$.

Soon after the discovery of gapped higher-order topological band structures,
it was realized that topological gapless phases may also have higher-order
anomalous boundary states [61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73,
74, 75, 76]. In particular, a second-order topological semimetal has anomalous
states on crystal hinges, extending between the projections of the nodal
points of the bulk band structure. No third-order topological semimetals can
exist, because a putative anomalous corner state can always hybridize with the
continuum of bulk states.

In this article, we consider the question, for which combinations of tenfold-
way and crystalline symmetries second-order topological gapless phases may
exist. Hereto, we note that, if we again view the Hamiltonian
$H(k_{x},k_{y},k_{z})$ as a one-parameter family of two-dimensional
Hamiltonians $H_{k_{z}}(k_{x},k_{y})$, a second-order topological nodal
semimetal or superconductor arises if, as a function of $k_{z}$,
$H_{k_{z}}(k_{x},k_{y})$ goes through a topological phase transition involving
a gapped second-order phase [62, 67]. Hence, to determine which combinations
of tenfold-way and crystalline symmetries allow for second-order nodal phases,
it suffices to inspect topological phase transitions of the two-dimensional
Hamiltonians $H_{k_{z}}(k_{x},k_{y})$. We here provide such a classification
for three crystalline symmetries: mirror, twofold rotation, or inversion
symmetry. These are the symmetries for which a full order-resolved
classification of gapped topological band structures exists [49, 51, 50, 52].

Specifically, we consider a pillar geometry, translation invariant in the $z$
direction and finite, with symmetry-compatible lattice termination, in the
$xy$ plane, see Fig. 1a. We set the unit cell size along $z$ to $1$, so that
$-\pi<k_{z}\leq\pi$. We view the Hamiltonian $H(k_{x},k_{y},k_{z})$ of such a
crystal as a $k_{z}$-dependent family $H_{k_{z}}(k_{x},k_{y})$, where each
Hamiltonian $H_{k_{z}}(k_{x},k_{y})$ describes a two-dimensional crystal with
symmetry-compatible boundaries. Throughout, we take $H_{0}(k_{x},k_{y})$ and
$H_{\pi}(k_{x},k_{y})$ to be gapped, so that any nodal points or loops in the
bulk band structure occur away from the high-symmetry planes $k_{z}=0$, $\pi$.
For definiteness, we assume that $H_{\pi}(k_{x},k_{y})$ is topologically
trivial, which is an assumption that can be made without loss of generality
for a classification of features directly associated with the nodes of the
bulk spectrum. In this geometry, anomalous second-order boundary states (if
any) exist between projections of bulk nodes on both sides of $k_{z}=0$ or
between projections of two bulk nodes on the same side of $k_{z}=0$.

The tenfold-way and crystalline symmetries imply constraints for the family of
two-dimensional Hamiltonians $H_{k_{z}}(k_{x},k_{y})$. As symmetry operations
may change the sign of $k_{z}$, in general the symmetry constraints at the
high-symmetry planes $k_{z}=0$, $\pi$ differ from those for generic
$0<|k_{z}|<\pi$. Because of the different symmetries involved, different types
of anomalous boundary states are possible around $k_{z}=0$ and for generic
$0<|k_{z}|<\pi$. Considering all combinations of tenfold-way and order-two
crystalline symmetries, we find three scenarios for topological semimetals or
nodal superconductors with second-order hinge states around $k_{z}=0$ and one
scenario for second-order hinge states in a region at finite $0<|k_{z}|<\pi$.
We now discuss these four scenarios, together with examples of lattice models
that realize them.

(i) Flat zero-energy hinge arcs around $k_{z}=0$.— If $H_{k_{z}}(k_{x},k_{y})$
is in a second-order topological phase for $k_{z}=0$ and remains so upon
(continuously) going to finite $|k_{z}|>0$, there are flat arcs of zero-energy
hinge states running between the projections of nodal points or loops on both
sides of $k_{z}=0$ onto the edge Brillouin zone. (The hinge states are the
zero-energy corner states of the two-dimensional Hamiltonians
$H_{k_{z}}(k_{x},k_{y})$.) An example is a band structure with a chiral
antisymmetry $\mathcal{C}$, corresponding to tenfold-way class AIII, and a
mirror symmetry ${\cal M}^{x}$ that commutes with ${\cal C}$. This symmetry
class may be realized in a time-reversal symmetric superconductor with $U(1)$
spin-rotation symmetry [24]. A concrete lattice model is the four-band model

$\begin{split}H(k_{x},k_{y},k_{z})=&\,t\tau_{1}\sigma_{1}[m(k_{z})+2-\cos
k_{x}-\cos k_{y}]\\\ &\,\mbox{}+t\tau_{1}\sigma_{3}\sin
k_{x}+t\tau_{1}\sigma_{2}\sin k_{y},\end{split}$ (1)

with Pauli matrices $\tau_{j}$ and $\sigma_{j}$, $j=0,1,2,3$. Here and below,
$m(k_{z})$ is defined as $m(k_{z})=\cos k_{0}-\cos k_{z}$. The symmetry
constraints are

$\displaystyle H(k_{x},k_{y},k_{z})=$ $\displaystyle\,-U_{\cal
C}H(k_{x},k_{y},k_{z})U_{\cal C}^{\dagger}$ $\displaystyle=$
$\displaystyle\,U_{\cal M}H(-k_{x},k_{y},k_{z})U_{\cal M}^{\dagger},$ (2)

with $U_{\mathcal{C}}=\tau_{3}\sigma_{0}$ and
$U_{\mathcal{M}}=\tau_{3}\sigma_{3}$. The Hamiltonian (1) has double band
crossings at $k_{z}=\pm k_{0}$, which are turned into nodal loops upon
addition of a generic symmetry-preserving perturbation. Seen as a two-
dimensional Hamiltonian, $H_{k_{z}}(k_{x},k_{y})$ is in a second-order phase
with non-degenerate zero-energy states at mirror-symmetric corners for
$-k_{0}<k_{z}<k_{0}$ and in a trivial phase for $|k_{z}|>k_{0}$, see App. S.1
for details. Hence, for a pillar geometry the three-dimensional Hamiltonian
$H(k_{x},k_{y},k_{z})$ describes a nodal-loop semimetal with a flat arc of
hinge states. Exact diagonalization, with symmetry-preserving terms added such
that eventual trivial surface states are gapped out, confirms the appearance
of a flat arc of zero-energy states on the two mirror-symmetric hinges of the
pillar, see Figs. 1a and b.

The presence of the antisymmetry ${\cal C}$ is essential for pinning the hinge
states to zero energy. Other antisymmetries, such as the antiunitary particle-
hole antisymmetry ${\cal P}$ associated with the Bogoliubov-de Gennes
description of superconductors or combinations of ${\cal C}$ or ${\cal P}$ and
crystalline symmetries can also stabilize hinge states at zero energy. Flat
arcs of hinge states also appear in proposals for second-order semimetals in
the literature [61, 62, 63, 66, 69, 67, 68, 64, 65], although the antisymmetry
required for the topological protection is not always recognized explicitly
[61, 62, 63, 66, 64, 65] or is allowed to be weakly broken [67, 69], so that
the hinge arc is no longer strictly pinned to zero energy. In that case, the
hinge arc no longer has a strict topological protection and may be removed by
a sufficiently strong local perturbation at the crystal hinge.

The tenfold-way class CII with a twofold rotation symmetry ${\cal R}^{z}$
allows for a variant of this scenario, in which the zero-energy hinge arcs do
not terminate at nodes of the bulk band structure, but at (topologically
protected) Dirac nodes of the surface band structure. In this case, the
presence of anomalous zero-energy hinge states at $k_{z}=0$ as well as their
absence at $k_{z}=\pi$ are protected by the topology of the bulk band
structure, whereas the zero-energy hinge states at generic $0<|k_{z}|<\pi$ are
“extrinsic”, i.e. unrelated to the bulk topology [49]. Such hinge states may
disappear at a $k_{z}$-dependent closing of the surface gap. We refer to App.
S.2 for details.

(ii) Flat zero-energy hinge arcs around $k_{z}=0$ and first-order surface
states at $k_{z}=0$.— With a suitable combination of tenfold-way and
crystalline symmetries, it is possible that $H_{k_{z}}(k_{x},k_{y})$ is in a
first-order topological phase at $k_{z}=0$, and in a second-order topological
phase at finite $0<|k_{z}|<k_{0}$. Then, the band structure is of a hybrid
order, with a flat arc of zero-energy states at the hinge coexisting with
first-order Dirac cone surface states on adjacent surfaces that are themselves
not invariant under the crystalline symmetry. An example is a spinful odd-
parity time-reversal symmetric superconductor (tenfold-way class DIII) with
mirror symmetry ${\cal M}$. A concrete model is the four-band Hamiltonian

$\begin{split}H(k_{x},k_{y},k_{z})=&\,t\tau_{3}\sigma_{0}[m(k_{z})+2-\cos
k_{x}-\cos k_{y}]\\\ &\,\mbox{}+t\tau_{2}\sigma_{0}\sin
k_{x}+t\tau_{1}\sigma_{3}\sin k_{y},\end{split}$ (3)

which has nodal points at $|k_{z}|=k_{0}$ and satisfies the symmetry
constraints

$\displaystyle H(k_{x},k_{y},k_{z})=$ $\displaystyle\,-U_{\cal
P}H(-k_{x},-k_{y},-k_{z})^{*}U_{\cal P}^{\dagger}$ $\displaystyle=$
$\displaystyle\,U_{\cal T}H(-k_{x},-k_{y},-k_{z})^{*}U_{\cal T}^{\dagger}$
$\displaystyle=$ $\displaystyle\,U_{\cal M}H(-k_{x},k_{y},k_{z})U_{\cal
M}^{\dagger},$ (4)

where $U_{\mathcal{T}}=\tau_{0}\sigma_{2}$,
$U_{\mathcal{P}}=\tau_{1}\sigma_{0}$, and
$iU_{\mathcal{M}}=i\tau_{3}\sigma_{1}$ represent particle-hole conjugation,
time reversal, and mirror, respectively. At generic $k_{z}$,
$H_{k_{z}}(k_{x},k_{y})$ is invariant under the products ${\cal P}{\cal T}$
and ${\cal T}{\cal M}$ that leave $k_{z}$ invariant, but not under ${\cal P}$,
${\cal T}$, or ${\cal M}$ separately. Because the symmetries differ for
$k_{z}=0$ or $\pi$ and generic $0<|k_{z}|<\pi$, $H_{k_{z}}(k_{x},k_{y})$ is in
different topological classes at $k_{z}=0$ and at $k_{z}\neq 0$: a first-order
phase with helical Majorana edge states for $k_{z}=0$, and a second-order
phase with zero-energy corner states for $0<|k_{z}|<k_{0}$. As shown in Fig.
1b, for the three-dimensional Hamiltonian $H(k_{x},k_{y},k_{z})$ this implies
a combination of Dirac cone surface states at $k_{z}=0$ and a flat hinge arc
ranging from $k_{z}=0$ to the projection of the nodal loop around
$|k_{z}|=k_{0}$ onto the hinge Brillouin zone.

(iii) Chiral or helical hinge states around $k_{z}=0$.— With rotation or
inversion symmetry, it is possible that $H_{0}(k_{x},k_{y})$ is in a second-
order phase with zero-energy corner states, whereas $H_{k_{z}}(k_{x},k_{y})$
is in a nontrivial atomic limit phase at generic $k_{z}$. In this case, the
bulk band crossing is protected by the transition between the nontrivial and
trivial atomic limits as a function of $k_{z}$. Around $k_{z}=0$, such a
system hosts chiral or helical hinge mode which are pinned to zero energy at
$k_{z}=0$.

As an example, we consider a crystal with twofold rotation symmetry ${\cal
R}^{z}$, which has co-propagating chiral hinge modes on rotation-related
hinges. Co-propagating chiral hinge modes are ruled out in gapped topological
phases, but with a gapless bulk, an equilibrium current carried by bulk states
may compensate the equilibrium current carried by the chiral hinges. This
scenario applies to a spinful even-rotation-parity time-reversal symmetry-
breaking superconductor (tenfold-way class D), which has the Bogoliubov-de
Gennes Hamiltonian

$\begin{split}H(k_{x},k_{y},k_{z})=&\,t\tau_{3}\sigma_{3}[m(k_{z})+2-\cos
k_{x}-\cos k_{y}]\\\ &\,\mbox{}+t\tau_{1}\sigma_{3}\sin
k_{x}+t\tau_{2}\sigma_{3}\sin k_{y},\end{split}$ (5)

with the representations $U_{\mathcal{P}}=\tau_{1}\sigma_{0}$ and
$iU_{\mathcal{R}}=i\tau_{3}\sigma_{3}$. Since particle-hole conjugation
changes the sign of $k_{z}$, for generic $k_{z}$, $H_{k_{z}}(k_{x},k_{y})$ is
constrained by ${\cal R}_{z}$ only. The spectrum, showing the linearly
dispersing chiral hinge states around $k_{z}=0$, is shown in Fig. 1b. In the
presence of time-reversal symmetry, this scenario leads to helical hinge
states around $k_{z}=0$ instead of chiral hinge states, as discussed in App.
S.2.

(iv) Flat zero-energy hinge arcs away from $k_{z}=0$.— If
$H_{k_{z}}(k_{x},k_{y})$ can be in a second-order phase for generic
$0<|k_{z}|<\pi$, flat zero-energy hinge arcs between projections of bulk nodes
on hinge Brillouin zones away from $k_{z}=0$ and $k_{z}=\pi$ are possible
irrespective of the topological phase of $H_{k_{z}}(k_{x},k_{y})$ at
$k_{z}=0$. Such flat arcs of zero-energy hinge states are the only possible
second-order boundary features associated with a bulk gap closing if the
topological classification of $H_{k_{z}}(k_{x},k_{y})$ at $k_{z}=0$ does not
allow for topological phases with anomalous boundary features or
$H_{k_{z}}(k_{x},k_{y})$ immediately becomes trivial upon going from $k_{z}=0$
to nonzero $k_{z}$.

An example is a spinful odd-rotation-parity time-reversal-symmetric
superconductor (class DIII)

$\displaystyle H(k_{x},k_{y},k_{z})=$ $\displaystyle\,t\tau_{0}\sigma_{3}\sin
k_{z}(m-1-\cos k_{x}-\cos k_{y})$
$\displaystyle\,\mbox{}+t\tau_{0}\sigma_{1}\sin k_{x}+t\tau_{3}\sigma_{2}\sin
k_{y}$ $\displaystyle\,\mbox{}+t^{\prime}\tau_{3}\sigma_{0}\cos^{2}k_{z},$ (6)

with the representations $U_{\mathcal{T}}=\tau_{0}\sigma_{2}$,
$U_{\mathcal{P}}=\tau_{1}\sigma_{0}$, and
$iU_{\mathcal{R}}=i\tau_{0}\sigma_{3}$. The Hamiltonian (6) is topologically
trivial at $k_{z}=0$, $\pi$ and is in a second-order phase around
$|k_{z}|=\pi/2$. It has a total of eight nodal loops. Four of these, which
appear closest to $k_{z}=0$ and $\pi$, are “fragile” and can be removed by the
addition of trivial bands (see App. S.1). The remaining four, which are closer
to $k_{z}=\pm\pi/2$, are associated with the topological phase transitions
between second-order and atomic-limit phases of $H_{k_{z}}(k_{x},k_{y})$.
There is a flat zero-energy hinge arc connecting projections of the nodal
loops around $k_{z}=\pm\pi/2$ onto the edge Brillouin zone. Figure 1b shows
the spectrum for this model for the pillar geometry.

Class | $\mathcal{T}^{2}$ | $\mathcal{P}^{2}$ | $\mathcal{C}^{2}$ | type | $\mathcal{M}^{x/y}$ | $\mathcal{R}^{x/y}$ | $\mathcal{R}^{z}$ | $\mathcal{I}$
---|---|---|---|---|---|---|---|---
AIII | 0 | 0 | 1 | (i) | $\mathcal{M}_{+}^{x/y}$ | $\mathcal{T}^{+}\mathcal{R}_{+}^{x/y}$ | $\mathcal{R}_{-}^{z}$ | $\mathcal{T}^{+}\mathcal{I}_{-}$
BDI | 1 | 1 | 1 | (i) | $\mathcal{M}_{++}^{x/y}$ | $\mathcal{R}_{++}^{x/y}$ | $\mathcal{R}_{+-}^{z}$ | $\mathcal{I}_{+-}$
|  |  |  | (iv) | $\mathcal{M}_{--}^{x/y}$ |  | $\mathcal{R}_{-+}^{z}$ |
D | 0 | 1 | 0 | (i) | $\mathcal{T}^{+}\mathcal{M}_{+}^{x/y}$ | $\mathcal{R}_{+}^{x/y}$ |  |
|  |  |  | (iii) |  |  | $\mathcal{R}_{-}^{z}$ | $\mathcal{T}^{+}\mathcal{I}_{+}$
|  |  |  | (iv) | $\mathcal{T}^{-}\mathcal{M}_{-}^{x/y}$ |  |  |
DIII | -1 | 1 | 1 | (i) | $\mathcal{M}_{--}^{x/y}$ |  |  |
|  |  |  | (ii) | $\mathcal{M}_{--}^{x/y}$ | $\mathcal{R}_{-+}^{x/y}$ | $\mathcal{R}_{+-}^{z}$ | $\mathcal{I}_{--}$
|  |  |  | (iii) |  |  | $\mathcal{R}_{+-}^{z}$ |
|  |  |  | (iv) | $\mathcal{M}_{++}^{x/y}$ |  | $\mathcal{R}_{-+}^{z}$ |
CII | -1 | -1 | 1 | (i) | $\mathcal{M}_{++}^{x/y}$ |  |  $\mathcal{R}_{+-}^{z}$* |
|  |  |  | (iv) | $\mathcal{M}_{\pm\pm}^{x/y}$ | $\mathcal{R}_{--}^{x/y}$ | $\mathcal{R}_{\pm\mp}^{z}$ | $\mathcal{I}_{-+}$
C | 0 | -1 | 0 | (i) | $\mathcal{T}^{-}\mathcal{M}_{+}^{x/y}$ |  |  |
|  |  |  | (iv) | $\mathcal{T}^{\mp}\mathcal{M}_{\pm}^{x/y}$ | $\mathcal{R}_{-}^{x/y}$ |  |
CI | 1 | -1 | 1 | (i) | $\mathcal{M}_{--}^{x/y}$ |  |  |
|  |  |  | (iv) | $\mathcal{M}_{\pm\pm}^{x/y}$ | $\mathcal{R}_{+-}^{x/y}$ | $\mathcal{R}_{\mp\pm}^{z}$ | $\mathcal{I}_{++}$
Table 1:  Combinations of tenfold-way symmetries ${\cal T}$, ${\cal P}$, or
${\cal C}={\cal P}{\cal T}$ and order-two crystalline symmetries
$\mathcal{S}=\mathcal{M}^{x/y}$, ${\cal R}^{x/y}$, $\mathcal{R}^{z}$,
$\mathcal{I}$ or magnetic symmetries ${\cal T}{\cal S}$ allowing for a second-
order topological gapless phase with anomalous hinge states around $k_{z}=0$
(of type (i), (ii), or (iii), as discussed in the text) or with flat hinge
arcs at generic $k_{z}$ (type (iv)). The subscripts ${\cal S}_{\pm}$ indicate,
whether ${\cal S}$ commutes ($+$) or anticommutes ($-$) with ${\cal T}$,
${\cal P}$, and/or ${\cal C}$, using the convention that the representation
matrix $U_{\cal S}^{2}=1$. Magnetic symmetries are indicated by the product
${\cal T}{\cal S}$, where the superscript ${\cal T}^{\pm}$ indicates the
square $({\cal TS})^{2}=\pm 1$ and the subscript ${\cal S}_{\pm}$ indicates,
whether ${\cal TS}$ commutes or anticommutes with ${\cal P}$ or ${\cal C}$. In
class CII with ${\cal R}^{z}_{+-}$, the zero-energy hinge arc ends at a
surface Dirac cone, see App. S.2 for details.

Classification.— In Table 1, we list all combinations of tenfold-way and
crystalline symmetries for which second-order nodal semimetal or
superconductor phases exist, for the twofold rotations ${\cal R}^{x}$ or
${\cal R}^{y}$, ${\cal R}^{z}$, mirror ${\cal M}^{x}$ or ${\cal M}^{y}$, or
inversion symmetry ${\cal I}$. We include magnetic crystalline symmetries,
which are the product of a crystalline symmetry ${\cal S}$ and time-reversal
${\cal T}$. The subscripts $\pm$ in the table indicate, whether the
crystalline symmetry ${\cal S}$ commutes or anticommutes with the tenfold-way
symmetries ${\cal T}$, ${\cal P}$, or ${\cal C}$, if applicable (also see App.
S.6). Of the symmetry classes shown in the table, a second-order semimetal
with a flat arc of zero-energy hinge states was previously proposed in Ref. 68
in the presence of the magnetic inversion symmetry ${\cal T}{\cal I}_{-}$. A
list of second-order gapless phases protected by crystalline antisymmetries is
given in App. S.3.

Nodal loops vs. Dirac points.— For the order-two crystalline symmetries
considered here, a bulk band structure with a “Dirac point”, a multiply
degenerate Weyl point, is unstable to a splitting into a nodal loop. This is
the case, e.g., in the model considered in Ref. [68], as well as for the
models considered here. Nodal loops for the four examples discussed here are
shown in App. S.1. Stabilizing Dirac points requires crystalline symmetries of
higher order than considered here, such as fourfold or sixfold rotation
symmetry. Topological semimetals with flat or approximately flat arcs of hinge
states and fourfold rotation symmetry were discussed in Refs. 61, 62, 69, 67.
Reference 70 proposes models for a nodal superconductor with inversion-
symmetry-protected helical hinge states around $k_{z}=0$ and nodal points in
the bulk band structure protected by fourfold and sixfold rotation symmetry.
In principle, degenerate Weyl points may also split into single Weyl points if
the classification group at finite $k_{z}$ admits both first-order and second-
order or obstructed atomic-limit phases under taking the direct sum. Such a
scenario does not occur, however, for the order-two crystalline symmetries
[52].

Hinge disorder. — As long as the bulk density of states vanishes at the nodal
point, zero-energy hinge arcs are robust to weak disorder [77, 78]. However,
potential disorder may broaden a flat hinge arc, even if it respects the
tenfold-way symmetries ${\cal T}$, ${\cal P}$, and ${\cal C}$ (if applicable).
Such local hybridization of hinge states is absent, however, if the hinge
states are protected by a chiral antisymmetry ${\cal C}$ and all hinge states
have the same ${\cal C}$-parity. This is the case for our example (1). If the
hinge states appear in pairs with opposite ${\cal C}$-parity or if they are
not pinned to zero energy by an antisymmetry ${\cal C}$, no such protection
exists. In our examples (3) and (6), the zero-energy hinge states appear in
Kramers pairs with opposite ${\cal C}$-parity, allowing them to hybridize. For
type (iii), the hinge states are chiral or helical Majorana modes, which are
prohibited from localization by disorder.

Conclusion. — We identified four types of second-order topological semimetals
and nodal superconductors distinguished by their boundary signatures and
associated bulk topology around the nodal manifold. For a strict protection,
all types require the presence of an antisymmetry, which suggests
superconductors or semimetals with an approximate sublattice antisymmetry as
natural candidates. We hope that our work may contribute to the topological
interpretation of nodal superconductors, such as doped Bi2Se3 [79, 80, 81],
iron-based [82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93], heavy-fermion
[94], or other compounds [95], and inspire further theoretical and
experimental investigation of the novel phenomenology.

We thank Luka Trifunovic and Michele Burrello for discussions. This research
was supported by the German Science Foundation (DFG) through the CRC-TR 183
“entangled states of matter” (projects A02, A03, and C01). MG acknowledges
support by the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation program under grant agreement No. 856526,
and from the Danish National Research Foundation, the Danish Council for
Independent Research — Natural Sciences.

##  S.1 Further discussion of examples from the main text

To decide, whether a certain combination of tenfold-way and crystalline
symmetries can protect a second-order topological semimetal phase and to find
the type of the boundary signatures, one needs to

  * (a)

determine the crystalline symmetry classes of the two-dimensional Hamiltonian
$H_{k_{z}}(k_{x},k_{y})$ at the high-symmetry planes $k_{z}=0$, $\pi$, and at
generic $0<|k_{z}|<\pi$, and

  * (b)

obtain the order-resolved classification corresponding to the crystalline
symmetry classes at $k_{z}=0$, $\pi$, and at generic $0<|k_{z}|<\pi$, together
with the mapping between these.

For the twofold crystalline symmetries, order-resolved classification
information can be found in Ref. 52, which classifies two-dimensional
topological band structures in terms of a “subgroup sequence”

$K^{\prime\prime}\subseteq K^{\prime}\subseteq K.$

Here $K$ is the full classifying group (restricted to strong topological
phases and subject to the rules of stable equivalence [42]), $K^{\prime}$
classifies all topological band structures that do not have first-order
boundary signatures, and $K^{\prime\prime}$ classifies all topological band
structures without boundary signatures. The quotient
$K^{\prime}/K^{\prime\prime}$ then describes the second-order phases. Since
the crystalline symmetry class at generic $0<|k_{z}|<\pi$ is obtained from the
crystalline symmetry class at the high-symmetry planes $k_{z}=0$, $\pi$ by
lifting symmetry requirements, in the mathematical literature the mapping
between the classifying groups at $k_{z}=0$, $\pi$ and at generic
$0<|k_{z}|<\pi$ is known as the “forgetful functor” [42]. To the best of our
knowledge, no complete classification information that includes the action of
the forgetful functor for the tenfold-way classes with a twofold crystalline
symmetry is available in the literature, but this information can be obtained
relatively straightforwardly by inspecting models for the generators of the
classifying groups.

We now discuss the crystalline symmetry classes of the two-dimensional
Hamiltonian $H_{k_{z}}(k_{x},k_{y})$ at the high-symmetry planes $k_{z}=0$,
$\pi$, and at generic $0<|k_{z}|<\pi$, the order-resolved classifications, the
action of the forgetful functor, and the resulting consequences for three-
dimensional topological gapless phases for the four examples discussed in the
main text.

Figure S.1:  Bulk band structures (top) and nodal lines in the three-
dimensional Brillouin zone (bottom) for the representative models of Eqs. (1),
(3), (5), and (6) (from left to right) from the main text. The same parameters
were used as in Fig. 1. In the depictions of the nodal lines, we include a
color gradient proportional to the $k_{z}$ coordinate of the nodal line.

Example of Eq. (1). —  Considering $k_{z}$ as a parameter, the three-
dimensional Hamiltonian $H(k_{x},k_{y},k_{z})$ of Eq. (1) describes a one-
parameter family of two-dimensional Hamiltonians $H_{k_{z}}(k_{x},k_{y})$. At
the high-symmetry planes $k_{z}=0$ and $k_{z}=\pi$, $H_{k_{z}}(k_{x},k_{y})$
is subject to the chiral antisymmetry ${\cal C}$ and the mirror symmetry
${\cal M}^{x}$ as

$\displaystyle H_{0,\pi}(k_{x},k_{y})=$ $\displaystyle\,-U_{\cal
C}H_{0,\pi}(k_{x},k_{y})U_{\cal C}^{\dagger}$ $\displaystyle=$
$\displaystyle\,U_{\cal M}H_{0,\pi}(-k_{x},k_{y})U_{\cal M}^{\dagger},$ (S.1)

with $U_{\cal C}=\tau_{3}\sigma_{0}$ and $U_{\cal M}=\tau_{3}\sigma_{3}$, see
Eq. (2) of the main text. In the notation of Ref. 52, this is the crystalline
symmetry class $\mbox{AIII}^{{\cal M}_{+}}$, where the Cartan symbol “AIII”
indicates the presence of the chiral antisymmetry ${\cal C}$ and the
superscript “${\cal M}_{+}$” the presence of the crystalline mirror symmetry
${\cal M}^{x}$ that commutes with ${\cal C}$. Since the mirror symmetry leaves
$z$ invariant, the crystalline symmetry class for generic $0<|k_{z}|<\pi$ is
the same.

The subgroup sequence classifying the crystalline symmetry class
$\mbox{AIII}^{{\cal M}_{+}}$ is given in Table II of Ref. [52],

$0\subseteq\mathbb{Z}\subseteq\mathbb{Z},$

from which one immediately concludes that all nontrivial topological phases in
the crystalline symmetry class $\mbox{AIII}^{{\cal M}_{+}}$ are second-order
phases with zero-energy corner states at mirror-symmetric corners. The four-
band Hamiltonian (1) is a generator of the nontrivial topology. One can see
this, e.g., by noting that it has the maximal number of anticommuting terms
and the minimal number of bands. Alternatively, that the four-band Hamiltonian
(1) with $|k_{z}|<k_{0}$ is a generator of the nontrivial topology can also be
inferred from the exact diagonalization, which gives precisely one zero-energy
corner state per mirror-symmetric corner if $|k_{z}|<k_{0}$, see Fig. 1b in
the main text.

To bring about the generic higher-order topological features of the model (2),
non-protected surface and hinge states need to be gapped out. In our numerical
calculations, we accomplish this by adding the term

$\delta H(k_{x},k_{y},k_{z})=b\tau_{2}\sigma_{2}$ (S.2)

to the Hamiltonian of Eq. (1). In addition to gapping out non-topological
surface states, the addition of this term also lifts the accidental degeneracy
of the nodal point at $k_{z}=\pm k_{0}$ and turns it in to a nodal loop. This
can be seen in Fig. S.1, where we have chosen the parameter $b=0.4t$.

Example of Eq. (3). —  When seen as a $k_{z}$-dependent family of two-
dimensional Hamiltonians $H_{k_{z}}(k_{x},k_{y})$, the model (3) is in
symmetry class $\mbox{DIII}^{{\cal M}_{--}}$ at the high-symmetry planes
$k_{z}=0$, $\pi$. For generic $0<|k_{z}|<\pi$, $H_{k_{z}}(k_{x},k_{y})$ is
constrained by the product ${\cal C}={\cal P}{\cal T}$ and by the mirror
symmetry ${\cal M}$ only. Since ${\cal M}$ commutes with ${\cal C}$,
$H_{k_{z}}(k_{x},k_{y})$ is in the symmetry class $\mbox{AIII}^{{\cal M}_{+}}$
for generic $0<|k_{z}|<\pi$. The subgroup sequences for $k_{z}=0$, $\pi$ and
for generic $0<|k_{z}|<\pi$ are given in Tables IV and II of Ref. 52,
respectively,

$\displaystyle\begin{array}[]{ll}0\subseteq
2\mathbb{Z}\subseteq\mathbb{Z}&\mbox{for $k_{z}=0$, $\pi$},\\\
0\subseteq\mathbb{Z}\subseteq\mathbb{Z}&\mbox{for
$0<|k_{z}|<\pi$}.\end{array}$ (S.5)

From these subgroup sequences we conclude that the generator of the
classifying group is a first-order phase for $k_{z}=0$, $\pi$ and a second-
order phase for $0<|k_{z}|<\pi$. That the four-band Hamiltonian (3) is a
generator for $k_{z}=0$ as well as $0<|k_{z}|<k_{0}$ can be seen by noting
that it has the maximal number of anticommuting terms and the minimal number
of bands or, alternatively, by inspection of the exact diagonalization results
of Fig. 1b, which show that the two-dimensional Hamiltonian
$H_{k_{z}}(k_{x},k_{y})$ corresponding to Eq. (3) has helical Majorana edge
states for $k_{z}=0$ and non-degenerate zero-energy corner states for
$0<k_{z}<k_{0}$.

As in the previous example, non-topological surface states and accidental
degeneracies are removed by including an additional term

$\delta H(k_{x},k_{y},k_{z})=b\tau_{0}\sigma_{1}\sin k_{z}$ (S.6)

to the model (3). For the results shown in Fig. 1b and S.1 we set $b=0.4t$.

Example of Eq. (5). —  The model of Eq. (5) satisfies the symmetry constraints

$\displaystyle H(k_{x},k_{y},k_{z})=$ $\displaystyle\,-U_{\cal
P}H(-k_{x},-k_{y},-k_{z})^{*}U_{\cal P}^{\dagger}$ $\displaystyle=$
$\displaystyle\,U_{\cal R}H(-k_{x},-k_{y},k_{z})U_{\cal R}^{\dagger},$ (S.7)

with $U_{\cal P}=\tau_{1}\sigma_{0}$ and $iU_{\cal R}=i\tau_{3}\sigma_{3}$. It
is in the crystalline symmetry classes $\mbox{D}^{{\cal R}_{-}}$ for
$k_{z}=0$, $\pi$ and $\mbox{A}^{{\cal R}}$ for $0<|k_{z}|<\pi$. The
corresponding subgroup sequences are [52]

$\begin{array}[]{ll}2\mathbb{Z}\subseteq\mathbb{Z}\subseteq\mathbb{Z}^{2}&\mbox{for
$k_{z}=0$, $\pi$},\\\
\mathbb{Z}\subseteq\mathbb{Z}\subseteq\mathbb{Z}^{2}&\mbox{for
$0<|k_{z}|<\pi$}.\end{array}$

The two-dimensional Hamiltonian $H_{k_{z}}(k_{x},k_{y})$ corresponding to the
model (5) is the generator “1” of $K^{\prime}$ for $|k_{z}|<k_{0}$, which is a
second-order phase with zero-energy corner states for $k_{z}=0$ and an
obstructed atomic-limit phase for $0<|k_{z}|<k_{0}$. At $|k_{z}|=k_{0}$ there
is a transition between two atomic-limit phases, which lends topological
protection to the gaplessness, but does not come with a boundary signature.
This explains the phenomenology of the three-dimensional model (5): dispersing
chiral hinge modes around $k_{z}=0$, pinned to zero energy at $k_{z}=0$.

For the numerical results of Fig. 1 and S.1 an additional term of the form

$\delta
H(k_{x},k_{y},k_{z})=b_{1}\sigma_{2}\tau_{2}+b_{2}\sigma_{1}\tau_{1}\sin
k_{z}$ (S.8)

is added to $H(k_{x},k_{y},k_{z})$, which gaps out non-protected boundary
states and removes accidental degeneracies. The parameter values used for the
numerical calculations are $b_{1}=b_{2}=0.4t$.

Example of Eq. (6). —  The model of Eq. (6) satisfies the symmetry constraints

$\displaystyle H(k_{x},k_{y},k_{z})=$ $\displaystyle\,-U_{\cal
P}H(-k_{x},-k_{y},-k_{z})^{*}U_{\cal P}^{\dagger}$ $\displaystyle=$
$\displaystyle\,U_{\cal T}H(-k_{x},-k_{y},-k_{z})^{*}U_{\cal T}^{\dagger}$
$\displaystyle=$ $\displaystyle\,U_{\cal R}H(-k_{x},-k_{y},k_{z})U_{\cal
R}^{\dagger},$ (S.9)

with $U_{\mathcal{T}}=\tau_{0}\sigma_{2}$,
$U_{\mathcal{P}}=\tau_{1}\sigma_{0}$, and
$iU_{\mathcal{R}}=i\tau_{0}\sigma_{3}$. At the high-symmetry momenta
$k_{z}=0$, $\pi$, the two-dimensional Hamitonian $H_{k_{z}}(k_{x},k_{y})$
corresponding to this model is in the crystalline symmetry class
$\mbox{DIII}^{{\cal R}_{-+}}$; for generic momenta $0<|k_{z}|<\pi$, the
crystalline symmetry class is $\mbox{AIII}^{{\cal R}_{-}}$. The corresponding
subgroup sequences are [52]

$\begin{array}[]{ll}0\subseteq\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}&\mbox{for
$k_{z}=0$, $\pi$},\\\
2\mathbb{Z}\subseteq\mathbb{Z}\subseteq\mathbb{Z}&\mbox{for
$0<|k_{z}|<\pi$}.\end{array}$

Although the symmetries allow for a nontrivial topological phase at $k_{z}=0$,
the corresponding Hamiltonian immediately turns trivial upon going to nonzero
$k_{z}$, ruling out a topological semimetal phase that derives its protection
from a non-trivial phase of $H_{k_{z}}(k_{x},k_{y})$ at $k_{z}=0$. (This
observation, as well as the converse, that any topologically non-trivial
gapped two-dimensional Hamiltonian $H_{k_{z}}(k_{x},k_{y})$ must be trivial
for $k_{z}\to 0$, $\pi$ follows from the observation that there is no
nontrivial map from $\mathbb{Z}_{2}$ to $\mathbb{Z}$.) Instead, the family of
two-dimensional Hamiltonians $H_{k_{z}}(k_{x},k_{y})$ as defined from Eq. (6)
corresponds to the trivial class for $k_{z}=0$ and $\pi$ — i.e., it
corresponds to the element “0” in the classifying group $K$ — and is
nontrivial — corresponding to the element “1” — in a region
$k_{01}<|k_{z}|<k_{02}$ around $|k_{z}|=\pi/2$. The resulting phenomenology of
the three-dimensional model is as shown in Fig. 1: flat arcs of zero-energy
hinge states in a finite region around $|k_{z}|=\pi/2$, but no boundary
signatures at $k_{z}=0$ or $\pi$. For the numerical calculations leading to
Fig. 1 and S.1 we added the additional term

$\delta H=b\tau_{2}\sigma_{0}\sin k_{z}$ (S.10)

with $b=0.4t$ to gap out non-protected surface states.

##  S.2 More examples

Below we discuss more examples that realize variants of the four main
scenarios for second-order gapless topological band structures discussed in
the main text.

Figure S.2:  Band structures in a rhombic pillar geometry for the example
systems as defined in Eqs. (S.11), (S.13), (S.16), (S.19), and (S.21) (from
left to right, top) and bulk band structures (middle) and nodal line (bottom).
In the pillar geometry (top), periodic boundary conditions are applied in the
$z$ direction, which is along the pillar axis. The full and dashed black
curves denote the excitation gap in the bulk and on the surface, respectively.
Symmetry-preserving perturbations have been added to the models given in the
text to turn accidentally degenerate nodal points into nodal loops and to gap
out non-anomalous surface states, as desribed the text. In the models of Eqs.
(S.11), (S.13), (S.16), and (S.21) we took the parameter values $t=1$,
$k_{0}=0.6\pi$; for the model of Eq. (S.19) we took the parameter values
$m=0$, $t=1$, $t^{\prime}=2$

Type (i) with degenerate flat hinge arcs. — The example of Eq. (1) describes a
second-order topological gapless phase with non-degenerate flat arcs of hinge
states. Since the hinge states are protected by the chiral antisymmetry ${\cal
C}$ and, hence, have a $\mathbb{Z}$ topological classification, examples with
degenerate flat arcs of hinge states can trivially be obtained by taking the
direct sum of multiple copies of the model (1). A nontrivial example with
twofold degenerate hinge arcs, for which the degeneracy at $k_{z}=0$ is
symmetry-enforced, can be found by considering the direct sum of two copies of
the example of Eq. (3) of the main text,

$\displaystyle H(k_{x},k_{y},k_{z})=$
$\displaystyle\,t\tau_{3}\sigma_{0}\rho_{0}[m(k_{z})+2-\cos k_{x}-\cos k_{y}]$
$\displaystyle\,\mbox{}+t\tau_{2}\sigma_{0}\rho_{0}\sin
k_{x}+t\tau_{1}\sigma_{3}\rho_{0}\sin k_{y}.$ (S.11)

Here, the $2\times 2$ identity matrix $\rho_{0}$ and the associated Pauli
matrices $\rho_{j}$, $j=1,2,3$, describe an additional two-component degree of
freedom. The tenfold-way and crystalline symmetries of this eight-band model
are the same in the example of Eq. (3) of the main text, see the previous
Section for details. As before, we consider the model (S.11) as a one-
parameter family of two-dimensional Hamiltonians $H_{k_{z}}(k_{x},k_{y})$.
Because the model (S.11) is the direct sum of two copies of a Hamiltonian that
generates the classification group $K$, it corresponds to the element “2” in
the sequence of classifying groups. According to the subgroup sequences (S.5),
the element “2” of the classying group is a second-order phase with Kramers-
degenerate zero-energy corner states for $k_{z}=0$ and a second-order phase
with twofold degenerate corner states for $0<|k_{z}|<k_{0}$. The resulting
three-dimensional band structure has a flat arc of twofold degenerate zero-
energy hinge modes for $-k_{0}<k_{z}<k_{0}$, whereby the twofold degeneracy at
$k_{z}=0$ is protected by time-reversal symmetry, whereas the twofold
degeneracy at $0<|k_{z}|<k_{0}$ is protected by the chiral antisymmetry ${\cal
C}={\cal P}{\cal T}$.

Figure S.2 shows the nodal region, the bulk band structure, and the dispersion
for a pillar geometry with translation invariance in the $z$-direction. To
remove non-topological surface states and accidental degeneracies in the
spectrum, the additional term

$\delta H=b_{1}\tau_{0}\sigma_{1}\rho_{2}+b_{2}\tau_{0}\sigma_{1}\rho_{0}\sin
k_{z}$ (S.12)

has been added to the Hamiltonian (S.11), with parameter values $b_{1}=0.4t,\
b_{2}=0.2t$.

Although this model has the smallest number of bands compatible with the
requirement that it has Kramers degenerate hinge modes at $k_{z}=0$, it is not
a minimal model. In the context of the classification, this follows from the
fact that the model (S.11) corresponds to the non-minimal element “2” in the
classifying group. It also follows from the phenomenology of the model: In
principle, one can add a perturbation that preserves time-reversal symmetry,
but causes the transition of $H_{k_{z}}(k_{x},k_{y})$ from the nontrivial
phase “2” at $k_{z}=0$ towards the trivial phase “0” at $k_{z}=\pi$ to proceed
in two steps. As a result, there are nodal points at two values of $k_{z}$,
each of which can be broadened into a nodal loop by a symmetry-preserving
perturbation. For $k_{z}$ between the projections of the nodal points/loops
there are non-degenerate zero-energy hinge states, as in the original model
(3) of the main text.

Type (i) with rotation symmetry $\mathcal{R}^{y}$. — Another example for a
second-order nodal superconductor with a flat arc of zero-energy hinge states
around $k_{z}=0$ is a time-reversal symmetry breaking superconductor with a
twofold rotation symmetry ${\cal R}^{y}$ around the $y$ axis and even-parity
superconductivity. A concrete lattice model is the four-band model

$\displaystyle H(k_{x},k_{y},k_{z})=$
$\displaystyle\,t\tau_{3}\sigma_{0}(m(k_{z})+2-\cos k_{x}-\cos k_{y})$
$\displaystyle+t\tau_{1}\sigma_{1}\sin k_{x}+t\tau_{1}\sigma_{3}\sin k_{y},$
(S.13)

which satisfies the symmetry constraints

$\displaystyle H(k_{x},k_{y},k_{z})=$ $\displaystyle\,-U_{\cal
P}H(-k_{x},-k_{y},-k_{z})^{*}U_{\cal P}^{\dagger}$ $\displaystyle=$
$\displaystyle\,U_{\cal R}H(-k_{x},k_{y},-k_{z})U_{\cal R}^{\dagger},$ (S.14)

with $U_{\mathcal{P}}=\tau_{1}\sigma_{0}$ and
$iU_{\mathcal{R}}=i\tau_{0}\sigma_{3}$. Since the twofold rotation ${\cal
R}^{y}$ acts like a mirror $k_{x}\to-k_{x}$ at the high-symmetry values
$k_{z}=0$ or $k_{z}=\pi$, the crystalline symmetry class of the two-
dimensional Hamiltonian $H_{k_{z}}(k_{x},k_{y})$ at $k_{z}=0$, $\pi$ is
$\mbox{D}^{{\cal M}_{+}}$. At generic $k_{z}$, $H_{k_{z}}(k_{x},k_{y})$ is
constrained by the product ${\cal P}{\cal R}$ only, which, at a fixed value of
$k_{z}$, effectively acts as an antiunitary mirror antisymmetry ${\cal P}{\cal
M}^{x}$, so that $H_{k_{z}}(k_{x},k_{y})$ is in the crystalline symmetry class
$\mbox{A}^{{\cal P}^{+}{\cal M}}$. From Tables III and IV of Ref. 52 we then
find that in both cases the order-resolved topological classifications given
by the subgroup sequence

$0\subseteq\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}.$

According to this subgroup sequence, the only topologically nontrivial band
structures are of second order. For $|k_{z}|<k_{0}$ the model (S.13) realizes
such a topological band structure, for $k_{z}=0$ as well as $0<|k_{z}|<k_{0}$.
It has zero-energy hinge states for $-k_{0}<k_{z}<k_{0}$ on hinges that are
mapped to themselves under the twofold rotation symmetry ${\cal R}^{y}$.

To remove non-topological surface states and accidental degeneracies, for the
numerical calculations we add the additional term

$\delta H=b_{1}\tau_{3}\sigma_{3}+b_{2}\tau_{0}\sigma_{1}\sin k_{z}$ (S.15)

with $b_{1}=b_{2}=0.4t$. The corresponding bulk band structure and the quasi-
one-dimensional band structure of a rhombic pillar geometry are shown in Fig.
S.2.

Type (ii) with inversion symmetry. —  The model of Eq. (3) of the main text,
which describes an odd-parity nodal superconductor with mixed-order boundary
signatures (surface Dirac cones pinned to $k_{z}=0$ and flat arcs of hinge
states around $k_{z}=0$) also satisfies an unusual inversion symmetry
$\mathcal{I}$ with representation $U_{\mathcal{I}}=\tau_{3}\sigma_{3}$. (This
inversion symmetry is unusual, because it anticommutes with time reversal.) If
${\cal I}$ is imposed instead of ${\cal M}^{x}$, the model of Eq. (3) is still
topological, with crystalline symmetry class $\mbox{DIII}^{{\cal R}_{--}}$ for
$k_{z}=0$, $\pi$ and $\mbox{AIII}^{{\cal T}^{+}{\cal R}_{-}}$ for
$0<|k_{z}|<\pi$. (Note that at fixed $k_{z}=0$, $\pi$, inversion effectively
acts as a twofold rotation ${\cal R}^{z}$. The same applies to the product
${\cal T}{\cal I}$ at fixed $0<|k_{z}|<\pi$.) The subgroup sequences follow
from Tables IV and III of Ref. 52,

$\begin{array}[]{ll}0\subseteq\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}^{2}&\mbox{for
$k_{z}=0$, $\pi$},\\\
0\subseteq\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}&\mbox{for
$0<|k_{z}|<\pi$}.\end{array}$

Hence, at $k_{z}=0$, $\pi$, there are first-order as well as second-order
phases, with separate $\mathbb{Z}_{2}$ classifications. At generic
$0<|k_{z}|<\pi$ there is only a second-order phase. Upon continuously going
from $k_{z}=0$ to generic $0<|k_{z}|<\pi$, the generator of the first-order
phase at $k_{z}=0$ becomes a second-order phase at nonzero $k_{z}$. That this
is the case is seen by inspecting the behavior of $H_{k_{z}}(k_{x},k_{y})$ for
the concrete model (3), which has first-order boundary states at $k_{z}=0$ and
zero-energy corner states for $0<k_{z}<k_{0}$ (see discussion in the main text
and in the previous Section). Hence, we conclude that the model (3) with
inversion symmetry ${\cal I}_{--}$ instead of the mirror symmetry ${\cal
M}^{x}$ also realizes a second-order nodal superconductor of type (ii). A
difference between the inversion-symmetric and mirror-symmetric versions of
the model (3) is that in the mirror-symmetric version the hinge states appear
on the mirror-symmetric hinges, whereas in the inversion-symmetric version the
hinge states can appear on any pair of inversion-related hinges. Since the
phenomenology of the inversion-symmetric version of the model (3) is the same
as that of the mirror-symmetric version, we refer to Fig. 1b of the main text
for numerical results.

Type (iii) with helical hinge modes. — A second-order topological nodal
superconductor of type (iii) with helical hinge modes exists in the (three-
dimensional) symmetry class $\mbox{DIII}^{\mathcal{R}_{+-}^{z}}$, which
describes an odd-rotation-parity time-reversal-symmetric superconductor with
an unconventional twofold rotation symmetry ${\cal R}^{z}$. (The rotation
symmetry is unconventional, because ${\cal R}^{2}=1$ and ${\cal R}$ commutes
with time reversal.) To construct a concrete lattice model, one may take the
direct sum $H(k_{x},k_{y},k_{z})\oplus H^{*}(-k_{x},k_{y},k_{z})$ of two time-
reversed copies of the four-band model $H(k_{x},k_{y},k_{z})$ of Eq. (5),

$\displaystyle H(k_{x},k_{y},k_{z})=$
$\displaystyle\,t\tau_{3}\sigma_{3}\rho_{0}[m(k_{z})+2-\cos k_{x}-\cos k_{y}]$
$\displaystyle\,\mbox{}+t\tau_{1}\sigma_{3}\rho_{3}\sin
k_{x}+t\tau_{2}\sigma_{3}\rho_{0}\sin k_{y},$ (S.16)

where the Pauli matrices $\rho_{j}$ describe an additional two-component
degree of freedom. The model has the symmetries

$\displaystyle H(k_{x},k_{y},k_{z})=$ $\displaystyle\,-U_{\cal
P}H(-k_{x},-k_{y},-k_{z})^{*}U_{\cal P}^{\dagger}$ $\displaystyle=$
$\displaystyle\,U_{\cal T}H(-k_{x},-k_{y},-k_{z})^{*}U_{\cal T}^{\dagger}$
$\displaystyle=$ $\displaystyle\,U_{\cal R}H(-k_{x},-k_{y},k_{z})U_{\cal
R}^{\dagger},$ (S.17)

with $U_{\mathcal{P}}=\tau_{1}\sigma_{0}\rho_{0}$,
$U_{\mathcal{T}}=\tau_{0}\sigma_{0}\rho_{2}$, and
$U_{\mathcal{R}^{z}}=\tau_{3}\sigma_{3}\rho_{0}$. For the high-symmetry
momenta $k_{z}=0$, $\pi$, the two-dimensional Hamiltonian
$H_{k_{z}}(k_{x},k_{y})$ corresponding to Eq. (S.16) is in crystalline
symmetry class $\mbox{DIII}^{{\cal R}_{+-}}$; and at generic $0<|k_{z}|<\pi$
it is in symmetry class $\mbox{AIII}^{{\cal R}_{-}}$. The corresponding
subgroup sequences follow from Tables IV and II of Ref. 52,

$\begin{array}[]{ll}4\mathbb{Z}\subseteq
2\mathbb{Z}\subseteq\mathbb{Z}&\mbox{for $k_{z}=0$, $\pi$},\\\
2\mathbb{Z}\subseteq\mathbb{Z}\subseteq\mathbb{Z}&\mbox{for
$0<|k_{z}|<\pi$}.\end{array}$

For $k_{z}=0$ the eight-band model (S.16) corresponds to the element “2” in
this subgroup sequence, which corresponds to a second-order band structure
with Kramers pairs of zero-energy corner states. For $0<k_{z}<k_{0}$ the
element “2” in the subgroup sequence corresponds to a topologically nontrivial
atomic limit.

Like the model of Eq. (S.11), the model (S.16) is not a minimal model. As a
function of $k_{z}$, the two-dimensional Hamiltonian $H_{k_{z}}(k_{x},k_{y})$
can transition from the nontrivial second-order phase “2” at $k_{z}=0$ to the
trivial phase at $k_{z}=\pi$ via an intermediate gapped phase corresponding to
the generator “1” of the classification group $K$. If the gap closings take
place at $k_{z}=\pm k_{01}$ and $k_{z}=\pm k_{02}$, the three-dimensional
system has helical dispersing hinge modes around $k_{z}=0$ and flat arcs of
zero modes for $|k_{z}|$ between $k_{01}$ and $k_{02}$. As for all other
models considered here, a symmetry-preserving perturbation may change the gap
closing points into nodal loops.

A numerical evaluation of the model (S.16), with an additional term

$\delta H=b_{1}\tau_{2}\sigma_{2}\rho_{0}+b_{2}\tau_{1}\sigma_{1}\rho_{3}\sin
k_{z}+b_{3}\tau_{2}\sigma_{2}\rho_{2}\sin k_{z}$ (S.18)

to gap out non-topological surface states and to separate the gap closing
points, is shown in Fig. S.2. The parameter values are $b_{1}=b_{2}=0.3t,\
b_{3}=0.6t$.

Type (iv) with four nodal points. — The model (6) of a second-order
topological superconductor in symmetry class
$\mbox{DIII}^{\mathcal{R}^{z}_{-+}}$ has eight nodal loops. Four of the nodal
loops are a consequence of a transition between obstructed atomic limits. This
obstruction is “fragile”, in the sense that these four nodal loops can be
removed under the direct sum with the corresponding atomic limit that cancels
the obstruction. This cancellation is realized, e.g., in an eight-band
generalization of the model of Eq. (6),

$\displaystyle H(k_{x},k_{y},k_{z})=$ $\displaystyle\,t\tau_{0}\sigma_{3}\sin
k_{z}$ $\displaystyle\,\ \ \mbox{}\times[\mu_{3}(m-1)-\mu_{11}(\cos k_{x}+\cos
k_{y})]$ $\displaystyle\,\mbox{}+t\tau_{0}\sigma_{1}\mu_{11}\sin
k_{x}+t\tau_{3}\sigma_{2}\mu_{11}\sin k_{y}$
$\displaystyle\,\mbox{}+t^{\prime}\tau_{3}\sigma_{0}\mu_{1}\cos^{2}k_{z},$
(S.19)

where $\mu_{11}=\frac{1}{2}(\mu_{0}+\mu_{3})$. The bulk band structure, the
manifold of nodal excitations in the three-dimensional Brillouin zone, and the
band structure in a pillar geometry are shown in Fig. S.2. In the numerical
calculations, we included an additional symmetry-allowed hybridization

$\delta H=b_{1}\tau_{2}\sigma_{0}\mu_{0}\sin
k_{z}+b_{2}\tau_{2}\sigma_{0}\mu_{3}\sin k_{z}$ (S.20)

with $b_{1}=0.4t,\ b_{2}=0.2t$ to gap out non-topological surface states and
remove accidental spectral degeneracies.

Type (i)* in class CII with rotation symmetry $\mathcal{R}^{z}_{+-}$. — In
symmetry class CII with rotation symmetry $\mathcal{R}^{z}_{+-}$, we encounter
a special case of a second-order topological semimetal with zero-energy hinge
arcs that, for generic $0<|k_{z}|<\pi$, are only protected by the boundary
gap, but not by the bulk gap. The subgroup sequences for this crystalline
symmetry class are [52]

$\begin{array}[]{ll}4\mathbb{Z}\subseteq 2\mathbb{Z}\subseteq
2\mathbb{Z}&\mbox{for $k_{z}=0$, $\pi$},\\\
2\mathbb{Z}\subseteq\mathbb{Z}\subseteq\mathbb{Z}&\mbox{for
$0<|k_{z}|<\pi$}.\end{array}$

The subgroup sequence for $k_{z}=0$ indicates that for this momentum, the
system permits a second-order topological phase hosting an anomalous,
protected Kramers pair of zero-energy corner states. In class CII, the two
partners in the pair have the same eigenvalue under the chiral antisymmetry
$\mathcal{C}=\mathcal{TP}$, because ${\cal T}$ commutes with ${\cal C}$ [49].
These hinge states must remain at zero energy upon going to nonzero $k_{z}$,
because the chiral antisymmetry prohibits them from acquiring a non-zero
energy. However, since ${\cal T}$ maps $k_{z}$ to $-k_{z}$, it no longer pairs
zero-energy hinge state at nonzero $k_{z}$. It is possible to move and
hybridize such zero-energy hinge states with the zero-energy modes at the
opposite hinge, while preserving rotation symmetry [49]. This transformation
closes only the gap on the surface, but not in the bulk. This is in agreement
with the subgroup sequence for finite $k_{z}$, according to which the
generator of the nontrivial topology at $k_{z}=0$, which corresponds to the
element “2”, maps to an obstructed atomic-limit phase at nonzero $k_{z}$. The
obstructed atomic limit $H_{k_{z}}(k_{x},k_{y})$ does not have in-gap corner
states that are protected by the bulk topology.

These findings are realized in the eight-band lattice model

$\displaystyle H(k_{x},k_{y},k_{z})=$
$\displaystyle\,t\tau_{3}\sigma_{1}\rho_{0}[m(k_{z})+2-\cos k_{x}-\cos k_{y}]$
$\displaystyle\,\mbox{}+t\tau_{0}\sigma_{3}\rho_{3}\sin
k_{x}+t\tau_{0}\sigma_{3}\rho_{1}\sin k_{y},$ (S.21)

which satisfies the symmetries

$\displaystyle H(k_{x},k_{y},k_{z})=$ $\displaystyle\,-U_{\cal
P}H(-k_{x},-k_{y},-k_{z})^{*}U_{\cal P}^{\dagger}$ $\displaystyle=$
$\displaystyle\,U_{\cal T}H(-k_{x},-k_{y},-k_{z})^{*}U_{\cal T}^{\dagger}$
$\displaystyle=$ $\displaystyle\,U_{\cal R}H(-k_{x},-k_{y},k_{z})U_{\cal
R}^{\dagger},$ (S.22)

with $U_{\mathcal{P}}=\tau_{2}\sigma_{0}\rho_{0}$,
$U_{\mathcal{T}}=\tau_{1}\sigma_{2}\rho_{0}$, and
$U_{\mathcal{R}}=\tau_{3}\sigma_{1}\rho_{0}$. We further include a symmetry-
allowed hybridization,

$\displaystyle\delta H=$ $\displaystyle
b_{1}\tau_{3}\sigma_{1}\rho_{1}+b_{2}\tau_{1}\sigma_{2}\mu_{1}\sin k_{z}$
$\displaystyle\,\mbox{}+b_{3}\tau_{2}\sigma_{2}\mu_{3}\sin
k_{z}+b_{4}\tau_{2}\sigma_{2}\mu_{0}\sin k_{z},$ (S.23)

with $b_{1}=0.2t$, $b_{2}=b_{3}=0.4t$, and $b_{4}=0.6t$, to remove accidental
degeneracies and boundary states. The bulk and rhombic pillar band structure
as well as the nodal manifold are shown in Fig. S.2. Around $k_{z}=0$, the
system hosts a zero-energy hinge arc. These states hybridize and disappear at
a Dirac node of the surface band structure. Since this model corresponds to
the element “2” in the classification sequence for generic $0<|k_{z}|<\pi$, as
a function of $k_{z}$ the transition from the second-order phase at $k_{z}=0$
to the trivial phase at $k_{z}=\pi$ involves two gap closings, separated by an
intermediate second-order phase. As shown in Fig. S.2, between the first and
the second nodal loop, a non-degenerate arc of zero-energy hinge states
appears, which corresponds to the second-order topological phase for the value
”1” of the crystalline topological invariant at finite $k_{z}$.

##  S.3 Classification

|  |  |  |  | $k_{z}=0,\pi$ | $0<|k_{z}|<\pi$ |
---|---|---|---|---|---|---|---
Class | $\mathcal{T}^{2}$ | $\mathcal{P}^{2}$ | $\mathcal{C}^{2}$ | $\mathcal{S}$ | Class | $\mathcal{K}^{\prime\prime}\subseteq\mathcal{K^{\prime}\subseteq\mathcal{K}}$ | Class | $\mathcal{K}^{\prime\prime}\subseteq\mathcal{K^{\prime}\subseteq\mathcal{K}}$ | type
AIII | 0 | 0 | 1 | $\mathcal{M}_{+}^{x/y}$ | $\text{AIII}^{\mathcal{M}_{+}}$ | $0\subseteq\mathbb{Z}\subseteq\mathbb{Z}$ | $\text{AIII}^{\mathcal{M}_{+}}$ | $0\subseteq\mathbb{Z}\subseteq\mathbb{Z}$ | (i)
A | 0 | 0 | 0 | $\mathcal{CM}^{x/y}$ | $\text{A}^{\mathcal{CM}}$ | $0\subseteq\mathbb{Z}\subseteq\mathbb{Z}^{2}$ | $\text{A}^{\mathcal{CM}}$ | $0\subseteq\mathbb{Z}\subseteq\mathbb{Z}^{2}$ | (i)
BDI | 1 | 1 | 1 | $\mathcal{M}_{++}^{x/y}$ | $\text{BDI}^{\mathcal{M}_{++}}$ | $0\subseteq\mathbb{Z}\subseteq\mathbb{Z}$ | $\text{AIII}^{\mathcal{M}_{+}}$ | $0\subseteq\mathbb{Z}\subseteq\mathbb{Z}$ | (i)
DIII | -1 | 1 | 1 | $\mathcal{M}_{++}^{x/y}$ | $\text{DIII}^{\mathcal{M}_{++}}$ | $0\subseteq\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}$ | $\text{AIII}^{\mathcal{M}_{+}}$ | $0\subseteq\mathbb{Z}\subseteq\mathbb{Z}$ | (iv)
CII | -1 | -1 | 1 | $\mathcal{M}_{++}^{x/y}$ | $\text{CII}^{\mathcal{M}_{++}}$ | $0\subseteq 2\mathbb{Z}\subseteq 2\mathbb{Z}$ | $\text{AIII}^{\mathcal{M}_{+}}$ | $0\subseteq\mathbb{Z}\subseteq\mathbb{Z}$ | (iv)/(i)
CI | 1 | -1 | 1 | $\mathcal{M}_{++}^{x/y}$ | $\text{CI}^{\mathcal{M}_{++}}$ | $0\subseteq 0\subseteq 0$ | $\text{AIII}^{\mathcal{M}_{+}}$ | $0\subseteq\mathbb{Z}\subseteq\mathbb{Z}$ | (iv)
AI | 1 | 0 | 0 | $\mathcal{CM}_{-}^{x/y}$ | $\text{AI}^{\mathcal{CM}_{-}}$ | $0\subseteq 0\subseteq 0$ | $\text{A}^{\mathcal{CM}}$ | $0\subseteq\mathbb{Z}\subseteq\mathbb{Z}^{2}$ | (iv)
D | 0 | 1 | 0 | $\mathcal{CM}_{+}^{x/y}$ | $\text{D}^{\mathcal{CM}_{+}}$ | $0\subseteq\mathbb{Z}\subseteq\mathbb{Z}^{2}$ | $\text{A}^{\mathcal{CM}}$ | $0\subseteq\mathbb{Z}\subseteq\mathbb{Z}^{2}$ | (i)
AII | -1 | 0 | 0 | $\mathcal{CM}_{-}^{x/y}$ | $\text{AII}^{\mathcal{CM}_{-}}$ | $0\subseteq\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}^{2}$ | $\text{A}^{\mathcal{CM}}$ | $0\subseteq\mathbb{Z}\subseteq\mathbb{Z}^{2}$ | (iv)
C | 0 | -1 | 0 | $\mathcal{CM}_{+}^{x/y}$ | $\text{C}^{\mathcal{CM}_{+}}$ | $0\subseteq 2\mathbb{Z}\subseteq 2\mathbb{Z}^{2}$ | $\text{A}^{\mathcal{CM}}$ | $0\subseteq\mathbb{Z}\subseteq\mathbb{Z}^{2}$ | (iv)/(i)
BDI | 1 | 1 | 1 | $\mathcal{M}_{--}^{x/y}$ | $\text{BDI}^{\mathcal{M}_{--}}$ | $0\subseteq 0\subseteq 0$ | $\text{AIII}^{\mathcal{M}_{+}}$ | $0\subseteq\mathbb{Z}\subseteq\mathbb{Z}$ | (iv)
DIII | -1 | 1 | 1 | $\mathcal{M}_{--}^{x/y}$ | $\text{DIII}^{\mathcal{M}_{--}}$ | $0\subseteq 2\mathbb{Z}\subseteq\mathbb{Z}$ | $\text{AIII}^{\mathcal{M}_{+}}$ | $0\subseteq\mathbb{Z}\subseteq\mathbb{Z}$ | (ii)/(i)
CII | -1 | -1 | 1 | $\mathcal{M}_{--}^{x/y}$ | $\text{CII}^{\mathcal{M}_{--}}$ | $0\subseteq\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}$ | $\text{AIII}^{\mathcal{M}_{+}}$ | $0\subseteq\mathbb{Z}\subseteq\mathbb{Z}$ | (iv)
CI | 1 | -1 | 1 | $\mathcal{M}_{--}^{x/y}$ | $\text{CI}^{\mathcal{M}_{--}}$ | $0\subseteq 2\mathbb{Z}\subseteq 2\mathbb{Z}$ | $\text{AIII}^{\mathcal{M}_{+}}$ | $0\subseteq\mathbb{Z}\subseteq\mathbb{Z}$ | (iv)/(i)
AI | 1 | 0 | 0 | $\mathcal{CM}_{+}^{x/y}$ | $\text{AI}^{\mathcal{CM}_{+}}$ | $0\subseteq 2\mathbb{Z}\subseteq 2\mathbb{Z}$ | $\text{A}^{\mathcal{CM}}$ | $0\subseteq\mathbb{Z}\subseteq\mathbb{Z}^{2}$ | (iv)/(i)
D | 0 | 1 | 0 | $\mathcal{CM}_{-}^{x/y}$ | $\text{D}^{\mathcal{CM}_{-}}$ | $0\subseteq 0\subseteq 2\mathbb{Z}$ | $\text{A}^{\mathcal{CM}}$ | $0\subseteq\mathbb{Z}\subseteq\mathbb{Z}^{2}$ | (iv)
AII | -1 | 0 | 0 | $\mathcal{CM}_{+}^{x/y}$ | $\text{AII}^{\mathcal{CM}_{+}}$ | $0\subseteq 2\mathbb{Z}\subseteq\mathbb{Z}$ | $\text{A}^{\mathcal{CM}}$ | $0\subseteq\mathbb{Z}\subseteq\mathbb{Z}^{2}$ | (ii)/(i)
C | 0 | -1 | 0 | $\mathcal{CM}_{-}^{x/y}$ | $\text{C}^{\mathcal{CM}_{-}}$ | $0\subseteq 0\subseteq 2\mathbb{Z}$ | $\text{A}^{\mathcal{CM}}$ | $0\subseteq\mathbb{Z}\subseteq\mathbb{Z}^{2}$ | (iv)
Table S.1:  Combination of tenfold-way symmetries ${\cal T}$, ${\cal P}$, or ${\cal C}={\cal P}{\cal T}$ and mirror symmetry $\mathcal{S}=\mathcal{M}^{x/y}$ that allow for a second-order semimetal or nodal superconductor with anomalous hinge states around $k_{z}=0$ (of type (i), (ii), or (iii), as discussed in the text) and/or with flat hinge arcs that occur only between nodal manifolds at finite $k_{z}$ and do not cross $k_{z}=0,\pi$ (type (iv)). The table also contains the symmetry class of the two-dimensional Hamiltonian $H_{k_{z}}(k_{x},k_{y})$ at $k_{z}=0$, $\pi$ and at $0<|k_{z}|<\pi$, together with the corresponding subgroup sequence $\mathcal{K}^{\prime\prime}\subseteq\mathcal{K}^{\prime}\subseteq\mathcal{K}$. The rightmost column displays the type of boundary phenomenology around $k_{z}=0$ that is induced by a nontrivial higher-order topology of $H_{k_{z}}(k_{x},k_{y})$ for generic $0<|k_{z}|<\pi$. If multiple types are listed, the first and second entries are for odd and even multiples of the generator of the nontrivial topology, respectively. For the types “(i)”, “(ii)”, and “(iii)”, it is assumed that $H_{k_{z}}(k_{x},k_{y})$ does not go through a topological phase transition if $k_{z}\to 0$, whereas “(iv)” refers to a topological phase of $H_{k_{z}}(k_{x},k_{y})$ at generic $k_{z}$ that does not come with higher-order boundary signatures upon going to $k_{z}=0$ or that is incompatible with the symmetry constraints at $k_{z}=0$, so that a gap closing is necessary upon going to $k_{z}=0$.  |  |  |  |  | $k_{z}=0,\pi$ | $0<|k_{z}|<\pi$ |
---|---|---|---|---|---|---|---
Class | $\mathcal{T}^{2}$ | $\mathcal{P}^{2}$ | $\mathcal{C}^{2}$ | $\mathcal{S}$ | Class | $\mathcal{K}^{\prime\prime}\subseteq\mathcal{K^{\prime}\subseteq\mathcal{K}}$ | Class | $\mathcal{K}^{\prime\prime}\subseteq\mathcal{K^{\prime}\subseteq\mathcal{K}}$ | type
AIII | 0 | 0 | 1 | $\mathcal{P}^{+}\mathcal{R}_{+}^{x/y}$ | $\text{AIII}^{\mathcal{P}^{+}\mathcal{M}_{+}}$ | $0\subseteq\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}$ | $\text{AIII}^{\mathcal{P}^{+}\mathcal{M}_{+}}$ | $0\subseteq\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}$ | (i)
A | 0 | 0 | 0 | $\mathcal{P}^{+}\mathcal{R}^{x/y}$ | $\text{A}^{\mathcal{P}^{+}\mathcal{M}}$ | $0\subseteq\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}$ | $\text{A}^{\mathcal{P}^{+}\mathcal{M}}$ | $0\subseteq\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}$ | (i)
BDI | 1 | 1 | 1 | $\mathcal{R}_{++}^{x/y}$ | $\text{BDI}^{\mathcal{M}_{++}}$ | $0\subseteq\mathbb{Z}\subseteq\mathbb{Z}$ | $\text{AIII}^{\mathcal{P}^{+}\mathcal{M}_{+}}$ | $0\subseteq\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}$ | (i)
D | 0 | 1 | 0 | $\mathcal{R}_{+}^{x/y}$ | $\text{D}^{\mathcal{M}_{+}}$ | $0\subseteq\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}$ | $\text{A}^{\mathcal{P}^{+}\mathcal{M}}$ | $0\subseteq\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}$ | (i)
DIII | -1 | 1 | 1 | $\mathcal{R}_{-+}^{x/y}$ | $\text{DIII}^{\mathcal{M}_{-+}}$ | $0\subseteq\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}^{2}$ | $\text{AIII}^{\mathcal{P}^{+}\mathcal{M}_{+}}$ | $0\subseteq\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}$ | (ii)
AII | -1 | 0 | 0 | $\mathcal{CR}_{-}^{x/y}$ | $\text{AII}^{\mathcal{CM}_{-}}$ | $0\subseteq\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}^{2}$ | $\text{A}^{\mathcal{P}^{+}\mathcal{M}}$ | $0\subseteq\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}$ | (ii)
CII | -1 | -1 | 1 | $\mathcal{R}_{--}^{x/y}$ | $\text{CII}^{\mathcal{M}_{--}}$ | $0\subseteq\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}$ | $\text{AIII}^{\mathcal{P}^{+}\mathcal{M}_{+}}$ | $0\subseteq\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}$ | (iv)
C | 0 | -1 | 0 | $\mathcal{R}_{-}^{x/y}$ | $\text{C}^{\mathcal{M}_{-}}$ | $0\subseteq 0\subseteq 0$ | $\text{A}^{\mathcal{P}^{+}\mathcal{M}}$ | $0\subseteq\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}$ | (iv)
CI | 1 | -1 | 1 | $\mathcal{R}_{+-}^{x/y}$ | $\text{CI}^{\mathcal{M}_{+-}}$ | $0\subseteq 0\subseteq 0$ | $\text{AIII}^{\mathcal{P}^{+}\mathcal{M}_{+}}$ | $0\subseteq\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}$ | (iv)
AI | 1 | 0 | 0 | $\mathcal{CR}_{+}^{x/y}$ | $\text{AI}^{\mathcal{CM}_{+}}$ | $0\subseteq\mathbb{Z}\subseteq\mathbb{Z}$ | $\text{A}^{\mathcal{P}^{+}\mathcal{M}}$ | $0\subseteq\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}$ | (i)
Table S.2:  Same as table S.1, but for rotation symmetry $\mathcal{S}=\mathcal{R}^{x/y}$.  |  |  |  |  | $k_{z}=0,\pi$ | $0<|k_{z}|<\pi$ |
---|---|---|---|---|---|---|---
Class | $\mathcal{T}^{2}$ | $\mathcal{P}^{2}$ | $\mathcal{C}^{2}$ | $\mathcal{S}$ | Class | $\mathcal{K}^{\prime\prime}\subseteq\mathcal{K^{\prime}\subseteq\mathcal{K}}$ | Class | $\mathcal{K}^{\prime\prime}\subseteq\mathcal{K^{\prime}\subseteq\mathcal{K}}$ | type
AIII | 0 | 0 | 1 | $\mathcal{R}_{-}^{z}$ | $\text{AIII}^{\mathcal{R}_{-}}$ | $2\mathbb{Z}\subseteq\mathbb{Z}\subseteq\mathbb{Z}$ | $\text{AIII}^{\mathcal{R}_{-}}$ | $2\mathbb{Z}\subseteq\mathbb{Z}\subseteq\mathbb{Z}$ | (i)/(iv)
BDI | 1 | 1 | 1 | $\mathcal{R}_{+-}^{z}$ | $\text{BDI}^{\mathcal{R}_{+-}}$ | $2\mathbb{Z}\subseteq\mathbb{Z}\subseteq\mathbb{Z}$ | $\text{AIII}^{\mathcal{R}_{-}}$ | $2\mathbb{Z}\subseteq\mathbb{Z}\subseteq\mathbb{Z}$ | (i)/(iv)
DIII | -1 | 1 | 1 | $\mathcal{R}_{-+}^{z}$ | $\text{DIII}^{\mathcal{R}_{-+}}$ | $0\subseteq\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}$ | $\text{AIII}^{\mathcal{R}_{-}}$ | $2\mathbb{Z}\subseteq\mathbb{Z}\subseteq\mathbb{Z}$ | (iv)
CII | -1 | -1 | 1 | $\mathcal{R}_{+-}^{z}$ | $\text{CII}^{\mathcal{R}_{+-}}$ | $4\mathbb{Z}\subseteq 2\mathbb{Z}\subseteq 2\mathbb{Z}$ | $\text{AIII}^{\mathcal{R}_{-}}$ | $2\mathbb{Z}\subseteq\mathbb{Z}\subseteq\mathbb{Z}$ | (iv)/(i)*/(iv)/(iv)
CI | 1 | -1 | 1 | $\mathcal{R}_{-+}^{z}$ | $\text{CI}^{\mathcal{R}_{-+}}$ | $0\subseteq 0\subseteq 0$ | $\text{AIII}^{\mathcal{R}_{-}}$ | $2\mathbb{Z}\subseteq\mathbb{Z}\subseteq\mathbb{Z}$ | (iv)
D | 0 | 1 | 0 | $\mathcal{R}_{-}^{z}$ | $\text{D}^{\mathcal{R}_{-}}$ | $2\mathbb{Z}\subseteq\mathbb{Z}\subseteq\mathbb{Z}^{2}$ | $\text{A}^{\mathcal{R}}$ | $\mathbb{Z}\subseteq\mathbb{Z}\subseteq\mathbb{Z}^{2}$ | (iii)/(iv)
BDI | 1 | 1 | 1 | $\mathcal{R}_{-+}^{z}$ | $\text{BDI}^{\mathcal{R}_{-+}}$ | $0\subseteq 0\subseteq 0$ | $\text{AIII}^{\mathcal{R}_{-}}$ | $2\mathbb{Z}\subseteq\mathbb{Z}\subseteq\mathbb{Z}$ | (iv)
DIII | -1 | 1 | 1 | $\mathcal{R}_{+-}^{z}$ | $\text{DIII}^{\mathcal{R}_{+-}}$ | $4\mathbb{Z}\subseteq 2\mathbb{Z}\subseteq\mathbb{Z}$ | $\text{AIII}^{\mathcal{R}_{-}}$ | $2\mathbb{Z}\subseteq\mathbb{Z}\subseteq\mathbb{Z}$ | (ii)/(iii)/(ii)/(iv)
CII | -1 | -1 | 1 | $\mathcal{R}_{-+}^{z}$ | $\text{CII}^{\mathcal{R}_{-+}}$ | $0\subseteq\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}$ | $\text{AIII}^{\mathcal{R}_{-}}$ | $2\mathbb{Z}\subseteq\mathbb{Z}\subseteq\mathbb{Z}$ | (iv)
CI | 1 | -1 | 1 | $\mathcal{R}_{+-}^{z}$ | $\text{CI}^{\mathcal{R}_{+-}}$ | $2\mathbb{Z}\subseteq 2\mathbb{Z}\subseteq 2\mathbb{Z}$ | $\text{AIII}^{\mathcal{R}_{-}}$ | $2\mathbb{Z}\subseteq\mathbb{Z}\subseteq\mathbb{Z}$ | (iv)
Table S.3:  Same as table S.1, but for rotation symmetry $\mathcal{S}=\mathcal{R}^{z}$. If four types are listed in the rightmost column, the $n$th element refers to the boundary phenomenology for the case that the topological equivalence class of $H_{k_{z}}(k_{x},k_{y})$ at generic $k_{z}$ is that of the ($4m+n$)-fold multiple of the generator of the nontrivial topology, with $m$ integer. The type (i)∗ is discussed in detail in the text, see the discussion preceding Eq. (S.21).  |  |  |  |  | $k_{z}=0,\pi$ | $0<|k_{z}|<\pi$ |
---|---|---|---|---|---|---|---
Class | $\mathcal{T}^{2}$ | $\mathcal{P}^{2}$ | $\mathcal{C}^{2}$ | $\mathcal{S}$ | Class | $\mathcal{K}^{\prime\prime}\subseteq\mathcal{K^{\prime}\subseteq\mathcal{K}}$ | Class | $\mathcal{K}^{\prime\prime}\subseteq\mathcal{K^{\prime}\subseteq\mathcal{K}}$ | type
AIII | 0 | 0 | 1 | $\mathcal{T}^{+}\mathcal{I}_{-}$ | $\text{AIII}^{\mathcal{T}^{+}\mathcal{R}_{-}}$ | $0\subseteq\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}$ | $\text{AIII}^{\mathcal{T}^{+}\mathcal{R}_{-}}$ | $0\subseteq\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}$ | (i)
CI | 1 | -1 | 1 | $\mathcal{I}_{++}$ | $\text{CI}^{\mathcal{R}_{++}}$ | $0\subseteq 0\subseteq 0$ | $\text{AIII}^{\mathcal{T}^{+}\mathcal{R}_{-}}$ | $0\subseteq\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}$ | (iv)
BDI | 1 | 1 | 1 | $\mathcal{I}_{+-}$ | $\text{BDI}^{\mathcal{R}_{+-}}$ | $2\mathbb{Z}\subseteq\mathbb{Z}\subseteq\mathbb{Z}$ | $\text{AIII}^{\mathcal{T}^{+}\mathcal{R}_{-}}$ | $0\subseteq\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}$ | (i)
D | 0 | 1 | 0 | $\mathcal{CI}_{+}$ | $\text{D}^{\mathcal{CR}_{+}}$ | $0\subseteq\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}$ | $\text{A}^{\mathcal{T}^{+}\mathcal{R}}$ | $\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}$ | (iii)
DIII | -1 | 1 | 1 | $\mathcal{I}_{--}$ | $\text{DIII}^{\mathcal{R}_{--}}$ | $0\subseteq\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}^{2}$ | $\text{AIII}^{\mathcal{T}^{+}\mathcal{R}_{-}}$ | $0\subseteq\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}$ | (ii)
CII | -1 | -1 | -1 | $\mathcal{I}_{-+}$ | $\text{CII}^{\mathcal{R}_{-+}}$ | $0\subseteq\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}$ | $\text{AIII}^{\mathcal{T}^{+}\mathcal{R}_{-}}$ | $0\subseteq\mathbb{Z}_{2}\subseteq\mathbb{Z}_{2}$ | (iv)
Table S.4:  Same as table S.1, but for inversion symmetry
$\mathcal{S}=\mathcal{I}$.

Tables S.1–S.4 contain a complete list of all combinations of tenfold-way
symmetries and an additional order-two crystalline symmetry, for which second-
order gapless topological phases are possible. The crystalline symmetries are
mirror ${\cal M}^{x/y}$, twofold rotation ${\cal R}^{x/y}$ around the $x$ or
$y$ axis, twofold rotation ${\cal R}^{z}$ around the $z$ axis, and inversion
${\cal I}$. The tables also contain unitary and antiunitary crystalline
antisymmetries, which are denoted as the product of the chiral conjugation
operation ${\cal C}$ or particle-hole conjugation ${\cal P}$ (squaring to one)
and a crystalline symmetry. The subscripts $\pm$ indicate, whether the
crystalline symmetry ${\cal S}$ commutes ($+$) or anticommutes $(-)$ with
${\cal T}$, ${\cal P}$, or ${\cal C}$. (If there are two subscripts, the first
subscript describes the commutation or anticommutation with ${\cal T}$,
whereas the second subscript refers to ${\cal P}$.) The tables also contain
the crystalline symmetry classes of the two-dimensional Hamiltonian
$H_{k_{z}}(k_{x},k_{y})$ at the high-symmetry momenta $k_{z}=0$, $\pi$ and at
generic $0<|k_{z}|<\pi$, together with the corresponding subgroup sequences
$\mathcal{K}^{\prime\prime}\subseteq\mathcal{K}^{\prime}\subseteq\mathcal{K}$.

Within the same symmetry class, the second-order boundary phenomenology may be
different for the generators of the nontrivial higher-order topology and for
direct sums of these generators. An example is the tenfold-way symmetry class
DIII with a mirror symmetry ${\cal M}^{x}_{--}$, see Eqs. (3) and (S.11), for
which the generator of the nontrivial topology has boundary states of type
(ii) and the direct sum of the generator with itself has boundary states of
type (i). The rightmost columns of Tables S.1–S.4 list other examples of
symmetry classes, for which taking the direct sums of the generator of the
nontrivial topology leads to a different boundary phenomenology. To be
precise, for the right columns of Tables S.1–S.4 we consider nodal semimetals
or superconductors obtained by taking multiple direct sums of the generator of
the nontrivial higher-order topology of $H_{k_{z}}(k_{x},k_{y})$ at a
reference value $k_{z}=k_{z,{\rm ref}}$, with $0<|k_{z,{\rm ref}}|<\pi$. If
$H_{k_{z}}(k_{x},k_{y})$ can be continuously (i.e., without gap closing)
extended to $k_{z}=0$, the three-dimensional Hamiltonian
$H(k_{x},k_{y},k_{z})$ corresponding to $H_{k_{z}}(k_{x},k_{y})$ has a unique
nontrivial boundary phenomenology at $k_{z}=0$, which is one of the types (i)
or (ii) discussed in the main text. If a continuous extension to $k_{z}=0$ is
possible, but there is no nontrivial boundary signature at $k_{z}=0$, or if a
continuous extension to $k_{z}=0$ without gap closing is not possible, we use
the label “(iv)” to indicate that higher-order boundary signatures occur away
from $k_{z}=0$. Additionally, right column of Tables S.3 and S.4 contain two
symmetry classes with boundary phenomenology of type (iii), which originates
from the generators of the nontrivial topology at $k_{z}=0$, while
$H_{k_{z}}(k_{x},k_{y})$ is an obstructed atomic limit at generic
$0<|k_{z}|<\pi$.

To identify all second-order topological semimetals and nodal superconductors
and their types as summarized in tables S.1–S.4, we made use of the order-
resolved classification of two-dimensional second-order topological insulators
and superconductors from Refs. 49, 52 and performed a few additional checks as
follows. First, a necessary and sufficient criterion of a second-order gapless
topological phase of type (i), (ii), or (iv) is a two-dimensional second-order
topological phase at $0<|k_{z}|<\pi$. All symmetry classes with an order-two
symmetry fulfilling this criteria are included in the tables S.1–S.4. To
identify the type, one needs to compute the forgetful functor from the
classifying group at $k_{z}=0$, $\pi$ to the classifying group at finite
$0<|k_{z}|<\pi$, which can be done either by inspecting the generator
Hamiltonians and/or by inspecting the boundary signatures of the corresponding
topological phases, e.g. with the help of Ref. 49. Second-order gapless phases
of type (iii) (with chiral or helical hinge states) may appear when the
Hamiltonian at $k_{z}=0$ or $\pi$ is in a second-order topological phase,
which becomes an obstructed atomic limit at $0<|k_{z}|<\pi$ under the
forgetful functor. The positive results for this type are included in the
tables S.1–S.4.

Table 1 in the main text contains the possible types of higher-order
topological semimetals and nodal superconductors with a unitary or antiunitary
order-two crystalline symmetry as extracted from tables S.1–S.4. While in
tables S.1–S.4 we followed the convention of Refs. 49, 52 for the labeling of
the real crystalline symmetry classes with a unitary symmetry $\mathcal{S}$ or
antisymmetry $\mathcal{CS}$ for mathematical simplicity, in table 1 in the
main text we preferred the more physically relevant notation using unitary
symmetries $\mathcal{S}$ or antiunitary symmetries
$\mathcal{T}^{\pm}\mathcal{S}$. The relation between the two schemes follows
by combining the crystalline symmetry operators with the tenfold-way
symmetries, i.e. in class D (C) we can label an additional order-two
crystalline symmetry either by $\mathcal{CS}_{\pm}$ or equivalently by
$\mathcal{P}\mathcal{CS}_{\pm}=\mathcal{T}^{\pm}\mathcal{S}_{\pm}$
($\mathcal{T}^{\mp}\mathcal{S}_{\pm}$), and in class AIII we have
$\mathcal{C}\mathcal{P}^{s}\mathcal{S}_{\pm}=\mathcal{T}^{\pm
s}\mathcal{S}_{\pm}$ with $s=\pm 1$.

##  S.4 Disorder

Disorder breaks translation symmetry and any crystalline symmetries that
protect the hinge states. However, if the bulk density of states at the nodal
points is sufficiently small, which requires the absence of trivial bulk bands
or type-II Weyl nodes, the hybridization of hinge states with bulk states is
strongly suppressed for weak disorder. In that case, the dominant effect of
disorder is a hybridization between hinge states, leading to a disorder-
broadened arc of localized hinge states.

As discussed in the main text, for a disorder potential that obeys the
tenfold-way symmetries, such disorder-induced hybridization of hinge states
can be forbidden by the symmetry constraints. Whether or not this is the case
can be read off from the tables for the “extrinsic” classification group
$\bar{\cal K}_{\rm e}$ of anomalous corner states of two-dimensional second-
order topological phases in the presence of disorder [49]. In Ref. 49, the
label “extrinsic” indicates that corner states may arise as a result of a
nontrivial bulk topology or as a result of a symmetry-compatible “decoration”
of the crystal boundaries with topologically nontrivial insulators or
superconductors. Since disorder breaks both the order-two crystalline symmetry
as well as the translation symmetry along $z$, hinge disorder “collapses” the
one-dimensional hinge states into a collection of zero-dimensional corner
states. Thus, it is the extrinsic classification group $\bar{\mathcal{K}}_{e}$
of Ref. 49 for the crystalline symmetry group appropriate for $k_{z}=0$ that
classifies possible hinge states of second-order topological gapless phases in
the presence of disorder. In case the extrinsic classification group
$\bar{\mathcal{K}}_{e}\simeq\mathbb{Z}$, zero-energy hinge states all must
have the same eigenvalue under a unitary antisymmetry $\mathcal{C}$, which
prohibits their hybridization. In case of
$\bar{\mathcal{K}}_{e}\simeq\mathbb{Z}_{2}$, the hinge states are generally
allowed to hybridize in pairs, such that only the number parity of (Kramers
pairs of) zero-energy states per hinge must remain after hybridization (in the
presence of time-reversal symmetry $\mathcal{T}^{2}=-1$, if present). In case
of $\bar{\mathcal{K}}_{e}\simeq 0$, the hinge states are not only allowed to
hybridize in pairs, but they generically acquire a finite energy themselves.
In the following we illustrate the consequences of this protection of hinge
states for the examples discussed in the main text and this appendix.

Model of Eq. (1). — In class AIII, the zero-energy hinge states are
eigenstates under the chiral conjugation $\mathcal{C}$. The eigenvalue $c=\pm
1$ assigns a “chirality” to the zero-energy states. Any perturbation
preserving the chiral antisymmetry cannot hybridize eigenstates with the same
chirality eigenvalue. In this crystalline symmetry class, hinge states on the
same hinge have the same chirality eigenvalue. Therefore, disorder cannot
hybridize hinge-states on the same hinge.

This stability of the edge states is consistent with the extrinsic
classification group $\bar{\mathcal{K}}_{e}\simeq\mathbb{Z}$ of mirror-
symmetry breaking corners of two-dimensional gapped systems in class AIII with
mirror symmetry $\mathcal{M}_{+}$, see Ref. 49.

Models of Eq. (3) and Eq. (S.11). — In class DIII, eigenstates come in Kramers
pairs, where the two partners in the Kramers pair have opposite chirality
under the chiral antisymmetry $\mathcal{C}=\mathcal{PT}$. For the hinge states
with finite $k_{z}$, time-reversal symmetry relates hinge states with opposite
$k_{z}$ and opposite chirality eigenvalues. Although time-reversal symmetry
forbids that such Kramers pairs can hybridize and gap out in pairs, nonzero
matrix elements may exist between hinge states at values of $k_{z}$ that
differ in sign and magnitude, leading to a (weak) hybridization of hinge
states.

This result is consistent with the extrinsic classification group
$\bar{\mathcal{K}}_{e}\simeq\mathbb{Z}_{2}$ [49]. With broken translation
symmetry, all Kramers pairs formed from modes with opposite $k_{z}$ hybridize
and gap out in pairs. After hybridization, a zero-energy hinge state remains
if the number of zero-energy Kramers pairs per hinge is odd. The parity of
Kramers pairs per hinge in general depends on the geometry and boundary
conditions at top and bottom of the hinge. Notice that there is no Kramers
pair at $k_{z}=0$, since the node of the Dirac cone on the adjacent surfaces
lies at $k_{z}=0$.

Notice that, because the chirality eigenvalues of hinge states on a single
hinge at $k_{z}>0$ and $k_{z}<0$ must be opposite, the surface states at
$k_{z}=0$ must be gapless in order to support the change in boundary topology
between $k_{z}>0$ and $k_{z}<0$. This change of boundary topology manifests
itself as the first-order surface states at $k_{z}=0$ in this higher-order
topological gapless phase of type (ii).

Model Eq. (S.13). — For this crystalline symmetry class, hinge states are
protected by a $\mathbb{Z}_{2}$ invariant, so that hybridization between hinge
states at different $k_{z}$ is always possible. The parity of zero-energy
modes per hinge, and thus the question whether a zero-energy hinge mode
remains after hybridization, depends on the sample geometry and boundary
conditions. This is in agreement with the extrinsic classifying group
$\bar{\mathcal{K}}_{e}\simeq\mathbb{Z}_{2}$ for this symmetry class [49].

Models of Eq. (5) and Eq. (S.16). — These examples have chiral or helical
hinge states on each hinge. There is only a single zero-energy mode on each
hinge, at $k_{z}=0$. Chiral/helical hinge states are protected from
localization. The single mode at zero-energy is protected from gapping by the
local symmetries, consistent with the boundary classifying group
$\mathcal{K}\simeq\mathbb{Z}_{2}$ with rotation symmetry, c.f. Table I in Ref.
49. Because chiral/helical hinge modes have nonzero energy at $k_{z}\neq 0$,
there is a stronger hybridization with bulk states than for hinges with flat
arcs of zero-energy states away from $k_{z}=0$.

Model of Eq. (6)/ and Eq. (S.19).— For this model hinge states are eigenstates
of the chiral conjugation ${\cal C}$. Time-reversal enforces that modes at
$k_{z}$ and $-k_{z}$ have opposite chirality. As discussed above, time-
reversal symmetry protects only modes with opposite $k_{z}$ from
hybridization. The parity of zero-energy Kramers pairs per hinge, and thus
whether one must remain after hybridization, depends on the geometry. This
results is in agreement with the corresponding boundary classifying group
$\mathcal{K}\simeq\mathbb{Z}_{2}$, c.f. Table I in Ref. 49.

Model Eq. (S.21). — In this symmetry class, the hinge states on a single hinge
have the same eigenvalue under chiral conjugation $\mathcal{C}$. This
prohibits their hybridization from hinge disorder. This results is in
agreement with the corresponding boundary classifying group
$\mathcal{K}\simeq\mathbb{Z}$, c.f. Table I in Ref. 49.

##  S.5 Boundary features that do not occur in the presence of a single
twofold crystalline symmetry

Symmetry-enforced degeneracy of flat hinge arcs for $k_{z}\neq 0$. — The flat
zero-energy hinge arcs of the examples in the main text are non-degenerate.
The appendix discusses an example in which a twofold degeneracy is protected
by time-reversal symmetry at $k_{z}=0$, but the degeneracy is not symmetry-
enforced for nonzero $k_{z}$. In fact, with only a single twofold crystalline
symmetry, it is impossible to achieve a symmetry-enforced degeneracy away from
the high-symmetry momenta $k_{z}=0$, $\pi$. To see this, note that the
antiunitary symmetries ${\cal P}$ and ${\cal T}$ that could enforce such a
degeneracy change the sign of $k_{z}$, so that they would have to be combined
with a mirror reflection ${\cal M}^{z}$ to obtain a symmetry constraint of the
two-dimensional Hamiltonian $H_{k_{z}}(k_{x},k_{y})$ at generic
$0<|k_{z}|<\pi$ that acts within a hinge. That means that, if only a single
crystalline symmetry is present, it must be ${\cal M}^{z}$. However, if the
crystalline symmetry is ${\cal M}^{z}$, all symmetries of the effective two-
dimensional Hamiltonian $H_{k_{z}}(k_{x},k_{y})$ are local and only first-
order topological phases are possible.

Notice, however, that a second-order topological nodal superconductor with
flat hinge arcs with symmetry-enforced degeneracy can be achieved in the
presence of two order-two crystalline symmetries. An example of such a second-
order phase exists in tenfold-way class DIII with the (unusual) combination of
two mirror symmetries ${\cal M}^{z}$ and ${\cal M}^{x}$ that both square to
one, mutually commute, and commute with $\mathcal{T}$ and $\mathcal{P}$.
(Alternatively, ${\cal M}^{z}$ and ${\cal M}^{x}$ square to minus one, commute
mutually, and anticommute with ${\cal T}$ and ${\cal P}$.) Then, the two-
dimensional Hamiltonian $H_{k_{z}}(k_{x},k_{y})$ at generic $0<|k_{z}|<\pi$
satisfies symmetry constraints corresponding to ${\cal M}^{z}{\cal P}$, ${\cal
M}^{z}{\cal T}$, and ${\cal M}^{x}$. This corresponds to the two-dimensional
crystalline symmetry class $\mbox{DIII}^{{\cal M}_{++}}$, which admits a
second-order phase with Majorana Kramers pairs as corner states [49, 52].
Returning to the three-dimensional model, one obtains a second-order
topological nodal superconductor with symmetry-enforced twofold degenerate
zero-energy hinge Fermi arcs on ${\cal M}^{x}$-symmetric hinges. A full
classification of such topological gapless phases with multiple order-two
crystalline symmetries is beyond the scope of this article.

Second-order gapless phases with counter-propagating chiral hinge modes. —
The examples of type (iii) second-order topological semimetals or nodal
superconductors in Cartan classes D or CII that were discussed in the main
text and in the appendix have chiral hinge modes on rotation- or inversion-
related hinges that propagate in the same direction. By checking combinations
of Cartan classes and order-two crystalline symmetries as shown in Tables 1
and S.1–S.4, we find that second-order gapless phases with co-propagating
hinge arcs also exist in Cartan class D with rotation symmetry
$\mathcal{R}^{z}_{-}$ or inversion antisymmetry $\mathcal{CI}_{+}$. In all
cases, both rotation symmetry and inversion antisymmetry require that that the
chiral hinge modes on rotation- or inversion-symmetry related hinges must be
co-propagating.

One may ask if there exist type (iii) second-order topological semimetals or
nodal superconductors where the crystalline symmetries require the chiral
hinge modes at different hinges to be counter-propagating. By checking all
symmetry classes, we find that all systems with counter-propagating chiral
hinge modes on symmetry related hinges are fully-gapped second order
topological phases in three dimensions. Thus, while we do not exclude that
type (iii) second-order topological semimetals or nodal superconductors with
counter-propagating chiral hinge modes exist for other combinations of
crystalline symmetries, they do not exist within the set of tenfold-way
classes with a single order-two crystalline symmetry that is considered here.

##  S.6 Crystalline symmetry classes

The presence of time reversal symmetry ${\cal T}$, particle-hole antisymmetry
${\cal P}$, or their product ${\cal C}={\cal P}{\cal T}$ and the squares
${\cal T}^{2}$ and ${\cal P}^{2}$ (if present) determine the tenfold-way
symmetry class of an insulator or superconductor, see Table 1. We here
consider crystalline symmetry classes, in which one twofold crystalline
symmetry ${\cal S}$ is present in addition to the tenfold-way symmetries
${\cal T}$, ${\cal P}$, and/or ${\cal C}$. A symmetry operation is said to be
of order two if its square is proportional to the identity operation. In three
dimensions, the twofold crystalline symmetries are mirror ${\cal M}$, twofold
rotation ${\cal R}$, or inversion ${\cal I}$. The resulting symmetry classes
have been enumerated by Shiozaki and Sato [25]. We now briefly explain the
construction of the symmetry classes of Ref. 25 and the notation of Ref. 52,
which is used throughout this article.

The crystalline symmetry classes are distinguished by the algebraic relations
of their symmetry elements. For order-two crystalline symmetries
$\mathcal{S}$, it is sufficient to distinguish symmetry operations that square
to $+1$ (labeled by $\eta_{\mathcal{S}}=1$) and $-1$
($\eta_{\mathcal{S}}=-1$). We introduce the labels $\eta_{\mathcal{T}}$,
$\eta_{\mathcal{P}}$ and $\eta_{\mathcal{C}}$ to denote the commutation
relations of the crystalline symmetry element $\mathcal{S}$ with the
antiunitary symmetry $\mathcal{T}$, antiunitary antisymmetry $\mathcal{P}$ and
unitary antisymmetry $\mathcal{C}$. Unitary crystalline symmetries
($\sigma_{\mathcal{S}}=1$) and antisymmetries ($\sigma_{\mathcal{S}}=-1$) are
represented by a unitary matrix $U_{\mathcal{S}}$ as

$H(\mathbf{k})=\sigma_{\mathcal{S}}U(\mathcal{S})H(\mathcal{S}\mathbf{k})U_{\mathcal{S}}^{\dagger}$
(S.24)

and the representations satisfy the relations

$\displaystyle U_{\mathcal{S}}^{2}$ $\displaystyle=\eta_{\mathcal{S}}$
$\displaystyle U_{\mathcal{S}}U_{\mathcal{T}}$
$\displaystyle=\eta_{\mathcal{T}}U_{\mathcal{T}}U^{*}_{\mathcal{S}}$
$\displaystyle U_{\mathcal{S}}U_{\mathcal{P}}$
$\displaystyle=\eta_{\mathcal{P}}U_{\mathcal{P}}U^{*}_{\mathcal{S}}$
$\displaystyle U_{\mathcal{S}}U_{\mathcal{C}}$
$\displaystyle=\eta_{\mathcal{C}}U_{\mathcal{C}}U_{\mathcal{S}}.$

where $\mathcal{S}\mathbf{k}$ denotes the action of the crystalline symmetry
on the momentum coordinates. Similarly, the representations for antiunitary
crystalline symmetries and antisymmetries are defined as

$H(\mathbf{k})=\sigma_{\mathcal{S}}U_{\mathcal{S}}H(-\mathcal{S}\mathbf{k})^{*}U_{\mathcal{S}}^{\dagger}$
(S.25)

and the representations satisfy the relations

$\displaystyle U_{\mathcal{S}}U^{*}_{\mathcal{S}}$
$\displaystyle=\eta_{\mathcal{S}}$ $\displaystyle
U_{\mathcal{S}}U^{*}_{\mathcal{T}}$
$\displaystyle=\eta_{\mathcal{T}}U_{\mathcal{T}}U^{*}_{\mathcal{S}}$
$\displaystyle U_{\mathcal{S}}U^{*}_{\mathcal{P}}$
$\displaystyle=\eta_{\mathcal{P}}U_{\mathcal{P}}U^{*}_{\mathcal{S}}$
$\displaystyle U_{\mathcal{S}}U^{*}_{\mathcal{C}}$
$\displaystyle=\eta_{\mathcal{C}}U_{\mathcal{C}}U_{\mathcal{S}}.$

Following this discussion, a tenfold-way class with the additional crystalline
symmetry ${\cal S}$ is fully characterized by the tenfold way class, the set
of numbers $\eta_{{\cal S}},\eta_{\cal T},\eta_{\cal P},\eta_{\cal C}$ and
$\sigma_{\cal S}$, and by specifying, whether the crystalline symmetry is
unitary or antiunitary [25]. For the purpose of topological classification,
this characterization is partially redundant, however. In particular, a
Hamiltonian that satisfies a crystalline symmetry $\mathcal{S}$ with
representation $U_{\mathcal{S}}$ also satisfies this crystalline symmetry with
representation $iU_{\mathcal{S}}$. Under this transformation
$U_{\mathcal{S}}\to iU_{\mathcal{S}}$, the sign of the square of a unitary
crystalline symmetry is flipped $\eta_{\mathcal{S}}\to-\eta_{\mathcal{S}}$
while at the same time the commutation relation with the antiunitary symmetry
elements changes between commuting and anticommuting
$\eta_{\mathcal{T}}\to-\eta_{\mathcal{T}},\eta_{\mathcal{P}}\to-\eta_{\mathcal{P}}$.
We use this freedom to require that ${\cal S}^{2}=1$ for a unitary crystalline
symmetry ${\cal S}$ and to write a unitary antisymmetry, antiunitary symmetry,
and antiunitary antisymmetry as the products ${\cal C}{\cal S}^{\prime}$,
${\cal T}^{\pm}{\cal S}^{\prime}$, and ${\cal P}^{\pm}{\cal S}^{\prime}$, with
${\cal S}^{\prime}$ a unitary symmetry and ${\cal T}$ and ${\cal P}$
antiunitary operators, where the superscript indicates the square of the
antiunitary operator $({\cal T}^{\pm}{\cal S}^{\prime})^{2}=\pm 1$, $({\cal
P}^{\pm}{\cal S}^{\prime})^{2}=\pm 1$. The combination of the tenfold-way
class and the crystalline symmetry ${\cal S}$ or ${\cal T}^{\pm}{\cal
S}^{\prime}$ or crystalline antisymmetry ${\cal C}{\cal S}^{\prime}$ or ${\cal
P}^{\pm}{\cal S}^{\prime}$ then uniquely specifies the crystalline symmetry
class. This is the notation used in Tables 1 and S.1–S.4 and throughout the
text.

## References

  * Volovik [2009] G. E. Volovik, _The universe in a helium droplet_ , vol. 117 of _International series of monographs on physics_ (Oxford University Press, 2009), URL https://oxford.universitypressscholarship.com/view/10.1093/acprof:oso/9780199564842.001.0001/acprof-9780199564842.
  * Vafek and Vishwanath [2014] O. Vafek and A. Vishwanath, Annu. Rev. Condens. Matter Phys. 5, 83 (2014), URL https://www.annualreviews.org/doi/abs/10.1146/annurev-conmatphys-031113-133841.
  * Burkov [2016] A. Burkov, Nat. Mater. 15, 1145 (2016), URL https://www.nature.com/articles/nmat4788.
  * Fang et al. [2016a] C. Fang, H. Weng, X. Dai, and Z. Fang, Chin. Phys. B 25, 117106 (2016a), URL https://iopscience.iop.org/article/10.1088/1674-1056/25/11/117106.
  * Yan and Felser [2017] B. Yan and C. Felser, Annu. Rev. Condens. Matter Phys. 8, 337 (2017), URL https://www.annualreviews.org/doi/abs/10.1146/annurev-conmatphys-031016-025458.
  * Armitage et al. [2018] N. P. Armitage, E. J. Mele, and A. Vishwanath, Rev. Mod. Phys. 90, 015001 (2018), URL https://link.aps.org/doi/10.1103/RevModPhys.90.015001.
  * Burkov [2018] A. A. Burkov, Annu. Rev. Condens. Matter Phys. 9, 359 (2018), URL https://www.annualreviews.org/doi/10.1146/annurev-conmatphys-033117-054129.
  * Gao et al. [2019] H. Gao, J. W. Venderbos, Y. Kim, and A. M. Rappe, Annual Review of Materials Research 49, 153 (2019), eprint https://doi.org/10.1146/annurev-matsci-070218-010049, URL https://doi.org/10.1146/annurev-matsci-070218-010049.
  * Haldane [2004] F. D. M. Haldane, Phys. Rev. Lett. 93, 206602 (2004), URL https://link.aps.org/doi/10.1103/PhysRevLett.93.206602.
  * Murakami [2007] S. Murakami, New Journal of Physics 9, 356 (2007), URL http://dx.doi.org/10.1088/1367-2630/9/9/356.
  * Wan et al. [2011] X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, Phys. Rev. B 83, 205101 (2011), URL https://link.aps.org/doi/10.1103/PhysRevB.83.205101.
  * Yang et al. [2011] K.-Y. Yang, Y.-M. Lu, and Y. Ran, Phys. Rev. B 84, 075129 (2011), URL https://link.aps.org/doi/10.1103/PhysRevB.84.075129.
  * Burkov and Balents [2011] A. A. Burkov and L. Balents, Phys. Rev. Lett. 107, 127205 (2011), URL https://link.aps.org/doi/10.1103/PhysRevLett.107.127205.
  * Xu et al. [2011] G. Xu, H. Weng, Z. Wang, X. Dai, and Z. Fang, Phys. Rev. Lett. 107, 186806 (2011), URL https://link.aps.org/doi/10.1103/PhysRevLett.107.186806.
  * Burkov et al. [2011] A. A. Burkov, M. D. Hook, and L. Balents, Phys. Rev. B 84, 235126 (2011), URL https://link.aps.org/doi/10.1103/PhysRevB.84.235126.
  * Young et al. [2012] S. M. Young, S. Zaheer, J. C. Y. Teo, C. L. Kane, E. J. Mele, and A. M. Rappe, Phys. Rev. Lett. 108, 140405 (2012), URL https://link.aps.org/doi/10.1103/PhysRevLett.108.140405.
  * Wang et al. [2012] Z. Wang, Y. Sun, X.-Q. Chen, C. Franchini, G. Xu, H. Weng, X. Dai, and Z. Fang, Phys. Rev. B 85, 195320 (2012), URL https://link.aps.org/doi/10.1103/PhysRevB.85.195320.
  * Wang et al. [2013] Z. Wang, H. Weng, Q. Wu, X. Dai, and Z. Fang, Phys. Rev. B 88, 125427 (2013), URL https://link.aps.org/doi/10.1103/PhysRevB.88.125427.
  * Hasan et al. [2017] M. Z. Hasan, S.-Y. Xu, I. Belopolski, and S.-M. Huang, Annual Review of Condensed Matter Physics 8, 289 (2017), eprint https://doi.org/10.1146/annurev-conmatphys-031016-025225, URL https://doi.org/10.1146/annurev-conmatphys-031016-025225.
  * Schnyder and Brydon [2015] A. P. Schnyder and P. M. R. Brydon, Journal of Physics: Condensed Matter 27, 243201 (2015), URL https://doi.org/10.1088/0953-8984/27/24/243201.
  * Volovik [2013] G. E. Volovik, in _Analogue Gravity Phenomenology: Analogue Spacetimes and Horizons, from Theory to Experiment_ , edited by D. Faccio, F. Belgiorno, S. Cacciatori, V. Gorini, S. Liberati, and U. Moschella (Springer International Publishing, Cham, 2013), pp. 343–383, URL https://doi.org/10.1007/978-3-319-00266-8_14.
  * Sato and Ando [2017] M. Sato and Y. Ando, Reports on Progress in Physics 80, 076501 (2017), URL https://doi.org/10.1088/1361-6633/aa6ac7.
  * Altland and Zirnbauer [1997] A. Altland and M. R. Zirnbauer, Phys. Rev. B 55, 1142 (1997), URL https://link.aps.org/doi/10.1103/PhysRevB.55.1142.
  * Schnyder et al. [2008] A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, Phys. Rev. B 78, 195125 (2008), URL https://link.aps.org/doi/10.1103/PhysRevB.78.195125.
  * Shiozaki and Sato [2014] K. Shiozaki and M. Sato, Phys. Rev. B 90, 165114 (2014), URL https://link.aps.org/doi/10.1103/PhysRevB.90.165114.
  * Chiu and Schnyder [2014] C.-K. Chiu and A. P. Schnyder, Phys. Rev. B 90, 205136 (2014), URL https://link.aps.org/doi/10.1103/PhysRevB.90.205136.
  * Shiozaki et al. [2017] K. Shiozaki, M. Sato, and K. Gomi, Phys. Rev. B 95, 235425 (2017), URL https://link.aps.org/doi/10.1103/PhysRevB.95.235425.
  * Shiozaki et al. [2018] K. Shiozaki, M. Sato, and K. Gomi, arXiv (2018), eprint 1802.06694, URL https://arxiv.org/abs/1802.06694.
  * Cornfeld and Chapman [2019] E. Cornfeld and A. Chapman, Phys. Rev. B 99, 075105 (2019), URL https://link.aps.org/doi/10.1103/PhysRevB.99.075105.
  * Shiozaki [2019] K. Shiozaki (2019), eprint 1907.09354, URL https://arxiv.org/abs/1907.09354.
  * Cornfeld and Carmeli [2021] E. Cornfeld and S. Carmeli, Phys. Rev. Research 3, 013052 (2021), URL https://link.aps.org/doi/10.1103/PhysRevResearch.3.013052.
  * Matsuura et al. [2013] S. Matsuura, P.-Y. Chang, A. P. Schnyder, and S. Ryu, New Journal of Physics 15, 065001 (2013), URL http://dx.doi.org/10.1088/1367-2630/15/6/065001.
  * Fang et al. [2015] C. Fang, Y. Chen, H.-Y. Kee, and L. Fu, Phys. Rev. B 92, 081201 (2015), URL https://link.aps.org/doi/10.1103/PhysRevB.92.081201.
  * Yu et al. [2015] R. Yu, H. Weng, Z. Fang, X. Dai, and X. Hu, Phys. Rev. Lett. 115, 036807 (2015), URL https://link.aps.org/doi/10.1103/PhysRevLett.115.036807.
  * Huang et al. [2016] H. Huang, J. Liu, D. Vanderbilt, and W. Duan, Phys. Rev. B 93, 201114 (2016), URL https://link.aps.org/doi/10.1103/PhysRevB.93.201114.
  * Schoop et al. [2016] L. M. Schoop, M. N. Ali, C. Straßer, A. Topp, A. Varykhalov, D. Marchenko, V. Duppel, S. S. P. Parkin, B. V. Lotsch, and C. R. Ast, Nature Communications 7, 11696 (2016), URL https://doi.org/10.1038/ncomms11696.
  * Chan et al. [2016] Y.-H. Chan, C.-K. Chiu, M. Y. Chou, and A. P. Schnyder, Phys. Rev. B 93, 205132 (2016), URL https://link.aps.org/doi/10.1103/PhysRevB.93.205132.
  * Fang et al. [2016b] C. Fang, H. Weng, X. Dai, and Z. Fang, Chinese Physics B 25, 117106 (2016b), URL https://doi.org/10.1088/1674-1056/25/11/117106.
  * Yu et al. [2017] R. Yu, Z. Fang, X. Dai, and H. Weng, Frontiers of Physics 12, 127202 (2017), URL https://doi.org/10.1007/s11467-016-0630-1.
  * Yang et al. [2018] S.-Y. Yang, H. Yang, E. Derunova, S. S. P. Parkin, B. Yan, and M. N. Ali, Advances in Physics: X 3, 1414631 (2018), URL https://doi.org/10.1080/23746149.2017.1414631.
  * Hughes et al. [2011] T. L. Hughes, E. Prodan, and B. A. Bernevig, Phys. Rev. B 83, 245132 (2011), URL https://link.aps.org/doi/10.1103/PhysRevB.83.245132.
  * Chiu et al. [2016] C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Rev. Mod. Phys. 88, 035005 (2016), URL https://link.aps.org/doi/10.1103/RevModPhys.88.035005.
  * Hořava [2005] P. Hořava, Phys. Rev. Lett. 95, 016405 (2005), URL https://link.aps.org/doi/10.1103/PhysRevLett.95.016405.
  * Schindler et al. [2018] F. Schindler, A. M. Cook, M. G. Vergniory, Z. Wang, S. S. P. Parkin, B. A. Bernevig, and T. Neupert, Sci Adv 4, eaat0346 (2018), URL http://advances.sciencemag.org/content/4/6/eaat0346.
  * Benalcazar et al. [2017] W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Phys. Rev. B 96, 245115 (2017), URL https://link.aps.org/doi/10.1103/PhysRevB.96.245115.
  * Langbehn et al. [2017] J. Langbehn, Y. Peng, L. Trifunovic, F. von Oppen, and P. W. Brouwer, Phys. Rev. Lett. 119, 246401 (2017), URL https://link.aps.org/doi/10.1103/PhysRevLett.119.246401.
  * Song et al. [2017] Z. Song, Z. Fang, and C. Fang, Phys. Rev. Lett. 119, 246402 (2017), URL https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.119.246402.
  * Fang and Fu [2019] C. Fang and L. Fu, Science Advances 5 (2019), URL https://www.science.org/doi/10.1126/sciadv.aat2374.
  * Geier et al. [2018] M. Geier, L. Trifunovic, M. Hoskam, and P. W. Brouwer, Phys. Rev. B 97, 205135 (2018), URL https://link.aps.org/doi/10.1103/PhysRevB.97.205135.
  * Khalaf [2018] E. Khalaf, Phys. Rev. B 97, 205136 (2018), URL https://link.aps.org/doi/10.1103/PhysRevB.97.205136.
  * Khalaf et al. [2018] E. Khalaf, H. C. Po, A. Vishwanath, and H. Watanabe, Phys. Rev. X 8, 031070 (2018), URL https://link.aps.org/doi/10.1103/PhysRevX.8.031070.
  * Trifunovic and Brouwer [2019] L. Trifunovic and P. W. Brouwer, Phys. Rev. X 9, 011012 (2019), URL https://link.aps.org/doi/10.1103/PhysRevX.9.011012.
  * Okuma et al. [2019] N. Okuma, M. Sato, and K. Shiozaki, Phys. Rev. B 99, 085127 (2019), URL https://link.aps.org/doi/10.1103/PhysRevB.99.085127.
  * Ahn et al. [2019] J. Ahn, S. Park, D. Kim, Y. Kim, and B.-J. Yang, Chinese Physics B 28, 117101 (2019), URL https://doi.org/10.1088%2F1674-1056%2Fab4d3b.
  * Roberts et al. [2020] E. Roberts, J. Behrends, and B. Béri, Phys. Rev. B 101, 155133 (2020), URL https://link.aps.org/doi/10.1103/PhysRevB.101.155133.
  * Kooi et al. [2019] S. H. Kooi, G. van Miert, and C. Ortix, Phys. Rev. B 100, 115160 (2019), URL https://link.aps.org/doi/10.1103/PhysRevB.100.115160.
  * Rasmussen and Lu [2020] A. Rasmussen and Y.-M. Lu, Phys. Rev. B 101, 085137 (2020), URL https://link.aps.org/doi/10.1103/PhysRevB.101.085137.
  * Trifunovic and Brouwer [2020] L. Trifunovic and P. W. Brouwer, Physica Stat. Solidi B 258 (2020), URL https://doi.org/10.1002/pssb.202000090.
  * Geier et al. [2020] M. Geier, P. W. Brouwer, and L. Trifunovic, Phys. Rev. B 101, 245128 (2020), URL https://link.aps.org/doi/10.1103/PhysRevB.101.245128.
  * Geier et al. [2021] M. Geier, I. C. Fulga, and A. Lau, SciPost Phys. 10, 92 (2021), URL https://scipost.org/10.21468/SciPostPhys.10.4.092.
  * Ezawa [2018] M. Ezawa, Phys. Rev. B 97, 155305 (2018), URL https://link.aps.org/doi/10.1103/PhysRevB.97.155305.
  * Lin and Hughes [2018] M. Lin and T. L. Hughes, Phys. Rev. B 98, 241103 (2018), URL https://link.aps.org/doi/10.1103/PhysRevB.98.241103.
  * Wang et al. [2019] Z. Wang, B. J. Wieder, J. Li, B. Yan, and B. A. Bernevig, Phys. Rev. Lett. 123, 186401 (2019), URL https://link.aps.org/doi/10.1103/PhysRevLett.123.186401.
  * Wang et al. [2020a] H.-X. Wang, Z.-K. Lin, B. Jiang, G.-Y. Guo, and J.-H. Jiang, Phys. Rev. Lett. 125, 146401 (2020a), URL https://link.aps.org/doi/10.1103/PhysRevLett.125.146401.
  * Rui et al. [2021] W. B. Rui, S.-B. Zhang, M. M. Hirschmann, Z. Zheng, A. P. Schnyder, B. Trauzettel, and Z. D. Wang, Phys. Rev. B 103, 184510 (2021), URL https://link.aps.org/doi/10.1103/PhysRevB.103.184510.
  * Zhao et al. [2020] Y. X. Zhao, Y. Lu, and S. A. Yang, arXiv (2020), eprint 2005.14500, URL https://arxiv.org/abs/2005.14500.
  * Wieder et al. [2020] B. J. Wieder, Z. Wang, J. Cano, X. Dai, L. M. Schoop, B. Bradlyn, and B. A. Bernevig, Nature Communications 11, 627 (2020), URL https://doi.org/10.1038/s41467-020-14443-5.
  * Wang et al. [2020b] K. Wang, J.-X. Dai, L. B. Shao, S. A. Yang, and Y. X. Zhao, Phys. Rev. Lett. 125, 126403 (2020b), URL https://link.aps.org/doi/10.1103/PhysRevLett.125.126403.
  * Ghorashi et al. [2020] S. A. A. Ghorashi, T. Li, and T. L. Hughes, Phys. Rev. Lett. 125, 266804 (2020), URL https://link.aps.org/doi/10.1103/PhysRevLett.125.266804.
  * Zhang et al. [2020] R.-X. Zhang, Y.-T. Hsu, and S. Das Sarma, Phys. Rev. B 102, 094503 (2020), URL https://link.aps.org/doi/10.1103/PhysRevB.102.094503.
  * Călugăru et al. [2019] D. Călugăru, V. Juričić, and B. Roy, Phys. Rev. B 99, 041301 (2019), URL https://link.aps.org/doi/10.1103/PhysRevB.99.041301.
  * Roy [2019] B. Roy, Phys. Rev. Research 1, 032048 (2019), URL https://link.aps.org/doi/10.1103/PhysRevResearch.1.032048.
  * Ghorashi et al. [2019] S. A. A. Ghorashi, X. Hu, T. L. Hughes, and E. Rossi, Phys. Rev. B 100, 020509 (2019), URL https://journals.aps.org/prb/abstract/10.1103/PhysRevB.100.020509.
  * Wei et al. [2021] Q. Wei, X. Zhang, W. Deng, J. Lu, X. Huang, M. Yan, G. Chen, Z. Liu, and S. Jia, Nat. Mater. 20, 812 (2021), URL https://www.nature.com/articles/s41563-021-00933-4.
  * Ni and Alu [2021] X. Ni and A. Alu, APL Photonics 6, 050802 (2021), URL https://aip.scitation.org/doi/full/10.1063/5.0041458.
  * Luo et al. [2021] L. Luo, H.-X. Wang, B. Jiang, Y. Wu, Z.-K. Lin, F. Li, and J.-H. Jiang, Nat. Materials 20, 794 (2021), URL https://www.nature.com/articles/s41563-021-00985-6.
  * Szabó and Roy [2020] A. L. Szabó and B. Roy, Phys. Rev. Research 2, 043197 (2020), URL https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.2.043197.
  * Zhang et al. [2021] Z.-Q. Zhang, B.-L. Wu, C.-Z. Chen, and H. Jiang, Phys. Rev. B 104, 014203 (2021), URL https://link.aps.org/doi/10.1103/PhysRevB.104.014203.
  * Smylie et al. [2016] M. P. Smylie, H. Claus, U. Welp, W.-K. Kwok, Y. Qiu, Y. S. Hor, and A. Snezhko, Phys. Rev. B 94, 180510 (2016), URL https://link.aps.org/doi/10.1103/PhysRevB.94.180510.
  * Smylie et al. [2017] M. P. Smylie, K. Willa, H. Claus, A. Snezhko, I. Martin, W.-K. Kwok, Y. Qiu, Y. S. Hor, E. Bokari, P. Niraula, et al., Phys. Rev. B 96, 115145 (2017), URL https://link.aps.org/doi/10.1103/PhysRevB.96.115145.
  * Smylie et al. [2018] M. P. Smylie, K. Willa, H. Claus, A. E. Koshelev, K. W. Song, W.-K. Kwok, Z. Islam, G. D. Gu, J. A. Schneeloch, R. D. Zhong, et al., Scientific Reports 8, 7666 (2018), URL https://doi.org/10.1038/s41598-018-26032-0.
  * Fletcher et al. [2009] J. D. Fletcher, A. Serafin, L. Malone, J. G. Analytis, J.-H. Chu, A. S. Erickson, I. R. Fisher, and A. Carrington, Phys. Rev. Lett. 102, 147001 (2009), URL https://link.aps.org/doi/10.1103/PhysRevLett.102.147001.
  * Paglione and Greene [2010] J. Paglione and R. L. Greene, Nature Physics 6, 645 (2010), URL https://doi.org/10.1038/nphys1759.
  * Nakai et al. [2010] Y. Nakai, T. Iye, S. Kitagawa, K. Ishida, S. Kasahara, T. Shibauchi, Y. Matsuda, and T. Terashima, Phys. Rev. B 81, 020503 (2010), URL https://link.aps.org/doi/10.1103/PhysRevB.81.020503.
  * Hashimoto et al. [2010] K. Hashimoto, M. Yamashita, S. Kasahara, Y. Senshu, N. Nakata, S. Tonegawa, K. Ikada, A. Serafin, A. Carrington, T. Terashima, et al., Phys. Rev. B 81, 220501 (2010), URL https://link.aps.org/doi/10.1103/PhysRevB.81.220501.
  * Zhang et al. [2012] Y. Zhang, Z. R. Ye, Q. Q. Ge, F. Chen, J. Jiang, M. Xu, B. P. Xie, and D. L. Feng, Nature Physics 8, 371 (2012), URL https://doi.org/10.1038/nphys2248.
  * Hashimoto et al. [2012] K. Hashimoto, S. Kasahara, R. Katsumata, Y. Mizukami, M. Yamashita, H. Ikeda, T. Terashima, A. Carrington, Y. Matsuda, and T. Shibauchi, Phys. Rev. Lett. 108, 047003 (2012), URL https://link.aps.org/doi/10.1103/PhysRevLett.108.047003.
  * Su et al. [2012] Y. Su, C. Setty, Z. Wang, and J. Hu, Phys. Rev. B 85, 184517 (2012), URL https://link.aps.org/doi/10.1103/PhysRevB.85.184517.
  * Hong et al. [2013] X. C. Hong, X. L. Li, B. Y. Pan, L. P. He, A. F. Wang, X. G. Luo, X. H. Chen, and S. Y. Li, Phys. Rev. B 87, 144502 (2013), URL https://link.aps.org/doi/10.1103/PhysRevB.87.144502.
  * Kim et al. [2013] J. S. Kim, L. Y. Xing, X. C. Wang, C. Q. Jin, and G. R. Stewart, Phys. Rev. B 87, 054504 (2013), URL https://link.aps.org/doi/10.1103/PhysRevB.87.054504.
  * Johnson et al. [2015] P. D. Johnson, G. Xu, and W.-G. Yin, _Iron-based superconductivity_ , vol. 211 (Springer, 2015), URL https://www.springer.com/gp/book/9783319112534.
  * Smidman et al. [2018] M. Smidman, F. K. K. Kirschner, D. T. Adroja, A. D. Hillier, F. Lang, Z. C. Wang, G. H. Cao, and S. J. Blundell, Phys. Rev. B 97, 060509 (2018), URL https://link.aps.org/doi/10.1103/PhysRevB.97.060509.
  * Zhang et al. [2019] R.-X. Zhang, W. S. Cole, X. Wu, and S. Das Sarma, Phys. Rev. Lett. 123, 167001 (2019), URL https://link.aps.org/doi/10.1103/PhysRevLett.123.167001.
  * Ikeda et al. [2015] H. Ikeda, M.-T. Suzuki, and R. Arita, Phys. Rev. Lett. 114, 147003 (2015), URL https://link.aps.org/doi/10.1103/PhysRevLett.114.147003.
  * Pang et al. [2015] G. M. Pang, M. Smidman, W. B. Jiang, J. K. Bao, Z. F. Weng, Y. F. Wang, L. Jiao, J. L. Zhang, G. H. Cao, and H. Q. Yuan, Phys. Rev. B 91, 220502 (2015), URL https://link.aps.org/doi/10.1103/PhysRevB.91.220502.