$a^{2}-4\text{det}H=0$ case: We find due to particle-hole symmetry
$a=-E_{1}^{2}-E_{2}^{2}$, so $a^{2}-4\text{det}H\geq 0$, since

$\displaystyle a^{2}$ $\displaystyle\geq 4\text{det}H$
$\displaystyle\iff(E_{1}^{2}+E_{2}^{2})^{2}$ $\displaystyle\geq
4E_{1}^{2}E_{2}^{2}$ $\displaystyle\iff(E_{1}^{2}-E_{2}^{2})^{2}$
$\displaystyle\geq 0.$ (151)

So the point where $a^{2}-4\text{det}H=0$ must be a minimum of
$a^{2}-4\text{det}H$. We find the following form of $a^{2}-4\text{det}H$

$\displaystyle a^{2}-4\text{det}H$ $\displaystyle=s_{1}(G,M)\cos(K)^{3}$
$\displaystyle+s_{2}(G,M)\cos(K)+s_{3}(G,M).$ (152)

An optimum can be found via

$\displaystyle\frac{\text{d}(a^{2}-4\text{det}H)}{\text{d}K}$
$\displaystyle=-\sin(K)(3s_{1}\cos(K)^{2}+s_{2})=0$ (153)
$\displaystyle\implies K$ $\displaystyle\in\\{0,\pi\\}$ (154)
$\displaystyle\lor\cos(K)$
$\displaystyle=\pm\sqrt{\frac{-s_{2}}{3s_{1}}}=\pm\frac{1}{2}$ (155)
$\displaystyle\implies K$
$\displaystyle\in\\{\frac{\pi}{3},\frac{2\pi}{3}\\}.$ (156)

So once again $\\{0,\pi,\frac{\pi}{3},\frac{2\pi}{3}\\}$ are the only $K$
values where $H$ can get gapless, now via $a^{2}-4\text{det}H=0$. With
$K=\frac{n\pi}{3}$ and $n\in\mathbb{N}$ we find

$\displaystyle a^{2}-4\text{det}H$
$\displaystyle=(r_{1}(n,M)\cos(G)^{2}+r_{2}(n,M))\cos(G)^{2}$ (157)

solutions to $\cos(G)^{2}=0$ and $\cos(G)^{2}=\frac{-r_{2}}{r_{1}}$ only exist
for $G\in\\{-\frac{\pi}{2},0,\frac{\pi}{2}\\}$. For $G=0$ we find a gap
closing of the outer bandpairs at $C_{2}$ invariant lines. For
$G=\pm\frac{\pi}{2}$ the gap closing of the same bandpair occurs for any $q$,
as we find that $a^{2}-4\text{det}H=0$ for any $M,K$.

We therefore showed that the Hamiltonian is only gapless for
$G\in\\{-\frac{\pi}{2},-\frac{\pi}{4},0,\frac{\pi}{4},\frac{\pi}{2}\\}$ at
points away from $k=(0,0,0)$, which translates to
$\alpha_{2}=\pm\sqrt{\alpha_{0}^{2}+\alpha_{1}^{2}}$, $\alpha_{2}=0$ or
$\sqrt{\alpha_{0}^{2}+\alpha_{1}^{2}}=0$.

###  F.2 SG 207 $\bar{\Gamma}_{8}$ model with SOC

The generator of this irrep contains a time-reversal symmetry $T$, three
2-fold symmetries $2_{001}$, $2_{010}$, $2_{110}$ and a 3-fold symmetry
$3_{111}$. The first-order Hamiltonian generated from these symmetries is

$\displaystyle H=$
$\displaystyle\alpha_{0}\biggl{[}\frac{k_{x}}{2}\left(\sqrt{3}\sigma_{x}\tau_{0}+\sigma_{x}\tau_{z}+\sigma_{y}\tau_{0}-\sqrt{3}\sigma_{y}\tau_{z}\right)+$
$\displaystyle\frac{k_{y}}{2}\left(-\sqrt{3}\sigma_{x}\tau_{0}-\sigma_{x}\tau_{z}+\sigma_{y}\tau_{0}-\sqrt{3}\sigma_{y}\tau_{z}\right)+$
$\displaystyle
k_{z}\left(\sigma_{x}\tau_{x}-\sigma_{x}\tau_{y}\right)\biggr{]}+$
$\displaystyle\frac{\alpha_{1}}{\sqrt{2}}\biggl{[}k_{x}\left(\sigma_{0}\tau_{x}+\sigma_{0}\tau_{y}+\sigma_{z}\tau_{x}-\sigma_{z}\tau_{y}\right)+$
$\displaystyle
k_{y}\left(\sigma_{0}\tau_{x}+\sigma_{0}\tau_{y}-\sigma_{z}\tau_{x}+\sigma_{z}\tau_{y}\right)+2k_{z}\sigma_{z}\tau_{z}\biggr{]}.$
(158)

The characteristic polynomial of this Hamiltonian is identical to the one of
model 71 from the main text when $\alpha_{1}=0$ and $\alpha_{2}\to\alpha_{1}$.
Therefore the topological phase separating points in parameter space of
$\alpha_{0}$ and $\alpha_{1}$ are reproduced. Computing the Chern numbers, one
finds that topological phase diagram as a whole stays the same, except now,
one axis corresponds to only $\alpha_{0}$, instead of
$\sqrt{\alpha_{0}^{2}+\alpha_{1}^{2}}$. The statements about the symmetry
eigenvalue phase jumps also stay the same.

This model also describes SG 207 $\bar{R}_{8}$, SG 208 $\bar{\Gamma}_{8}$,
$\bar{R}_{8}$, SG 209 $\bar{\Gamma}_{8}$, SG 210 $\bar{\Gamma}_{8}$, SG 211
$\bar{\Gamma}_{8}$, $\bar{H}_{8}$, SG 212 $\bar{\Gamma}_{8}$, SG 213
$\bar{\Gamma}_{8}$, SG 214 $\bar{\Gamma}_{8}$ with SOC.

The SG 214 $\bar{H}_{8}$ model with SOC can be found by applying
$U=\sigma_{0}\tau_{x}$ and $\alpha_{1}\to-\alpha_{1}$ to Hamiltonian 158.

###  F.3 SG 198 $\bar{M}_{5}\bar{M}_{5}$ model with SOC

The generator of this irrep contains a time-reversal symmetry $T$ and two
2-fold symmetries $2_{001}$, $2_{010}$. The generated Hamiltonian is

$\displaystyle H=$
$\displaystyle\alpha_{0}k_{z}\sigma_{0}\tau_{y}+\alpha_{1}k_{x}\sigma_{z}\tau_{z}+\alpha_{2}k_{x}\sigma_{x}\tau_{z}+\alpha_{3}k_{x}\sigma_{y}\tau_{z}+$
$\displaystyle\alpha_{4}k_{y}\sigma_{z}\tau_{x}+\alpha_{5}k_{y}\sigma_{x}\tau_{x}+\alpha_{6}k_{y}\sigma_{y}\tau_{x}$
(159)

Its spectrum is particle-hole symmetric as the characteristic polynomial is
$\chi(E)=E^{4}+aE^{2}+\text{det}H$.

We can rewrite $H$ with

$\displaystyle{\bm{\alpha}}_{1}=(\alpha_{1},\alpha_{2},\alpha_{3})^{T}$ (160)
$\displaystyle{\bm{\alpha}}_{2}=(\alpha_{4},\alpha_{5},\alpha_{6})^{T}$ (161)

such that

$\displaystyle H=$
$\displaystyle\alpha_{0}k_{z}\tau_{y}\sigma_{0}+k_{x}\tau_{z}({\bm{\alpha}}_{1}\cdot{\bm{\sigma}})+k_{y}\tau_{x}({\bm{\alpha}}_{2}\cdot{\bm{\sigma}}).$
(162)

Due to the two nodal planes on the $k_{x}=0$ and $k_{y}=0$ planes, the Chern
number of single bands are always undefined. The Chern number for a filling of
2 can still be computed when there are no 4-fold degeneracies away from ${\bf
k}=(0,0,0)^{T}$. Therefore we only need to find these 4-fold degenerate points
in parameter space, that is when $\text{det}H=0$.

If $\alpha_{0}=0$, then $H$ is 4-fold degenerate on the $k_{z}$ line. If
$\alpha_{0}\neq 0$, ${\bm{\alpha}}_{1}\neq 0$ and ${\bm{\alpha}}_{2}\neq 0$,
we can normalize $|{\bm{\alpha}}_{1/2}|=1$ by using scale invariance of $H$.
We rotate ${\bm{\sigma}}$ with a unitary transformation
$U^{\dagger}{\bm{\sigma}}U=R{\bm{\sigma}}$ with $R$ being a rotation matrix,
such that

$\displaystyle U^{\dagger}({\bm{\alpha}}_{1}\cdot{\bm{\sigma}})U$
$\displaystyle={\bm{\alpha}}_{1}\cdot(R{\bm{\sigma}})$
$\displaystyle=(R^{-1}{\bm{\alpha}}_{1})\cdot{\bm{\sigma}}$
$\displaystyle=\sigma_{z}$ (163)

where we chose $R$ such that $R^{-1}{\bm{\alpha}}_{1}=(1,0,0)^{T}$. Then

$\displaystyle H^{\prime}=$ $\displaystyle U^{\dagger}HU$ $\displaystyle=$
$\displaystyle\alpha_{0}k_{z}\tau_{y}\sigma_{0}+k_{x}\tau_{z}\sigma_{z}+k_{y}\tau_{x}(\tilde{\alpha}_{2}\cdot{\bm{\sigma}}).$
(164)

with $\tilde{\alpha}_{2}=R^{-1}{\bm{\alpha}}_{2}$. We parameterize without
loss of generality

$\displaystyle\tilde{\alpha}_{2}$ $\displaystyle=(\cos P\cos R,\cos P\sin
R,\sin P)^{T}$ (165) $\displaystyle{\bf k}$ $\displaystyle=(\cos G\cos Z,\cos
G\sin Z,\sin G)^{T}$ (166)

We find

$\displaystyle|H^{\prime}|=\text{det}H=A(P,R,Z)\cos(G)^{4}+1$ (167)

and $\text{det}H\geq 0$. The point $\text{det}H=0$ must be a minimum of
$\text{det}H$. We find optima at $G\in\\{-\frac{\pi}{2},0,\frac{\pi}{2}\\}$.
$G=\pm\frac{\pi}{2}$ can be excluded, since there $\text{det}H=1\neq 0$. This
leaves $G=0$. With this, we find the constraint for $A$ such that
$\text{det}H=0$

$\displaystyle A(P,R,Z)$
$\displaystyle=4(\cos^{4}Z-\cos^{2}Z)(1-\cos^{2}R\cos^{2}P)$
$\displaystyle=-1$ (168)

We find optima of $\text{det}H$ with
$\frac{\text{d}\text{det}H}{\text{d}Z}|_{G=0}=\frac{\text{d}A}{\text{d}Z}=0$
at $Z=n\frac{\pi}{4}$ with $n\in\mathbb{Z}_{8}$. We can exclude
$Z=\frac{\pi}{2},\pi$ and $\frac{3\pi}{2}$, since there $A=0$. Inserting the
remaining $Z$ values, we get

$\displaystyle\cos^{2}R\cos^{2}P=0.$ (169)

Note that

$\displaystyle{\bm{\alpha}}_{1}\cdot{\bm{\alpha}}_{2}$
$\displaystyle=(R^{-1}{\bm{\alpha}}_{1})\cdot(R^{-1}{\bm{\alpha}}_{2})$
$\displaystyle=(1,0,0)^{T}\cdot\tilde{\alpha}_{2}$ $\displaystyle=\cos P\cos
R.$ (170)

Therefore ${\bm{\alpha}}_{1}\cdot{\bm{\alpha}}_{2}=0$ represents a surface in
parameter space separating different topological phases. This is also true
when $|{\bm{\alpha}}_{1/2}|=0$.

If $|{\bm{\alpha}}_{1}|=0$ we find the characteristic polynomial of $H$ with
the same parametrization of ${\bf k}$ as above and $|{\bm{\alpha}}_{2}|=1$

$\displaystyle\chi(E)=(E^{2}+\cos^{2}G\cos^{2}Z-1)^{2}.$ (171)

Note that $\chi(E)$ is independent of ${\bm{\alpha}}_{2}$ since it can always
be rotated by a unitary transformation, such that $H$ is rotationally
invariant in ${\bm{\alpha}}_{2}$. We find a 4-fold point at $E_{1,2,3,4}=0$.
This is the case, when $G=0$ and $Z\in\\{0,\pi\\}$, so $H$ is 4-fold
degenerate on the $k_{x}$ line.

If $|{\bm{\alpha}}_{2}|=0$ we find the characteristic polynomial of $H$ with
$|{\bm{\alpha}}_{1}|=1$

$\displaystyle\chi(E)=(E^{2}+\cos^{2}G\sin^{2}Z-1)^{2}.$ (172)

We find a 4-fold point at $G=0$ and $Z\in\\{\frac{\pi}{2},\frac{3\pi}{2}\\}$,
so $H$ is 4-fold degenerate on the $k_{y}$ line.

In conclusion, we found that the topology of this model is entirely dependent
on the sign of $\alpha_{0}$ and ${\bm{\alpha}}_{1}\cdot{\bm{\alpha}}_{2}$. We
find the Chern number of the lowest two bands to be
$\nu=2(-1)^{\theta(\alpha_{0})+\theta({\bm{\alpha}}_{1}\cdot{\bm{\alpha}}_{2})}$.

For both 2-fold symmetries, the constraint of Eq. (18) amounts to
$\nu_{12}=0\,\text{mod}\,2$ by direct calculation. This can be seen directly
on the $(001)$ line, where the same symmetry eigenvalues are paired by
$2_{001}$. This way, a symmetry eigenvalue jump can be defined for the two
degenerate bands and is equal to $\pi$ everywhere in the phase diagram. Using
Eq. (19) one arrives once again at
$\nu_{12}=\frac{2}{2\pi}(\pi+\pi)\,\text{mod}\,2=0\,\text{mod}\,2$.

This Hamiltonian also describes SG 18 $\bar{S}_{5}\bar{S}_{5}$,
$\bar{R}_{5}\bar{R}_{5}$, SG 19 $\bar{S}_{5}\bar{S}_{5}$ with SOC. By applying
a Chern number preserving $k_{x}\to k_{y},k_{y}\to k_{z},k_{z}\to k_{x}$
rotation, one arrives at the SG 19 $\bar{T}_{5}\bar{T}_{5}$, SG 92
$\bar{R}_{5}\bar{R}_{5}$ and SG 96 $\bar{R}_{5}\bar{R}_{5}$ Hamiltonian. By
applying a $k_{y}\to k_{z},k_{z}\to k_{y}$ reflection, during which Chern
numbers are flipped, one arrives at the SG 19 $\bar{U}_{5}\bar{U}_{5}$ model.

###  F.4 SG 212 $\bar{M}_{6}\bar{M}_{7}$ model with SOC

The little groups generator contains a time-reversal symmetry $T$ and three
2-fold symmetries $2_{001}$, $2_{010}$, $2_{110}$. The low-energy Hamiltonian
generated from these symmetries is

$\displaystyle H=$ $\displaystyle\alpha_{0}k_{z}\sigma_{z}\tau_{z}+$
$\displaystyle\alpha_{1}\biggl{[}k_{x}\sigma_{0}\tau_{y}-k_{y}\sigma_{z}\tau_{x}\biggr{]}+$
$\displaystyle\alpha_{2}\biggl{[}k_{x}\sigma_{x}\tau_{z}-k_{y}\sigma_{y}\tau_{0}\biggr{]}+$
$\displaystyle\alpha_{3}\biggl{[}k_{x}\sigma_{y}\tau_{z}+k_{y}\sigma_{x}\tau_{0}\biggr{]}$
(173)

The lower and upper two bands of this Hamiltonian always have doubly
degenerate points, such that the Chern number of the lower and upper band is
undefined. This means, we only have to look at all points in parameter space
where $\text{det}H=0$. This is the case when
$|(\alpha_{1},\alpha_{2},\alpha_{3})|=0$. There, $H=0$ on the $k_{z}=0$ plane.
We also find $H=0$ for $\alpha_{0}=0$ on the $k_{z}$ line. Due to these
considerations and scaling properties of $H$, we first consider the
$\alpha_{0}=1$ case, which corresponds to $\alpha_{0}>0$.

We also set $k_{z}=1$, such that all now reachable $k$ points correspond under
rescaling to the upper half of the unit sphere in $k$ space. We parameterize

$\displaystyle\alpha_{1}$ $\displaystyle=r_{1}\cos P\cos R$
$\displaystyle\alpha_{2}$ $\displaystyle=r_{1}\cos P\sin R$
$\displaystyle\alpha_{3}$ $\displaystyle=r_{1}\sin P$ (174)

with $r_{1}\geq 0$. $\text{det}H$ takes on the following form

$\displaystyle\text{det}H$
$\displaystyle=r_{1}^{4}(k_{x}^{4}+k_{y}^{4}+(16F(R,P)+2)k_{x}^{2}k_{y}^{2})$
$\displaystyle+2r_{1}^{2}(k_{x}^{2}+k_{y}^{2})+1$ (175)

where we find $-\frac{1}{4}\leq F(R,P)\leq 0$ with
$F(R,P)=\cos^{2}P\cos^{2}R(\cos^{2}P\cos^{2}R-1)=\alpha_{1}^{2}(\alpha_{1}^{2}-r_{1}^{2})/r_{1}^{4}$.
Inserting the minimal value of $F$ in $\text{det}H$, we get

$\displaystyle\text{det}H=r_{1}^{4}(k_{x}^{2}-k_{y}^{2})^{2}+2r_{1}^{2}(k_{x}^{2}+k_{y}^{2})+1.$
(176)

There $\text{det}H=0$ has no real solution, as $\text{det}H>0$. Therefore, for
all $F\in[-\frac{1}{4},0]$, $\text{det}H>0$. Due to symmetry, this means, that
also on the $k_{z}=-1$ plane, no degeneracies of the middle bandpair can
occur.

Figure 13: Topological phase diagram of the lower two bands Chern number
$\nu_{12}$ of the SG 212 $\bar{M}_{6}\bar{M}_{7}$ Model with $\alpha_{0}>0$.

The only way left, is the $k_{z}=0$ with $r_{1}>0$ and $|(k_{x},k_{y})|\neq 0$
case. We enforce this, by parameterizing
$k_{x}=r_{2}\cos(G),k_{y}=r_{2}\sin(G)$ with $r_{2}>0$. The determinant with
$k_{z}=0$ becomes

$\displaystyle\text{det}H=$ $\displaystyle
r_{1}^{4}r_{2}^{4}(1-4(1-\cos(P)^{2}\cos(R)^{2})$
$\displaystyle\cos(P)^{2}\cos(R)^{2}\sin(2G)^{2}).$ (177)

At $\text{det}H=0$, this becomes

$\displaystyle|\sin(2G)|=\frac{1}{2\sqrt{1-\left(\frac{\alpha_{1}}{r_{1}}\right)^{2}}|\frac{\alpha_{1}}{r_{1}}|}=K\left(\alpha_{1}/r_{1}\right)$
(178)

The only solutions with $0\leq K\leq 1$ and $K\in\mathbb{R}$, are
$\alpha_{1}=\pm\frac{r_{1}}{\sqrt{2}}$ with $K=1$, such that
$G\in\\{\frac{\pi}{4},\frac{3\pi}{4},\frac{5\pi}{4},\frac{7\pi}{4}\\}$. The
topological phase diagram is therefore rotationally invariant in the
$(\alpha_{1},\alpha_{2},\alpha_{3})$ parameter space around the $\alpha_{1}$
axis. The phase separating lines are
$\alpha_{1}=\pm\sqrt{\alpha_{2}^{2}+\alpha_{3}^{2}}$. The topological phase
diagram can be determined the same way as in the previous models, see figure
13. For $\alpha_{0}<0$, the Chern numbers in figure 13 switch signs.

By direct calculation, we find $\nu_{12}=0\,\text{mod}\,2$. The symmetry
eigenvalue phase jump for $2_{001}$ is defined, since on the $(001)$ the same
symmetry eigenvalues are paired, and $\pi$ for all bands. The symmetry
eigenvalue jump of $2_{110}$ is also defined and $\pi$. Both lead to the
$\nu_{12}=0\,\text{mod}\,2$ constraint once again. Symmetry eigenvalue jumps
are undefined for $2_{010}$, as there different symmetry eigenvalues are
paired.

We get the SG 213 $\bar{M}_{6}\bar{M}_{7}$ Model with SOC by mapping
$k_{y}\to-k_{y}$. This flips the sign of the Chern numbers in the topological
phase diagram of this model.

###  F.5 SG 198 $R_{1}R_{3}$ model without SOC

This model is symmetric under a time-reversal symmetry $T$, two 2-fold
symmetries $2_{001}$, $2_{010}$ and a 3-fold symmetry $3_{111}$.

$\displaystyle H=$
$\displaystyle\alpha_{0}\left[-k_{x}\sigma_{z}\tau_{x}+k_{y}\sigma_{0}\tau_{y}+k_{z}\sigma_{z}\tau_{z}.\right]$
(179)

Due to the nodal plane, the Chern number of the lower and upper bands are
undefined. The Chern number with a filling of 2 is
$\nu_{12}=2(-1)^{\theta(\alpha_{0})}$. Evaluating Eq. (18) yields the
constraints $\nu_{12}=0\,\text{mod}\,2$, $\nu_{12}=1\,\text{mod}\,3$ for
$\alpha_{0}>0$ and $\nu_{12}=2\,\text{mod}\,3$ for $\alpha_{0}<0$, which are
all fulfilled in this model. This model also describes SG 198 $R_{2}R_{2}$, SG
212 $R_{1}R_{2}$, SG 213 $R_{1}R_{2}$ without SOC.

###  F.6 SG 212 $R_{3}$ model without SOC

This model is symmetric under a time-reversal symmetry $T$, three 2-fold
symmetries $2_{001}$, $2_{010}$,$2_{110}$ and a 3-fold symmetry $3_{111}$. The
low-energy Hamiltonian is

$\displaystyle H=$
$\displaystyle\alpha_{0}\biggl{[}k_{x}\left(-\sigma_{0}\tau_{x}+\sigma_{0}\tau_{y}-\sigma_{z}\tau_{x}-\sigma_{z}\tau_{y}\right)+$
$\displaystyle
k_{y}\left(-\sigma_{0}\tau_{x}+\sigma_{0}\tau_{y}+\sigma_{z}\tau_{x}+\sigma_{z}\tau_{y}\right)+$
$\displaystyle 2k_{z}\sigma_{z}\tau_{z}\biggr{]}$ (180)

Due to the nodal plane, the Chern number of the lower and upper bands are
undefined. The Chern number with a filling of 2 is
$\nu_{12}=2(-1)^{\theta(\alpha_{0})}$. From Eq. (18) we get the fulfilled
constraints $\nu_{12}=0\,\text{mod}\,2$, $\nu_{12}=1\,\text{mod}\,3$ for
$\alpha_{0}>0$ and $\nu_{12}=2\,\text{mod}\,3$ for $\alpha_{0}<0$. This model
also describes SG 213 $R_{3}$ without SOC.

###  F.7 SG 90 $\bar{A}_{6}\bar{A}_{7}$ model with SOC

This model is symmetric under a time-reversal symmetry $T$, two 2-fold
symmetries $2_{001}$, $2_{010}$ and a 4-fold symmetry $4_{001}$. The
Hamiltonian is

$\displaystyle H=$ $\displaystyle\alpha_{0}k_{z}\sigma_{z}\tau_{z}+$
$\displaystyle\alpha_{1}\biggl{[}k_{x}\left(\sigma_{0}\tau_{y}+\sigma_{z}\tau_{x}\right)+k_{y}\left(\sigma_{0}\tau_{y}-\sigma_{z}\tau_{x}\right)\biggr{]}+$
$\displaystyle\frac{\alpha_{2}}{\sqrt{2}}\biggl{[}k_{x}\left(-\sigma_{x}\tau_{0}-\sigma_{x}\tau_{z}+\sigma_{y}\tau_{0}-\sigma_{y}\tau_{z}\right)+$
$\displaystyle
k_{y}\left(-\sigma_{x}\tau_{0}+\sigma_{x}\tau_{z}+\sigma_{y}\tau_{0}+\sigma_{y}\tau_{z}\right)\biggr{]}+$
$\displaystyle\frac{\alpha_{3}}{\sqrt{2}}\biggl{[}k_{x}\left(\sigma_{x}\tau_{0}-\sigma_{x}\tau_{z}+\sigma_{y}\tau_{0}+\sigma_{y}\tau_{z}\right)+$
$\displaystyle
k_{y}\left(\sigma_{x}\tau_{0}+\sigma_{x}\tau_{z}+\sigma_{y}\tau_{0}-\sigma_{y}\tau_{z}\right)\biggr{]}$
(181)

We only need to look at the $\text{det}H=0$ points, due to the double
degeneracy of the first two bands at some k lines. When $\alpha_{0}=0$, $H=0$
on the $k_{z}$ line, so $\alpha_{0}=0$ divides the topological phase diagram.
Therefore we set $\alpha_{0}=1$, which corresponds to the $\alpha_{0}>0$
region. The Chern numbers we get will either remain unchanged or flip for
$\alpha_{0}<0$. We also first look at the $k_{z}=1$ plane. Then we get

$\displaystyle\text{det}H$
$\displaystyle=A(k_{x},k_{y},r)\sin(P)^{4}+B(k_{x},k_{y},r)\sin(P)^{2}$
$\displaystyle+C(k_{x},k_{y},r)$ (182)

where we parameterized
$\alpha_{1}=r\sin(P),\alpha_{2}=r\cos(P)\cos(R),\alpha_{3}=r\cos(P)\sin(R)$
and $r>0$. We find that the determinant fulfills $B^{2}-4AC\leq 0$. So the
only possible real solutions of $\text{det}H=0$ are when $B^{2}-4AC=0$, which
is the case when $k_{x}=0\lor k_{y}=0$. At those points $\text{det}H=0$ can
not be fulfilled, since
$A(0,k_{y},r)=A(k_{x},0,r)=B(0,k_{y},r)=B(k_{x},0,r)=0$ and $C(0,k_{y},r)\geq
1$ and $C(k_{x},0,r)\geq 1$. The only place left is the $k_{z}=0$ plane. We
parameterize $k_{x}=\cos(G),k_{y}=\sin(G)$, where we used the scaling
properties of $H$. We get

$\displaystyle\text{det}H$
$\displaystyle=r^{4}(-64\sin(G)^{4}\sin(P)^{4}+64\sin(G)^{4}\sin(P)^{2}$
$\displaystyle+64\sin(G)^{2}\sin(P)^{4}-64\sin(G)^{2}\sin(P)^{2}+4)$ (183)

The only real solutions are
$G\in\\{\frac{\pi}{4},\frac{3\pi}{4},\frac{5\pi}{4},\frac{7\pi}{4}\\}$ with
$P=\pm\frac{\pi}{4}$. The topological phase diagram is identical to the one of
the SG 212 $\bar{M}_{6}\bar{M}_{7}$ model, see figure 13, with $\alpha_{0}>0$.
The Chern numbers flip when $\alpha_{0}<0$.

The symmetry eigenvalue jump of $2_{001}$ is always $\pi$. The one of
$4_{001}$ and $2_{010}$ can not be defined, since different eigenvalues are
paired. So we need to use Eq. (18) to derive constraints. We get
$\nu_{12}=0\,\text{mod}\,2$ and $\nu_{12}=2\,\text{mod}\,4$, both fulfilled in
all topological phases of this model.

This model also describes SG 90 $\bar{M}_{6}\bar{M}_{7}$, SG 92
$\bar{M}_{6}\bar{M}_{7}$, SG 94 $\bar{M}_{6}\bar{M}_{7}$, SG 96
$\bar{M}_{6}\bar{M}_{7}$ with SOC. The reflection $k_{x}\leftrightarrow k_{y}$
can be applied, during which Chern numbers are flipped, to get the SG 94
$\bar{A}_{6}\bar{A}_{7}$ model with SOC. Apply the Chern number preserving
rotation $k_{x}\to k_{y}$ and $k_{y}\to-k_{x}$ and set $\alpha_{2}=0$ and
$\alpha_{3}=0$ to get the SG 92 $A_{1}A_{2}$ model without SOC, where the
Chern number with a filling of 2 is just $\nu=-2(-1)^{\theta(\alpha_{0})}$.
The SG 96 $A_{1}A_{2}$ model is found by applying a $k_{y}\to-k_{y}$
reflection and also by setting $\alpha_{2}=0$ and $\alpha_{3}=0$. This flips
the sign of the Chern number.

###  F.8 SG 92 $\bar{A}_{7}\bar{A}_{7}$ model with SOC

This model is symmetric under a time-reversal symmetry $T$, two 2-fold
symmetries $2_{001}$, $2_{010}$ and a 4-fold symmetry $4_{001}$. The generated
Hamiltonian is

$\displaystyle H=$
$\displaystyle\alpha_{0}k_{z}\sigma_{z}\tau_{z}+\alpha_{1}k_{z}\sigma_{x}\tau_{0}+\alpha_{2}k_{z}\sigma_{y}\tau_{0}$
(184)

No Chern numbers can be defined. This Hamiltonian also describes SG 96
$\bar{A}_{7}\bar{A}_{7}$ with SOC. This Hamiltonian must be expanded to
$k^{2}$ to extract Chern numbers. This was done in [103, 40]. One arrives at
$\nu=\pm 4$ for a filling of 2. Other Chern numbers can not be defined due to
degeneracies. See [103] for the exact topological phase diagram.

###  F.9 SG 19 $R_{1}R_{1}$ model without SOC

This model is symmetric under a time-reversal symmetry $T$ and two 2-fold
symmetries $2_{001}$, $2_{010}$. The Hamiltonian is

$\displaystyle H=$ $\displaystyle\alpha k_{x}\sigma_{z}\tau_{z}+\beta
k_{y}\sigma_{z}\tau_{x}+\gamma k_{z}\sigma_{0}\tau_{y}$ (185)

$H$ is always doubly degenerate due to the nodal planes. Therefore we only
need to look for $\text{det}H=0$.

$\displaystyle\text{det}H=(k_{x}^{2}\alpha^{2}+k_{y}^{2}\beta^{2}+k_{z}^{2}\gamma^{2})^{2}$
(186)

This leads to the condition

$\displaystyle k_{x}^{2}\alpha^{2}+k_{y}^{2}\beta^{2}+k_{z}^{2}\gamma^{2}=0$
(187)

There exist ${\bf k}$ points away from ${\bf k}=0$ where this is the case when
$\alpha=0\lor\beta=0\lor\gamma=0$. We get
$\nu_{12}=-2(-1)^{\theta(\alpha)+\theta(\beta)+\theta(\gamma)}$ for the lower
two bands. The symmetry eigenvalue phase jump is always $\pi$ for both
$2_{001}$ and $2_{010}$. Therefore we get the condition
$\nu_{12}=0\,\text{mod}\,2$ from Eq. (19), which is fulfilled.

##  APPENDIX G 3-fold models

Introduce $\kappa=\alpha+i\beta$. Normalize it without loss of generality
$\kappa=e^{i\gamma}$. Then

$\displaystyle H=\begin{pmatrix}0&k_{z}e^{-i\gamma}&-k_{y}e^{i\gamma}\\\
&0&k_{x}e^{-i\gamma}\\\ \ldots&&0\\\ \end{pmatrix}$ (188)

The characteristic polynomial is

$\displaystyle\chi(\lambda)$
$\displaystyle=-\lambda^{3}+\lambda(k_{x}^{2}+k_{y}^{2}+k_{z}^{2})-2k_{x}k_{y}k_{z}\cos(3\gamma)$
$\displaystyle=-\lambda^{3}+3\lambda-2k_{x}k_{y}k_{z}\cos(3\gamma)$ (189)

where we set normalized $k_{x}^{2}+k_{y}^{2}+k_{z}^{2}=3$ using the scale
invariance of $H$. The characteristic polynomial with two degenerate energies
is

$\displaystyle\chi(\lambda)$ $\displaystyle=-(\lambda-E_{1})^{2}(\lambda-
E_{2})$
$\displaystyle=-\lambda^{3}+\lambda^{2}(E_{2}+2E_{1})-\lambda(E_{1}^{2}+2E_{1}E_{2})+E_{1}^{2}E_{2}$
(190)

We see that

$\displaystyle E_{2}$ $\displaystyle=-2E_{1}$ (191) $\displaystyle 3$
$\displaystyle=-E_{1}^{2}-2E_{1}E_{2}=3E_{1}^{2}$ $\displaystyle\implies
E_{1}=\pm 1\land E_{2}=\mp 2.$ (192)

The only solutions to

$\displaystyle k_{x}^{2}k_{y}^{2}k_{z}^{2}\cos(3\gamma)^{2}=1\land
k_{x}^{2}+k_{y}^{2}+k_{z}^{2}=3$ (193)

are

$\displaystyle k_{x/y/z}=\pm 1\land\gamma=\frac{\pi
n}{3}\text{\,with\,}n\in\mathbb{Z}$ (194)

so all lines in $\alpha\beta$ parameter space going from the origin and in
$\frac{\pi}{3}$ angle to each other, starting at the $\beta=0$ line are the
only lines separating different topological phases. We get the phase diagram
for $\nu_{1}$ shown in figure 14, and $\nu_{2}=-\nu_{1}$. The $\nu_{1}=-2$
phases coincide with a symmetry eigenvalue phase jump of
$\Delta\varphi_{1}=2\pi/3$, $\Delta\varphi_{2}=0$ and
$\Delta\varphi_{3}=4\pi/3$ for the 3-fold rotation. The first bands phase jump
leads with Eq. (1) to the constraint $\nu_{1}=1\,\text{mod}\,3$, consistent
with $\nu_{1}=-2$. We get phase jumps of $\Delta\varphi_{1}=4\pi/3$,
$\Delta\varphi_{2}=0$ and $\Delta\varphi_{3}=2\pi/3$ for the $\nu_{1}=2$
phase, which is consistent with the constraint $\nu_{1}=2\,\text{mod}3$
derived from the $\Delta\varphi_{1}$.

Figure 14: Topological phase diagram of band 1 of the SG 199 $\bar{P}_{7}$
model.

##  APPENDIX H 6-fold models

The topological phase diagram for the 6-fold model 76 of irrep
$\bar{R}_{7}\bar{R}_{7}$ is invariant under rescaling
$\alpha_{n}\to\alpha_{n}/r$ with $r>0$. So we can reduce the number of
parameters of the 6-fold model to 3 by choosing an $r$ such that
$(\alpha_{0}+i\alpha_{1})/r=e^{i\phi}$. We rename $\alpha_{2/3}/r$ as
$\alpha_{2/3}$. The parameters $\phi$, $b=\alpha_{2}+i\alpha_{3}$ correspond
to the ones found in [16].

The characteristic polynomial is of the form

$\displaystyle\chi(E)=E^{6}+A_{1}E^{4}+A_{2}E^{2}-\text{det}H.$ (195)

We see that the spectrum must be particle hole symmetric. Due to the nodal
planes, Chern numbers for odd fillings can not be defined. One way a
topological phase transition can happen is by a 6-fold degeneracy. We first
show that the only places, where this can happen is on the nodal planes. The
determinant of $H$ looks like this

$\displaystyle\text{det}H$
$\displaystyle=B(\alpha_{n})k_{x}^{2}k_{y}^{2}k_{z}^{2}.$ (196)

$\text{det}H=0$ is a necessary condition for a 6-fold degeneracy. When $B\neq
0$, this is only the case on the nodal planes. A 6-fold degeneracy and
particle hole symmetry implies $E_{n}=0$. Therefore $A_{1}=0\land A_{2}=0$ on
the nodal plane. We parameterize $k_{x}=\cos(P),k_{y}=\sin(P),k_{z}=0$. Then

$\displaystyle A_{1}$ $\displaystyle=-2(\alpha_{2}^{2}+\alpha_{3}^{2}+1)$
(197) $\displaystyle A_{2}$
$\displaystyle=(\alpha_{2}^{2}+\alpha_{3}^{2}+1)^{2}$ (198)

$A_{1}=0\land A_{2}=0$ is not possible. This is also the case for all other
nodal planes.

Another way a topological phase transition can happen is by a two-fold
degeneracy. At the energy of a two-fold degeneracy, $\chi$ must have a maximum
or minimum with $\chi=0$. With $k_{x}^{2}+k_{y}^{2}+k_{z}^{2}=1$ we find that

$\displaystyle A_{1}$ $\displaystyle=-2(\alpha_{2}^{2}+\alpha_{3}^{2}+1)$
(199) $\displaystyle A_{2}$
$\displaystyle=(\alpha_{2}^{2}+\alpha_{3}^{2}+1)^{2}$ (200)

remains true. This means, $\chi$ has a local maximum at

$\displaystyle
E^{\text{max}}_{\pm}=\pm\sqrt{\frac{\alpha_{2}^{2}+\alpha_{3}^{2}+1}{3}}$
(201)

independent of $a_{0}$ and ${\bf k}$. $\text{det}H$ is tuning the value of
this maximum. Therefore $\text{det}H$ can be tuned such that
$\chi(E^{\text{max}}_{\pm})=0$, where $E^{\text{max}}_{\pm}$ are the energies
of the double degeneracies, in this case of band pairs $(2,3)$ and $(4,5)$. We
get the condition

$\displaystyle\text{det}H=\frac{4}{27}(\alpha_{2}^{2}+\alpha_{3}^{2}+1)^{3}.$
(202)

We find that $\text{det}H$ is

$\displaystyle\text{det}H=$ $\displaystyle
k_{x}^{2}k_{y}^{2}k_{z}^{2}(2\cos(2\phi)-1+\alpha_{2}^{2}+\alpha_{3}^{2})^{2}$
$\displaystyle(2\cos(2\phi)+2+4\alpha_{2}^{2}+4\alpha_{3}^{2})$ (203)

Figure 15: Analytical topological phase diagram for a filling of 2 of the
$\bar{R}_{7}\bar{R}_{7}$ model. All phases separated by a closing gap are
numerated from 1 to 6.

With $F=\alpha_{2}^{2}+\alpha_{3}^{2}+1$ ($1\leq F$), $S=2\cos(2\phi)-2$
($0\geq S\geq-4$) and $A^{-1}=27k_{x}^{2}k_{y}^{2}k_{z}^{2}$ ($1\leq A$) we
have

$\displaystyle(S+F)^{2}(S+4F)=4AF^{3}.$ (204)

The only solutions with $1\leq A$ are $A=1\land S=-3F$ or $A=1\land S=0$.
$A=1$ implies $k_{x}=k_{y}=k_{z}=\pm\frac{1}{\sqrt{3}}$ again. We get
$\cos(2\phi)=\frac{-3\alpha_{2}^{2}-3\alpha_{3}^{2}-1}{2}$ and
$\phi\in\\{0,\pi\\}$ as surfaced in parameter space separating different
topological phases.

Further, $\chi$ has minima at $E^{\text{min}}=0$ and
$E^{\text{min}}_{\pm}=\pm(\alpha_{2}^{2}+\alpha_{3}^{2}+1)$ with
$\chi(E^{\text{min}})=-\text{det}H$. So we once again have the condition
$\text{det}H=0$. The only unexplored way this condition can be fulfilled is by
$B(\alpha_{n})=0$. This leads to another phase separating surface with
$\cos(2\phi)=\frac{-\alpha_{2}^{2}-\alpha_{3}^{2}+1}{2}$, which corresponds to
degeneracies of band pairs $(1,2)$, $(3,4)$ and $(5,6)$. Since $\chi$ is
independent of $k_{n}$ at these parameters, this degeneracy occurs at all $k$
points. It turns out, that this gap closing does not lead to a change in Chern
number. This can be seen in figure 15, where the topological phase diagram of
this model is show and every phase is numerated from 1 to 6. Additionally, for
small off diagonal terms ($\alpha_{2/3}<<1$) the topological phase diagram is
equivalent to a double 3-fold point, as expected, since at $b=0$, $H$ is a
direct sum of two 3-fold points. At large off diagonal terms, this is no
longer the case. This is the only topological phase diagram out of all
multifold crossings, where the topological phase depends on the parameters
relative magnitude, in this case the ratio
$\frac{|\alpha_{0}+i\alpha_{1}|}{|\alpha_{2}+i\alpha_{3}|}$. The Chern number
for the two middle bands is always zero. Following [16], we can get the SG
212/213 $\bar{R}_{7}\bar{R}_{8}$ model by setting $\alpha_{0}=\frac{\pi}{2}$.

The band 1 symmetry eigenvalue jump of the 3-fold rotation is 0 in phases 3
and 6, $\frac{4\pi}{3}$ in phases 1 and 2 and $\frac{2\pi}{3}$ in phases 4 and
5. The band 2 symmetry eigenvalue jump of the 3-fold rotation is 0 in phases 2
and 5, $\frac{4\pi}{3}$ in phases 1 and 3 and $\frac{2\pi}{3}$ in phases 4 and
6. For band 3, we get symmetry eigenvalue jumps of 0 in phases 1 and 4,
$\frac{4\pi}{3}$ in phases 2 and 6 and $\frac{2\pi}{3}$ in phases 3 and 5. For
the remaining bands, particle hole symmetry interchanges
$\frac{2\pi}{3}\leftrightarrow\frac{4\pi}{3}$. We see that we can still
distinguish between phases 1 and 2 (4 and 5) by symmetry eigenvalue jumps.

Bandpairs are degenerate on rotation axis of the two-fold rotation, so Eq. 1
can not be applied here. Considering Eq. 18, all jumps of
$\log\det\mathcal{B}_{C_{2}}$ are zero for all bandpairs. This puts
constraints $\nu_{1,2}=0\,\text{mod }2$ and $\nu_{3,4}=0\,\text{mod }2$ on the
nonabelian Chern numbers. Further, the 3-fold rotation symmetry eigenvalue
jumps lead with Eq. 19 to the following constraints $\nu_{1,2}=1\,\text{mod
}3$ in phases 1, 5 and 6 and $\nu_{1,2}=2\,\text{mod }3$ in phases 2, 3 and 4.
These constraints are consistent with the Chern numbers found in figure 15.
For bands 3 and 4 we get $\nu_{1,2}=0\,\text{mod }3$, which is also fulfilled.

##  APPENDIX I Gellmann matrices

We use the following definition of the Gellmann matrices.

$\displaystyle\lambda_{0}$ $\displaystyle=\begin{pmatrix}1&0&0\\\ 0&-1&0\\\
0&0&0\end{pmatrix}$ (205)

$\displaystyle\lambda_{1}$ $\displaystyle=\begin{pmatrix}0&-i&0\\\ i&0&0\\\
0&0&0\end{pmatrix}$ (206)

$\displaystyle\lambda_{2}$ $\displaystyle=\begin{pmatrix}0&0&-i\\\ 0&0&0\\\
i&0&0\end{pmatrix}$ (207)

$\displaystyle\lambda_{3}$ $\displaystyle=\begin{pmatrix}0&1&0\\\ 1&0&0\\\
0&0&0\end{pmatrix}$ (208)

$\displaystyle\lambda_{4}$
$\displaystyle=\begin{pmatrix}\frac{1}{\sqrt{3}}&0&0\\\
0&\frac{1}{\sqrt{3}}&0\\\ 0&0&-\frac{2}{\sqrt{3}}\end{pmatrix}$ (209)

$\displaystyle\lambda_{5}$ $\displaystyle=\begin{pmatrix}0&0&-i\\\ 0&0&0\\\
i&0&0\end{pmatrix}$ (210)

$\displaystyle\lambda_{6}$ $\displaystyle=\begin{pmatrix}0&0&1\\\ 0&0&0\\\
1&0&0\end{pmatrix}$ (211)

$\displaystyle\lambda_{7}$ $\displaystyle=\begin{pmatrix}0&0&0\\\ 0&0&1\\\
0&1&0\end{pmatrix}$ (212)

$\displaystyle\lambda_{8}$ $\displaystyle=\mathbb{1}$ (213)

##  APPENDIX J Details on the calculation for TaO2 and NbO2

Figure 16: a) Crystal structure of the parent compound NbO2 in SG $\\#136$ and
b) of TaO2 with increased oxygen-deficiency in SG $\\#80$. c) - d) DFT band
structures for the reported oxygen-deficient NbO2, the same structure with Nb
substituted by Ta and the structure with enhanced oxygen-deficiency modeled by
an increased distortion. The topological gaps are colored in red. NbO2 has a
small topological gap that can be slightly enhanced by substitution.
Distorting the material (by a factor of $1.5$ w.r.t. the distortion of the
parent compound in SG $\\#136$ to the reported oxygen-deficient structure in
SG $\\#80$) leads to a band inversion and a significant enhancement of the
topological band gap.

To enlarge the topological band gap, we distort the reported oxygen-deficient
NbO2 structure further, modeling a stronger oxygen-deficiency, as well as
substituting Nb with Ta. We assume that the enhanced deficiency leads to a
distortion that is larger in magnitude but preserves the ratio of the modes it
is composed of. In general, this is a complicated linear combination of many
distortion modes. First, we identify which linear combination of modes leads
to the above-mentioned structural phase transition by using the ISODISTORT
tool.[97] By comparing the parent and the reported oxygen-deficient compound,
we can identify and then exaggerate the distortion by a factor of $1.5$. The
band structure of that structure has a much larger topological band gap while
preserving the symmetry of the reported oxygen-deficient crystal. The
crystallographic axes of the distorted cell are $a_{1}=(-4.847,4.847,2.967)$,
$a_{2}=(4.847,-4.847,2.967)$ and $a_{3}=(4.847,4.847,-2.967)$ in units of
Angstrom. The positions of the atoms are

Atom | $x$ | $y$ | $z$
---|---|---|---
Ta | 0.276080 | 0.033580 | 0.217160
Ta | 0.816420 | 0.058920 | 0.782840
Ta | 0.308920 | 0.026080 | 0.742500
Ta | 0.283580 | 0.566420 | 0.257500
Ta | 0.206250 | 0.448580 | 0.727330
Ta | 0.721250 | 0.478920 | 0.272670
Ta | 0.728920 | 0.956250 | 0.257670
Ta | 0.698580 | 0.471250 | 0.742330
O | 0.097210 | 0.840550 | 0.951600
O | 0.888950 | 0.145610 | 0.048400
O | 0.395610 | 0.847210 | 0.756660
O | 0.090550 | 0.638950 | 0.243340
O | 0.618220 | 0.372050 | 0.977430
O | 0.394620 | 0.640790 | 0.022570
O | 0.890790 | 0.368220 | 0.746170
O | 0.622050 | 0.144620 | 0.253830
O | 0.111550 | 0.357720 | 0.449770
O | 0.907950 | 0.661780 | 0.550230
O | 0.911780 | 0.861550 | 0.253830
O | 0.607720 | 0.657950 | 0.746170
O | 0.595550 | 0.852220 | 0.484930
O | 0.367290 | 0.110620 | 0.515070
O | 0.360620 | 0.345550 | 0.243330
O | 0.102220 | 0.117290 | 0.756670
Table 5: Atomic positions of the distorted $\beta$-TaO2 compound. The
coordinates are given in lattice vectors $a_{1}=(-4.847,4.847,2.967)$,
$a_{2}=(4.847,-4.847,2.967)$ and $a_{3}=(4.847,4.847,-2.967)$ in units of
Angstrom.

##  APPENDIX K Details on DFT calculations

The DFT calculations have been performed using the VASP software[104, 105,
106] and Quantum Espresso [87, 107] with optimized norm-conserving Vanderbilt
pseudopotentials [108] from PseudoDojo [109] within the PBE approximation[110]
of the exchange-correlation functional. For wannierization Wannier90[111] has
been employment and the surface simulation was carried out using
Wanniertools.[112]

## References

  * Chiu  _et al._ [2016] C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Classification of topological quantum matter with symmetries, Rev. Mod. Phys. 88, 035005 (2016).
  * Armitage  _et al._ [2018] N. P. Armitage, E. J. Mele, and A. Vishwanath, Weyl and dirac semimetals in three-dimensional solids, Rev. Mod. Phys. 90, 015001 (2018).
  * Bernevig  _et al._ [2018] A. Bernevig, H. Weng, Z. Fang, and X. Dai, Recent progress in the study of topological semimetals, Journal of the Physical Society of Japan 87, 041001 (2018).
  * Bernevig  _et al._ [2022] B. A. Bernevig, C. Felser, and H. Beidenkopf, Progress and prospects in magnetic topological materials, Nature 603, 41 (2022).
  * Gao  _et al._ [2019] H. Gao, J. W. Venderbos, Y. Kim, and A. M. Rappe, Topological semimetals from first principles, Annual Review of Materials Research 49, 153 (2019), https://doi.org/10.1146/annurev-matsci-070218-010049 .
  * Lv  _et al._ [2021] B. Q. Lv, T. Qian, and H. Ding, Experimental perspective on three-dimensional topological semimetals, Rev. Mod. Phys. 93, 025002 (2021).
  * Wan  _et al._ [2011] X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, Topological semimetal and fermi-arc surface states in the electronic structure of pyrochlore iridates, Phys. Rev. B 83, 205101 (2011).
  * Xu  _et al._ [2011] G. Xu, H. Weng, Z. Wang, X. Dai, and Z. Fang, Chern semimetal and the quantized anomalous hall effect in hgcr 2 se 4, Physical review letters 107, 186806 (2011).
  * Soluyanov  _et al._ [2015] A. A. Soluyanov, D. Gresch, Z. Wang, Q. Wu, M. Troyer, X. Dai, and B. A. Bernevig, Type-ii weyl semimetals, Nature 527, 495 (2015).
  * Yan and Felser [2017] B. Yan and C. Felser, Topological materials: Weyl semimetals, Annual Review of Condensed Matter Physics 8, 337 (2017), https://doi.org/10.1146/annurev-conmatphys-031016-025458 .
  * Chang  _et al._ [2018] G. Chang, B. J. Wieder, F. Schindler, D. S. Sanchez, I. Belopolski, S.-M. Huang, B. Singh, D. Wu, T.-R. Chang, T. Neupert, _et al._ , Topological quantum properties of chiral crystals, Nature materials 17, 978 (2018).
  * Yu  _et al._ [2019] Z.-M. Yu, W. Wu, Y. Zhao, and S. A. Yang, Circumventing the no-go theorem: A single weyl point without surface fermi arcs, Physical Review B 100, 041118 (2019).
  * Xiao  _et al._ [2020] M. Xiao, L. Ye, C. Qiu, H. He, Z. Liu, and S. Fan, Experimental demonstration of acoustic semimetal with topologically charged nodal surface, Science advances 6, eaav2360 (2020).
  * Wilde  _et al._ [2021] M. A. Wilde, M. Dodenhöft, A. Niedermayr, A. Bauer, M. M. Hirschmann, K. Alpin, A. P. Schnyder, and C. Pfleiderer, Symmetry-enforced topological nodal planes at the fermi surface of a chiral magnet, Nature 594, 374 (2021).
  * Huber  _et al._ [2022] N. Huber, K. Alpin, G. L. Causer, L. Worch, A. Bauer, G. Benka, M. M. Hirschmann, A. P. Schnyder, C. Pfleiderer, and M. A. Wilde, Network of topological nodal planes, multifold degeneracies, and weyl points in cosi, Physical Review Letters 129, 026401 (2022).
  * Bradlyn  _et al._ [2016] B. Bradlyn, J. Cano, Z. Wang, M. Vergniory, C. Felser, R. J. Cava, and B. A. Bernevig, Beyond dirac and weyl fermions: Unconventional quasiparticles in conventional crystals, Science 353 (2016).
  * Tang  _et al._ [2017] P. Tang, Q. Zhou, and S.-C. Zhang, Multiple types of topological fermions in transition metal silicides, Physical review letters 119, 206402 (2017).
  * Schröter  _et al._ [2019] N. Schröter, D. Pei, M. G. Vergniory, Y. Sun, K. Manna, F. De Juan, J. Krieger, V. Süss, M. Schmidt, P. Dudin, _et al._ , Chiral topological semimetal with multifold band crossings and long fermi arcs, Nature Physics 15, 759 (2019).
  * Cano  _et al._ [2019] J. Cano, B. Bradlyn, and M. Vergniory, Multifold nodal points in magnetic materials, APL Materials 7, 101125 (2019).
  * Ong and Liang [2021] N. P. Ong and S. Liang, Experimental signatures of the chiral anomaly in dirac–weyl semimetals, Nature Reviews Physics 3, 394 (2021).
  * Hosur and Qi [2013] P. Hosur and X. Qi, Recent developments in transport phenomena in weyl semimetals, Comptes Rendus Physique 14, 857 (2013), topological insulators / Isolants topologiques.
  * Burkov [2015] A. A. Burkov, Chiral anomaly and transport in weyl metals, Journal of Physics: Condensed Matter 27, 113201 (2015).
  * Liang  _et al._ [2015] T. Liang, Q. Gibson, M. N. Ali, M. Liu, R. J. Cava, and N. P. Ong, Ultrahigh mobility and giant magnetoresistance in the dirac semimetal cd3as2, Nature Materials 14, 280 (2015).
  * de Juan  _et al._ [2017] F. de Juan, A. G. Grushin, T. Morimoto, and J. E. Moore, Quantized circular photogalvanic effect in weyl semimetals, Nature Communications 8, 15995 (2017).
  * Flicker  _et al._ [2018] F. Flicker, F. de Juan, B. Bradlyn, T. Morimoto, M. G. Vergniory, and A. G. Grushin, Chiral optical response of multifold fermions, Phys. Rev. B 98, 155145 (2018).
  * Ni  _et al._ [2021] Z. Ni, K. Wang, Y. Zhang, O. Pozo, B. Xu, X. Han, K. Manna, J. Paglione, C. Felser, A. G. Grushin, F. de Juan, E. J. Mele, and L. Wu, Giant topological longitudinal circular photo-galvanic effect in the chiral multifold semimetal cosi, Nature Communications 12, 154 (2021).
  * Liu  _et al._ [2019] P. Liu, J. R. Williams, and J. J. Cha, Topological nanomaterials, Nature Reviews Materials 4, 479 (2019).
  * Burkov [2016] A. A. Burkov, Topological semimetals, Nature Materials 15, 1145 (2016).
  * Schoop  _et al._ [2018] L. M. Schoop, A. Topp, J. Lippmann, F. Orlandi, L. Müchler, M. G. Vergniory, Y. Sun, A. W. Rost, V. Duppel, M. Krivenkov, S. Sheoran, P. Manuel, A. Varykhalov, B. Yan, R. K. Kremer, C. R. Ast, and B. V. Lotsch, Tunable weyl and dirac states in the nonsymmorphic compound cesbte, Science Advances 4, eaar2317 (2018), https://www.science.org/doi/pdf/10.1126/sciadv.aar2317 .
  * Hirschmann  _et al._ [2022] M. M. Hirschmann, A. S. Gibbs, F. Orlandi, D. Khalyavin, P. Manuel, V. Abdolazimi, A. Yaresko, J. Nuss, H. Takagi, A. P. Schnyder, and A. W. Rost, Creating and controlling dirac fermions, weyl fermions, and nodal lines in the magnetic antiperovskite ${\mathrm{eu}}_{3}\mathrm{PbO}$, Phys. Rev. Mater. 6, 114202 (2022).
  * Zhao and Schnyder [2016] Y. X. Zhao and A. P. Schnyder, Nonsymmorphic symmetry-required band crossings in topological semimetals, Phys. Rev. B 94, 195109 (2016).
  * Nielsen and Ninomiya [1983] H. Nielsen and M. Ninomiya, The adler-bell-jackiw anomaly and weyl fermions in a crystal, Physics Letters B 130, 389 (1983).
  * Watanabe  _et al._ [2016] H. Watanabe, H. C. Po, M. P. Zaletel, and A. Vishwanath, Filling-enforced gaplessness in band structures of the 230 space groups, Phys. Rev. Lett. 117, 096404 (2016).
  * Young and Kane [2015] S. M. Young and C. L. Kane, Dirac semimetals in two dimensions, Phys. Rev. Lett. 115, 126803 (2015).
  * Takahashi  _et al._ [2017] R. Takahashi, M. Hirayama, and S. Murakami, Spinless hourglass nodal-line semimetals, Phys. Rev. B 96, 155206 (2017).
  * Young and Wieder [2017] S. M. Young and B. J. Wieder, Filling-enforced magnetic dirac semimetals in two dimensions, Phys. Rev. Lett. 118, 186401 (2017).
  * Zhang  _et al._ [2018] J. Zhang, Y.-H. Chan, C.-K. Chiu, M. G. Vergniory, L. M. Schoop, and A. P. Schnyder, Topological band crossings in hexagonal materials, Phys. Rev. Mater. 2, 074201 (2018).
  * Chan  _et al._ [2019] Y.-H. Chan, B. Kilic, M. M. Hirschmann, C.-K. Chiu, L. M. Schoop, D. G. Joshi, and A. P. Schnyder, Symmetry-enforced band crossings in trigonal materials: Accordion states and weyl nodal lines, Phys. Rev. Mater. 3, 124204 (2019).
  * Leonhardt  _et al._ [2021] A. Leonhardt, M. M. Hirschmann, N. Heinsdorf, X. Wu, D. H. Fabini, and A. P. Schnyder, Symmetry-enforced topological band crossings in orthorhombic crystals: Classification and materials discovery, Phys. Rev. Mater. 5, 124202 (2021).
  * Hirschmann  _et al._ [2021] M. M. Hirschmann, A. Leonhardt, B. Kilic, D. H. Fabini, and A. P. Schnyder, Symmetry-enforced band crossings in tetragonal materials: Dirac and weyl degeneracies on points, lines, and planes, Phys. Rev. Materials 5, 054202 (2021).
  * Yu  _et al._ [2022] Z.-M. Yu, Z. Zhang, G.-B. Liu, W. Wu, X.-P. Li, R.-W. Zhang, S. A. Yang, and Y. Yao, Encyclopedia of emergent particles in three-dimensional crystals, Science Bulletin 67, 375 (2022).
  * Xie  _et al._ [2021] Y.-M. Xie, X.-J. Gao, X. Y. Xu, C.-P. Zhang, J.-X. Hu, J. Z. Gao, and K. T. Law, Kramers nodal line metals, Nature communications 12, 1 (2021).
  * Tang and Wan [2021a] F. Tang and X. Wan, Exhaustive construction of effective models in 1651 magnetic space groups, Phys. Rev. B 104, 085137 (2021a).
  * Tang and Wan [2022] F. Tang and X. Wan, Complete classification of band nodal structures and massless excitations, Phys. Rev. B 105, 155156 (2022).
  * Knoll and Timm [2022] A. Knoll and C. Timm, Classification of weyl points and nodal lines based on magnetic point groups for spin-$\frac{1}{2}$ quasiparticles, Phys. Rev. B 105, 115109 (2022).
  * Fang  _et al._ [2012a] C. Fang, M. J. Gilbert, X. Dai, and B. A. Bernevig, Multi-weyl topological semimetals stabilized by point group symmetry, Phys. Rev. Lett. 108, 266802 (2012a).
  * Tsirkin  _et al._ [2017] S. S. Tsirkin, I. Souza, and D. Vanderbilt, Composite weyl nodes stabilized by screw symmetry with and without time-reversal invariance, Phys. Rev. B 96, 045102 (2017).
  * González-Hernández  _et al._ [2021] R. González-Hernández, E. Tuiran, and B. Uribe, Chiralities of nodal points along high-symmetry lines with screw rotation symmetry, Phys. Rev. B 103, 235143 (2021).
  * Schröter  _et al._ [2020] N. B. M. Schröter, S. Stolz, K. Manna, F. de Juan, M. G. Vergniory, J. A. Krieger, D. Pei, T. Schmitt, P. Dudin, T. K. Kim, C. Cacho, B. Bradlyn, H. Borrmann, M. Schmidt, R. Widmer, V. N. Strocov, and C. Felser, Observation and control of maximal chern numbers in a chiral topological semimetal, Science 369, 179 (2020), https://www.science.org/doi/pdf/10.1126/science.aaz3480 .
  * Yao  _et al._ [2020] M. Yao, K. Manna, Q. Yang, A. Fedorov, V. Voroshnin, B. Valentin Schwarze, J. Hornung, S. Chattopadhyay, Z. Sun, S. N. Guin, _et al._ , Observation of giant spin-split fermi-arc with maximal chern number in the chiral topological semimetal ptga, Nature communications 11, 1 (2020).
  * Hughes  _et al._ [2011] T. L. Hughes, E. Prodan, and B. A. Bernevig, Inversion-symmetric topological insulators, Phys. Rev. B 83, 245132 (2011).
  * Fang  _et al._ [2012b] C. Fang, M. J. Gilbert, and B. A. Bernevig, Bulk topological invariants in noninteracting point group symmetric insulators, Phys. Rev. B 86, 115112 (2012b).
  * Alexandradinata  _et al._ [2014] A. Alexandradinata, X. Dai, and B. A. Bernevig, Wilson-loop characterization of inversion-symmetric topological insulators, Phys. Rev. B 89, 155114 (2014).
  * Bouhon and Black-Schaffer [2017] A. Bouhon and A. M. Black-Schaffer, Global band topology of simple and double dirac-point semimetals, Phys. Rev. B 95, 241101 (2017).
  * Vanderbilt [2018] D. Vanderbilt,  _Berry phases in electronic structure theory: electric polarization, orbital magnetization and topological insulators_ (Cambridge University Press, 2018).
  * Soluyanov and Vanderbilt [2012] A. A. Soluyanov and D. Vanderbilt, Smooth gauge for topological insulators, Physical Review B 85, 115415 (2012).
  * Xiao  _et al._ [2010] D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on electronic properties, Reviews of modern physics 82, 1959 (2010).
  * Furusaki [2017] A. Furusaki, Weyl points and dirac lines protected by multiple screw rotations, Science Bulletin 62, 788 (2017).
  * Huang  _et al._ [2016] S.-M. Huang, S.-Y. Xu, I. Belopolski, C.-C. Lee, G. Chang, T.-R. Chang, B. Wang, N. Alidoust, G. Bian, M. Neupane, D. Sanchez, H. Zheng, H.-T. Jeng, A. Bansil, T. Neupert, H. Lin, and M. Z. Hasan, New type of weyl semimetal with quadratic double weyl fermions, Proceedings of the National Academy of Sciences 113, 1180 (2016), https://www.pnas.org/doi/pdf/10.1073/pnas.1514581113 .
  * Tan  _et al._ [2022] W. Tan, X. Jiang, Y. Li, X. Wu, J. Wang, and B. Huang, A unified understanding of diverse spin textures of kramers–weyl fermions in nonmagnetic chiral crystals, Advanced Functional Materials , 2208023 (2022).
  * Chen  _et al._ [2022] Q. Chen, F. Chen, Y. Pan, C. Cui, Q. Yan, L. Zhang, Z. Gao, S. A. Yang, Z.-M. Yu, H. Chen, _et al._ , Discovery of a maximally charged weyl point, Nature Communications 13, 7359 (2022).
  * Feng  _et al._ [2016] B. Feng, Y.-H. Chan, Y. Feng, R.-Y. Liu, M.-Y. Chou, K. Kuroda, K. Yaji, A. Harasawa, P. Moras, A. Barinov, _et al._ , Spin texture in type-ii weyl semimetal wte 2, Physical Review B 94, 195134 (2016).
  * Marković  _et al._ [2019] I. Marković, C. Hooley, O. J. Clark, F. Mazzola, M. D. Watson, J. M. Riley, K. Volckaert, K. Underwood, M. Dyer, P. Murgatroyd, _et al._ , Weyl-like points from band inversions of spin-polarised surface states in nbgesb, Nature Communications 10, 5485 (2019).
  * Xu  _et al._ [2016] S.-Y. Xu, I. Belopolski, D. S. Sanchez, M. Neupane, G. Chang, K. Yaji, Z. Yuan, C. Zhang, K. Kuroda, G. Bian, _et al._ , Spin polarization and texture of the fermi arcs in the weyl fermion semimetal taas, Physical review letters 116, 096801 (2016).
  * Gatti  _et al._ [2020] G. Gatti, D. Gosálbez-Martínez, S. Tsirkin, M. Fanciulli, M. Puppin, S. Polishchuk, S. Moser, L. Testa, E. Martino, S. Roth, _et al._ , Radial spin texture of the weyl fermions in chiral tellurium, Physical review letters 125, 216402 (2020).
  * He  _et al._ [2021] W.-Y. He, X. Y. Xu, and K. T. Law, Kramers weyl semimetals as quantum solenoids and their applications in spin-orbit torque devices, Communications Physics 4, 66 (2021).
  * Hagiwara  _et al._ [2022] K. Hagiwara, P. Rüßmann, X. L. Tan, Y.-J. Chen, K. Ueno, V. Feyer, G. Zamborlini, M. Jugovac, S. Suga, S. Blügel, _et al._ , Link between weyl-fermion chirality and spin texture, arXiv preprint arXiv:2205.15252 (2022).
  * Zhang  _et al._ [2020] T. Zhang, R. Takahashi, C. Fang, and S. Murakami, Twofold quadruple weyl nodes in chiral cubic crystals, Phys. Rev. B 102, 125148 (2020).
  * Li  _et al._ [2021] H. Li, T. Zhang, A. Said, Y. Fu, G. Fabbris, D. G. Mazzone, J. Zhang, J. Lapano, H. N. Lee, H. C. Lei, M. P. M. Dean, S. Murakami, and H. Miao, Observation of a chiral wave function in the twofold-degenerate quadruple weyl system baptge, Phys. Rev. B 103, 184301 (2021).
  * Cui  _et al._ [2021] C. Cui, X.-P. Li, D.-S. Ma, Z.-M. Yu, and Y. Yao, Charge-four weyl point: Minimum lattice model and chirality-dependent properties, Phys. Rev. B 104, 075115 (2021).
  * Luo  _et al._ [2022] L. Luo, W. Deng, Y. Yang, M. Yan, J. Lu, X. Huang, and Z. Liu, Observation of quadruple weyl point in hybrid-weyl phononic crystals (2022).
  * Elcoro  _et al._ [2017] L. Elcoro, B. Bradlyn, Z. Wang, M. G. Vergniory, J. Cano, C. Felser, B. A. Bernevig, D. Orobengoa, G. Flor, and M. I. Aroyo, Double crystallographic groups and their representations on the bilbao crystallographic server, Journal of Applied Crystallography 50, 1457 (2017).
  * Chiu  _et al._ [2018] P.-M. Chiu, C.-Y. Huang, W.-J. Li, and T.-K. Lee, Emergence of topological phases by stacking of two-dimensional lattices with nonsymmorphic symmetry, Journal of Physics: Condensed Matter 31, 035501 (2018).
  * González-Hernández  _et al._ [2020] R. González-Hernández, E. Tuiran, and B. Uribe, Topological electronic structure and weyl points in nonsymmorphic hexagonal materials, Physical Review Materials 4, 124203 (2020).
  * Türker and Moroz [2018] O. b. u. Türker and S. Moroz, Weyl nodal surfaces, Phys. Rev. B 97, 075120 (2018).
  * Wu  _et al._ [2018a] W. Wu, Y. Liu, S. Li, C. Zhong, Z.-M. Yu, X.-L. Sheng, Y. Zhao, and S. A. Yang, Nodal surface semimetals: Theory and material realization, Physical Review B 97, 115125 (2018a).
  * Aroyo  _et al._ [2006] M. I. Aroyo, A. Kirov, C. Capillas, J. Perez-Mato, and H. Wondratschek, Bilbao crystallographic server. ii. representations of crystallographic point groups and space groups, Acta Crystallographica Section A: Foundations of Crystallography 62, 115 (2006).
  * Tang and Wan [2021b] F. Tang and X. Wan, Exhaustive construction of effective models in 1651 magnetic space groups, Physical Review B 104, 085137 (2021b).
  * Luttinger [1956] J. M. Luttinger, Quantum theory of cyclotron resonance in semiconductors: General theory, Physical review 102, 1030 (1956).
  * Gresch [2018] D. Gresch,  _Identifying Topological Semimetals_ , Ph.D. thesis, ETH Zurich (2018).
  * Jiang  _et al._ [2021] Y. Jiang, Z. Fang, and C. Fang, A k· p effective hamiltonian generator, Chinese Physics Letters 38, 077104 (2021).
  * Rarita and Schwinger [1941] W. Rarita and J. Schwinger, On a theory of particles with half-integral spin, Physical Review 60, 61 (1941).
  * Ezawa [2016] M. Ezawa, Pseudospin-$\frac{3}{2}$ fermions, type-ii weyl semimetals, and critical weyl semimetals in tricolor cubic lattices, Phys. Rev. B 94, 195205 (2016).
  * Liang and Yu [2016] L. Liang and Y. Yu, Semimetal with both rarita-schwinger-weyl and weyl excitations, Phys. Rev. B 93, 045113 (2016).
  * Chang  _et al._ [2017] G. Chang, S.-Y. Xu, B. J. Wieder, D. S. Sanchez, S.-M. Huang, I. Belopolski, T.-R. Chang, S. Zhang, A. Bansil, H. Lin, _et al._ , Unconventional chiral fermions and large topological fermi arcs in rhsi, Physical review letters 119, 206401 (2017).
  * Jain  _et al._ [2013] A. Jain, S. P. Ong, G. Hautier, W. Chen, W. D. Richards, S. Dacek, S. Cholia, D. Gunter, D. Skinner, G. Ceder, and K. a. Persson, The Materials Project: A materials genome approach to accelerating materials innovation, APL Materials 1, 011002 (2013).
  * Giannozzi  _et al._ [2009] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, _et al._ , Quantum espresso: a modular and open-source software project for quantum simulations of materials, Journal of physics: Condensed matter 21, 395502 (2009).
  * Brauer [1941] G. Brauer, Die oxyde des niobs, Zeitschrift für anorganische und allgemeine Chemie 248, 1 (1941).
  * Magneli  _et al._ [1955] A. Magneli, G. Andersson, and G. Sundkvist, Note on the crystal structure of niobium dioxide, Acta Chemica Scandinavica (Denmark) Divided into Acta Chem. Scand., Ser. A and Ser. B 9 (1955).
  * O’Hara  _et al._ [2014] A. O’Hara, T. N. Nunley, A. B. Posadas, S. Zollner, and A. A. Demkov, Electronic and optical properties of nbo2, Journal of Applied Physics 116, 213705 (2014).
  * Seta and Naito [1982] K. Seta and K. Naito, Calorimetric study of the phase transition in nbo2, The Journal of Chemical Thermodynamics 14, 921 (1982).
  * Shapiro  _et al._ [1974] S. Shapiro, J. Axe, G. Shirane, and P. Raccah, Neutron scattering study of the structural phase transition in nbo2, Solid State Communications 15, 377 (1974).
  * Rao  _et al._ [1973] C. Rao, G. R. Rao, and G. S. Rao, Semiconductor-metal transitions in nbo2 and nb1- xvxo2, Journal of Solid State Chemistry 6, 340 (1973).
  * Jacob  _et al._ [2010] K. Jacob, C. Shekhar, M. Vinay, and Y. Waseda, Thermodynamic properties of niobium oxides, Journal of Chemical & Engineering Data 55, 4854 (2010).
  * Schweizer and Gruehn [1982] H.-J. Schweizer and R. Gruehn, Zur darstellung und kristallstruktur von $\beta$-nbo2/synthesis and crystal structure of $\beta$-nbo2, Zeitschrift für Naturforschung B 37, 1361 (1982).
  * Schulz  _et al._ [2017] K. J. Schulz, N. M. Piatak, and J. F. Papp,  _Niobium and tantalum_ , Tech. Rep. (US Geological Survey, 2017).
  * Stokes  _et al._ [2006] H. T. Stokes, D. M. Hatch, B. J. Campbell, and D. E. Tanner, Isodisplace: a web-based tool for exploring structural distortions, Journal of Applied Crystallography 39, 607 (2006).
  * Setyawan and Curtarolo [2010] W. Setyawan and S. Curtarolo, High-throughput electronic band structure calculations: Challenges and tools, Computational materials science 49, 299 (2010).
  * Wang  _et al._ [2016] Z. Wang, D. Gresch, A. A. Soluyanov, W. Xie, S. Kushwaha, X. Dai, M. Troyer, R. J. Cava, and B. A. Bernevig, ${\mathrm{mote}}_{2}$: A type-ii weyl topological metal, Phys. Rev. Lett. 117, 056805 (2016).
  * Belopolski  _et al._ [2017] I. Belopolski, P. Yu, D. S. Sanchez, Y. Ishida, T.-R. Chang, S. S. Zhang, S.-Y. Xu, H. Zheng, G. Chang, G. Bian, _et al._ , Signatures of a time-reversal symmetric weyl semimetal with only four weyl points, Nature communications 8, 942 (2017).
  * Fukui  _et al._ [2005] T. Fukui, Y. Hatsugai, and H. Suzuki, Chern numbers in discretized brillouin zone: efficient method of computing (spin) hall conductances, Journal of the Physical Society of Japan 74, 1674 (2005).
  * Aroyo  _et al._ [2014] M. I. Aroyo, D. Orobengoa, G. de la Flor, E. S. Tasci, J. M. Perez-Mato, and H. Wondratschek, Brillouin-zone database on the Bilbao Crystallographic Server, Acta Crystallographica Section A 70, 126 (2014).
  * Wu  _et al._ [2020] W. Wu, Z.-M. Yu, X. Zhou, Y. Zhao, and S. A. Yang, Higher-order dirac fermions in three dimensions, Physical Review B 101, 205134 (2020).
  * Kresse and Hafner [1993] G. Kresse and J. Hafner, Ab initio molecular dynamics for liquid metals, Phys. Rev. B 47, 558 (1993).
  * Kresse and Furthmüller [1996] G. Kresse and J. Furthmüller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set, Computational materials science 6, 15 (1996).
  * Kresse and Furthmüller [1996] G. Kresse and J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54, 11169 (1996).
  * Giannozzi  _et al._ [2017] P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. B. Nardelli, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, M. Cococcioni, _et al._ , Advanced capabilities for materials modelling with quantum espresso, Journal of physics: Condensed matter 29, 465901 (2017).
  * Hamann [2013] D. Hamann, Optimized norm-conserving vanderbilt pseudopotentials, Physical Review B 88, 085117 (2013).
  * van Setten  _et al._ [2018] M. J. van Setten, M. Giantomassi, E. Bousquet, M. J. Verstraete, D. R. Hamann, X. Gonze, and G.-M. Rignanese, The pseudodojo: Training and grading a 85 element optimized norm-conserving pseudopotential table, Computer Physics Communications 226, 39 (2018).
  * Perdew  _et al._ [1996] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996).
  * Mostofi  _et al._ [2014] A. A. Mostofi, J. R. Yates, G. Pizzi, Y.-S. Lee, I. Souza, D. Vanderbilt, and N. Marzari, An updated version of wannier90: A tool for obtaining maximally-localised wannier functions, Computer Physics Communications 185, 2309 (2014).
  * Wu  _et al._ [2018b] Q. Wu, S. Zhang, H.-F. Song, M. Troyer, and A. A. Soluyanov, Wanniertools : An open-source software package for novel topological materials, Computer Physics Communications 224, 405 (2018b).