††thanks: These two authors contributed equally.††thanks: These two authors
contributed equally.

# Classification of chiral band crossings: local and global constraints

Kirill Alpin  Max Planck Institute for Solid State Research, Heisenbergstrasse
1, D-70569 Stuttgart, Germany  Moritz M. Hirschmann  Max Planck Institute for
Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany  Niclas
Heinsdorf  Max Planck Institute for Solid State Research, Heisenbergstrasse 1,
D-70569 Stuttgart, Germany  Stewart Blusson Quantum Matter Institute,
University of British Columbia, Vancouver, BC V6T 1Z4, Canada  Andreas
Leonhardt  Max Planck Institute for Solid State Research, Heisenbergstrasse 1,
D-70569 Stuttgart, Germany  Wan Yee Yau  Max Planck Institute for Solid State
Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany  Institute for
Theoretical Physics III, University of Stuttgart, D-70550 Stuttgart, Germany
Xianxin Wu  Max Planck Institute for Solid State Research, Heisenbergstrasse
1, D-70569 Stuttgart, Germany  Institute for Theoretical Physics, Chinese
Academy of Sciences, Beijing, China  Andreas P. Schnyder  Max Planck Institute
for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany

(August 28, 2024)

# Fundamental laws of chiral band crossings: local constraints,
global constraints, and topological phase diagrams

Kirill Alpin  Max Planck Institute for Solid State Research, Heisenbergstrasse
1, D-70569 Stuttgart, Germany  Moritz M. Hirschmann  Max Planck Institute for
Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany  Niclas
Heinsdorf  Max Planck Institute for Solid State Research, Heisenbergstrasse 1,
D-70569 Stuttgart, Germany  Stewart Blusson Quantum Matter Institute,
University of British Columbia, Vancouver, BC V6T 1Z4, Canada  Andreas
Leonhardt  Max Planck Institute for Solid State Research, Heisenbergstrasse 1,
D-70569 Stuttgart, Germany  Wan Yee Yau  Max Planck Institute for Solid State
Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany  Institute for
Theoretical Physics III, University of Stuttgart, D-70550 Stuttgart, Germany
Xianxin Wu  Max Planck Institute for Solid State Research, Heisenbergstrasse
1, D-70569 Stuttgart, Germany  Institute for Theoretical Physics, Chinese
Academy of Sciences, Beijing, China  Andreas P. Schnyder  Max Planck Institute
for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany

(August 28, 2024)

###### Abstract

We derive two fundamental laws of chiral band crossings: (i) a local
constraint relating the Chern number to phase jumps of rotation eigenvalues;
and (ii) a global constraint determining the number of chiral crossings on
rotation axes. Together with the fermion doubling theorem, these laws describe
all conditions that a network of chiral band crossing must satisfy. We apply
the fundamental laws to prove the existence of enforced double Weyl points,
nodal planes, and generic Weyl points, among others. In addition, we show that
chiral space-group symmetries cannot stabilize nodal lines with finite Chern
numbers. Combining the local constraint with explicit low-energy models, we
determine the generic topological phase diagrams of all multi-fold crossings.
Remarkably, we find a four-fold crossing with Chern number 5, which exceeds
the previously conceived maximum Chern number of 4. We identify BaAsPt as a
suitable material with this four-fold crossing exhibiting Chern number 5 near
the Fermi energy.

##  I Introduction

Recent theoretical and experimental efforts have uncovered a huge number of
materials with chiral band crossings near the Fermi level [1, 2, 3, 4, 5, 6].
These include Weyl semimetals [7, 8, 9, 10], where two bands cross at isolated
points, nodal plane materials with two-fold degeneracies on planes [11, 12,
13, 14, 15], as well as materials with multifold band crossings [16, 17, 18,
19], such as threefold, fourfold, or sixfold Weyl points. Common to these band
crossings is that their little groups are chiral (i.e., do not contain
inversion or mirror symmetries), such that they act as sources or sinks of
Berry flux with quantized Chern numbers. As a consequence, chiral Weyl
materials exhibit a number of unusual phenomena, e.g., the chiral anomaly [20,
21, 22], large negative magnetoresistance [23], or a quantized circular
photogalvanic effect [24, 25, 26], which might be utilized for the development
of new devices and technologies [27, 28]. In combination with magnetic order,
chiral band crossings are highly tunable via an external magnetic field [29,
30, 14], a property that is vital for future applications.

Importantly, the Chern numbers of chiral band crossings, as well as their
multiplicities and arrangements in the Brillouin zone (BZ) is not only
constrained by crystallographic symmetries, but must obey further topological
conditions that originate from the periodicity of the BZ. For example, screw
rotations lead to symmetry-enforced chiral crossings, due to a nontrivial
winding of the symmetry eigenvalues along the BZ torus [31]. In addition,
there are in a band structure, in general, also chiral crossings at high-
symmetry points, due to higher-dimensional irreducible representations of the
little groups, and moreover there can exist accidental crossings at generic
positions in the BZ. All of these crossings carry a Chern number, forming an
interrelated network of band topologies [15]. The Chern numbers of this
topological network are restricted by the fermion doubling theorem [32], which
dictates that the sum of the Chern numbers over the entire BZ must add up to
zero. The values of the individual Chern numbers, in turn, are constrained by
the crystalline symmetries and, moreover, each chiral crossing has symmetry
related partners with the same Chern number. Hence, there is a complicated
interplay between crystallography and topology that determines the possible
network of band crossings that can exist in a given space group (SG).

These conditions due to symmetry and topology can be used to systematically
classify networks of chiral band crossings. For example, Refs. [33, 34, 35,
36, 37, 38, 39, 40, 41, 42] classified symmetry-enforced topological features
of periodic band structures, Refs. [16, 19, 43, 44, 45] studied band
topologies of low-energy models, and Refs. [46, 47, 48] investigated the
connection between rotation eigenvalues and Chern numbers of Weyl points.
While these works uncovered a number of interesting band topologies, a deeper
understanding for why these arise is lacking, and a fundamental theory of
chiral band crossings still needs to be developed.

In this article, we close this gap by deriving two fundamental laws of chiral
band crossings (Secs. II and III), namely, (i) a local constraint, Eq. (1),
which relates the Chern number of a chiral crossing to the phase jumps of the
rotation eigenvalues; and (ii) a global constraint, Eq. (2), which specifies
the number and types of band crossings existing on a rotation axis. Together
with the famous fermion doubling theorem [32], these two fundamental laws
describe all the conditions that a chiral topological network must satisfy. To
demonstrate the usefulness of these concepts, we present in Sec. IV several
applications of the local and global constraints. For example, we use the
fundamental theorems to prove the existence of enforced double Weyl points on
a twofold rotation axis away from time-reversal invariant momenta (TRIMs)
(Sec. IV.1.1), we derive the existence of topological nodal planes (Sec.
IV.2.1), and we deduce the existence of Weyl points at generic positions in
the BZ (Sec. IV.2.2). In addition, we show that chiral nodal lines with finite
Chern numbers cannot be stabilized by space-group symmetries, but only by
internal (artificial) symmetries. We present a low-energy model that realizes
such a chiral nodal line (Sec. IV.1.3).

We complement these fundamental considerations by an explicit construction of
low-energy models of all multi-fold crossings (Sec. V). Combining these with
the non-Abelian local constraint, we determine the generic topological phase
diagrams of these multi-fold crossings. Remarkably, we find that there exist
four-fold crossings with Chern number 5 (Sec. V.2.1), which exceeds the
previously conceived maximum Chern number of 4 for multi-fold crossings [49,
50]. We perform a database search for materials with multi-fold crossings
exhibiting Chern number 5 and identify BaAsPt in SG 198 as a suitable compound
(Sec. VI.1). We also briefly discuss the materials NbO2 and TaO2 in SG 80,
which realize double Weyl points at TRIMs and away from TRIMs (Sec. VI.2). In
Sec. VII we conclude with a discussion and provide directions for future work.
Technical details are presented in several appendices.

##  II Two symmetry constraints on chiral crossings

A chiral band crossing point, commonly referred to as a Weyl point, acts as a
monopole of Berry curvature $\Omega({\bf k})$. Each Weyl point can be
characterized by a topological charge, the chirality, which is given by the
Chern number $\nu$ calculated on a closed manifold of gapped bands surrounding
the crossing point. Previously, it has been found that a crossing at ${\bf
k}_{c}$ is always topologically charged, if the little group $G_{{\bf k}_{c}}$
is chiral, i.e., if there are neither inversion nor mirror symmetries [11].
Without fine-tuning or additional symmetries the charge of a Weyl point is
$\nu=\pm 1$.

If one considers one or more rotation symmetries, the Chern numbers $\nu$ of
all chiral crossing in the BZ as well as their multiplicity are subject to
local and global constraints, respectively. In this section we formulate these
two constraints on the existence and the topological charge of chiral
crossings, generalizing previous works [46, 47, 48]. The proofs are then given
in Sec. III.

###  II.1 The local constraint

We find a simple relation how the charge $\nu_{b,c_{b}}$ of a crossing $c_{b}$
between the bands numbered by $b$ and $b+1$ is related to the change of
complex phase $\Delta\varphi_{b,c_{b}}$ of an $n$-fold rotation eigenvalue
$\lambda_{b}$. Here and in what follows, we sort the bands by their energy,
i.e., $E_{b+1}({\bf k})>E_{b}({\bf k})\;\;\forall{\bf k}$. For a given band
$b$ the eigenvalue $\lambda_{b}({\bf k})$ is generally a function of the
crystal momentum ${\bf k}$, which is restricted to the rotation axis. The
eigenvalue may, but does not need to jump at each crossing on the axis,
yielding $\Delta\varphi_{b,c_{b}}=\arg\left(\lambda_{b}({\bf
k}_{c_{b}}+\epsilon{\bf\hat{z}})/\lambda_{b}({\bf
k}_{c_{b}}-\epsilon{\bf\hat{z}})\right)$ with the unit vector along the
rotation axis ${\bf\hat{z}}$ and $\epsilon>0$ is sent to 0, see Fig. 1(a).

With these definitions we will show that

$\displaystyle\nu_{b,c_{b}}=\Delta\varphi_{b,c_{b}}\frac{n}{2\pi}\mod n,$ (1)

where the complex phase is only determined up to the order $n$ of the rotation
axis. Equation (1) includes previous results obtained for low-energy models
subject to one rotation symmetry and a time-reversal symmetry [47, 46], and
agrees with the expression derived by classifying equivariant line bundles
[48]. If one recalls that at generic low-symmetry positions in the vicinity of
the rotation axis the number of singly-charged Weyl points is also restricted
by symmetry to be equal $n$, one finds that larger $|\nu_{b,c_{b}}|>n/2$ would
actually be fine-tuned. Therefore, although the relation is only valid $\mod
n$, it is expected that real systems are restricted to crossings with
$|\nu_{b,c_{b}}|<n/2$. Equation (1) is valid even with time-reversal symmetry
or other crystalline symmetries as long as the crossing $c_{b}$ is point-like,
e.g., also if time-reversal enables a gapless crossings between equal rotation
eigenvalues. With this insight a recently discovered type of unusual twofold
double Weyl point, which occurs on a twofold instead of a fourfold or sixfold
rotation axis away from time-reversal invariant momenta (TRIMs), can be
understood, see Sec. IV.1.1. But a caveat is in order here: If time-reversal
and screw symmetries appear together not only can equal eigenvalues be paired
but in several cases this enforces nodal planes, in which case Eq. (1) does
not apply. Nevertheless, we will see that Eq. (1) is a central tool to
identify the topology of nodal planes. Since our results can be applied to
more than one rotation symmetry at a time, it provides a handle to study
higher-fold crossings, where more than two bands intersect. In such crossings
every band $b$ is subject to Eq. (1) for each rotation symmetry. We will see
in Sec. IV.1.2 that this not only explains the observed topological charges,
but results in more than one possible configuration of topological charges.

###  II.2 The global constraint

The second implication of rotation (and mirror) symmetries that governs the
qualitative band topology of topological semimetals, is a global constraint on
the number and type of required crossings $c_{b}$ per band $b$. Generally, two
types of crystalline symmetries can be distinguished, those with symmorphic
operations, which leave at least one point in space invariant, and those with
nonsymmorphic operations that leave no point invariant, e.g., screw rotations
and glide mirror operations. Since the BZ is periodic, nonsymmorphic
symmetries lead to an exchange of bands, due to the ${\bf k}$-dependence of
their eigenvalues $\lambda_{b}({\bf k})$, which implies the existence of at
least one band crossing on a nonsymmorphic rotation axis [31]. Conversely, for
bands along a symmorphic rotation axis it must be possible to undo all band
crossings via pair annihilation, due to the BZ periodicity. These constraints
can be formalized with complex phase differences $\Delta\varphi_{b,c_{b}}$. We
consider an $n$-fold symmetry comprising a translation $(a,b,\tfrac{m}{n})$,
e.g., for a rotation this corresponds to $\\{C^{z}_{n}|a,b,\tfrac{m}{n}\\}$ in
Seitz notation. If a band $b$ is not part of a multifold crossings, one finds
for each of the rotation axes

$\displaystyle\sum_{b,c_{b}}\Delta\varphi_{b,c_{b}}=-b\cdot
2\pi\frac{m}{n}\mod 2\pi,$ (2)

which only depends on the band index $b$, the translation part $\frac{m}{n}$,
and the phase difference $\Delta\varphi_{b,c_{b}}$ for crossings $c_{b}$
between the bands $b$ and $b+1$. If there is a multifold crossing for band
$b$, a similar relation has to be considered, where $c_{b}$ comprises crossing
to higher and lower bands, see Eq. (50). Equation (2) constrains the complex
phase that must be accumulated as one moves through the BZ, up to multiples of
$2\pi$. If the right side of Eq. (2) is non-zero up to $2\pi$, it is clear
that there must be at least one crossing, which contributes to the summation
on the left side. We note that a glide mirror symmetry can be treated
analogously, by considering crossings on any path within the mirror plane that
crosses the entire BZ, such that it is closed due the periodicity in ${\bf
k}$. The usefulness of this formalization for rotation symmetric systems
becomes evident in conjunction with our first result, Eq. (1), which relates
each $\Delta\varphi_{b,c_{b}}\neq 0$ to a topological charge. Therefore, Eq.
(2) states that the _total chirality_ of all crossings on a rotation axis is
given by the band index and the translation part of the screw, up to multiples
of the order of the rotation axis. This implies that accidental crossings on
the rotation axis may change the total charge only by multiples of the order
of the rotation axis, which is reminiscent of what happens at generic
positions, where a symmetry imposes certain multiplicities of topological
crossings.

##  III Derivation of the two constraints

In this section we derive the local and global constraints, which were
discussed in the previous section. For pedagogical reasons, we first present
the proof for nondegenerate bands in Sec. III.1, and then generalize it to
degenerate bands in Sec. III.2. In Sec. III.3 we discuss properties of the
sewing matrix with anti-unitary symmetries. The global constraint is derived
in Sec. III.4.

###  III.1 Abelian Chern numbers and eigenvalue jumps

In the following we derive the constraint, Eq. (1), on the Chern number $\nu$
of a crossing $c_{b}$ in band $b$, which is protected by an $n$-fold rotation
symmetry $C_{n}$. A related proof is given in Ref. [48], where the Picard
group of complex line bundles is computed over a sphere subject to a cyclic
group action. To give self-contained proofs, we calculate the Chern number by
generalizing the formalism used in Ref. [51, 52, 53, 54] to spherical
integration surfaces. The Chern number for a nondegenerate band is defined
using the flux of Berry curvature ${\bf\Omega}({\bf k})$ through a surface $S$
enclosing the crossing as

$\displaystyle\nu=\frac{1}{2\pi}\int_{S}\differential{\bf
n}\cdot{\bf\Omega}({\bf k}),$ (3)

where the surface $S$ is assumed to be a sphere in reciprocal space, without
loss of generality, and ${\bf n}$ is the vector normal to the sphere, see Fig.
1(a). For ease of presentation we have excluded here the case of bands that
are degenerate also away from the crossing point, for which a non-Abelian
Berry curvature must be considered, see Sec. III.2. To calculate the Chern
number, Eq. (3), we split the sphere $S$ into $n$ spherical wedges $S_{W}$,
which are related by the $C_{n}$ rotation symmetry.

Figure 1:  (a) _Local constraint_ : Integration area to determine the Chern
number around a point crossing (red sphere). In the presence of a $C_{n}$
rotation the full sphere of radius $\epsilon$ (green + red) comprises $n$
symmetry-related copies of the spherical wedge $S_{W}$ (red). By Stokes
theorem the flux of Berry curvature ${\bf\Omega}$ through $S_{W}$ is equal to
the line integral of the Berry connection ${\bf A}$ along its boundary
$\partial S_{W}$ (dashed lines). The value of the Berry phase depends on
$\Delta\varphi_{b,c_{b}}=\varphi(+\epsilon{\bf\hat{z}})-\varphi(-\epsilon{\bf\hat{z}})$.
(b) _Global constraint_ : The phase $\varphi_{b}$ of screw rotation symmetry
eigenvalues enforces a band crossing in band $b$ with chirality $\nu_{b}$. The
points $\Gamma(0,0,0)$ and $\Gamma_{2}(0,0,2\pi)$ are related by a reciprocal
lattice vector.

The Abelian Berry curvature transforms as a vector under rotations, i.e.,
$D(C_{n}){\bf\Omega}({\bf k})={\bf\Omega}(D(C_{n}){\bf k})$ [52], where
$D(C_{n})$ is the spatial representation of the rotation $C_{n}$. Noting that
the scalar product ${\bf n}\cdot{\bf\Omega}({\bf k})$ is left invariant under
the introduction of the orthogonal matrix $D(C_{n}){\bf\Omega}$, one obtains

$\displaystyle\nu=\frac{n}{2\pi}\int_{S_{W}}\differential{\bf
n}\cdot{\bf\Omega}({\bf k}).$ (4)

Further, by using that the curvature, ${\bf\Omega}({\bf k})=\nabla\times{\bf
A}$, is the rotation of the Berry connection $({\bf A}({\bf
k}))_{ab}=i\matrixelement{u_{a}({\bf k})}{\nabla}{u_{b}({\bf k})}$, where
$\ket{u_{b}({\bf k})}$ is the orbital part of a Bloch eigenfunction of the
considered Hamiltonian. For a sufficiently small $S$ the only relevant
divergence of the Berry curvature ${\bf\Omega}({\bf k})$ occurs at the
crossing $c_{b}$, i.e., ${\bf\Omega}({\bf k})$ has continuous derivatives on
$S$. We can thus apply Stokes theorem and find

$\displaystyle\nu=\frac{n}{2\pi}\int_{\partial S_{W}}\differential{\bf
s}\cdot{\bf A}({\bf k})\mod n.$ (5)

We note, that the integration in Eq. (5) corresponds to the Berry phase.
Hence, Stokes theorem holds up to multiples of $2\pi$, which, when taking the
factor $\frac{n}{2\pi}$ into account, amounts to an equation valid $\mod n$.
In other words, the $U(1)$ gauge freedom of eigenstates implies that the
integration of ${\bf A}({\bf k})$ in Eq. (5) can be changed by any integer
multiple of $2\pi$, whereas the Chern number $\nu$ is gauge-invariant. When we
want to determine the value of the Chern number, the corresponding gauge
choice is not known, and thus Eq. (5) holds modulo $n$.

The closed path $\partial S_{W}$ can be split into two open and
$C_{n}$-symmetry related paths $\partial S_{W}^{1}$ and $\partial S_{W}^{2}$,
i.e., $\partial S_{W}=\partial S_{W}^{1}+\partial S_{W}^{2}$. But since a non-
zero Berry flux through the surface $S_{W}$ implies that no single-valued
phase convention can be found on the full edge $\partial S_{W}$, we need to
account for a mismatch in the phase convention. For this purpose we consider
the sewing matrix $\mathcal{B}$, which is defined as [52]

$\displaystyle[\mathcal{B}_{C_{n}}({\bf
k})]_{ab}=\matrixelement{u_{a}(D(C_{n}){\bf k})}{U(C_{n})}{u_{b}({{\bf k}})},$
(6)

where $U(C_{n})$ describes the action of the rotation on the eigenstates of
the Hamiltonian $\ket{u_{b}({\bf k})}$. For nondegenerate bands the sewing
matrix is simply a complex phase factor $[\mathcal{B}_{C_{n}}({\bf
k})]_{ab}=\delta_{ab}\mathrm{e}^{i\phi_{b}({\bf k})}$. Specifically, at
symmetry invariant momenta ${\bf k}_{\text{inv}}$ with $D(C_{n}){\bf
k}_{\text{inv}}={\bf k}_{\text{inv}}$ the sewing matrix, Eq. (6), reduces to
the symmetry eigenvalue $\lambda_{b}=\mathrm{e}^{i\varphi_{b}({\bf
k}_{\text{inv}})}$ of $U(C_{n})$ for band $b$, i.e., $\phi_{b}({\bf
k}_{\text{inv}})=\varphi_{b}({\bf k}_{\text{inv}})$. More generally, at ${\bf
k}_{\text{inv}}$ the sewing matrix becomes a diagonal matrix for an
appropriate basis within a degenerate subspace. The Berry connection is then
given by [52]

$\displaystyle{\bf A}(D(C_{n}){\bf k})$
$\displaystyle=D(C_{n})\big{[}\mathcal{B}_{C_{n}}({\bf k}){\bf A}({\bf
k})\mathcal{B}^{-1}_{C_{n}}({\bf k})$ $\displaystyle+i\mathcal{B}_{C_{n}}({\bf
k})\nabla\mathcal{B}^{-1}_{C_{n}}({\bf k})\big{]},$ (7)

see appendix A for details. For nondegenerate bands $i\mathcal{B}_{C_{n}}({\bf
k})\nabla\mathcal{B}^{-1}_{C_{n}}({\bf k})=\nabla\phi_{b}({\bf k})$ and
$\mathcal{B}_{C_{n}}({\bf k}){\bf A}({\bf k})\mathcal{B}^{-1}_{C_{n}}({\bf
k})={\bf A}({\bf k})$. Using the fact that the path $\partial S_{W}^{2}$
corresponds to the rotated path $\partial S_{W}^{1}$ but traversed in the
reversed direction, we perform an integral substitution with ${\bf
k}=(C_{n}){\bf k}^{\prime}$ in the line integral over $\partial S_{W}^{2}$,
which turns the integration path $\partial S_{W}^{2}$ into $-\partial
S_{W}^{1}$. The integral substitution has a unity Jacobian determinant, such
that the term ${\bf A}({\bf k})$ cancels leaving only the sewing matrix term.
Using Eq. (5) we complete the proof of Eq. (1),

$\displaystyle\nu$ $\displaystyle=\frac{n}{2\pi}\left(\int_{\partial
S_{W}^{1}}\kern-10.00002pt\differential{\bf s}\cdot{\bf A}({\bf
k})+\int_{\partial S_{W}^{2}}\kern-10.00002pt\differential{\bf s}\cdot{\bf
A}({\bf k})\right)\kern-5.0pt$ $\displaystyle\text{mod }n$ (8)
$\displaystyle=-\frac{n}{2\pi}\int_{\partial S_{W}^{1}}\differential{\bf
s}\cdot\nabla\phi_{b}({\bf k})$ $\displaystyle\text{mod }n$ (9)
$\displaystyle=\frac{n}{2\pi}\left(\varphi({\bf
k}_{c_{b}}+\epsilon{\bf\hat{z}})-\varphi({\bf
k}_{c_{b}}-\epsilon{\bf\hat{z}})\right)$ $\displaystyle\text{mod }n$ (10)
$\displaystyle=\frac{n}{2\pi}\Delta\varphi$ $\displaystyle\text{mod }n,$ (11)

where ${\bf k}_{c_{b}}+\epsilon{\bf\hat{z}}$ and ${\bf
k}_{c_{b}}-\epsilon{\bf\hat{z}}$ are the north and south pole of the original
sphere, respectively, see Fig. 1(a). The difference in complex phases
$\Delta\varphi_{b,c_{b}}$ of the enclosed crossing is only meaningful up to
multiplies of $2\pi$, which is consistent with the equality up to $\mod n$. A
comment on the used gauge is in order. Here, we have used the cell-periodic
part of the Bloch functions in the calculation of the Chern number and Berry
phase [55], the ${\bf k}$-dependence of the phases $\varphi_{b}({\bf k})$
originates only from the exchange of symmetry eigenvalues and the wave
functions $u_{b{\bf k}}({\bf r})$ that correspond to $\ket{u_{b}({\bf k})}$
are periodic in ${\bf r}$ in agreement with the crystal lattice. In the next
section, we will consider Bloch functions $\psi_{b{\bf k}}({\bf
r})=\mathrm{e}^{i{\bf k}\cdot{\bf r}}u_{b{\bf k}}({\bf r})$, which capture the
periodicity of the Brillouin zone, i.e., $\psi_{b{\bf k}+{\bf K}}({\bf
r})=\psi_{b{\bf k}}({\bf r})$ for all reciprocal lattice vectors ${\bf K}$.
The symmetry action for the periodic gauge $\psi_{b{\bf k}}({\bf r})$ captures
the global symmetry constraints on the band structure, because the symmetry
eigenvalues of nonsymmorphic symmetries gain a phase factor that represents
the translation part of the screw and glide symmetry operations. Nevertheless,
Eq. (1) holds independently of the gauge choice, because all symmetry
eigenvalues on a rotation axis obtain the same additional ${\bf k}$-dependence
in $\varphi_{b}({\bf k})$. In other words, practically we think of
$\Delta\varphi_{b,c_{b}}$ in the limit of $\epsilon\rightarrow 0$ [see, e.g.,
Eq. (10)], whereby $\Delta\varphi_{b,c_{b}}$ becomes the same for both
conventions.

###  III.2 Non-abelian Chern numbers
and eigenvalue jumps

For bands with degeneracies on $S$, for example pairs of bands forming a nodal
plane, Eq. (1) is not applicable, since Chern numbers can either become
undefined or assume non-integer values. But in these cases, a non-abelian
Chern number can still be defined [56]

$\displaystyle\nu_{b_{1}b_{2}}=\frac{1}{2\pi}\int_{S}\differential{\bf
n}\cdot\text{tr\,}{\bf\Omega}^{b_{1}b_{2}}({\bf k}),$ (12)

where the trace runs over band indices $b$ with $b_{1}\leq b\leq b_{2}$ and
the non-abelian Berry curvature and connection [57] are

$\displaystyle{\bf\Omega}^{b_{1}b_{2}}$ $\displaystyle=\gradient_{{\bf
k}}\crossproduct{\bf A}-i{\bf A}\crossproduct{\bf A},$ (13) $\displaystyle{\bf
A}^{n,m}$ $\displaystyle=i\innerproduct{n}{\gradient_{{\bf k}}|m},$ (14)

respectively, with $b_{1}\leq n,m\leq b_{2}$. The band index range
$b_{1},...,b_{2}$ must be chosen such that these bands have a non-zero bandgap
to bands $b_{1}-1$ and $b_{2}+1$ on the surface $S$.

A similar equation as Eq. (1) can be derived for non-abelian Chern numbers.
Using Eq. (12) and (13) we have

$\displaystyle\nu_{b_{1}b_{2}}$
$\displaystyle=\frac{1}{2\pi}\int_{S}\differential{\bf
n}\cdot\text{tr}(\gradient_{{\bf k}}\crossproduct{\bf A}-i{\bf
A}\crossproduct{\bf A})$
$\displaystyle=\frac{1}{2\pi}\int_{S}\differential{\bf
n}\cdot\text{tr}(\gradient_{{\bf k}}\crossproduct{\bf A})$
$\displaystyle=\frac{n}{2\pi}\int_{\partial S_{W}}\differential{\bf
s}\cdot\text{tr}{\bf A}({\bf k})\mod n,$ (15)

where we used the fact that $\text{tr}({\bf A}\crossproduct{\bf A})=0$ since
$\text{tr}(A_{i}A_{j}-A_{j}A_{i})=0$. In going from the second to the third
line we reduced the integration area using symmetry and applied Stokes theorem
just like in the proof for the abelian case. Splitting $\partial S_{W}$ into
$\partial S^{1}_{W}$ and $\partial S^{2}_{W}$ and mapping the latter to the
former with Eq. (III.1) we get

$\displaystyle\nu_{b_{1}b_{2}}$
$\displaystyle=\biggl{[}\frac{n}{2\pi}\int_{\partial
S^{1}_{W}}\differential{\bf s}\cdot\text{tr}{\bf A}({\bf k})$
$\displaystyle-\frac{n}{2\pi}\int_{\partial S^{1}_{W}}\differential{\bf
s}\cdot\text{tr}(\mathcal{B}_{C_{n}}({\bf k}){\bf A}({\bf
k})\mathcal{B}^{-1}_{C_{n}}({\bf k}))$
$\displaystyle-i\frac{n}{2\pi}\int_{\partial S^{1}_{W}}\differential{\bf
s}\cdot\text{tr}(\mathcal{B}_{C_{n}}({\bf
k})\nabla\mathcal{B}^{-1}_{C_{n}}({\bf k}))\biggr{]}$ $\displaystyle\mod n$
$\displaystyle=-i\frac{n}{2\pi}\int_{\partial S^{1}_{W}}\differential{\bf
s}\cdot\text{tr}(\mathcal{B}_{C_{n}}({\bf
k})\nabla\mathcal{B}^{-1}_{C_{n}}({\bf k}))$ $\displaystyle\mod n,$ (16)

where we used $\text{tr}(\mathcal{B}_{C_{n}}({\bf k}){\bf A}({\bf
k})\mathcal{B}^{-1}_{C_{n}}({\bf k}))=\text{tr}{\bf A}({\bf k})$. Using
Jacobi’s formula we have

$\displaystyle\frac{\differential}{\differential\lambda}\log\det\mathcal{B}^{-1}_{C_{n}}(\lambda)$
$\displaystyle=\frac{1}{\det\mathcal{B}^{-1}_{C_{n}}(\lambda)}\frac{\differential\det\mathcal{B}^{-1}_{C_{n}}(\lambda)}{\differential\lambda}$
$\displaystyle=\text{tr}\left(\mathcal{B}_{C_{n}}(\lambda)\frac{\differential\mathcal{B}^{-1}_{C_{n}}(\lambda)}{\differential\lambda}\right).$
(17)

Combining this with Eq. (16) we obtain

$\displaystyle\nu_{b_{1}b_{2}}$ $\displaystyle=-i\frac{n}{2\pi}\int_{\partial
S^{1}_{W}}\differential{\bf s}\cdot\gradient_{{\bf
k}}\log\text{det}\mathcal{B}^{-1}_{C_{n}}({\bf k})$ $\displaystyle\mod n$
$\displaystyle=-i\frac{n}{2\pi}[\log\text{det}\mathcal{B}_{C_{n}}({\bf
k}_{c_{b}}+\epsilon{\bf\hat{z}})$
$\displaystyle-\log\text{det}\mathcal{B}_{C_{n}}({\bf
k}_{c_{b}}-\epsilon{\bf\hat{z}})]$ $\displaystyle\mod n.$ (18)

This is equivalent to

$\displaystyle\nu_{b_{1}b_{2}}$
$\displaystyle=\frac{n}{2\pi}\sum_{b=b_{1}}^{b_{2}}\Delta\varphi_{b,c_{b}}$
$\displaystyle\text{mod }n,$ (19)

when the bands $b_{1},...,b_{2}$ are nondegenerate, consistent with the
abelian case, although here this is also true if the bands are degenerate
somewhere on the sphere except at the poles. When they are degenerate at the
poles, one must either resort to using Eq. (18) or choose an eigenbasis in the
degenerate subspace, such that the sewing matrix $\mathcal{B}_{C_{n}}$ is
diagonal and use Eq. (19).

###  III.3 Sewing matrices of anti-unitary symmetries

In this section we derive similar expressions for the anti-unitary symmetries
as in section III.1. Applying these to generic crossings, we find that single
band Chern numbers of crossings with time-reversal symmetry have even (odd)
Chern numbers without (with) SOC. In the following, $W$ is either just the
time-reversal symmetry $W=T$ with $R=1$ or $W$ is a combination of time-
reversal and crystalline symmetry. We start with the derivation of the sewing
matrix $\alpha$ for degenerate bands

$\displaystyle WH({\bf k})W^{-1}$ $\displaystyle=H(-R{\bf k})$ (20)
$\displaystyle\implies WH({\bf k})\ket{u({\bf k})}$ $\displaystyle=H(-R{\bf
k})W\ket{u({\bf k})}$ (21) $\displaystyle\implies E({\bf k})W\ket{u({\bf k})}$
$\displaystyle=H(-R{\bf k})W\ket{u({\bf k})}$ (22)

so $W\ket{u({\bf k})}$ must be an eigenstate of $H(-R{\bf k})$. Therefore

$\displaystyle\alpha({\bf k})\ket{u(-R{\bf k})}=W\ket{u({\bf k})},$ (23)

which leads to the sewing matrix for anti-unitary symmetries

$\displaystyle\alpha({\bf k})=\bra{u(-R{\bf k})}W\ket{u({\bf k})}.$ (24)

The Berry connection transforms under $W$ like so

$\displaystyle A({\bf k})$ $\displaystyle=i\bra{u({\bf k})}\partial_{\bf
k}\ket{u({\bf k})}$ $\displaystyle=-i\bra{u({\bf k})}W^{\dagger}\partial_{\bf
k}W\ket{u({\bf k})}$ $\displaystyle=-i\alpha^{*}({\bf k})\bra{u(-R{\bf
k})}\partial_{\bf k}\alpha({\bf k})\ket{u(-R{\bf k})}$
$\displaystyle=iR^{-1}\bra{u(-R{\bf k})}\partial_{-R{\bf k}}\ket{u(-R{\bf
k})}-i\alpha^{*}({\bf k})\partial_{\bf k}\alpha({\bf k})$
$\displaystyle=R^{-1}A(-R{\bf k})-i\alpha^{*}({\bf k})\partial_{\bf
k}\alpha({\bf k}),$ (25)

where we used

$\displaystyle\bra{u({\bf k})}W^{\dagger}\partial_{\bf k}W\ket{u({\bf k})}$
$\displaystyle=$ $\displaystyle\lim_{\epsilon\to
0}\frac{1}{\epsilon}(\bra{u({\bf k})}W^{\dagger}W\ket{u({\bf
k}+\epsilon)}-\bra{u({\bf k})}W^{\dagger}W\ket{u({\bf k})})$ $\displaystyle=$
$\displaystyle\lim_{\epsilon\to 0}\frac{1}{\epsilon}(\bra{u({\bf
k}+\epsilon)}\ket{u({\bf k})}-\bra{u({\bf k})}\ket{u({\bf k})})$
$\displaystyle=$ $\displaystyle\innerproduct{\partial_{\bf k}u({\bf
k})}{u({\bf k})}=-\bra{u({\bf k})}\partial_{\bf k}\ket{u({\bf k})}$ (26)

together with the anti-unitarity of $W$

$\displaystyle\bra{\Psi}W^{\dagger}W\ket{\Phi}=\bra{W\Psi}\ket{W\Phi}=\bra{\Phi}\ket{\Psi}$
(27)

and

$\displaystyle\innerproduct{u({\bf k})}{u({\bf k})}$ $\displaystyle=1$ (28)
$\displaystyle\implies\partial_{\bf k}\innerproduct{u({\bf k})}{u({\bf k})}$
$\displaystyle=0$ (29) $\displaystyle\implies\innerproduct{\partial_{\bf
k}u({\bf k})}{u({\bf k})}$ $\displaystyle=-\innerproduct{u({\bf
k})}{\partial_{\bf k}u({\bf k})}.$ (30)

####  III.3.1 Chern number constraints from $C_{n}T$ symmetries

Using Eq.(25), we can derive expressions similar to Eq.(1) for $C_{4}T$

$\displaystyle\nu_{c_{b}}$ $\displaystyle=\frac{2}{\pi}\left[\phi({\bf
S})-\phi({\bf N})\right]\mod 4$
$\displaystyle=-\frac{2}{\pi}\Delta\phi_{b,c_{b}}\mod 4,$ (31)

and for $C_{6}T$

$\displaystyle\nu_{c_{b}}=-\frac{3}{\pi}\Delta\phi_{b,c_{b}}\mod 3,$ (32)

with $\alpha({\bf k})=e^{-i\phi({\bf k})}$. The constraint for $C_{6}T$ is
only defined mod $3$ instead of mod $6$, since $C_{6}T$ relates the Berry
curvature of wedges spanning $1/3$ of a sphere to each other, instead of $1/6$
wedges. The main difference of Eq. (31) and (32) to Eq. (1) is that
$\Delta\phi_{b,c_{b}}$ is no longer the change of a symmetry eigenvalue but
the phase change of the anti-unitary symmetry sewing matrix (24).

$C_{2}T$ relates the Berry curvature of the upper to the lower hemisphere, so
to derive a local constraint we need to consider a path $S_{\text{equator}}$
on the equator

$\displaystyle\nu_{c_{b}}$
$\displaystyle=\frac{1}{\pi}\int_{S_{\text{equator}}}{\bf A}({\bf
k})\differential{\bf k}\mod 2$ (33)
$\displaystyle=\frac{1}{\pi}\int_{S_{\text{equator}}}C_{2}{\bf A}({\bf
k})\differential{\bf k}$
$\displaystyle-\frac{1}{\pi}\int_{S_{\text{equator}}}i\alpha^{*}({\bf
k})\partial_{\bf k}\alpha({\bf k})\differential{\bf k}\mod 2$ (34)
$\displaystyle=-\frac{1}{\pi}\int_{S_{\text{equator}}}{\bf A}({\bf
k})\differential{\bf k}$
$\displaystyle-\frac{1}{\pi}\int_{S_{\text{equator}}}i\alpha^{*}({\bf
k})\partial_{\bf k}\alpha({\bf k})\differential{\bf k}\mod 2$ (35)
$\displaystyle\implies\nu_{c_{b}}$
$\displaystyle=-\frac{1}{2\pi}\int_{S_{\text{equator}}}\partial_{\bf
k}\phi({\bf k})\differential{\bf k}\mod 2$ (36)

where we used Eq. (25), ${\bf A}(-C_{2}{\bf k})={\bf A}({\bf k})$ and
$(C_{2}{\bf v})\differential{\bf k}=-{\bf v}\differential{\bf k}$ with any
vector ${\bf v}$ on the equator. So $\nu_{c_{b}}$ is even (odd) when
$\phi({\bf k})$ winds an even (odd) number of times around the equator.

A $C_{3}T$ constraint is redundant, since $C_{3}T$ implies $T=(C_{3}T)^{3}$
and therefore also $C_{3}$ to exist separately.

####  III.3.2 Chern number constraint of crossings at TRIMs

Next we would like to evaluate a single band Chern number of a crossing with
time-reversal symmetry, $W=T$ ($R=1$) and $T^{2}=\gamma$, where $\gamma=1$ for
spinless and $\gamma=-1$ for spinful fermions. We split the integration-sphere
around the crossing into two halves,

$\displaystyle\nu$ $\displaystyle=\frac{1}{\pi}\biggl{(}\int_{\partial
S_{1}}A({\bf k})\text{d}{\bf k}+\int_{\partial S_{2}}A({\bf k})\text{d}{\bf
k}\biggr{)}\mod 2$ (37) $\displaystyle=\frac{1}{\pi}\biggl{(}\int_{\partial
S_{1}}A({\bf k})\text{d}{\bf k}+\int_{\partial S_{2}}A(-{\bf k})\text{d}{\bf
k}$ $\displaystyle-\int_{\partial S_{2}}i\alpha^{*}({\bf k})\partial_{\bf
k}\alpha({\bf k})\text{d}{\bf k}\biggr{)}\mod 2$ (38)
$\displaystyle=-\frac{1}{\pi}\int_{\partial S_{2}}i\alpha^{*}({\bf
k})\partial_{\bf k}\alpha({\bf k})\text{d}{\bf k}\mod 2,$ (39)

where $\partial S_{1}$ and $\partial S_{2}$ are paths at the halves edge
running on opposite sides. Using $\alpha({\bf k})=e^{-i\phi({\bf k})}$ we have

$\displaystyle\nu$ $\displaystyle=-\frac{1}{\pi}\int_{\partial
S_{2}}\partial_{\bf k}\phi({\bf k})\text{d}{\bf k}\mod 2$ (40)
$\displaystyle=\frac{1}{\pi}(\phi({\bf S})-\phi({\bf N}))\mod 2$ (41)

with ${\bf N}$ and ${\bf S}$ being the north and southpole. To evaluate this
expression, consider Eq. (23). We can reinsert itself with a replacement ${\bf
k}\to-{\bf k}$ to get

$\displaystyle\alpha({\bf k})\ket{u(-{\bf k})}$
$\displaystyle=T\alpha^{*}(-{\bf k})T\ket{u(-{\bf k})}$ (42)
$\displaystyle=\alpha(-{\bf k})\gamma\ket{u(-{\bf k})}$ (43)

so

$\displaystyle\gamma$ $\displaystyle=\alpha({\bf k})\alpha^{*}(-{\bf k})$ (44)
$\displaystyle=e^{-i(\phi({\bf k})-\phi(-{\bf k}))}$ (45)

which can be applied to Eq. (41) to arrive at $\nu=0\mod 2$ for the spinless
case ($\gamma=1$) and $\nu=1\mod 2$ for the spinful one ($\gamma=-1$). So any
crossing at TRIMs, which include also multifold ones, without further
degeneracies away from the crossing, must have even (odd) Chern numbers
without (with) SOC. We see that this constraint is explicitly fulfilled in all
models found in this paper, for example in section V.2 and in all low-energy
Weyl point Hamiltonian at TRIMs in [47].

###  III.4 Global constraint on band topology

For chiral band crossings global constraints on the band topology arise due to
conditions on the sum of the topological charges of nodal points. One such
global constraint is the fermion doubling theorem by Nielsen and Ninomiya,
which states that for each band the sum of all chiralities has to vanish [32].
Here, we prove a global constraint on the rotation eigenvalues, which
ultimately follows from the periodicity of the BZ, i.e., the compactness of
the BZ. To do so, we employ symmetry representations along the full rotation
axis, which can be obtained by taking powers of the symmetry [58]. For
concreteness we consider a screw rotation symmetry $C_{n}(x,y,\tfrac{m}{n})$,
which describes an $n$-fold rotation around the $z$ axis followed by a
translation with the vector $(x,y,\tfrac{m}{n})$. Taking the $n$-th power of
the screw rotation we obtain

$\displaystyle\big{[}C_{n}(x,y,\tfrac{m}{n})\big{]}^{n}=\mathrm{e}^{i\pi
s}\,T(0,0,m)=\mathrm{e}^{i\pi s+imk_{z}},$ (46)

where $m,n\in\mathbb{Z}$, $|m|<|n|$, and $s=0$ ($s=1$) for spinless (spinful)
systems. In the second step the translation by a full lattice vector
$T(0,0,m)$ is replaced by the usual one-dimensional representation of the
translation group. Notably, the above and all following steps apply
analogously to glide mirror operations, which would correspond to an operation
with $n=2$ and either $m=0$ or $m=1$ for mirror and glide mirror symmetry,
respectively. The symmetry eigenvalues of the $C_{n}$ screw rotation is found
as the complex root of Eq. (46) yielding

$\displaystyle\lambda_{C_{n}}=\exp(i\frac{2\pi p+s\pi+mk_{z}}{n}),$ (47)

where $p\in\\{0,1,\ldots,n-1\\}$ distinguishes the $n$ different complex
roots. On the rotation axes invariant under the rotation
$C_{n}(x,y,\tfrac{m}{n})$, we label the bands using $\lambda_{C_{n}}$ or
rather, equivalently, we consider the complex phase
$\varphi(k_{z})=\arg\lambda_{C_{n}}$. To label a specific band $b$ that is
identified by sorting the eigenvalues of the Hamiltonian according to their
energy, we have to consider that Eq. (47) does not yet include band crossings.
The phase $\varphi_{b}(k_{z})$ for a specific band is given by

$\displaystyle\varphi_{b}(k_{z})=\frac{2\pi p+s\pi+mk_{z}}{n}+\sum_{k_{c}\leq
k_{z}}\Delta\varphi_{b,c},$ (48)

which must include all phase jumps $\Delta\varphi_{b,c}$ at $k_{c}$
corresponding to all crossings $c$ up to $k_{z}$, which may be, for example,
with the bands $b-1$ or $b+1$. The essential step to identify the global
constraints on $\varphi_{b}(k_{z})$, $\Delta\varphi_{b,c}$, and in extension
also on all chiral crossings is the periodicity of the Brillouin zone. Thus,
we compare the phase $\varphi_{b}(k^{0}_{z})$ at some position $k^{0}_{z}$
with the phase $\varphi_{b}(k^{0}_{z}+2\pi)$ after traversing the Brillouin
and accumulating phase jumps $\Delta\varphi_{b,c}$ at $k_{c}$ as well as a
contribution from $\tfrac{mk_{z}}{n}$,

$\displaystyle\varphi_{b}(k^{0}_{z}+2\pi)-\varphi_{b}(k^{0}_{z})$
$\displaystyle=0\mod 2\pi$ (49)
$\displaystyle\sum_{c_{b}}\Delta\varphi_{b,c}+2\pi\frac{m}{n}$
$\displaystyle=0\mod 2\pi.$ (50)

The phase jumps $\Delta\varphi_{b,c}$ in Eq. (50) are not independent for
different bands $b$. For every phase jump there should be the reverse exchange
of eigenvalues in a higher or lower band. Suppose we consider a system, where
all band crossings are twofold, then one may iteratively substitute Eq. (50)
for band $b-1$ into the equation for band $b$. The induction process leads to
Eq. (2)

$\displaystyle b=1$ $\displaystyle\sum_{c_{1}}\Delta\varphi_{1,c_{1}}$
$\displaystyle=-2\pi\frac{m}{n}\mod 2\pi$ $\displaystyle b=2$
$\displaystyle\sum_{c_{2}}\Delta\varphi_{2,c_{2}}-\sum_{c_{1}}\Delta\varphi_{1,c_{1}}$
$\displaystyle=-2\pi\frac{m}{n}\mod 2\pi$
$\displaystyle\Leftrightarrow\sum_{c_{2}}\Delta\varphi_{2,c_{2}}$
$\displaystyle=-2\cdot 2\pi\frac{m}{n}\mod 2\pi$ $\displaystyle\vdots\quad$
$\displaystyle\vdots$ $\displaystyle\text{any }b$
$\displaystyle\sum_{c_{b}}\Delta\varphi_{b,c_{b}}$ $\displaystyle=-b\cdot
2\pi\frac{m}{n}\mod 2\pi.$ (51)

This result contains the notion of filling-enforced semimetals, namely, if
$b\tfrac{m}{n}\not\in\mathbb{Z}$, then there must be at least one symmetry-
enforced band crossing [33]. Once the filling, i.e., the considered band $b$,
is a multiple of $\tfrac{n}{m}$, band crossings do not need to exist.

##  IV Applications and extensions

To demonstrate the power of the local and global constraints, we present a
number of applications and discuss some extensions.

###  IV.1 Applications and extensions
of the local constraint

In the following we use the local constraint, Eq. (1), to prove the existence
of enforced double Weyl points away from TRIMs (Sec. IV.1.1). We generalize
the local constraint to multiple rotation symmetries in Sec. IV.1.2, which
enables us to infer conditions for the Chern numbers for all types of (higher-
fold) chiral crossings. Finally, we use the local constraint to show that
nodal lines with nonzero Chern numbers cannot be stabilized by chiral space-
group symmetries (Sec. IV.1.3).

####  IV.1.1 Chiral crossings between identical symmetry eigenvalues

In this section we use the local constraint, Eq. (1), to explain the existence
of unusual enforced double Weyl points away from TRIMs [46, 47]. First, we
clarify why these Weyl points pose an open question in the understanding of
chiral crossings. According to conventional wisdom, a stable band degeneracy
can only occur if at least one of the three following conditions is fulfilled:
(i) The two bands forming the crossing belong to different symmetry
representations, which prevents the introduction of gap opening terms, (ii)
there is a higher-dimensional representation of the little group, or (iii)
there exists an anti-unitary symmetry that leaves the degeneracy point
invariant, leading to Kramers degeneracy. However, in space groups 80, 98, and
210 there exist band crossings away from TRIMs between bands with identical
representations of dimension one [40]. So at first glance, all of the above
three conditions for a crossing seem violated. Yet, the combination of time-
reversal and fourfold rotation symmetry generates, due to Kramers theorem,
point-like degeneracies at high-symmetry points of certain non-primitive
Brillouin zones that are not TRIMs [40]. Interestingly, with SOC these
crossings are known to be double Weyl points with Chern number $\pm 2$ [41],
but could until now not be understood in terms of symmetry eigenvalues [46,
47].

For concreteness, let us now focus on the body-centered tetragonal SG 80
($I4_{1}$), whose P point can host two-fold degeneracies both with and without
SOC [e.g., see Figs. 2(a) and 12]. As we will see, this band crossing can be
understood by noting that the combined symmetry $TC_{4z}$, comprising time-
reversal $T$ and fourfold screw rotation $C_{4z}$, leaves the P point
invariant. Other than that, the only unitary symmetry that leaves P invariant
is the rotation $C_{2z}$ whose symmetry eigenvalues can be used to label the
bands. We now need to distinguish the case with and without SOC, which differ
slightly for SG 80. Without SOC different eigenvalues are paired by the anti-
unitary operation $TC_{4z}$. In our notation this corresponds to
$\Delta\varphi=\pm\pi$ for the Weyl point at P which implies by Eq. (1) a
Chern number of $\nu_{P,\text{SG}80}=1\mod 2$. With SOC the representation is
doubled compared to before and splits into two one-dimensional and one two-
dimensional representation at P, because the Kramers theorem only applies to
the latter representation, see Ref. [40] for details. Since for the two-
dimensional representation one eigenvalue of $C_{2z}$ is paired to itself, one
finds $\Delta\varphi=0$ implying $\nu_{P,\text{SG}80,\text{SOC}}=0\mod 2$.
Taking into account that the crossing at P has been $\nu_{P,\text{SG}80}=1$
without spin, it follows from the conservation of topological charge that
$\nu_{P,\text{SG}80,\text{SOC}}=\pm 2$. We have thus reached an explanation
for the double Weyl point at P in terms of symmetry eigenvalues.

Figure 2:  _Band structure, Berry curvature and spin texture for SG 80._ (a)
Tight-binding model with SOC of SG 80 as defined in Eq.
(LABEL:Eq_SG80fullHamiltonian). The body-centered tetragonal Brillouin zone is
shown in the inset of Fig. 11(b). The double Weyl point at P is characterized
by its Berry curvature and spin textures as shown in (b) and (c),
respectively, which are given for the lower band of the crossing on a sphere
enclosing it. For comparison, (d) and (e) show the Berry curvature and spin
textures of a regular double Weyl point. For all vector plots the north pole
of the sphere is labelled by “N”.

The discussed double Weyl point at P in SG 80 has a different origin and
symmetry than any other twofold double Weyl point, which occur either on
fourfold or sixfold rotation axes or at TRIMs in the presence of spinless
time-reversal symmetry [59, 47, 46]. Hence, we expect that also the spin
texture [60, 61, 62, 63, 64, 65, 66, 67] and Berry curvature are distinct from
the conventional double Weyl points. To demonstrate this, we compute the Berry
curvature and spin texture of the double Weyl point in SG 80. For this
purpose, we derive in Appendix C a tight-binding model including SOC for SG
80. Figure 2(a) shows the band structure of this model defined by Eq.
(LABEL:Eq_SG80fullHamiltonian). As expected we find a double Weyl point of
charge $\nu_{\text{P}}=2$ at each P point, which is compensated by a pair of
double Weyl points on the fourfold rotation axis $\Gamma$-Z-M with
$\nu_{\Gamma\text{-Z-M}}=-2$. Figures 2(b) and 2(c) show the Berry curvature
and spin texture around the P point, respectively. To contrast this with
conventional double Weyl points we plot in Figs. 2(d) and 2(e) the Berry
curvature and spin texture of a conventional double Weyl point defined by
$H(\mathbf{k})=k_{z}\sigma_{z}+(\sigma_{+}(k_{x}-ik_{y})^{2}+H.C.)$, where
$\sigma_{+}=1/2(\sigma_{x}+i\sigma_{y})$ with the Pauli matrices $\sigma_{j}$
[68, 61]. While the details of these textures are parameter-dependent, their
symmetry properties are generic and dictated by the local little groups. In
general the spin texture at P in SG 80 is anisotropic and symmetric under the
_anti-unitary_ $C_{4}T$ symmetry [see regions of similar color shading in Fig.
2(c)]. In contrast, the texture of a conventional double Weyl point is
symmetric under an _unitary_ (e.g., fourfold) rotaiton symmetry, see Fig.
2(d,e). Another difference is that the spin texture around the equator of Fig.
2(c) has a unit winding, whereas the one of Fig. 2(e) has a winding of two.
These differences in spin texture could be measured experimentally, using,
e.g., spin- and angle-resolved photoemission spectroscopy [65, 60, 67].

Using a database search (see Sec. VI) we have identified NbO2 and TaO2 as
candidate materials in SG 80 realizing the double Weyl points away from TRIMs.
The band structure and surface states of these compounds are presented in Sec.
VI.2. Notably, we find that for surface terminations perpendicular to any of
the crystal axes there appear four Fermi arcs. This is because for these
terminations the P point is projected onto a symmetry related copy of itself
with the same Chern number $\pm 2$, such that there emerge four Fermi arcs
from the projeted P point in the surface BZ.

The above arguments for SG 80 apply in a similar manner also to SG 98
($I4_{1}22$) and SG 210 ($F4_{1}32$), for which the double Weyl points appaer
at the P and W points, respectively. In addition, related arguments using the
local constraint can be employed to understand the charge $\nu=\pm 2$ of the
threefold crossings in SG 199 ($I2_{1}3$) and SG 214 ($I4_{1}32$) at the point
P, see the discussion in Sec. V.2.2.

####  IV.1.2 Chiral crossings with multiple rotation axes

Band crossing points symmetric under little groups that contain more than one
rotation symmetry often exhibit larger topological charges than in the case of
a single rotation symmetry [16, 17, 49]. Also in this case the local
constraint, Eq. (1), can be used to understand the observed topological
charges. In the following, we extend the above arguments to multiple rotation
axes and consider, for concreteness, a twofold quadruple Weyl point at
$\Gamma$ in SG 195, for which a Chern number of $\nu=4$ has been reported [68,
69, 70, 71]. Other non-trivial examples of nodal points with multiple rotation
symmetries are discussed in Secs. IV.2.2 and V in the context of multifold
band crossings.

For a single rotation axis one usually finds that
$\nu_{c_{b}}=\Delta\varphi_{b,c_{b}}\frac{n}{2\pi}$ holds without the modulus
operation, although the local constraint, Eq. (1), restricts the possible
charges only up to the order of the rotation $n$. This is because higher
topological charges would require fine tuning. To see this, consider a
crossing point of charge $\nu=\nu_{c_{b}}+mn$, where $m$ is some non-zero
integer. If this crossing is perturbed by some symmetry-allowed perturbation,
the crossing may split into one with charge $\nu_{c_{b}}$ and $m$ sets of each
$n$ Weyl points. In fact, generally exactly this happens, because placing $mn$
Weyl points on the rotation axis is a fine-tuned situation. In other words, to
achieve higher topological charges $\nu$, more lower orders in the low-energy
expansion need to be set to zero, which would require fine tuning.

In the presence of multiple rotation symmetries, however, there are more
symmetry constraints that can lead to higher topological charges, such that
the smallest possible value given by the local constraint (1) is not realized.
To demonstrate this, let us consider SG 195 ($P23$) with time-reversal
symmetry, where a twofold quadruple Weyl point is enforced to occur at the
TRIM $\Gamma$ (and also at R), if, for example, the ${}^{1}E^{2}E$
representation is placed on the Wyckoff position 1a [72]. The corresponding
little group at $\Gamma$ consist of the point group 23 together with time-
reversal, which does contain a twofold and a threefold rotation, but no
fourfold rotation, different from the conventional twofold quadruple Weyl
points of Refs. [68, 69, 70, 71]. From the representation of this little
group, one finds that there is no exchange of the twofold rotation
eigenvalues, while the threefold rotation eigenvalue switch. Thus, the local
constraint (1) on the charge $\nu_{\Gamma}$ is

$\displaystyle\nu_{\Gamma}=0\mod 2\quad\text{and}\quad\nu_{\Gamma}=1\mod 3.$
(52)

Thus, both $\nu_{\Gamma}=4$ and $\nu_{\Gamma}=-2$ are in agreement with Eqs.
(52). To resolve this ambiguity we construct a low-energy model
$H_{\text{T}}(\mathbf{k})$ around $\Gamma$, which is symmetric under the point
group 32 and time-reversal, see Appendix D. The energy bands of this model
exhibit quadratic and cubic dispersions along different directions away from
the crossing point, see Fig. 3. The topological phase diagram of this low-
energy model contains only one phase with $\nu_{\Gamma}=4$, in agreement with
Eq. (52). However, the lowest possible topological charge of
$\nu_{\Gamma}=-2$, cf. Eq. (52), is not realized, in contrast to the
conventional twofold quadruple Weyl points with fourfold rotation symmetry
[68, 69, 70, 71]. This raises the question, why can the charge
$\nu_{\Gamma}=-2$ not be realized in our low-energy model, even though it
would be consistent with the local constraint?

Figure 3:  _Quasi-symmetry-enforced Chern number in twofold quadruple Weyl
points._ Band structure around a twofold band crossing symmetric under point
group 23 and time reversal, as given by Eq. (LABEL:Eq_App_PG23) with
$d_{1}=0$. The splitting of the bands along the threefold (twofold) rotation
axes in red (purple) is cubic (quadratic). A fourfold quasi-symmetry enforces
a charge of $\nu=4$.

There are two ways to answer this question. First, a closer look at the low-
energy model presented in Appendix D reveals that there is a fourfold _quasi-
symmetry_ , i.e., a symmetry of the low-energy model that is broken by terms
of higher order in $k$. Namely, the low-energy Hamiltonian
$H_{\text{T}}(\mathbf{k})$, Eq. (LABEL:Eq_App_PG23), is left invariant by
$U(C_{4}^{z})^{\dagger}H_{\text{T}}(\mathbf{k})U(C_{4}^{z})=H_{\text{T}}(R^{-1}\mathbf{k})$,
where $U(C_{4}^{z})$ is the representation of a fourfold rotation symmetry and
$R$ is the corresponding transformation in real space, see Eq. (D).
$U(C_{4}^{z})$ is a quasi-symmetry, as it is a symmetry only of the lowest-
order terms, but not of the full Hamiltonian. Yet, since the Chern number is
determined exclusively by the lowest orders in $k$, at which the point
crossing is well-defined, this quasi-symmetry forces the charge to be
$\nu_{\Gamma}=+4$ for the crossing by adding the local constraint
$\nu_{\Gamma}=0\mod 4$.

Second, the Chern number $\nu_{\Gamma}=0\mod 4$ can be understood by
considering how the symmetries act on the Berry curvature integration. For
this purpose we need to consider the role of the time-reversal symmetry
together with the twofold rotation, to be specific the combination $C_{2}T$.
The Berry curvature flux through the northern $S_{W}(N)$ and southern
$S_{W}(S)$ half of the spherical wedge $S_{W}$ when considering $C_{2}$ are
identical due to $C_{2}T$ and thus

$\displaystyle\nu_{\Gamma}$
$\displaystyle=2\tfrac{2}{2\pi}\int_{S_{W}(N)}\differential{\bf
n}\cdot{\bf\Omega}({\bf k})$ (53) $\displaystyle=\frac{2}{\pi}\int_{S_{{\bf
N}{\bf P_{1}}}+S_{\text{equator}}+S_{{\bf P_{2}}{\bf N}}}{\bf
A}\differential{\bf s}$ $\displaystyle\text{mod }4$ (54)

where we split the path $S_{W}(N)$ into $S_{{\bf N}{\bf
P_{1}}}+S_{\text{equator}}+S_{{\bf P_{2}}{\bf N}}$ and $P_{1}$ and $P_{2}$
being endpoints of $S_{\text{equator}}$. The Berry connection integration on
paths $S_{{\bf N}{\bf P_{1}}}$ and $S_{{\bf P_{2}}{\bf N}}$ are related by
symmetry and can be evaluated in a similar way as in Eq. 8,

$\displaystyle\int_{S_{{\bf N}{\bf P_{1}}}+S_{{\bf P_{2}}{\bf N}}}{\bf
A}\differential{\bf s}$ $\displaystyle=\phi_{b}({\bf P_{1}})-\phi_{b}({\bf
N}).$ (55)

Regarding the $S_{\text{equator}}$ integration, consider Eq. (III.1) applied
on this path

$\displaystyle\int_{S_{\text{equator}}}{\bf A}(C_{2}{\bf k})\differential{\bf
s}$ $\displaystyle=\int_{S_{\text{equator}}}C_{2}({\bf A}({\bf
k})-\nabla\phi_{b}({\bf k}))\differential{\bf s}$
$\displaystyle\implies\int_{S_{\text{equator}}}{\bf A}(-{\bf
k})\differential{\bf s}$ $\displaystyle=-\int_{S_{\text{equator}}}{\bf A}({\bf
k})-\nabla\phi_{b}({\bf k})\differential{\bf s}$
$\displaystyle\implies\int_{S_{\text{equator}}}{\bf A}({\bf
k})\differential{\bf s}$ $\displaystyle=-\int_{S_{\text{equator}}}{\bf A}({\bf
k})-\nabla\phi_{b}({\bf k})\differential{\bf s}$
$\displaystyle\implies\int_{S_{\text{equator}}}{\bf A}({\bf
k})\differential{\bf s}$
$\displaystyle=\frac{1}{2}\int_{S_{\text{equator}}}\nabla\phi_{b}({\bf
k})\differential{\bf s}$ (56) $\displaystyle=\frac{1}{2}(\phi_{b}({\bf
P_{2}})-\phi_{b}({\bf P_{1}}))$ (57)

where all equations are valid up to mod $2\pi$ and with ${\bf A}(C_{2}{\bf
k})={\bf A}(-{\bf k})$ and $(C_{2}{\bf v})\differential s=-{\bf
v}\differential s$ with any vector ${\bf v}$ on the equator. We also used Eq.
(25) relating ${\bf A}({\bf k})$ and ${\bf A}(-{\bf k})$ via time-reversal
symmetry

$\displaystyle\int_{S_{\text{equator}}}{\bf A}(-{\bf k})\differential{\bf
s}\mod 2\pi=$ $\displaystyle\int_{S_{\text{equator}}}{\bf A}({\bf
k})\differential{\bf s}-i\int_{S_{\text{equator}}}\alpha^{*}({\bf
k})\partial_{\bf k}\alpha({\bf k})\differential{\bf s}\mod 2\pi=$
$\displaystyle\int_{S_{\text{equator}}}{\bf A}({\bf k})\differential{\bf
s}+\phi_{T}({\bf P_{1}})-\phi_{T}({\bf P_{2}})\mod 2\pi=$
$\displaystyle\int_{S_{\text{equator}}}{\bf A}({\bf k})\differential{\bf
s}\mod 2\pi,$ (58)

where the time-reversal symmetry sewing matrix has the form $\alpha({\bf
k})=e^{-i\phi_{T}({\bf k})}$ and $\phi_{T}({\bf P_{1}})=\phi_{T}(-{\bf
P_{1}})=\phi_{T}({\bf P_{2}})$ in the spinless case (see Eq. (45)). In total,
we get

$\displaystyle\nu_{\Gamma}$ $\displaystyle=\frac{2}{\pi}\left(-\phi_{b}({\bf
N})+\frac{1}{2}(\phi_{b}({\bf P_{1}})+\phi_{b}({\bf P_{2}}))\right)$
$\displaystyle\text{mod }4.$ (59)

Explicitly calculating $\phi_{b}({\bf P_{1}})+\phi_{b}({\bf P_{2}})$ in the
irrep $\Gamma_{2}\Gamma_{3}$ by plugging in the Bloch wavefunctions of the
low-energy Hamiltonian into Eq.(6) yields $0$. So

$\displaystyle\nu_{\Gamma}$ $\displaystyle=\frac{2}{\pi}\varphi({\bf N})$
$\displaystyle\text{mod }4.$ (60)

The irrep $\Gamma_{2}\Gamma_{3}$ in ${}^{1}E^{2}E$ of $C_{2}$ is $\sigma_{0}$,
so $\varphi({\bf N})=0$, which means we get

$\displaystyle\nu_{\Gamma}$ $\displaystyle=0\,\text{mod }4.$ (61)

A material implementation of this WP with charge 4 was found in BaIrP, see
[68]. There it is shown that, upon introducing SOC, this crossing evolves into
12 WPs and a 4-fold crossing at $\Gamma$ with $C=\nu_{1,2}=\pm 4$, which is in
section V.2.1 revealed to be the $\nu_{n}=\\{\pm 3,\pm 1,\mp 1,\mp 3\\}$ phase
of the model described there.

####  IV.1.3 Chiral nodal lines

Nodal lines protected by crystalline symmetries are commonly discussed in the
context of mirror symmetries, which leave a plane in the Brillouin zone
invariant where they provide two distinct representations. The presence of two
distinct representations is sufficient to obtain accidental nodal lines.
Furthermore, there can be symmetry-enforced line crossings, for example, if
another symmetry operation anticommutes with the mirror symmetry, one finds
nodal lines pinned to high-symmetry paths. Alternatively, if the original
mirror symmetry is nonsymmorphic this is already enough to conclude in analogy
to Eq. (2) that there must be an odd number of nodal lines crossing every
other gap, which are movable in the sense that their position is parameter-
dependent [34, 35]. Other cases of nodal lines include higher-fold nodal lines
or almost movable nodal lines, which are only pinned to a finite number of
high-symmetry points [42, 40].

For all of these nodal lines the Chern number vanishes because of the mirror
symmetry, when calculated on a surface that fully encloses the nodal line. It
comes to no surprise that despite the extensive research on various types of
nodal lines, no example of a stable _chiral_ nodal line, i.e., a nodal line
with Chern number, has been discussed so far. Nevertheless, there are some
reports of such nodal lines without mirror symmetry in the literature, which
are either of unclear symmetry protection [73] or as in the case of the nodal
lines in hexagonal AgF3 [74] have found to be actually weakly gapped [48].
Whether a chiral nodal line can exist is not only of interest due to its
unique topology, but also important for the study of enforced topological
nodal planes. To rigorously deduce the existence of the latter, one needs to
assume that a chiral nodal line does not exist. In this case a non-zero sum of
Weyl point chiralities within the Brillouin zone, implies a charged nodal
plane, see Sec. IV.2.1.

In this section we aim to answer, whether chiral nodal lines can be stabilized
by crystalline symmetries, and we will extensively apply the rotation symmetry
constraint of Eq. (1). Doing so we consider points in reciprocal space lying
away from any (glide) mirror planes. To approach the first goal, let us assume
that we have obtained a nodal line at a generic position in the Brillouin zone
with a chirality $\nu_{\text{line}}<n$, where $n$ is the order of the highest
rotation symmetry. Suppose in this gedankenexperiment that we introduce all
symmetry-allowed perturbations to the system to gap out the chiral nodal line.
Since the nodal line is assumed to be chiral, its topological charge has to
persist in the form of Weyl points. But as long as the original rotation
symmetry is preserved, the condition $\nu_{\text{line}}<n$ implies that a
nodal line cannot be gapped, because the number of resulting Weyl points at
generic positions would be equal to $\nu_{\text{line}}$ and thus incompatible
with the required multiplicity $n$. Unlike nodal lines protected by a
$\mathbb{Z}_{2}$ invariant, shrinking the nodal line to a point would not
remove it, but leave a Weyl point with the same topological charge behind.

Yet, despite this argument to stabilize a chiral nodal line, we will discuss
in the following that the relation between rotation symmetry eigenvalues and
the chirality, see Eq. (1), strongly limit the possibility to find any nodal
band feature fulfilling $\nu_{\text{line}}<n$.

First, suppose there is a nodal line encircling an n-fold rotation axis. Then,
we can enclose the whole line by a sphere analogously to Fig. 1(a), which
implies by the arguments given in Sec. III.1 that the chirality of all band
crossings enclosed by the sphere is related to the change of rotation
eigenvalues on the north and south pole of the rotation axis
$\Delta\varphi_{b}$. Several cases must be distinguished. If there is no
additional point crossing on the rotation axes, then $\Delta\varphi_{b}=0$
leading to $\nu_{\text{line}}=0\mod n$, implying that the nodal line is
trivial or at least unstable. If there are indeed additional point crossings
on the rotation axis, then $\Delta\varphi_{b}\neq 0$ and one may choose the
sphere to enclose only the point crossings, which implies that these crossings
by themselves are responsible for the charge of
$\Delta\varphi_{b}\tfrac{n}{2\pi}\mod n$, which would be observed on the
original sphere. In both cases the chiral nodal line is unstable.

To circumvent the objections, one may consider more intricate configurations
of nodal lines. If one examines a nodal line that is sufficiently extended
such that it cannot be enclosed by a sphere, it is generally still possible to
find a surface to enclose the line and a section of the rotation axis. The
proof of Eq. (1) can then be repeated for this new surface, after the
subdivision of the integration surface for the Berry curvature, the edges must
be related by symmetry, see also Ref. [52], where the integration surface
intersects more than one rotation axis. Ultimately, one finds an expression
depending on the changes of eigenvalues of the different rotation axes, but
the symmetry representation only changes when traversing the integration
contour if a crossing on a rotation axes has been enclosed. Thus, either the
nodal line itself has crossed a rotation axes and is responsible for the
exchange of symmetry eigenvalues, such that the nodal line can be gapped out
except at a set of corresponding point crossings on the rotation axes, or
there will be no exchange of symmetry eigenvalues and hence at most a trivial
charge. In Appendix B we discuss the case of antiunitary symmetries of higher
multiplicity and show that they do not circumvent the result obtained above
from Eq. (1). In summary, we find that all configurations of chiral nodal
lines discussed here do not fit to the original proposal of a topological
charge $\nu_{\text{line}}<n$, hence we find that no _crystalline_ symmetry is
able to protect a chiral nodal line. Note, a chiral nodal line may still be
found by considering systems with internal symmetries.

To conclude this section we propose a low-energy model of a chiral nodal line
to illustrate how our above symmetry argument can be circumvented. In this
construction we place two Weyl points, $W({\bf k},\epsilon)$, of different
energy at the origin and couple them by the matrix $A({\bf k})$ in a way that
preserves an internal symmetry $U_{\text{line}}$. We define

(62) (63)

where we set the energy offset to $\epsilon=1$, ${\bf\sigma}$ is the vector of
Pauli matrices, and $\sigma_{0}$ denotes the two-dimensional unit matrix. The
bands of the Weyl points intersect in a nodal sphere [75] and are gapped by
$A({\bf k})$ except at $k_{z}=0$, see Fig. 4 for the resulting band structure.
This model exhibits a chiral nodal line with a Chern number of
$\nu_{\text{line}}=2$, which is inherited from the interplay of two $\nu=+1$
Weyl points. It has to be noted that the charge of such a nodal line does also
dependent on the hybridization away from $k_{z}=0$, e.g., for $A({\bf
k})=k_{z}\sigma_{3}$ the nodal line is not charged. Interestingly, there is a
fourfold rotation symmetry $U(C_{4})=\sigma_{0}\otimes\sigma_{3}$, which is
broken by $A({\bf k})$ for $k_{z}\neq 0$. Yet, perturbations that preserve the
$U(C_{4})$ symmetry gap out the nodal line, because the nodal line is not
pinned to the $k_{z}=0$ plane and loses its symmetry protection once moved
away despite $\nu_{\text{line}}<n$. Nevertheless, there is an orbital symmetry
in our model, namely,

which fulfills $[H({\bf k}),U_{\text{line}}]=0$. The matrix is a unitary
operation with the eigenvalues $\\{-1,-1,+1,+1\\}$ that exchange at the chiral
nodal line. Thus, any perturbation that respects the symmetry
$U_{\text{line}}$ may deform the nodal line, but cannot introduce a gap. Such
a chiral nodal line could be realizable for example in optical metamaterials
or other synthetic systems.

Figure 4:  _Chiral nodal line._ Band structure for the model chiral nodal line
(red highlighting) described by Eq. (62). The dispersion of the nodal line is
linear (quadratic) in radial ($k_{z}$) direction.

###  IV.2 Applications and extensions of the global constraint

The global constraint contains the information on the possible numbers of
crossings on rotation axes or mirror planes. This is for example a guide to
the search for semimetals with few point or line crossings [40]. In this
section we combine both constraints and the Nielsen-Ninomiya theorem [32].
First, we will discuss a _paramagnetic_ space group with an enforced
topological nodal plane duo. Secondly, we illustrate the constraints with a
real band structure including accidental Weyl points, nodal planes, and multi-
fold crossings.

####  IV.2.1 Symmetry-enforced topological nodal planes

In the following section we apply local and global constraints to the theory
of symmetry-enforced topological nodal planes. After a brief summary of the
basic arguments that lead to (topological) nodal planes, we consider the
nontrivial case of SG 94 $P4_{2}2_{1}2$. This space group is the only known
case with two symmetry-enforced topological nodal planes in a paramagnetic
space group, i.e., in a grey group including time reversal as a symmetry
element.

We consider nodal planes as two-fold degeneracies on the surface of the
Brillouin zone. Such degenerate planes can be symmetry-enforced by the
combined symmetry comprising time-reversal $T$ and a two-fold screw rotation
$\tilde{C}_{2}$ [76, 11, 12, 14]. In short, the anti-unitary symmetry
$T\tilde{C}_{2}$ fulfills Kramers theorem at every point on a plane in the
Brillouin zone. Regions that host nodal planes are described by $k_{i}=\pi$ in
units of the corresponding inverse lattice constant and have to be at the
surface of the Brillouin zone. This gives rise to a natural distinction based
on the number nodal planes (one, two, or three) or equivalently distinct
symmetries $T\tilde{C}_{2}$ with eligible planes in the Brillouin zone. We
refer two the case of two (three) nodal planes as nodal plane duo (trio) to
highlight that these nodal planes form a single connected object that can only
be assigned a single Chern number.

The whole gapless structure of nodal planes may exhibit a non-zero Chern
number on a surface that encloses the plane, if mirror and inversion
symmetries are absent. For nodal planes trios, i.e., nodal planes at
$k_{i}=\pi$ with $i\in\\{x,y,z\\}$, a single Kramers Weyl point at the TRIM
$\Gamma$ can only be compensated by an opposite charge on the nodal planes,
where one needs to consider the case of spinful time-reversal symmetry [11,
12]. For nodal plane duos, a similar argument would result in two Kramers Weyl
points that might cancel, hence it is a priori unclear, whether nodal plane
duos may be nontrivial.

A topological nodal plane duo can, for example, occur due to the global
constraint in a time-reversal broken state. The simplest case is realized in
ferromagnetic MnSi with the magnetic space group 19.27
$P2_{1}^{\prime}2_{1}^{\prime}2_{1}$. While the planes $k_{x}=\pi$ and
$k_{y}=\pi$ exhibit a nodal plane duo, there remains only the two-fold
rotation axis through $\Gamma$ that is not part of nodal planes. On this axes
the global constraint takes the form

$\displaystyle\sum_{c_{b}}\Delta\varphi_{b,c_{b}}=-b\cdot\pi\mod 2\pi,$ (64)

which results in an odd number of crossings for bands with odd $b$. Since each
crossing exhibits a charge of $\nu=\pm 1$, cf. Eq. (1), there is an odd
overall charge within the Brillouin zone that cannot be compensated by generic
crossings of even multiplicity. Thus, the nodal plane duo is topological with
a charge of $\nu_{\text{NP}}=1\mod 2$, see [14].

Finally, we consider the nodal plane duo enforced by SG 94 $P4_{2}2_{1}2$ for
a spinful description with time-reversal symmetry. Again, the global
constraint gives a non-zero sum of phase jumps for odd $b$ along the fourfold
rotation axis Z-$\Gamma$-Z. Here, one needs the local constraint, because it
is insufficient to count the number of Weyl points, which may occur as single
and double Weyl points. One finds

$\displaystyle\nu_{\text{Z-}\Gamma\text{-Z}}\equiv\sum_{c_{b}}\nu_{c_{b}}=\sum_{c_{b}}\frac{n}{2\pi}\Delta\varphi_{b,c_{b}}=-2b\mod
4,$ (65)

where the global and local constraints, Eqs. (1) and (2), have been
substituted into the sum of all crossings on the fourfold rotation axis, and
$m=2$, $n=4$. Thus, for odd $b$ the Chern number of the nodal plane duo
$\nu_{\text{NP}}=-\nu_{\text{Z-}\Gamma\text{-Z}}\neq 0$ independently of the
details of the system.

To illustrate these results we have devised a Hamiltonian of SG 94, see
Appendix E. This model has minimal set of four connected bands with symmetry-
enforced hourglass band structures along $\Gamma$-Z and $\Gamma$-X, see Fig
5(a), and two nodal planes covering the surfaces defined by $k_{x}=\pi$ or
$k_{y}=\pi$, Fig 5(c). The chiralities of Weyl pointson the
$\Gamma$-Z-$\Gamma_{2}$ follow our local constraint Eq. (1), see Fig 5(b). As
predicted the nodal planes are topologically charged. For example, for the
lowest band the chiralities $\nu_{1,1}=\nu_{1,2}=-1$ at $\Gamma$ and Z,
respectively, add up and are also not compensated by charges at generic
positions. Thus, the lower nodal planes carries the opposite Chern number
$\nu_{1,\text{NP}}=+2$. This concludes our discussion of SG 94, which is the
only known space group that enforces a pair of enforced topological nodal
planes without magnetism.

Figure 5:  _Tight-binding model of SG 94._ (a) Band structure of the model
defined in Eq. (E). (b) Symmetry eigenvalues of the fourfold screw rotation
$\tilde{C}_{4}$ along the full rotation axis $\Gamma-Z-\Gamma_{2}$. The
chiralities $\nu$ are compensated by an opposite charge contained within the
nodal plane shaded in red in (a) and in the Brillouin zone (c). In (c) the
arrows show exemplarily the Berry curvature associated to the point crossings
at $\Gamma$ and Z.

####  IV.2.2 Global constraints for multi-fold degeneracies

While in the previous example the absence of a multi-fold crossing simplified
the exposition, we are going to consider in the following the opposite case,
where several multi-fold crossings occur and local and global constraints may
not directly substituted into each other. The salient difference is that at
multi-fold crossings the exchange of bands described by $\Delta\varphi$ may
occur with lower (or higher bands) that are not necessarily adjacent to the
considered band, i.e., not only bands $b-1$ (or $b+1$).

To illustrate the constraints with multi-fold crossings in a real band
structure we discuss the cubic compound BaAsPt (SG 198 $P2_{1}3$) exhibiting
an unusual multi-fold crossing point including a $\nu=5$ band. The material
will be closer examined in Sec. VI.1.

Here, our goal is to give some intuition on how the global constraint is
fulfilled, while respecting the Nielsen-Ninomyia theorem. For the latter one
may pick in principle any subset of bands to determine the relevant
chiralities of crossings by the non-Abelian generalization of the Chern number
for this set of bands. Here, we consider for simplicity only the band
$b^{\prime}$ that bounds the red shaded area in Fig. 9 from below.

Since we encounter multi-band crossings, e.g., at $\Gamma$, M, and R, we have
to use the general form of the global constraint introduced in Eq. (50). It
implies that each two-fold rotation axis should exhibit a total phase
$\sum_{c_{b^{\prime}}}\Delta\varphi_{b^{\prime},c_{b^{\prime}}}=\pi\mod 2\pi$,
whereas the symmorphic three-fold rotations require a phase change of
$\sum_{c_{b^{\prime}}}\Delta\varphi_{b^{\prime},c_{b^{\prime}}}=0\mod 2\pi$.
Note that in Eq. (50) the summations include crossings to lower and to higher
bands on the considered rotation axis.

One half of the threefold rotation axis $\Gamma$-R, cf.points Fig. 9, exhibits
three Weyl points to higher and three to lower bands together with multi-fold
crossings at $\Gamma$ and R (one of the latter is in close proximity to
$\Gamma$). Since between each two crossing on $\Gamma$-R there is a crossing
in the next lower gap, all crossings to the higher (lower) band have an
identical phase jump $\Delta\varphi$ and the numerical calculation determines
$\Delta\varphi=+2\pi/3$ ($-2\pi/3$). When taking the position relative to band
$b^{\prime}$ into account all five Weyl points thus contribute $\nu=+1$. As a
side remark, the lower crossing of charge $\nu=-1$ in the inset appears
together with three generic Weyl points with $\nu=+1$ in its close proximity.
Due to their close proximity, we have labeled the crossing on the $\Gamma$-R
axis with the total charge of the four crossings on at next to the rotation
axis, i.e., $\nu=+2$ for the band that bounds the red-shaded region from
below. While these generic crossings are not symmmetry-enforced similar
arrangements of crossings around a threefold rotation axis have been predicted
before in an analysis of CoSi, which has the same SG 198 [15]. For the full
threefold axis $\Gamma$-R there are 12 phase jumps adding up to a phase shift
of $\Delta\varphi=+24\pi/3=0\mod 2\pi$. The multi-fold crossings at $\Gamma$
and R exhibit $\Delta\varphi=\pm 2\pi/3$ such that in total the phase equals
$0\mod 2\pi$ on each threefold rotation axis. In total the band $b^{\prime}$
comprises thus 48 Weyl points of charge $\nu=+1$ on generic points of its
threefold rotation axes.

On the two-fold rotation axis along $\Gamma$-X there is one crossing
contributing $\Delta\varphi=\pi$. Since $\Gamma$ and X are time reversal
invariant and the twofold rotation eigenvalues are complex at $\Gamma$ and
real at X, it is clear prior to any calculation that the phase changes at
$\Gamma$ but not at X. Thus, a full twofold rotation axis X-$\Gamma$-X
exhibits an odd number of phases as expected. Overall there are 6 Weyl points
of $\nu=+1$ on the twofold axes through $\Gamma$. The two-fold bands on the
nodal planes along the R-M line exhibit two distinct representations that are
characterized by two-fold rotation eigenvalues like on $\Gamma$-X, thus also
here the global constraint applies. On R-M there is a crossing to a lower band
as well as a pinned crossing at M, both exhibit an exhange of bands, whereas
none occurs at R. Along R-M-R there is an odd number of crossings fulfilling
the global constraint Although we encounter chiral crossings on R-M, these do
not contribute an Abelian Chern number to the band $b^{\prime}$, because a
surface enclosing them is gapless due to the presence of the nodal planes. But
for the Nielsen-Ninomiya theorem applied to band $b^{\prime}$, one has to
consider the contribution to the nodal plane of $\nu=3$.

In summary, the chiral charges on the band that bounds the red-shaded region
from below are $6\nu_{b^{\prime},\Gamma\text{-X}}=6$,
$8\nu_{b^{\prime},\Gamma\text{-R}}=48$, as well as
$\nu_{b^{\prime},\Gamma}=-5$, $\nu_{b^{\prime},\text{R}}=-4$ and
$\nu_{b^{\prime},\text{NP},b^{\prime}}=+3$, which adds up to 48. Note, that
despite the relevance of the sixfold crossing at R to the band $b^{\prime}$,
one needs to use the non-Abelian Chern number calculation to determine the
charge contributed to the red-shaded gap, see Ref. [15] for the details of
such a calculation. By using the Nielsen-Ninomiya theorem for band
$b^{\prime}$ we can infer that there are at least two set of Weyl points at
generic positions. Indeed, by a closer inspection of the band structure we
find that there are additional Weyl points close to the $\Gamma$-R axis axes.
As mentioned before there are 24 Weyl points of charge $\nu=-1$ in the
vicinity R as well as another set of Weyl points with also $\nu=-1$ close to
$\Gamma$.

While we had to consider the charge of the nodal plane explicitly, in absence
of nodal planes it is possible to infer the existence of Weyl points at
generic positions based on symmetries alone, e.g., in a spinful representation
of SG 19 or the magnetic SG 19.27 for the movable fourfold double Weyl points
as noticed for a tight-binding model in Ref. [11, 14]. It is thus possible to
use the local and global constraints together with the Nielsen-Ninomiya
theorem to deduce the existence of Weyl points at generic positions within the
Brillouin zone.

##  V Generation and classification of low-energy Hamiltonians for the
multifold crossing case

As we have already seen in a previous section IV.1.2, combinations of
different symmetries, including time-reversal, can lead to surprising results.
Up until now we considered only Weyl points. So the next question is how the
non-abelian constraints affect multifold crossings in this regard. Here we do
not only want to restrict ourselves on just the evaluation of constraints, but
to explicitly calculate Chern numbers in all topological phases of all multi-
fold crossings, as the solution to constraints derived for the non-abelian
case (see section III.2) are not unique and larger Chern numbers than the
minimal ones fulfilling the given constraints can, due to the higher symmetry,
be no longer excluded. We can see these cases directly when such a topological
classification is carried out explicitly.

This complete topological classification of all multifold crossings in all
space groups follows a three phase approach. First all irreducible
representations (irreps) with dimensions higher than 2 were found at all high
symmetry points using the Bilbao Crystallographic Server [77]. Since we
included time-reversal symmetry in all of our analysis, the search can be
restricted to double space groups with broken inversion symmetry, since only
there topological charges are allowed to be nonzero in presence of time-
reversal symmetry. Then, low energy Hamiltonians were generated for all irreps
found in the last step, such that these Hamiltonians respect all symmetries at
the given high symmetry points. Finally, the whole parameter space of these
Hamiltonian are topologically classified.

We note that there is an alternative approach for generating low-energy
Hamiltonians than the one shown in this section based on [78], where all
possible Hamiltonian terms are tabulated. We used the method described here,
since we found it more convenient to lookup a small number of symmetry
generators and their representations instead of all possible Hamiltonian
terms. See also [79, 80, 81] for more alternative algorithms.

###  V.1 Automatic generation of low-energy Hamiltonians from irreps

A general low energy Hamiltonian up to second order in wave-vector $k$ has the
following form

$\displaystyle H_{nm}({\bf k})$
$\displaystyle=\sum_{hp_{1}}\alpha_{h,1}H^{1}_{hp_{1}nm}k_{p_{1}}$
$\displaystyle\quad+\sum_{hp_{1}p_{2}}\alpha_{h,2}H^{2}_{hp_{1}p_{2}nm}k_{p_{1}}k_{p_{2}}+\mathcal{O}(k^{3}).$
(66)

$n$ and $m$ enumerate the orbital degrees of freedom. $\alpha_{h,q}$ are the
free parameters of $H$ at order $q$. $H^{q}_{hp_{1}...p_{q}nm}$ are the
linearly independent terms in $H$. The goal of the following algorithm is to
compute these terms.

The starting point of the automatic generation are symmetry generators and the
corresponding irrep at a given high symmetry point. With these generators we
build up the whole little group $G$ at his high symmetry point and the
representation $D(g)$ of those symmetries $g\in G$. The only constraint of a
low energy Hamiltonian at this point must be that it is symmetric

$\displaystyle\forall g\in G:H(g{\bf k})$ $\displaystyle=D(g)H({\bf
k})D(g^{-1}).$ (67)

We can symmetrize the Hamiltonian in Eq. (66) via

$\displaystyle\tilde{H}^{1}_{hv_{1}ij}$ $\displaystyle=\frac{1}{|G|}\sum_{g\in
G}g^{-1}_{vp_{1}}D(g)_{in}H^{1}_{hp_{1}nm}D(g^{-1})_{mj}$ (68)
$\displaystyle\tilde{H}^{2}_{hv_{1}v_{2}ij}$
$\displaystyle=\frac{1}{|G|}\sum_{g\in
G}g^{-1}_{v_{1}p_{1}}g^{-1}_{v_{2}p_{2}}D(g)_{in}H^{2}_{hp_{1}p_{2}nm}D(g^{-1})_{mj}$
(69) $\displaystyle...$

where $g_{ab}$ is the real space representation of $g$ and Einstein notation
was used. Then $\tilde{H}^{q}_{hp_{1}...p_{q}nm}$ are the new terms of a
symmetric $H$. Note that $D(g)$ can be anti-unitary, which is the case when
$g$ is for example the time-reversal symmetry. In this case,
$D(g)=U\mathcal{K}$ with $U$ unitary and $\mathcal{K}$ being the complex
conjugation operator. The latter one can be eliminated by commuting it through
all term in Eq. 68 and 69 until we can use $\mathcal{K}^{2}=1$.

The algorithm starts by generating a set of random complex
$H^{q}_{hp_{1}...p_{q}nm}$ terms, with $h\in\\{1,...,N\\}$ and $N$ being the
total amount of randomly generated terms. These are then symmetrized via 68 to
produce $N$ symmetrized $\tilde{H}^{q}_{hp_{1}...p_{q}nm}$ terms. Only the
linearly independent terms are kept, which is done using a Gram-Schmidt
orthogonalization, during which the terms are treated as vectors by flattening
them to a single index. This also reduces the number of terms $\tilde{N}\leq
N$ to the maximal set of symmetric and linearly independent terms. The number
of free parameters of this Hamiltonian at order $q$ is also $\tilde{N}$.

For better handling of these terms, we would like to normalize the real or
imaginary part of as many of their entries to 1, since they are still filled
with random numerical values of arbitrary magnitude. We can not normalize all
entries to 1, since not all are linearly independent. This normalization is
done by first gathering all nonzero columns in
$\text{Re}(\tilde{H}^{q}_{hp_{1}...p_{q}nm})$ and
$\text{Im}(\tilde{H}^{q}_{hp_{1}...p_{q}nm})$ in a new matrix $M_{hr}$ with
size $(\tilde{N},P)$, with $P$ the number of nonzero columns. Rows that are
linearly dependent on other rows are removed in $M$, such that $M$ is
quadratic and invertible. The final terms of $H$ are then computed with

$\displaystyle H^{\prime
q}_{h^{\prime}p_{1}...p_{q}nm}=\sum_{h}M^{-1}_{h^{\prime}h}\tilde{H}^{q}_{hp_{1}...p_{q}nm}.$
(70)

Due to the inversion of $M$, the real or imaginary part of all nonzero entries
in $\tilde{H}$ which are chosen to build up the $M$ matrix are normalized to 1
in only one of the terms while they are set to 0 in all other. Entries that
are not part of the final $M$ are either a fraction or a fraction consisting
of square roots. The last step of the algorithm is to convert the numerical
values of $H^{\prime}$ into analytical expressions by comparing the entries to
the values of those analytical expressions and also to project $H^{\prime}$ to
Pauli or Gellmann matrices. To test if this conversion worked, the symmetry of
the resulting Hamiltonian is checked.

###  V.2 Classification of all multifold crossings at high-symmetry points

Using the algorithm described in the previous section, a Hamiltonian for each
entry in the compiled list of all irreps with dimension $>2$ was generated.
Since we only want to study the topological charge of the crossing at the high
symmetry point in question, it is sufficient, with only one exception as we
will see later, to generate only the terms up to linear order in $k$, since
higher orders could only produce additional crossings away from the high
symmetry point and do not alter the topological charge of the multifold
crossing. Some of the generated Hamiltonian are equivalent or equivalent up to
a transformation, so these cases can be grouped and classified together. The
transformations either have no effect on or flip the topological charge.

The determination of every bands topological phase diagrams of the
Hamiltonians all follow the same idea of first finding all points in parameter
space where the current band in question of the given Hamiltonian becomes
gapless. These are the only points where topological phase transitions can
happen, i.e. the topological charge of the multifold crossing can change.
These points make up subspaces in parameter space which separate different
topological phases, which were found by considering the characteristic
polynomial of $H$ and comparing it to a characteristic polynomial describing a
Hamiltonian in a gapless phases.

So after finding these subspaces it is possible to determine the topological
charges of every phase by evaluating it numerically deep in a given phase.
This way one can color in the whole phase by the determined topological
charge. Since no other topological phases are possible, we can enumerate all
possible topological charges for all multifold crossings.

During this topological classification, the Chern number of single bands is
sometimes undefined. This happens due to band degeneracies, for example nodal
planes, which by symmetry persist to all orders in ${\bf k}$. In most of these
cases, one can still define a non-abelian Chern number, see Eq. (12). In the
case of 4-fold crossings on nodal planes, we compute non-abelian Chern numbers
$\nu_{b,b+1}$, where bands $b$ and $b+1$ are part of the nodal plane.

####  V.2.1 4-fold crossings

The main results for all 4-fold crossings are summarized in tables 1 and 2.
The topological charge of the lowest band $\nu_{1}$ is undefined in most
irreps, since there the lowest two bands can be shown to be always twofold
degenerate at some k-points away from $0$ at all orders in ${\bf k}$ due to
symmetry constraints. Where this is not the case, an unusually high Chern
number of $\pm 5$ can be observed.

As this result is quite unexpected, we explicitly show the topological phase
diagram and its derivation of one of the two Hamiltonians, the model for the
$\bar{\Gamma}_{6}\bar{\Gamma}_{7}$ irrep, that describe these cases. This
irrep can be found in SG 195-199. The little group contains $C_{2}^{x,y,z}$,
$C_{3}^{(\pm 1,\pm 1,\pm 1)}$ and time-reversal symmetry. In the following,
all used representations are equivalent to the ones on the Bilbao
Crystallographic Server [77]. The Hamiltonian generated by the algorithm
described in the previous section is

$\displaystyle H=$
$\displaystyle\alpha_{0}\biggl{[}2k_{x}\sigma_{x}\tau_{z}+k_{y}\left(-\sqrt{3}\sigma_{x}\tau_{0}-\sigma_{y}\tau_{0}\right)+$
$\displaystyle
k_{z}\left(\sigma_{x}\tau_{x}+\sqrt{3}\sigma_{y}\tau_{x}\right)\biggr{]}+$
$\displaystyle\alpha_{1}\biggl{[}-2k_{x}\sigma_{y}\tau_{z}+k_{y}\left(-\sigma_{x}\tau_{0}+\sqrt{3}\sigma_{y}\tau_{0}\right)+$
$\displaystyle
k_{z}\left(\sqrt{3}\sigma_{x}\tau_{x}-\sigma_{y}\tau_{x}\right)\biggr{]}+$
$\displaystyle
2\alpha_{2}\biggl{[}k_{x}\sigma_{z}\tau_{x}+k_{y}\sigma_{0}\tau_{y}+k_{z}\sigma_{z}\tau_{z}\biggr{]}$
(71)

with $\tau$ and $\sigma$ being Pauli matrices. It is possible to show (see
Appendix F.1) that the Hamiltonian is only gapless for
$\alpha_{2}=\pm\sqrt{\alpha_{0}^{2}+\alpha_{1}^{2}}$, $\alpha_{2}=0$ or
$\sqrt{\alpha_{0}^{2}+\alpha_{1}^{2}}=0$ at points away from $k=(0,0,0)$. We
can assign the spaces in between gapless planes in parameter space
$\alpha_{n}$ with precomputed Chern numbers to arrive at the topological phase
diagram of the $\bar{\Gamma}_{6}\bar{\Gamma}_{7}$ irrep model. See figure 6
for the phase diagram for band 2. We find that band 1 has two phases, for
$\alpha_{2}<0$ the Chern number is $\nu_{1}=-3$, for $\alpha_{2}>0$ it is
$\nu_{1}=3$. For bands 3 and 4 use $\nu_{3}=-\nu_{2}$ and $\nu_{4}=-\nu_{1}$.

Figure 6: Topological phase diagram of the 4-fold crossing described by
Hamiltonian 71 with three parameters $\alpha_{0},\alpha_{1}$ and $\alpha_{2}$.
Depicted is the Chern number of band 2 $\nu_{2}$. An unusually high absolute
Chern number of $5$ can be found in regions where
$\alpha_{2}>\sqrt{\alpha_{0}^{2}+\alpha_{1}^{2}}$ or
$\alpha_{2}<-\sqrt{\alpha_{0}^{2}+\alpha_{1}^{2}}$.

In the appendix of [68], a symmetry equivalent Hamiltonian to Eq. (71) has
been derived, although there the whole topological phase diagram has not been
mapped out. Previously [17], this 4-fold crossing has been described by a
Rarita-Schwinger-Weyl spin-3/2 Hamiltonian [82, 83, 84] $H({\bf k})=\alpha{\bf
S}\cdot{\bf k}$, which only supports the $\nu_{n}=\\{\pm 3,\pm 1,\mp 1,\mp
3\\}$ phase, or a distinction was made [85] but only this phase was
considered. Here we see that this description is incomplete. To our knowledge,
the $\nu_{n}=\\{\pm 3,\mp 5,\pm 5,\mp 3\\}$ topological phase has not been
observed yet.

We find no 3-fold symmetry eigenvalue phase jumps for the lowest/highest band.
A phase jump of $2\pi/3$ for $\alpha_{2}<0$ and of $4\pi/3$ for $\alpha_{2}>0$
was observed for band 2. For all bands, a phase jump of $\pi$ was found for
both 2-fold symmetries, which constraints all Chern numbers to be odd
$\nu_{n}=1\,\text{mod }2$. Further, the 3-fold symmetry constraints the Chern
number of the lowest and highest band to $\nu_{1/4}=0\,\text{mod }3$, which is
consistent with $\nu_{1/4}=\pm 3$. Then, for $\alpha_{2}<0$ we have the
constraint $\nu_{2/3}=1\,\text{mod }3$ and for $\alpha_{2}>0$ we have
$\nu_{2/3}=2\,\text{mod }3$, which are fulfilled in all phases in figure 6.

The transition between $\nu_{2}=\pm 1$ and $\nu_{2}=\mp 5$ phases is
facilitated (see Appendix F.1) by a gap closing of the middle two bands. This
suggests, that the mechanism behind this Chern number switch is an
absorption/emission of 6 Weyl points on the $C_{2}$ invariant axes into/out of
the multifold crossing. This implies an exchange of $C_{2}$ symmetry
eigenvalues of bands $2$ and $3$ between these two phases, which we confirmed
by a direct calculation.

The phase transition $\nu_{n}$ to $-\nu_{n}$ at $\alpha_{2}=0$ takes place by
a simultaneous gap closing on $C_{2}$ invariant lines of the outer band pairs
$(1,2)$ and $(3,4)$ as well as on $C_{3}$ invariant lines of band pair
$(2,3)$. Since corresponding symmetry eigenvalues switch on both invariant
lines, 6 WPs on the outer band pairs and 8 WPs on the middle bandpair fuse
with or emerge from the 4-fold crossing. The WPs on $C_{2}$ invariant lines
with total charge $\pm 6$ switch the sign of the lowest/highest Chern number
$\nu_{1/4}=\mp 3\to\pm 3$. For the middle two bands a combined total Chern
number of $\mp 6\pm 8=\pm 2$ switches the sign of the middle Chern numbers
$\nu_{2/3}=\mp 1\to\pm 1$.

Figure 7: Starting from a singe 4-fold crossing described by the
$\bar{\Gamma}_{6}\bar{\Gamma}_{7}$ irrep model (see Eq. 71) in the $\nu_{2}=5$
phase, a phase transition to $\nu_{2}=1$ over $\nu_{2}=-1$ produces a total of
$26$ WPs across the 3 bandpairs. WP and multifold point charges are color-
coded.

This means, consecutive topological phase transitions over
$\nu_{2}=5\rightarrow-1\rightarrow 1$ produce a total of $26$ WPs distributed
across the 3 bandpairs, $6$ in the lower and upper bandpair respectively and
$6+8=14$ in the middle bandpair, provided there are no other crossings at the
start in the $\nu_{2}=5$ phase, since these could also be merged into the
multi-fold point to carry out the phase transition. This process is visualised
in figure 7.

Band 1  $\nu_{2}=-1$ $\nu_{2}=5$ Band 2

Figure 8:  Spin texture from the expectation value of $\sigma_{a}\tau_{0}$ for
the 4-fold crossing described by Eq. (71) for bands 1 and 2 in the
$\nu_{2}=-1$ and $\nu_{2}=5$ phase. The arrow size and color depend on the
magnitude of the spin, red being maximal.

In figure 8 the spin texture of this 4-fold crossing in the $\nu_{2}=-1$ and
$\nu_{2}=5$ phase are compared for bands 1 and 2. For this, the spin
expectation value $\langle{\bm{\sigma}}\tau_{0}\rangle$ is used, since the
time-reversal irrep $-i\sigma_{y}\tau_{0}$ suggests that ${\bm{\sigma}}$ is
the spin degree of freedom of this crossing. The parameters chosen for the
$\nu_{2}=-1$ are $\alpha_{0}=0.1$, $\alpha_{1}=0$ and $\alpha_{1}=1$, while
for the $\nu_{2}=5$ phase the parameters are $\alpha_{0}=1$, $\alpha_{1}=0$
and $\alpha_{1}=0.1$. We note that the spin texture is not symmetric under the
little group at this crossing, since SOC mixes the spin and orbital degrees of
freedom in the irrep $\bar{\Gamma}_{6}\bar{\Gamma}_{7}$. The spin texture
differences especially in the first band might be measurable in a spin-
resolved ARPES experiment [65, 67]. In this way, these two topological phases
can be distinguished.

####  V.2.2 3 and 6-fold crossings

The Hamiltonians for 3 and 6-fold crossings found by the procedure described
above reproduce the ones listed in [16]. We also find the Hamiltonians for all
3-fold crossings

$\displaystyle
H=\begin{pmatrix}0&k_{z}(\alpha-i\beta)&-k_{y}(\alpha+i\beta)\\\
&0&k_{x}(\alpha-i\beta)\\\ \ldots&&0\\\ \end{pmatrix}$ (75)

to be equivalent up to transformations. Since these transformations and the
explicit dependence of the topological charge were left out of [16], we
included them here in the tables 3 and 4 and in the Appendix G.

The 6-fold crossing Hamiltonian of SG 198 $\bar{R}_{7}\bar{R}_{7}$ irrep is
equal to

$\displaystyle H=$
$\displaystyle\alpha_{0}\biggl{[}k_{x}\sigma_{z}\lambda_{7}-k_{y}\sigma_{z}\lambda_{6}+k_{z}\sigma_{z}\lambda_{3}\biggr{]}+$
$\displaystyle\alpha_{1}\biggl{[}k_{x}\sigma_{0}\lambda_{5}+k_{y}\sigma_{0}\lambda_{2}+k_{z}\sigma_{0}\lambda_{1}\biggr{]}+$
$\displaystyle\alpha_{2}\biggl{[}k_{x}\sigma_{x}\lambda_{7}-k_{y}\sigma_{x}\lambda_{6}+k_{z}\sigma_{x}\lambda_{3}\biggr{]}+$
$\displaystyle\alpha_{3}\biggl{[}k_{x}\sigma_{y}\lambda_{7}-k_{y}\sigma_{y}\lambda_{6}+k_{z}\sigma_{y}\lambda_{3}\biggr{]}$
(76)

with $\sigma_{n}$ and $\lambda_{n}$ being Pauli and Gellmann matrices (see the
Appendix I for a definition). This Hamiltonian is equivalent up to a unitary
transformation to the one found in [16], with
$e^{i\phi}=\alpha_{0}+i\alpha_{1}$ and $b=\alpha_{2}+i\alpha_{3}$. There it
was also shown that you can arrive at the Hamiltonian for the SG 212 and 213
$\bar{R}_{7}\bar{R}_{8}$ irrep by setting $\phi=\frac{\pi}{2}$. Due to nodal
planes crossing these points, Chern numbers for odd fillings can not be
defined. The non-abelian Chern number for the middle two bands $\nu_{34}=0$
remains trivial, while the Chern number for the remaining bands are
$\nu_{12}=-\nu_{56}=\pm 4$. The exact phase diagram and its derivation can be
found in the Appendix H.

SG | Irrep | $\nu_{1}$ | $\nu_{2}$ | $\nu_{12}$ | Model | Transformation
---|---|---|---|---|---|---
19 | $R_{1}R_{1}$ | - | - | $\pm 2$ | F.9 | -
92 | $A_{1}A_{2}$ | - | - | $\pm 2$ | F.7 |  $k_{x}\to k_{y}$,$k_{y}\to-k_{x}$,$\alpha_{2}=0$,$\alpha_{3}=0$
96 | $A_{1}A_{2}$ | - | - | $\pm 2$ | F.7 |  $k_{y}\to-k_{y}$,$\alpha_{2}=0$,$\alpha_{3}=0$
198 |  $R_{1}R_{3}$/$R_{2}R_{2}$ | - | - | $\pm 2$ | F.5 | -
212/213 | $R_{1}R_{2}$ | - | - | $\pm 2$ | F.5 | -
212/213 | $R_{3}$ | - | - | $\pm 2$ | F.6 | -
Table 1: All possible topological charges of all 4-fold crossings in the spinless case (without SOC). Their Hamiltonians are either listed under the Model column or are obtained after a possible transformation applied on the given Hamiltonian.  SG | Irrep | $\nu_{1}$ | $\nu_{2}$ | $\nu_{12}$ | Model | Transformation
---|---|---|---|---|---|---
18 |  $\bar{S}_{5}\bar{S}_{5}$/$\bar{R}_{5}\bar{R}_{5}$ | - | - | $\pm 2$ | F.3 | -
19 | $\bar{S}_{5}\bar{S}_{5}$ | - | - | $\pm 2$ | F.3 | -
19 | $\bar{T}_{5}\bar{T}_{5}$ | - | - | $\pm 2$ | F.3 | $k_{x}\to k_{y},k_{y}\to k_{z},k_{z}\to k_{x}$
19 | $\bar{U}_{5}\bar{U}_{5}$ | - | - | $\pm 2$ | F.3 | $k_{y}\to k_{z},k_{z}\to k_{y}$
90 |  $\bar{A}_{6}\bar{A}_{7}$/$\bar{M}_{6}\bar{M}_{7}$ | - | - | $\pm 2$ | F.7 | -
92/94/96 | $\bar{M}_{6}\bar{M}_{7}$ | - | - | $\pm 2$ | F.7 | -
92/96 | $\bar{R}_{5}\bar{R}_{5}$ | - | - | $\pm 2$ | F.3 | $k_{x}\to k_{y},k_{y}\to k_{z},k_{z}\to k_{x}$
92/96 | $\bar{A}_{7}\bar{A}_{7}$ | - | - | $\pm 4$ | F.8 | -
94 | $\bar{A}_{6}\bar{A}_{7}$ | - | - | $\pm 2$ | F.7 | $k_{x}\leftrightarrow k_{y}$
195/196/197/198/199 | $\bar{\Gamma}_{6}\bar{\Gamma}_{7}$ | $\pm 3$ |  $\pm 1$,$\mp 5$ |  $\pm 4$,$\pm 2$ | V.2.1 | -
195 | $\bar{R}_{6}\bar{R}_{7}$ | $\pm 3$ |  $\pm 1$,$\mp 5$ |  $\pm 4$,$\pm 2$ | V.2.1 | -
197 | $\bar{H}_{6}\bar{H}_{7}$ | $\pm 3$ |  $\pm 1$,$\mp 5$ |  $\pm 4$,$\pm 2$ | V.2.1 | -
198 | $\bar{M}_{5}\bar{M}_{5}$ | - | - | $\pm 2$ | F.3 | -
199 | $\bar{H}_{6}\bar{H}_{7}$ | $\pm 3$ |  $\pm 1$,$\mp 5$ |  $\pm 4$,$\pm 2$ | V.2.1 | $U=\sigma_{z}\tau_{x}$
207/208/209/210 | $\bar{\Gamma}_{8}$ | $\pm 3$ |  $\pm 1$,$\mp 5$ |  $\pm 4$,$\pm 2$ | F.2 | -
/211/212/213/214 |  |  |  |  |  |
207/208 | $\bar{R}_{8}$ | $\pm 3$ |  $\pm 1$,$\mp 5$ |  $\pm 4$,$\pm 2$ | F.2 | -
211 | $\bar{H}_{8}$ | $\pm 3$ |  $\pm 1$,$\mp 5$ |  $\pm 4$,$\pm 2$ | F.2 | -
212 | $\bar{M}_{6}\bar{M}_{7}$ | - | - | $\pm 2$ | F.4 | -
213 | $\bar{M}_{6}\bar{M}_{7}$ | - | - | $\pm 2$ | F.4 | $k_{y}\to-k_{y}$
214 | $\bar{H}_{8}$ | $\pm 3$ |  $\pm 1$,$\mp 5$ |  $\pm 4$,$\pm 2$ | F.2 |  $U=\sigma_{0}\tau_{x}$,$\alpha_{1}\to-\alpha_{1}$
Table 2: All possible topological charges of 4-fold crossings in all spinful SGs and their Hamiltonians.  SG | Irrep | Transformation
---|---|---
195…199 | $\Gamma_{4}$ |  $\beta\to-\beta$,$k_{x}\to k_{z}$,$k_{y}\to-k_{x}$,$k_{z}\to k_{y}$
195 | $R_{4}$ |  $\beta\to-\beta$,$k_{x}\to k_{z}$,$k_{y}\to-k_{x}$,$k_{z}\to k_{y}$
197 |  $H_{4}$, $P_{4}$ |  $\beta\to-\beta$,$k_{x}\to k_{z}$,$k_{y}\to-k_{x}$,$k_{z}\to k_{y}$
199 | $H_{4}$ |  $\beta\to-\beta$,$k_{y}\to-k_{y}$
207…214 |  $\Gamma_{4}$, $\Gamma_{5}$ |  $\alpha=0$,$\beta\to-\beta$,$k_{x}\to k_{z}$,$k_{y}\to-k_{x}$,$k_{z}\to k_{y}$
207,208 |  $R_{4}$, $R_{5}$ |  $\alpha=0$,$\beta\to-\beta$,$k_{x}\to k_{z}$,$k_{y}\to-k_{x}$,$k_{z}\to k_{y}$
211,214 |  $H_{4}$, $H_{5}$ |  $\alpha=0$,$\beta\to-\beta$,$k_{x}\to k_{z}$,$k_{y}\to-k_{x}$,$k_{z}\to k_{y}$
Table 3: All 3-fold crossings without SOC and corresponding transformations, which generate their Hamiltonian from Eq. (75).  SG | Irrep | Transformation
---|---|---
199,214 | $\bar{P}_{7}$ | -
Table 4: All spinful 3-fold crossings. The Hamiltonian is described in Eq.
(75).

##  VI Materials

Here we discuss two material examples. Details on the calculations can be
found in the Appendix K.

###  VI.1 BaAsPt and related compounds (SG 198)

Figure 9: Bandstructure of BaAsPt and all band crossings on high symmetry
lines. The bandgap between the bands with a Chern number of $\pm 5$ at the
4-fold $\Gamma$ crossings is indicated by the pink region. This topological
bandgap is responsible for the Fermi arcs in figure 10. The blue dashed line
is the energy slice taken there. We also observe NPs with Chern numbers up to
$21$ at energies below $E_{f}$. Figure 10: Surface density of states (DOS) of
BaAsPt at $0.1\,$eV. Also shown are the projected topological charges,
resulting from a bandfilling in between the bandpair with charge $\nu=\pm 5$
at $\Gamma$ from the bulk. This is the bandfilling fulfilled in the parts of
the surface DOS without bulk states and where the Fermi arcs reside. With this
filling, the charge of the $\Gamma$ point is $3-5=-2$, which is compensated by
two WPs on the $\Gamma X$-line. Further, there are 8 WPs on the $\Gamma
R$-line very close to the $\Gamma$ point which we included into the total
charge of $8$ of the projected $\bar{\Gamma}$ point. The total charge of bulk
states around $\bar{\Gamma}$ is $4$, so there are 4 Fermi arcs connecting
these states to the $\bar{R}$ point with topological charge $\nu=-4$. Note
that the $\nu=2$ charges near $\bar{R}$ are slightly away from $\bar{R}$, so
there are 4 copies at every corner. These charges are connected with 8 Fermi
arcs to bulk states with $\nu=-2$ points between $\bar{\Gamma}$ and $\bar{R}$.

A material search for a 4-fold crossing with a Chern number of $5$
sufficiently close to the Fermi energy was done in space groups 195-199 and
207-214. First materials from the materials project [86] are screened for
4-fold crossings near the Fermi energy. The Chern number of this point was
directly computed [15] using density-functional theory, in particular Quantum
Espresso [87]. This search was stopped at the first material found, which was
BaAsPt in SG 198. There a 4-fold point with $\nu=\pm 5$ was found at $\Gamma$
at $-100\,$meV, see figure 9. We note that BaAsPt belongs to a whole class of
materials in SG 198, referred to as LaIrSi-type materials in [68], consisting
of three elements with similar bandstructures, as seen on the materials
project [86], and likely similar orbital characteristics, such that $\nu=\pm
5$ might also be found in those, though at different distances to $E_{F}$, due
to variations of the Fermi energy in these compounds.

A full topological classification [15] of the 6 bands crossing $E_{F}$ has
been carried out. We enumerate these bands from 1 to 6 in descending order in
energy. The charge of the nodal planes, which occur in SG 198 at the BZ
boundary, is shown in figure 9 in solid colored lines. The figure also shows
all crossings on high symmetry lines. WPs at generic positions have been found
for band pair $(4,5)$ (the band pair with $\nu=\pm 5$ at $\Gamma$ and whose
bandgap is shaded red) at ${\bf k}=2\pi(-0.0289,-0.2699,-0.2988)^{T}$ and all
symmetry related points with $\nu_{5}=-1$. Another generic WP was found at
${\bf k}=2\pi(0,-0.2085,-0.291)^{T}$ with $\nu_{4}=-1$ for band pair $(3,4)$.
The fermion doubling theorem is fulfilled when counting in the topological
charges of the WPs, multifold crossings and NPs found by the full topological
classification of the 6 bands crossing $E_{F}$.

Further, a large topological band gap shaded in red separating the two bands
with $\nu=\pm 5$ can be seen. A surface DOS calculation at $100\,$meV shows a
large number of Fermi arcs, see figure 10, despite a screening of the
topological charge from the 4-fold point, which due to the filling of this
topological band gap is $-3+5=-2$, by charges on $\Gamma-X$, which sum up to
$2$. 4 copies of these WPs appear on the projection $\bar{\Gamma}-\bar{M}$.
Very close to the $\Gamma$ point on the $\Gamma-R$ line there are 8 WPs, which
we included into the charge of $\bar{\Gamma}$. The total charge of the bulk
bands surrounding $\bar{\Gamma}$ of 4 give rise to 4 Fermi arcs emerging from
the bulk states at $\bar{\Gamma}$ and running to the $\bar{R}$ point with
charge $-4$. The remaining Fermi arcs are entirely explained by projected
topological crossings of the band pair $(4,5)$, namely a small pocket between
$\bar{\Gamma}$ and $\bar{R}$ with containing a charge of $-2$ connecting via 2
Fermi arcs to bulk bands with a charge of $2$ near $\bar{R}$. In total, we are
counting 12 Fermi arcs.

###  VI.2 NbO2 and TaO2 (SG 80)

Figure 11: a) DFT bulk band structure of the distorted TaO2 compound with the
topological band gap colored in red and an arrow pointing to the Weyl point
(WP). The inset is showing a zoom of the double Weyl point. b) Surface
spectral density at $-0.297$ eV (blue dashed line in a)) with a termination
projecting $M$ onto $\Gamma$ ($\bar{\Gamma}$) and $P$ onto $N$ ($\bar{M}$) as
shown in the inset. Two Fermi arcs connect the projection of the Weyl point at
$P$ with the bulk bands.

Niobium dioxide was first synthesized in 1941 and was shown to crystallize in
a rutile structure with tetragonal space group symmetry 136[88]. Further
research revealed the existence of a distorted lower-symmetry phase
$\beta$-NbO2[89]. During the structural transition, pairs of niobium atoms
dimerize along the c-axis, and although the nature of the transition is
believed to be of Peierls type, the specifics have been the subject of an
extensive amount of research throughout the years[90, 91, 92, 93, 94].
Slightly sub-stochiometric single crystals of $\beta$-NbO2 can be synthesized
in oxygen-deficient environments, and its crystal structure has space group
symmetry 80[95]. Much later, $\beta$-NbO2 was proposed as a potential
realization of a topological chiral crystal with Kramers-Weyl fermions in its
bulk and the corresponding boundary modes on its surface[11].

Since the topological band gap in $\beta$-NbO2 is small and the crossing of
interest is overshadowed by spectral weight of other bands in its vicinity, we
propose two alterations to the compound to improve its usefulness as a
topological semimetal. First, to increase the effect of spin-orbit coupling,
we consider the hypothetical compound $\beta$-TaO2, which is expected to have
the same crystal structure since tantalum and niobium have very similar ionic
radii and electron configurations[96]. Second, we enhance the distortion mode
that connects the rutile and the reported lower-symmetry phase of NbO2. To do
this, we compare the crystal structures of the parent and the distorted
compound, calculate the irreducible representations of the distortions and
identify the linear combination of modes that connects the two configurations
using the ISODISTORT tool[97]. The computed distortion is then exaggerated by
a factor of 1.5, retaining the space group symmetry of $\beta$-NbO2. Potential
routes to synthesize the proposed crystal include growing it at higher
temperatures or in a more oxygen-deficient environment[95].

The band structure and the surface states of $\beta$-TaO2 are shown in Fig.
11. In the vicinity of the Fermi energy there are two time-reversal-related
double Weyl points protected by fourfold rotation symmetry, one of which is
seen on the line $\Gamma$-M, as well as two double Weyl points pinned to the
points labeled P. Our calculation shows that Weyl points on $\Gamma$-M with
charge $\nu=+2$ compensate the ones at P with $\nu=-2$. To our knowledge this
is the first example, where double Weyl points are enforced away from a TRIM
but pinned to a lower symmetry point. These doubly charged degeneracies on a
two-fold rotation axis contradict previous suggestions that double Weyl points
require four- or six-fold rotational symmetry[46], and can only be understood
from our argument relating symmetry eigenvalues.

##  VII Conclusion

In this paper we have derived two fundamental laws of chiral band crossings: A
local constraint relating the Chern number to phase jumps of rotation
eigenvalues (Sec. II.1), and a global constraint that restricts the number of
chiral crossings on rotation axes (Sec. II.2). To demonstrate the strength of
these laws, we have applied them to determine the existence of enforce double
Weyl points, nodal planes, and other band topologies (Sec. IV). Complementing
these arguments by an exhaustive classification of low-energy models, we have
determined the generic topological phase diagrams of all multifold crossings
(Sec. V). Our analysis reveals, among others, that there are fourfold crossing
points with Chern number 5 (Sec. V.2.1). To illustrate some of the derived
topological band features, we have discussed two material examples (Sec. VI):
BaAsPt in SG 198 with fourfold crossing of Chern number 5 and NbO2/TaO2 in SG
80 with double Weyl points.

There are several directions for future work. First, the local and global
constraints can be applied in a straightforward manner to magnetic space
groups. For example, the local constraint can be used to infer the existence
of double Weyl points away from TRIMs in magnetic space groups, similar to
Sec. IV.1.1. Second, our fundamental laws can be employed to study (multifold)
nodal points and nodal planes of bosonic band structures, e.g., phonon or
magnon bands. Third, our results have implications for topological response
functions that are influenced by the Berry curvature, e.g., anomalous Hall
currents, photogalvanic effects, and magnetooptic Kerr effects. Working out
signatures of the discussed band topologies (e.g., the nodal planes or the
fourfold crossings with $\nu=5$) in these response functions would be an
interesting task for future study.

## Acknowledgements

We are thankful to Douglas Fabini, and Johannes Mitscherling for enlightening
discussions. We acknowledges the support by the Max Planck-UBC-UTokyo Center
for Quantum Materials.

##  APPENDIX A Proof of Eq. (III.1)

Since we chose a different sign in the definition of the Berry connection (see
Eq. (14)) as compared to [52], we here present the derivation of Eq. (III.1)
(Eq. (83) in [52]).

$\displaystyle{\bf A}(R\cdot{\bf k})=i\innerproduct{n(R\cdot{\bf
k})}{R\cdot\gradient_{{\bf k}}|m(R\cdot{\bf k})}$
$\displaystyle=iR\sum_{c,d}\mathcal{B}_{C_{n}}({\bf
k})_{nc}\innerproduct{c({\bf k})}{\hat{R}^{-1}\gradient_{{\bf
k}}\mathcal{B}_{C_{n}}^{*}({\bf k})_{md}\hat{R}\biggl{|}d({\bf k})}$
$\displaystyle=iR\sum_{c,d}\mathcal{B}_{C_{n}}({\bf
k})_{nc}\innerproduct{c({\bf k})}{\gradient_{{\bf
k}}\mathcal{B}_{C_{n}}^{*}({\bf k})_{md}\biggl{|}d({\bf k})}$
$\displaystyle=R(\mathcal{B}_{C_{n}}{\bf A}({\bf
k})\mathcal{B}_{C_{n}}^{-1}+i\mathcal{B}_{C_{n}}\gradient_{{\bf
k}}\mathcal{B}_{C_{n}}^{-1}).$ (116)

##  APPENDIX B Chiral nodal lines from magnetic symmetries

In Sec. IV.1.3 we discussed the possibility of a nodal line characterized by a
non-zero Chern number. In the following we generalize this discussion to
symmetries, which comprise both time reversal $T$ and an $n$-fold rotation
$C_{n}$ around the z direction. The arguments excluding the possibility of
chiral nodal lines for $C_{4}T$ and $C_{6}T$ follow from the constraints in
Eqs. (31) and (32) in the same way as in the main text.

If there is a nonzero sewing matrix phase difference $\Delta\phi_{c_{b}}$, the
relations imply a point-like band crossing on the axis. Since this implies
that the only possible chiral charges of the line are equal to the
multiplicity of Weyl points, such lines would be unstable. To see this,
consider a case where $\Delta\phi_{c_{b}}$ is nonzero and does not change when
the size of the sphere surrounding the $C_{n}T$ invariant point shrinks to
zero. $\Delta\phi_{c_{b}}\neq 0$ implies with the local constraints
$\nu_{c_{b}}\neq 0$. If $\nu_{c_{b}}$ is nonzero, only a point-like crossing
can carry the charge implied by the arbitrary small integration sphere.

Alternatively, if $\Delta\phi_{c_{b}}$ changes discontinuously when the sphere
shrinks to zero, the crossings carrying the charge difference implied by the
constraints, Eqs. (31) or (32), must lie on the rotation axis and can not be
attributed to a chiral nodal line. To see this, one must deform the
integration sphere into an spheroid while keeping the intersection points on
the axis constant, such that $\Delta\phi_{c_{b}}$ remains unchanged. Thereby,
the equatorial radius of the spheroid can be reduced to zero to exclude any
finite size nodal line from the enclosed region, such that the topological
charges may only lie on the rotation axis in form of WPs located where
$\Delta\phi_{c_{b}}$ changes.

The case of $C_{2}T$ differs, as the constraint derived from it does not
include a term in the form of $\Delta\phi_{c_{b}}$, see Eq. (36). Instead it
involves the winding of $\phi$ around the rotation axis on a $C_{2}T$
invariant path. A nonzero winding, which results in a $\nu_{c_{b}}=1\mod 2$
constraint, does not imply a charged nodal line of charge 1, since this nodal
line is able to gap out into just a single WP on the $C_{2}T$ invariant plane.

##  APPENDIX C Tight-binding model for SG 80

To illustrate the band topology induced by SG 80, we give a minimal tight-
binding model for the spinless and spinful case, as discussed in Sec. IV.1.1.
We consider a generic model for the 2a Wyckoff position and take all symmetry-
allowed terms up to 2nd nearest neighbors into account. We use the phase
convention of Bloch functions for the tight-binding orbitals [55] and the
primitive vectors as basis for $\mathbf{k}$ [98]. Our model takes the form

$\displaystyle(H_{\text{SG80},\uparrow})_{11}$
$\displaystyle=2t^{\prime}_{1}\big{(}\cos(k_{1}+k_{2}+k_{3})+\cos(k_{3})\big{)}$
$\displaystyle+2t^{\prime}_{2}\big{(}\cos(k_{1})+\cos(k_{2})\big{)},$ (117)
$\displaystyle(H_{\text{SG80},\uparrow})_{22}$
$\displaystyle=2t^{\prime}_{1}\big{(}\cos(k_{1})+\cos(k_{2})\big{)}$
$\displaystyle+2t^{\prime}_{2}\big{(}\cos(k_{1}+k_{2}+k_{3})+\cos(k_{3})\big{)},$
(118) $\displaystyle(H_{\text{SG80},\uparrow})_{12}$
$\displaystyle=(H_{\uparrow})_{21}^{*}$ (119)
$\displaystyle=t_{1}\left(1+\mathrm{e}^{ik_{1}}+\mathrm{e}^{i(k_{1}+k_{3})}+\mathrm{e}^{i(k_{1}+k_{2}+k_{3})}\right)$
$\displaystyle+t_{2}\left(\mathrm{e}^{i(k_{1}+k_{2})}+\mathrm{e}^{-ik_{2}}+\mathrm{e}^{i(2k_{1}+k_{2}+k_{3})}+\mathrm{e}^{ik_{3}}\right),$
(120)

where the parameters $t_{1},t_{2},t^{\prime}_{1},t^{\prime}_{2}\in\mathbb{R}$.
With spin-orbit coupling the full Hamiltonian becomes where
$H_{\text{SG80},\uparrow}^{\text{SOC}}$ and
$H_{\text{SG80},\downarrow}^{\text{SOC}}$ is obtained by
$H_{\text{SG80},\uparrow}$ by adding hopping terms that differ for the two
spin directions. We introduce this by

$\displaystyle H_{\text{SG80},\uparrow}^{\text{SOC}}$
$\displaystyle=H_{\text{SG80},\uparrow}+\Delta H_{\text{SG80},\uparrow},$
(121) $\displaystyle(\Delta H_{\text{SG80},\uparrow})_{11}$
$\displaystyle=\big{(}\tilde{t}^{\prime}_{1}(\mathrm{e}^{i(-k_{1}-k_{2}-k_{3})}+\mathrm{e}^{ik_{3}})$
$\displaystyle+\tilde{t}^{\prime}_{2}(\mathrm{e}^{-ik_{1}}+\mathrm{e}^{-ik_{2}})\big{)}+c.c.,$
(122) $\displaystyle(\Delta H_{\text{SG80},\uparrow})_{22}$
$\displaystyle=\big{(}\tilde{t}^{\prime}_{1}(\mathrm{e}^{-ik_{1}}+\mathrm{e}^{-ik_{2}})$
$\displaystyle+\tilde{t}^{\prime}_{2}(\mathrm{e}^{i(-k_{1}-k_{2}-k_{3})}+\mathrm{e}^{ik_{3}})\big{)}+c.c.,$
(123) $\displaystyle(\Delta H_{\text{SG80},\uparrow})_{12}$
$\displaystyle=(\Delta H_{\text{SG80},\uparrow})_{21}^{*}$
$\displaystyle=\tilde{t}_{1}(\mathrm{e}^{ik_{1}}+\mathrm{e}^{i(k_{1}+k_{2}+k_{3})})$
$\displaystyle+\tilde{t}_{2}(\mathrm{e}^{ik_{1}+ik_{2}}+\mathrm{e}^{i(2k_{1}+k_{2}+k_{3})})$
$\displaystyle+\tilde{t}_{1}^{*}(1+\mathrm{e}^{i(k_{1}+k_{3})})$
$\displaystyle+\tilde{t}_{2}^{*}(\mathrm{e}^{-ik_{2}}+\mathrm{e}^{ik_{3}}),$
(124)

where $\tilde{t}^{\prime}_{1},\tilde{t}^{\prime}_{2}\in\mathbb{C}$. The
corresponding matrix elements in $H_{\text{SG80},\downarrow}^{\text{SOC}}$ can
be obtained from $H_{\text{SG80},\uparrow}^{\text{SOC}}$ by replacing in the
diagonal entries $\mathbf{k}\rightarrow-\mathbf{k}$ and in the off-diagonal
entries
$\tilde{t}_{1}(\tilde{t}_{2})\rightarrow\tilde{t}_{1}^{*}(\tilde{t}_{2}^{*})$
for all occurrences. The coupling between spins takes the form

$\displaystyle(H_{\text{SG80},\uparrow\downarrow})_{11}$
$\displaystyle=2il^{\prime}_{1}\left(\sin(k_{1}+k_{2}+k_{3})+\sin(k_{3})\right)$
$\displaystyle+2il^{\prime}_{2}\left(\sin(k_{1})-\sin(k_{2})\right),$ (125)
$\displaystyle(H_{\text{SG80},\uparrow\downarrow})_{22}$
$\displaystyle=2l^{\prime}_{1}\left(\sin(k_{1})-\sin(k_{2})\right)$
$\displaystyle+2l^{\prime}_{2}\left(-\sin(k_{1}+k_{2}+k_{3})-\sin(k_{3})\right),$
(126) $\displaystyle(H_{\text{SG80},\uparrow\downarrow})_{12}$
$\displaystyle=l_{1}\left(1+i\mathrm{e}^{ik_{1}}-\mathrm{e}^{i(k_{1}+k_{3})}-i\mathrm{e}^{i(k_{1}+k_{2}+k_{3})}\right)$
$\displaystyle+l_{2}\left(-i\mathrm{e}^{i(k_{1}+k_{2})}+\mathrm{e}^{-ik_{2}}+i\mathrm{e}^{i(2k_{1}+k_{2}+k_{3})}-\mathrm{e}^{ik_{3}}\right),$
(127) $\displaystyle(H_{\text{SG80},\uparrow\downarrow})_{21}$
$\displaystyle=-(H_{\text{SG80},\uparrow\downarrow}(\mathbf{k}\rightarrow-\mathbf{k}))_{12}$
(128)

where the parameters $l_{1},l_{2},l^{\prime}_{1},l^{\prime}_{2}\in\mathbb{C}$.

For the remaining discussion we use the parameters defined in units of $t_{1}$
as $t_{2}=0.2,\,t^{\prime}_{1}=0.3,\,t^{\prime}_{2}=-0.2$ and if spin-orbit
coupling is included we add
$\tilde{t}_{1}=-0.1+0.2i,\,\tilde{t}_{2}=0.05-0.05i,\,l_{1}=0.4i,\,l_{2}=0.05-0.15i,\,\tilde{t}^{\prime}_{1}=0.1,\,\tilde{t}^{\prime}_{2}=0.05,\,l^{\prime}_{1}=l^{\prime}_{2}=0.1i$.
A possible minimal set of parameters that realizes the same band topology is
given by $\tilde{t}_{1}=0.2+0.4i,\,l_{1}=0.8i,\,\tilde{t}^{\prime}_{1}=0.4$,
where all other parameters (except $t_{1}=1$) are set to 0.

Figure 12:  Band structure for the model defined in Eq.
(LABEL:Eq_SG80fullHamiltonian) respecting SG 80 _without SOC_. The Brillouin
zone is shown in the in-set of Fig. 11(b).

In the limit of vanishing spin-orbit coupling our model exhibits only three
Weyl points for each spin sector, one at each of the two distinct points P
with $\nu=+1$ and a double Weyl point at the TRIM M with $\nu=-2$, see Fig.
12. Without the spin degeneracy the unity charge at the point P is different
from the possible charges of twofold crossings at TRIMs, which are always
$\nu=\pm 2$ in spinless systems [47]. The double Weyl point at M shows on the
$\Gamma$-M path, i.e., where $k_{z}=0$ is constant, the typical quadratic
dispersion expected perpendicular to the rotation axis $\Gamma$-Z-M [68].
Note, our model for SG 80 without SOC is a counter example to the commonly
conceived notion that there have to be at least four Weyl points in the
presence of time-reversal symmetry [99, 100]. Here, the number of Weyl points
in our model is lower than four without additional band crossings [40].

Once spin-orbit coupling is taken into account the bands at P (M) split into a
double Weyl point and two non-degenerate bands (two double Weyl points on the
$\Gamma$-Z-M path), a detailed description is given in Sec. IV.1.1.

For the numerical determination of Chern numbers we use the Wilson loop
approach on a discretized Brillouin zone as described in Ref. [101].

##  APPENDIX D Local constraints in the presence of quasi-symmetries

In the following we derive a low-energy model for a twofold crossing that
appears among the spinless representations of the point group T 23, describing
the rotation symmetries of a tetrahedron. The resulting model applies to SGs
195-199 at the TRIMs $\Gamma$ and R, L or H within the primitive, face-
centered, or body-centered cubic unit cells in the nomenclature of the Bilbao
crystallographic server [102]. All possible terms up to third order that are
compatible with the twofold and threefold rotation of the cubic point groups
symmetry are given in the model $H_{\text{T}}({\bf k})$ as

where the Pauli matrices $\sigma_{i}$ have been used and the first term is
diagonal that shifts and bends both bands with $d_{0},d_{1}\in\mathbb{R}$. The
chirality $\nu$ of this twofold crossing is $\nu=4\text{sgn}(a_{0})$ and
$a_{1}\in\mathbb{C}$. In the given basis, the twofold and threefold rotations
and time-reversal are represented by

(129)

respectively.

As discussed in the main text, $\nu=\pm 4$ is a peculiar value. For example
for $a_{0}>0$, the local constraints, Eq. (1), derived from the twofold and
threefold rotations imply $\nu=0\mod 2=1\mod 3$, which is consistent with
$\nu=4$ but would imply the simpler possibility of $\nu=-2$. This can not only
be understood by a refined local constraint, Eq. (61), but also by the
emergence of a yet recognized quasi-symmetry that mimics a fourfold rotation.
The existence of a fourfold symmetry, can be motivated pictorially by noticing
that the symmetries $C^{z}_{2}$ and $C^{z}_{2}T$ together reduce the
integration surface of the Chern number into a quarter of the full sphere, as
discussed in Sec. IV.1.2. To gap out the bands away from ${\bf k}=0$ and to
obtain a well-defined chirality, it is sufficient to take all the symmetry-
allowed terms up to cubic order in $k_{i}$, as we did in $H_{\text{T}}({\bf
k})$. Interestingly, the fourfold rotation symmetry of the model is only
broken at the fourth order in $k_{i}$, which is irrelevant for the value of
the Chern number.

The model, Eq. (LABEL:Eq_App_PG23), exhibits a parameter-dependent fourfold
rotation symmetry $U(C^{z}_{4})$ defined as

fulfilling

$\displaystyle
U(C^{z}_{4})^{\dagger}H_{\text{T}}(k_{x},k_{y},k_{z})U(C^{z}_{4})=H_{\text{T}}(k_{y},-k_{y},k_{z}).$
(130)

The symmetry eigenvalues of this operation are independent of $a_{1}$ and
equal to $\lambda(C^{z}_{4})\in\\{+1,-1\\}$. We made a choice choice of
complex phase in Eq. (D) that also does not affect the eigenvalues, but has
been used to ensure that $U(C^{z}_{4})$ commutes with time-reversal symmetry
$T$. The commuting property together with the reality of eigenvalues implies
that at the twofold degeneracy the symmetry eigenvalues of $U(C^{z}_{4})$ do
not exchange, i.e., the phase jump is $\Delta\varphi=0$. And thus we can apply
our local constraint according to Eq. (1) to the fourfold rotation to obtain
$\nu=0\mod 4$ resulting in $\nu=+4$ for $a_{0}>0$.

Let us compare this result to the low-energy model, again for a spinless
system, described by the octahedral point group O 432, where the twofold axes
of tetrahedral group T are replaced by fourfold rotations. We can turn
$H_{\text{T}}({\bf k})$ into a model of point group O by setting
$a_{1}=-2\pi/3$ in Eq. (LABEL:Eq_App_PG23). This choice turns the quasi-
symmetry $U(C^{z}_{4})$ into a representation of the actual fourfold rotation
of the octahedral point group. Naturally, for this real fourfold rotation
$U(C^{z}_{4})$ our argument on the local constraint it opposes on the
chirality is unchanged and one finds $\nu=\pm 4$ as well.

In summary, we find that the existence of a fourfold quasi-symmetry in the
low-energy model explains the charge $\nu=\pm 4$ of the twofold degeneracies
with the tetrahedral point group T.

##  APPENDIX E Tight-binding model for SG 94

In the following we define the model of SG 94 including spin and with time-
reversal symmetry, which is used to create Fig. 5. We take the 2a Wyckoff
position with the sites $(0,\,0,\,0),\,(1/2,\,1/2,\,1/2)$ and a spin-1/2 as
internal degree of freedom on each site. To keep the model simple, we pick
three independent terms that are sufficient to avoid any accidental band
degeneracies from the set of hopping terms between up to next-nearest
neighbors. Our 4$\times$4 Hamiltonian thus takes the form

where the matrix blocks are

$\displaystyle(H_{\text{SG94},\uparrow})_{11}$
$\displaystyle=(H_{\text{SG94},\uparrow})_{22}=t_{2}\mathrm{e}^{-ik_{z}}+t_{2}^{*}\mathrm{e}^{ik_{z}},$
(131) $\displaystyle(H_{\text{SG94},\uparrow})_{12}$
$\displaystyle=(1+\mathrm{e}^{ik_{x}})(1+\mathrm{e}^{ik_{y}})(t_{1}+t_{1}^{*}\mathrm{e}^{ik_{z}}),$
(132) $\displaystyle H_{\text{SG94},\downarrow}$
$\displaystyle=H_{\text{SG94},\downarrow}(t_{1}\leftrightarrow
t_{1}^{*},t_{2}\leftrightarrow t_{2}^{*})$ (133)

for hopping that preserves spin and

$\displaystyle(H_{\text{SG94,SOC}})_{11}=(H_{\text{SG94,SOC}})_{22}=0,$ (134)
$\displaystyle(H_{\text{SG94,SOC}})_{12}=$ $\displaystyle\quad
t_{\text{SOC}}^{*}(-i\mathrm{e}^{ik_{x}}+i\mathrm{e}^{ik_{y}}-\mathrm{e}^{ik_{z}}+\mathrm{e}^{i(k_{x}+k_{y}+k_{z})})$
$\displaystyle\quad+t_{\text{SOC}}(i-i\mathrm{e}^{i(k_{x}+k_{y})}-\mathrm{e}^{i(k_{x}+k_{z})}+\mathrm{e}^{i(k_{y}+k_{z})}),$
(135)
$\displaystyle(H_{\text{SG94,SOC}})_{21}=\mathrm{e}^{-i(k_{x}+k_{y}+k_{z})}\big{(}$
$\displaystyle\quad
t_{\text{SOC}}^{*}(-1+\mathrm{e}^{i(k_{x}+k_{y})}-i\mathrm{e}^{i(k_{x}+k_{z})}+i\mathrm{e}^{i(k_{y}+k_{z})})$
$\displaystyle\quad+t_{\text{SOC}}(-\mathrm{e}^{ik_{x}}+\mathrm{e}^{ik_{y}}+i\mathrm{e}^{ik_{z}}-i\mathrm{e}^{i(k_{x}+k_{y}+k_{z})})\big{)}$
(136)

for spin-orbit coupling terms. For Fig. 5 we use the parameters
$t_{1}=1+i,\,t_{2}=0.5+0.8i,\,t_{\text{SOC}}=-0.2+0.5i$.

##  APPENDIX F 4-fold models

###  F.1 SG 198 $\bar{\Gamma}_{6}\bar{\Gamma}_{7}$ model with SOC

In the following we show that the phase transitions of the low-energy
Hamiltonian (Eq. (71)) describing the $\bar{\Gamma}_{6}\bar{\Gamma}_{7}$ irrep
are the ones shown in figure 6. We first apply a transformation into a basis
diagonal in the 3-fold rotation

$\displaystyle\begin{pmatrix}q_{+}\\\ q_{-}\\\
q_{z}\end{pmatrix}=\frac{1}{2\sqrt{3}}\begin{pmatrix}2&-1+i\sqrt{3}&-1-i\sqrt{3}\\\
2&-1-i\sqrt{3}&-1+i\sqrt{3}\\\ -2&-2&-2\end{pmatrix}\cdot{\bf k}.$ (137)

Due to scale invariance, we can parameterize the following way

$\displaystyle\alpha_{0}$ $\displaystyle=\cos(G)\cos(Z),$ (138)
$\displaystyle\alpha_{1}$ $\displaystyle=\cos(G)\sin(Z),$ (139)
$\displaystyle\alpha_{2}$ $\displaystyle=\sin(G).$ (140)

with $0\leq Z<2\pi$ and $-\frac{\pi}{2}\leq G\leq\frac{\pi}{2}$. We find all
points in parameter space $\alpha_{n}$ where the energy levels of $H$ become
degenerate away from ${\bf k}=0$ by considering the characteristic polynomial
of $H$

$\displaystyle\chi(E)=E^{4}+a(q_{+},q_{z})E^{2}+\text{det}H.$ (141)

Since there are only even powers of $E$ in $\chi$, the spectrum is particle-
hole symmetric. It is gapless when $\text{det}H=0$ or $a^{2}-4\text{det}H=0$.
We find that $\chi$ and $\text{det}H$ do not depend on $Z$, so the topological
phase diagram must be rotationaly invariant. Further we parameterize

$\displaystyle q_{+}$ $\displaystyle=\exp(iK)\cos(M)$ (142) $\displaystyle
q_{z}$ $\displaystyle=\sin(M)$ (143)

due to scale-invariance without loss of generality.

$\text{det}H=0$ case: We find

$\displaystyle\text{det}H$ $\displaystyle=p_{1}(G,M)\cos(K)^{3}$
$\displaystyle+p_{2}(G,M)\cos(K)+p_{3}(G,M).$ (144)

Particle-hole symmetry implies $\text{det}H\geq 0$, so $\text{det}H=0$ must be
a minimum. Therefore

$\displaystyle\frac{\text{d\,}\text{det}H}{\text{d}K}$
$\displaystyle=-\sin(K)(3p_{1}\cos(K)^{2}+p_{2})=0$ (145)
$\displaystyle\implies(\sin(K)$ $\displaystyle=0\implies K\in\\{0,\pi\\}$
(146) $\displaystyle\lor\cos(K)$
$\displaystyle=\pm\sqrt{\frac{-p_{2}}{3p_{1}}}=\pm\frac{1}{2}$ (147)
$\displaystyle\implies K$
$\displaystyle\in\\{\frac{\pi}{3},\frac{2\pi}{3}\\}).$ (148)

So $\\{0,\pi,\frac{\pi}{3},\frac{2\pi}{3}\\}$ are the only $K$ values where
$H$ can get gapless via $\text{det}H=0$.

With $K=0$, $\text{det}H$ has the form

$\displaystyle\text{det}H=f_{4}(M)\sin(G)^{4}+f_{2}(M)\sin(G)^{2}+f_{0}(M).$
(149)

It is zero when

$\displaystyle\sin(G)^{2}=\frac{-f_{2}\pm\sqrt{f_{2}^{2}-4f_{4}f_{0}}}{2f_{4}}=U_{\pm}(M).$
(150)

A solution for $G$ is found, when $U_{\pm}$ is real and $0\leq U_{\pm}\leq 1$.
We find this is the case for
$M\in\\{-\frac{\pi}{2},\arctan\left(\frac{1}{2}\right),\frac{\pi}{2}\\}$,
where $U_{+}=0\implies G=0$, and $M=-\frac{\pi}{4}$ with
$U_{+}=\frac{1}{2}\implies G=\pm\frac{\pi}{4}$. $U_{-}=\sin(G)^{2}$ only
reproduces the $M=-\frac{\pi}{4}$ solution.

The $M\in\\{-\frac{\pi}{2},\arctan\left(\frac{1}{2}\right),\frac{\pi}{2}\\}$
solutions with $K=0$ correspond (under symmetry and scale transformations) to
degeneracies on the $k=(t,t,t)$ line, while $M=-\frac{\pi}{4}$ corresponds to
$k=(t,0,0)$.

For the case $K=\pi$ we find after a transformation with $M\rightarrow-M$ and
$U_{\pm}\rightarrow U_{\mp}$ the same solutions. All other cases can be found
by applying the $3$-fold symmetry found in the little group at this $k$-point.
Therefore, with $\text{det}H=0$, degeneracies only happen on high symmetry
lines and there are gapless points at $G\in\\{0,\pm\frac{\pi}{4}\\}$. We find
for $G=0$ a gap closing of the bandpair $(2,3)$ on $C_{3}$ invariant lines and
at $G=\pm\frac{\pi}{4}$ a gap closing on $C_{2}$ invariant lines for the same
bandpair.