# Weyl-point teleportation

György Frank  Department of Theoretical Physics, Budapest University of
Technology and Economics, Hungary  MTA-BME Exotic Quantum Phases Group,
Budapest University of Technology and Economics, Hungary  Dániel Varjas
Department of Physics, Stockholm University, AlbaNova University Center, 106
91 Stockholm, Sweden  Gergő Pintér  Department of Theoretical Physics,
Budapest University of Technology and Economics, Hungary  MTA-BME Exotic
Quantum Phases Group, Budapest University of Technology and Economics, Hungary
András Pályi  Department of Theoretical Physics, Budapest University of
Technology and Economics, Hungary  MTA-BME Exotic Quantum Phases Group,
Budapest University of Technology and Economics, Hungary

###### Abstract

In this work, we describe the phenomenon of Weyl-point teleportation. Weyl
points usually move continuously in the configuration parameter space of a
quantum system when the control parameters are varied continuously. However,
there are special transition points in the control space where the continuous
motion of the Weyl points is disrupted. In such transition points, an extended
nodal structure (nodal line or nodal surface) emerges, serving as a wormhole
for the Weyl points, allowing their teleportation in the configuration space.
A characteristic side effect of the teleportation is that the motional
susceptibility of the Weyl point diverges in the vicinity of the transition
point, and this divergence is characterized by a universal scaling law. We
exemplify these effects via a two-spin model and a Weyl Josephson circuit
model. We expect that these effects generalize to many other settings
including electronic band structures of topological semimetals.

_Introduction._ Electronic band structures of crystalline materials often
exhibit band touching points [1, 2, 3], leading to characteristic phenomena
such as surface Fermi arcs [4], chiral anomaly, anomalous Hall effect, quantum
oscillations, and quantized circular photogalvanic effect [5, 6]. The generic
pattern for band touching is the Weyl point, where the dispersion relation in
the vicinity of the degeneracy point starts linearly in all directions.

However, for many materials, the spatial symmetry of the crystal structure
implies that band touching can happen in the form of extended nodal
structures, such as nodal lines or nodal surfaces [7, 8, 9, 10, 11, 12, 13,
14, 15, 16, 17, 18, 19, 20, 21]. Furthermore, nodal lines and nodal surfaces
can also arise in parameter-dependent quantum systems [22, 23, 24] in the
presence of fine-tuning or symmetries. For brevity, we will use ‘fine-tuned’
to describe both of these scenarios.

Generic perturbations with respect to the fine-tuned setting will necessarily
dissolve the extended nodal structures into a number of Weyl points. For
example, mechanical strain can break the symmetry of a crystal and thereby
split a nodal line into Weyl points [14]. Details of such a dissolution
process have various physical implications, e.g., a qualitative change in the
density of states and the properties of the surface states, etc. In this work
we uncover general properties of this dissolution process.

In the above setting, the six independent components of the mechanical strain
tensor provide examples of _control parameters_. We will refer to the fine-
tuned control parameter point where the extended nodal structure appears as
the _transition point_. Furthermore, we will call the parameter space in which
the degeneracy structures live, i.e., the three-dimensional momentum space in
the above example, the _configuration space_.

In this work, our main observation is that a nodal loop or nodal surface
emerging at a transition point can be thought of as a ‘wormhole’ for Weyl
points, allowing for the ‘teleportation’ of Weyl points. A side effect of
teleportation is that the motion of the Weyl points become singular as the
control parameters approach the transition point: an infinitesimal change in
the control parameters induces a macroscopic displacement of the Weyl point in
the configuration space.

Figure 1: Teleportation of magnetic Weyl points of a spin-orbit-coupled double
quantum dot. (a) Double quantum dot hosting two interacting electrons, in the
presence of a Zeeman field $B$, and spin-orbit interaction induced by the
electric field $E$. (b) Trajectory of the control parameter vector, i.e., the
electric field in the xy plane. The trajectory is parametrized by $t\in[-1,1]$
with $\epsilon=1\text{V}/\mu\text{m}$ (see text). (c,d,e) Evolution of ground-
state Weyl points in the configuration space ($B$ space), as the control
parameter vector is varied according to panel (b). Parameters: $g=2$,
$J=11.6\,\mu\text{eV}$. (c) Weyl points for $t<0$. (d) Nodal surface (sphere)
at $t=0$ (black circle in panel b). Teleportation of Weyl points happens at
this point of the control-parameter trajectory. (e) Weyl points for $t>0$.

_Teleportation of magnetic Weyl points of a spin-orbit-coupled double quantum
dot._ We illustrate Weyl-point teleportation using an elementary model of a
two-electron double quantum dot with spin-orbit interaction. The setup is
shown in Fig. 1a. It consists of a cylindrically symmetric semiconducting
nanowire (blue) where gate electrodes create a double-well potential (solid
red), with each well capturing a single spinful electron (red clouds). The two
localized spins interact with each other via exchange interaction.

We assume that an external homogeneous electric field
$\text{\boldmath{$E$}}=(E_{x},E_{y},0)$ breaks the cylindrical symmetry of the
system, and induces an anisotropy in the Heisenberg interaction among the
spins via the spin-orbit interaction of the Rashba type. The two-dimensional
space of $E$ serves as the control space. A homogeneous external magnetic
field $B$ is also present in the setup, taken into account by a Zeeman
interaction term in the model. The three-dimensional space of $B$ serves as
the configuration space.

The two-electron system is described by the following $4\times 4$ Hamiltonian:

$\hat{H}_{\text{DD}}=g\mu_{B}\text{\boldmath{{\hbox{ B
}}}}\cdot(\hat{\text{\boldmath{{\hbox{ S
}}}}}_{L}+\hat{\text{\boldmath{{\hbox{ S
}}}}}_{R})+J\hat{\text{\boldmath{{\hbox{ S }}}}}_{L}\cdot
R(\text{\boldmath{{\hbox{ E }}}})\hat{\text{\boldmath{{\hbox{ S }}}}}_{R}.$
(1)

Here $\hat{\text{\boldmath{$S$}}}_{L/R}$ is the spin operator of the
left/right electron, $g$ is the $g$-factor of the electrons, $\mu_{B}$ is the
Bohr magneton, $B$ is the external magnetic field, $J$ is the exchange
interaction strength, and $R(\text{\boldmath{$E$}})$ is a $3\times 3$ rotation
matrix describing the effect of spin-orbit interaction on the exchange
interaction [25, 26].

The exchange term in Eq. (1) can be derived from the Rashba spin-orbit
Hamiltonian and the corresponding two-site Hubbard model. Here, we outline the
simple physical picture that predicts the qualitative dependence of the
rotation matrix $R$ on the electric field $E$. Assume that the electric field
creates a spin-orbit term of the Rashba type,
$H_{\text{so}}=\alpha\hat{\text{\boldmath{$S$}}}\cdot\left(\text{\boldmath{$E$}}\times\hat{\text{\boldmath{$p$}}}\right)$
[27]. When an electron tunnels from the right dot to the left
($\text{\boldmath{$p$}}||-\text{\boldmath{$e$}}_{z}$), it feels a spin-orbit
magnetic field
$\text{\boldmath{$B$}}_{\text{so}}\propto\text{\boldmath{$e$}}_{z}\times\text{\boldmath{$E$}}$,
and hence its spin rotates around $\text{\boldmath{$B$}}_{\text{so}}$ with an
angle $\theta=\gamma E$ proportional to the electric field causing the spin-
orbit coupling. This effect is incorporated in Eq. (1) as the matrix $R$:

$\displaystyle R(\text{\boldmath{$E$}})$ $\displaystyle=$
$\displaystyle\exp{\left[\gamma\begin{pmatrix}0&0&E_{x}\\\ 0&0&E_{y}\\\
-E_{x}&-E_{y}&0\end{pmatrix}\right]}.$ (2)

Without spin-orbit interaction (that is, $\text{\boldmath{$E$}}=0$), the
system is isotropic. At zero magnetic field the ground state is a singlet, and
the excited states are three degenerate triplets. Switching on an external
magnetic field leaves the energy of the singlet state unchanged, but lowers
the energy of one triplet state. At $B_{0}=J/\mu_{g\text{B}}$ the ground state
becomes degenerate. In the magnetic parameter space, the ground state
degeneracy points form a nodal surface, i.e., a spherical surface with radius
$B_{0}$, see Fig. 1d.

With spin-orbit interaction ($\text{\boldmath{$E$}}\neq 0$), the ground state
degeneracies split everywhere except the two points where the external
magnetic field $B$ is parallel to the spin-orbit magnetic field
$\text{\boldmath{$B$}}_{\text{so}}$. These two remaining degeneracy points are
located at

$\displaystyle\text{\boldmath{$B$}}_{W,\pm}(\text{\boldmath{$E$}})=\pm
B_{0}\frac{\text{\boldmath{$e$}}_{z}\times\text{\boldmath{$E$}}}{E},$ (3)

where $\text{\boldmath{$e$}}_{z}$ is the z-directional unit vector.

The above observations naturally combine into the Weyl-point teleportation
effect, as illustrated by Fig. 1b-e. Consider the continuous trajectory of the
control vector shown in Fig. 1b, parametrized by $t\in[-1,1]$. This trajectory
is defined as $\text{\boldmath{$E$}}(t)=-t\,\epsilon\,(1,0,0)$ for $t<0$ and
$\text{\boldmath{$E$}}(t)=t\,\epsilon\,(0,1,0)$ for $t\geq 0$, with
$\epsilon>0$ having the dimension of electric field. This trajectory contains
the transition point $\text{\boldmath{$E$}}(0)=\text{\boldmath{$0$}}$. For
$t<0$, Weyl points are located in $\pm B_{0}\text{\boldmath{$e$}}_{y}$ (panel
b), whereas for $t>0$, Weyl points are located in $\pm
B_{0}\text{\boldmath{$e$}}_{x}$ (panel e), and for $t=0$ there are no Weyl
points but a degeneracy sphere (panel d) that serves as a ‘wormhole’, allowing
the ‘teleportation’ of the Weyl points as the control-space trajectory
traverses the transition point.

Note that in our model, the Weyl points do not move ‘before’ ($t<0$) and
‘after’ ($t>0$) the teleportation. This changes in more realistic models,
e.g., taking into account the electric-field dependence of the $g$-factor; see
also our discussion on Weyl Josephson circuits below.

_Divergence of motional susceptibility._ Here, we exemplify that the motion of
Weyl points in the configuration space is singular as the control parameters
approach the transition point. We use the same two-spin model as above.

First, we introduce the _motional susceptibility matrix_ $\chi$ of a Weyl
point, which is the quantity characterizing the motion of the Weyl point in
configuration space in response to a small change of the control parameters.
This matrix depends on the control parameters, and it is defined everywhere in
the control space except in the transition point. For a selected Weyl point
having the location $\text{\boldmath{$B$}}_{\text{W}}(\text{\boldmath{$E$}})$
in the configuration space (see, e.g., Eq. (3)), it is defined as

$\chi_{ik}(\text{\boldmath{{\hbox{ E }}}})=\frac{\partial
B_{\text{W},i}(\text{\boldmath{{\hbox{ E }}}})}{\partial E_{k}},$ (4)

where $i\in\\{x,y,z\\}$ and $k\in\\{x,y\\}$.

Hence for our two-spin model, this susceptibility matrix has dimension
$3\times 2$. It can be obtained analytically from Eq. (3), e.g, by selecting
the Weyl point $\text{\boldmath{$B$}}_{W,+}$, via the definition (4) as

$\chi(\text{\boldmath{{\hbox{ E
}}}})=\frac{B_{0}}{E^{3}}\begin{pmatrix}-E_{x}E_{y}&E_{x}^{2}\\\
-E_{y}^{2}&E_{x}E_{y}\\\ 0&0\end{pmatrix}.$ (5)

The singular value decomposition of this matrix has the form $\chi=U\Sigma
V^{\text{T}}$, where

$\displaystyle U(\text{\boldmath{$E$}})$ $\displaystyle=$
$\displaystyle\frac{1}{E}\begin{pmatrix}-E_{x}&E_{y}&0\\\ -E_{y}&-E_{x}&0\\\
0&0&E\end{pmatrix}$ (6) $\displaystyle\Sigma(\text{\boldmath{$E$}})$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{cc}\Sigma_{1}&0\\\
0&\Sigma_{2}\\\ 0&0\end{array}\right)$ (10) $\displaystyle
V(\text{\boldmath{$E$}})$ $\displaystyle=$
$\displaystyle\frac{1}{E}\begin{pmatrix}-E_{y}&-E_{x}\\\
E_{x}&-E_{y}\end{pmatrix}$ (11)

Here, the two singular values are $\Sigma_{1}=B_{0}/E$, $\Sigma_{2}=0$.

Remarkably, the largest singular value $\Sigma_{1}$ diverges as $1/E$ as the
control parameters approach the transition point $\text{\boldmath{$E$}}=0$.
This shows that for paths that go close, but not through the transition point,
an infinitesimal change of the control parameter yields a large movement of
the Weyl point in the configuration space. This becomes a macroscopic jump
when the path passes through the transition point and teleportation happens.

_Weyl Josephson circuits._ To show the generic nature of the effects discussed
above, we now identify them in a different setup: a Weyl Josephson circuit
[24]. The inset of Fig. 2a shows the schematic arrangement of a multi-terminal
Josephson circuit, originally proposed in Fig. 1 of [24]. It is built from
four superconducting terminals (black circles), where terminal 0 is grounded,
and terminals 1, 2, and 3 are floating and controlled by local gate
electrodes. The corresponding dimensionless gate voltages are denoted by
$n_{\text{g}\alpha}$ with $\alpha\in\\{1,2,3\\}$. We regard these gate
voltages as the control parameters. Furthermore, the three loops formed by the
terminals, denoted as $x$, $y$, $z$ in Fig. 2a, are pierced by controllable
magnetic fluxes $\varphi_{x}$, $\varphi_{y}$, $\varphi_{z}$; we consider the
3D space of these fluxes as the configuration space. We denote the Josephson
energy (capacitance) associated to the junction between terminals $\alpha$ and
$\beta$ as $E_{\text{J},\alpha\beta}$ ($C_{\alpha\beta}$).

The Hamiltonian of this Josephson circuit reads

$\displaystyle\hat{H}$ $\displaystyle=$ $\displaystyle
E_{\text{C}}\left(\hat{\text{\boldmath{$n$}}}-\text{\boldmath{$n$}}_{\text{g}}\right)\cdot
c^{-1}\left(\hat{\text{\boldmath{$n$}}}-\text{\boldmath{$n$}}_{\text{g}}\right)$
(12) $\displaystyle-$
$\displaystyle\sum\limits_{\begin{subarray}{c}\alpha,\beta=0\\\
\alpha<\beta\end{subarray}}^{3}E_{\text{J},\alpha\beta}\cos\left[\hat{\varphi}_{\alpha}-\hat{\varphi}_{\beta}+\gamma_{\alpha\beta}(\varphi_{x},\varphi_{y},\varphi_{z})\right].$

Here, $\hat{n}_{\alpha}$ is the number operator counting the Cooper pairs on
terminal $\alpha\in\\{0,1,2,3\\}$, and
$\hat{\text{\boldmath{$n$}}}=(\hat{n}_{1},\hat{n}_{2},\hat{n}_{3})$.
Furthermore, $\hat{\varphi}_{\alpha}$ are the phase operators canonically
conjugated to $\hat{n}_{\alpha}$, $E_{\text{C}}=(2e)^{2}/(2C_{0})$ is a
capacitance scale characteristic of the network of terminals, $c=C/C_{0}$ is
the dimensionless capacitance matrix defined from the capacitance matrix [28]
$C$ (see Appendix), and $\gamma_{\alpha\beta}$ are control angles depending on
the fluxes as $\gamma_{0\beta}=0$, $\gamma_{12}=\varphi_{x}$,
$\gamma_{13}=-\varphi_{z}$, and $\gamma_{23}=\varphi_{y}$.

If all three gate voltages $n_{\text{g}\alpha}$ have integer or half-integer
values, then the Hamiltonian $\hat{H}$ above has an effective time-reversal
symmetry (see Appendix), implying that the ground-state degeneracy points, if
they exist, form a loop in the configuration space. This is exemplified in
Fig. 2 (a) and (b), where the black circle in panel (a) shows a gate-voltage
vector with half-integer components, whereas the black solid loop in panel (b)
shows the corresponding ground-state degeneracy pattern, i.e., a nodal loop.
The black circle in panel (a) is a transition point, and the corresponding
nodal loop is an extended degeneracy pattern, analogous to the sphere in the
magnetic field space seen in Fig. 1c. The circuit parameters used to obtain
this result are
$(E_{J,01},E_{J,02},E_{J,03},E_{J,12},E_{J,13},E_{J,23})/h=(2,4,6,3,3,6)\,\text{GHz}$,
and $(C_{01},C_{02},C_{03},C_{12},C_{13},C_{23})/h=(2,1,2,3,4,3)\,\text{fF}$.
Furthermore, capacitances between terminals 1, 2, 3 and their respective
control gates are
$C_{\text{g},1}=C_{\text{g},2}=C_{\text{g},3}=0.1\,\text{fF}$. (Numerical
techniques used to obtain this result can be found in the Appendix.)

Figure 2: Weyl points and their teleportation in a Weyl Josephson circuit. (a)
Continuous trajectory $\blacksquare\to\circ\to+$ of the control parameter
vector, i.e., the vector $\text{\boldmath{$n$}}_{\text{g}}$ of gate voltages
that controling the Weyl Josephson circuit proposed in [24] (inset). (b)
Evolution of four Weyl points in the configuration space, i.e., the space of
the flux vector $\varphi$, as the control parameter is varied on the path
$\blacksquare\to\circ\to+$ in (a). Teleportation of the Weyl points happens at
$\text{\boldmath{$n$}}_{\text{g}}=(0.5,0.5,0.5)$, marked by the black circle
in (a), when the degeneracy points in the configuration space form a nodal
loop (black) in panel (b). See text for parameters. (c) Scaling of Weyl-point
motion in a Weyl Josephson circuit. The greatest singular value $\Sigma_{1}$
of the motional susceptibility matrix $\chi$ exhibits a $1/t$ divergence as
the control vector approaches the transition point as
$\text{\boldmath{$n$}}_{\text{g}}(t)=\text{\boldmath{$n$}}_{\text{g}0}+t\text{\boldmath{$e$}}$.
The two data sets correspond to
$\text{\boldmath{$e$}}=\text{\boldmath{$e$}}_{\blacksquare}$ and
$\text{\boldmath{$e$}}=\text{\boldmath{$e$}}_{+}$ (see text), and the motion
of the top left Weyl point in panel b). The straight lines in c) are single-
parameter fits to the data, see inset.

The nodal loop serves as a wormhole for the Weyl points, enabling their
teleportation. This is illustrated in Fig. 2 (a) and (b). In (a), a continuous
trajectory $\blacksquare\to\circ\to+$ including the transition point ($\circ$)
is shown (green line). Here, the incoming and outgoing paths are straight,
enclosing a finite angle. Formally, the trajectory in control space is defined
as
$\text{\boldmath{$n$}}_{\text{g}}(t)=\text{\boldmath{$n$}}_{\text{g}0}-t\text{\boldmath{$e$}}_{\blacksquare}$
for $t<0$,
$\text{\boldmath{$n$}}_{\text{g}}(t)=\text{\boldmath{$n$}}_{\text{g}0}+t\text{\boldmath{$e$}}_{+}$
for $t\geq 0$, with
$\text{\boldmath{$e$}}=\text{\boldmath{$e$}}_{\blacksquare}=(3,8,6)/\sqrt{109}$,
$\text{\boldmath{$e$}}=\text{\boldmath{$e$}}_{+}=(3,8,-6)/\sqrt{109}$, and
$t\in[-0.1,0.1]$. The corresponding motion of 4 ground-state Weyl points in
the configuration space is shown in (b), where the red (blue) color denotes
the $+1$ ($-1$) topological charge (ground-state Chern number) of the Weyl
point. Even though the path in the control space is continuous (Fig. 2a), the
Weyl-point positions in the configuration space suffer a sudden jump as the
control-space trajectory crosses the transition point (Fig. 2b).

In the two-spin model described above, we have demonstrated the $1/E$
divergence of motional susceptibility of the Weyl points. We now identify the
same type of divergence in this Weyl Josephson circuit model, using the
results in Fig. 2c. For each of the two directions of Fig. 2a, i.e., the ones
denoted by the green square and the green cross, we numerically compute the
greatest singular value of the motional susceptibility matrix of a selected
Weyl point as the control vector is changed according to
$\text{\boldmath{$n$}}_{\text{g}}(t)=\text{\boldmath{$n$}}_{\text{g}0}+t\text{\boldmath{$e$}}$
with $t\in[10^{-4},0.01]$, for two cases:
$\text{\boldmath{$e$}}=\text{\boldmath{$e$}}_{\blacksquare}$ and
$\text{\boldmath{$e$}}=\text{\boldmath{$e$}}_{+}$. Figure 2c shows the
obtained data points, along with the two single-parameter fits to the data
(solid lines). These results reveal that in the vicinity of the transition
point (that is, for $t\to 0$), all of these singular values show a $1/t$
divergence. We have checked that this behavior holds for all Weyl points
created from the nodal loop, for various Hamiltonian parameter values and
directions in control space.

Based on our results for the two-spin model and the Weyl Josephson circuit
model, we conjecture that this $1/t$ divergence of Weyl-point motional
susceptibility is a generic feature in the vicinity of a nodal loop and a
nodal surface, irrespective of the specific physical setup. We pose it as an
open problem to rigorously identify the preconditions of this scaling law.

In fact, the divergent motional susceptibility of the Weyl points in the
vicinity of the nodal surface or nodal line can be regarded as a side effect
of teleportation. A simple argument is as follows. Consider a circular path
connecting the control points
$\text{\boldmath{$n$}}_{\text{g}0}+t\text{\boldmath{$e$}}_{\blacksquare}$ and
$\text{\boldmath{$n$}}_{\text{g}0}+t\text{\boldmath{$e$}}_{+}$, centered
around the transition point $\text{\boldmath{$n$}}_{\text{g}0}$. The path
length approaches zero as $t\to 0$, being proportional to $t$. We compare this
with the corresponding Weyl-point path length in the configuration space, by
inspecting, e.g., the top left Weyl point in Fig. 2b. The Weyl-point path
length does not converge to zero as $t\to 0$; instead, it converges to the
path length between the meeting points of the Weyl-point trajectory and the
nodal loop (i.e., the _teleportation length_), which is finite. This simple
consideration explains that in the vicinity of the transition point, a small
generic change in the control vector implies a large jump of the Weyl points;
furthermore, it provides an interpretation of the $1/t$ type divergence of the
motional susceptibility.

_Conclusions._ In conclusion, we have argued that nodal loops and nodal
surfaces of parameter-dependent quantum systems serve as wormholes for Weyl
points, allowing their teleportation. Furthermore, as the control parameters
approach the transition point where the degeneracy points form a loop or a
surface, the pace of motion of the Weyl points in the configuration space
diverges, following a simple scaling law. A special case of the teleportation
effect was found in superfluid 3He-A [29] where two oppositely charged Weyl
points switch places through a nodal loop. We exemplified more general
teleportation patterns on a two-spin model and a Weyl Josephson circuit, and
we expect that they generalize naturally to many other settings, including
electronic [3, 11, 14, 19, 30], photonic [31, 32], phononic [33], magnonic
[34], and synthetic [35] band structures.

_Author contributions._ Gy. F. performed analytical and numerical
calculations, and produced the figures. Gy. F. and A. P. wrote the initial
draft of the manuscript. A. P. acquired funding, and managed the project. G.
P. consulted on differential-geometric aspects of the work. All authors
contributed to the formulation of the project, discussed the results and took
part in writing the manuscript.

###### Acknowledgements.

We acknowledge fruitful discussions and correspondence with J. Asbóth, L.
Bretheau, V. Fatemi, Y. Qian, A. Schnyder, and H. Weng. This research was
supported by the Ministry of Innovation and Technology (MIT) and the National
Research, Development and Innovation Office (NKFIH) within the Quantum
Information National Laboratory of Hungary and the Quantum Technology National
Excellence Program (Project No. 2017-1.2.1-NKP-2017-00001), by the NKFIH fund
TKP2020 IES (Grant No. BME-IE-NAT) under the auspices of the MIT, and by the
NKFIH through the OTKA Grants FK 124723 and FK 132146. D. V. was supported by
the Swedish Research Council (VR) and the Knut and Alice Wallenberg
Foundation.

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##  Appendix A Exchange rotation in the interacting two-spin model

Here, we provide a heuristic estimate for the exchange rotation angle $\gamma
E$ included in the interacting two-spin model of the main text.

The exchange interaction in Eq. (2) contains a rotation matrix of angle
$\gamma E$. To estimate $\gamma$ in a realistic setting, we consider electrons
in an InAs nanowire ($m_{\text{eff}}=0.023\,m_{\text{e}}$), subject to an
electric field that is assumed to be homogeneous. We identify the exchange
rotation angle with the spin rotation angle via

$\gamma E=2\pi\ell_{\text{d}}/\ell_{\text{so}}$ (13)

of a free conduction electron in the nanowire as it traverses the interdot
distance $\ell_{\text{d}}$ of the double quantum dot system. The spin-orbit
length $\ell_{\text{so}}$, according to [27], can be expressed as

$\ell_{\text{so}}=\frac{\hbar^{2}}{m_{\text{eff}}e\alpha_{0}E},$ (14)

with $\alpha_{0}=1.17\,\text{nm}^{2}$.

The above relations imply

$\gamma=\frac{2\pi l_{\text{d}}m_{\text{eff}}e\alpha_{0}}{\hbar^{2}}\approx
222\,\text{nm}/\text{V},$ (15)

where an interdot distance of $\ell_{\text{d}}=100\,\text{nm}$ was used. That
is, an electric field $E=1\,\mu\text{m}/\text{V}$ implies an exchange rotation
of $\gamma E=0.222$ radians, approximately 13 degrees.

##  Appendix B The capacitance matrix of the Weyl Josephson circuit

The capacitance matrix [28] of the Weyl Josephson circuit described in the
main text reads:

$\displaystyle C=\begin{pmatrix}C_{1}&-C_{12}&-C_{13}\\\
-C_{12}&C_{2}&-C_{23}\\\ -C_{13}&-C_{23}&C_{3}\end{pmatrix},$ (16)

where

$\displaystyle C_{1}$ $\displaystyle=$ $\displaystyle
C_{01}+C_{12}+C_{13}+C_{\text{g1}},$ (17a) $\displaystyle C_{2}$
$\displaystyle=$ $\displaystyle C_{02}+C_{12}+C_{23}+C_{\text{g2}},$ (17b)
$\displaystyle C_{3}$ $\displaystyle=$ $\displaystyle
C_{03}+C_{13}+C_{23}+C_{\text{g3}}.$ (17c)

##  Appendix C Effective $\mathcal{PT}$ symmetry protecting the nodal loops of
the Weyl Josephson circuit

The Josephson-circuit Hamiltonian
$\hat{H}(\text{\boldmath{$n$}}_{g},\text{\boldmath{$\varphi$}})$ has time-
reversal symmetry in the absence of magnetic fluxes:

$\displaystyle\mathcal{T}\hat{H}(\text{\boldmath{$n$}}_{\text{g}},0)\mathcal{T}^{-1}=\hat{H}(\text{\boldmath{$n$}}_{\text{g}},0),$
(18)

where $\mathcal{T}=\mathcal{K}$ is the complex conjugation in the charge basis
[24], and hence $\mathcal{T}^{2}=1$. Furthermore, for nonzero magnetic flux,
the relation (18) generalizes as

$\displaystyle\mathcal{T}\hat{H}(\text{\boldmath{$n$}}_{\text{g}},\text{\boldmath{$\varphi$}})\mathcal{T}^{-1}=\hat{H}(\text{\boldmath{$n$}}_{\text{g}},-\text{\boldmath{$\varphi$}}).$
(19)

In the charge basis, this is translated to

$\displaystyle
H^{\ast}_{\text{\boldmath{$n$}},\text{\boldmath{$n$}}^{\prime}}(\text{\boldmath{$n$}}_{\text{g}},\text{\boldmath{$\varphi$}})=H_{\text{\boldmath{$n$}},\text{\boldmath{$n$}}^{\prime}}(\text{\boldmath{$n$}}_{\text{g}},-\text{\boldmath{$\varphi$}}).$
(20)

Other symmetries of $\hat{H}$ are charge inversion

$\displaystyle
H(\text{\boldmath{$n$}}_{\text{g}},\text{\boldmath{$\varphi$}})_{\text{\boldmath{$n$}},\text{\boldmath{$n$}}^{\prime}}=H(-\text{\boldmath{$n$}}_{\text{g}},-\text{\boldmath{$\varphi$}})_{-\text{\boldmath{$n$}},-\text{\boldmath{$n$}}^{\prime}},$
(21)

and discrete charge translation symmetry

$\displaystyle
H(\text{\boldmath{$n$}}_{\text{g}},\text{\boldmath{$\varphi$}})_{\text{\boldmath{$n$}},\text{\boldmath{$n$}}^{\prime}}=H(\text{\boldmath{$n$}}_{\text{g}}+\text{\boldmath{$m$}},\text{\boldmath{$\varphi$}})_{\text{\boldmath{$n$}}+\text{\boldmath{$m$}},\text{\boldmath{$n$}}^{\prime}+\text{\boldmath{$m$}}}$
(22)

where $m$ is an arbitrary integer offset charge vector.

The charge inversion and charge translation together gives

$\displaystyle
H(\text{\boldmath{$n$}}_{\text{g}},\text{\boldmath{$\varphi$}})_{\text{\boldmath{$n$}},\text{\boldmath{$n$}}^{\prime}}=H(\text{\boldmath{$m$}}-\text{\boldmath{$n$}}_{\text{g}},-\text{\boldmath{$\varphi$}})_{\text{\boldmath{$m$}}-\text{\boldmath{$n$}},\text{\boldmath{$m$}}-\text{\boldmath{$n$}}^{\prime}}.$
(23)

This relation implies that for fixed integer or a half-integer offset charge
vector $\text{\boldmath{$n$}}_{g}=\frac{\text{\boldmath{$m$}}}{2}$, it holds
that

$\displaystyle
H\left(\frac{\text{\boldmath{$m$}}}{2},\text{\boldmath{$\varphi$}}\right)_{\text{\boldmath{$n$}},\text{\boldmath{$n$}}^{\prime}}=H\left(\frac{\text{\boldmath{$m$}}}{2},-\text{\boldmath{$\varphi$}}\right)_{\text{\boldmath{$m$}}-\text{\boldmath{$n$}},\text{\boldmath{$m$}}-\text{\boldmath{$n$}}^{\prime}}.$
(24)

This relation can be expressed by the charge inversion operator
$\mathcal{P}^{(\text{\boldmath{$m$}}/2)}$, whose matrix elements in the charge
basis are

$\displaystyle
P^{(\text{\boldmath{$m$}}/2)}_{\text{\boldmath{$n$}},\text{\boldmath{$n$}}^{\prime}}=\delta_{\text{\boldmath{$n$}},\text{\boldmath{$m$}}-\text{\boldmath{$n$}}^{\prime}}.$
(25)

With this definition, we rewrite Eq. (24) as

$\displaystyle\mathcal{P}^{(\text{\boldmath{$m$}}/2)}\hat{H}\left(\frac{\text{\boldmath{$m$}}}{2},\text{\boldmath{$\varphi$}}\right)\left(\mathcal{P}^{(\text{\boldmath{$m$}}/2)}\right)^{-1}=\hat{H}\left(\frac{\text{\boldmath{$m$}}}{2},-\text{\boldmath{$\varphi$}}\right).$
(26)

inverting the charge with respect to $\frac{\text{\boldmath{$m$}}}{2}$. This
is an exact symmetry even when the charge basis is restricted to a finite
interval symmetrically around $\frac{\text{\boldmath{$m$}}}{2}$.

Henceforth, we fix the charge inversion point as
$\frac{\text{\boldmath{$m$}}}{2}$, and suppress it in the formulas below.
Then, the combination of charge inversion and time-reversal symmetry results
in

$\displaystyle(\mathcal{PT})\hat{H}\left(\text{\boldmath{$\varphi$}}\right)\mathcal{(PT)}^{-1}=\hat{H}\left(\text{\boldmath{$\varphi$}}\right),$
(27)

i.e., we have identifyed an antiunitary symmetry $\mathcal{PT}$ with
$(\mathcal{PT})^{2}=1$ that restricts the Hamiltonian at every value of
$\varphi$ for a fixed
$\text{\boldmath{$n$}}_{g}=\frac{\text{\boldmath{$m$}}}{2}$ integer or a half-
integer offset charge vector. Carrying out a basis transformation from the
charge basis by the unitary $U=\sqrt{\mathcal{P}}$, $\mathcal{PT}$ is
represented as complex conjugation, making the transformed
$H(\text{\boldmath{$\varphi$}})$ real. The co-dimension for real valued
symmetric matrices to be two-fold degenerate is 2 (see, e.g., Ref. [1]),
meaning that the level crossings generally appear as 1 dimensional space
curves in the 3 dimensional $\varphi$ parameter space.

To show a specific example, we consider the truncated $8\times 8$ Hamiltonian
of the Josephson circuit in the charge basis
$\\{\ket{000},\ket{100},\ket{010},\ket{001},\ket{111},\ket{011},\ket{101},\ket{110}\\}$,
which reads

$\displaystyle H_{8\times
8}(\text{\boldmath{$n$}}_{g},\text{\boldmath{$\varphi$}})=$
$\displaystyle=\begin{pmatrix}\epsilon_{000}&h_{01}&h_{02}&h_{03}&0&0&0&0\\\
h_{01}&\epsilon_{100}&h_{12}^{\ast}&h_{31}&0&0&h_{03}&h_{02}\\\
h_{02}&h_{12}&\epsilon_{010}&h_{23}^{\ast}&0&h_{03}&0&h_{01}\\\
h_{03}&h_{31}^{\ast}&h_{23}&\epsilon_{001}&0&h_{02}&h_{01}&0\\\
0&0&0&0&\epsilon_{111}&h_{01}&h_{02}&h_{03}\\\
0&0&h_{03}&h_{02}&h_{01}&\epsilon_{011}&h_{12}&h_{31}^{\ast}\\\
0&h_{03}&0&h_{01}&h_{02}&h_{12}^{\ast}&\epsilon_{101}&h_{23}\\\
0&h_{02}&h_{01}&0&h_{03}&h_{31}&h_{23}^{\ast}&\epsilon_{110}\end{pmatrix},$
(28)

where

$\displaystyle\epsilon_{abc}$ $\displaystyle=$ $\displaystyle
E_{\text{C}}[(a,b,c)^{\text{T}}-\text{\boldmath{$n$}}_{\text{g}}]\cdot
c^{-1}[(a,b,c)^{\text{T}}-\text{\boldmath{$n$}}_{\text{g}}]$ (29)
$\displaystyle h_{\alpha\beta}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}E_{\text{J},\alpha\beta}e^{-i\gamma_{\alpha\beta}}.$
(30)

At the time-reversal offset charge point
$\text{\boldmath{$n$}}_{\text{g}}=\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)^{\text{T}}\equiv\text{\boldmath{$\frac{1}{2}$}}$,
the diagonal elements have the property

$\displaystyle\epsilon_{a,b,c}=\epsilon_{1-a,1-b,1-c},$ (31)

yielding the $\mathcal{PT}$ symmetry

$\displaystyle PH^{\ast}_{8\times
8}\left(\text{\boldmath{$\varphi$}}\right)P^{-1}=H_{8\times
8}\left(\text{\boldmath{$\varphi$}}\right),$ (32)

with $P=\sigma_{x}\otimes\mathds{1}_{4\times 4}$. The corresponding unitary
matrix $U=\sqrt{i/2}\left(\mathds{1}_{2\times
2}-i\sigma_{x}\right)\otimes\mathds{1}_{4\times 4}$ transforms the Hamiltonian
matrix to be real-valued

$\displaystyle UH_{8\times
8}\left(\text{\boldmath{$\varphi$}}\right)U^{-1}=\left[UH_{8\times
8}\left(\text{\boldmath{$\varphi$}}\right)U^{-1}\right]^{\ast}.$ (33)

##  Appendix D Weyl Josephson circuit: Numerical techniques

The calculations of the Weyl Josephson circuits were done numerically, with
the aid of analytical techniques. The starting point of the calculation was
the truncated matrix representation of the Hamiltonian of Eq. (10) of the main
text. The matrix representation was obtained by projecting the Hamiltonian
onto the subspace spanned by the charge basis states
$\ket{n_{1},n_{2},n_{3}}$, with $n_{1},n_{2},n_{3}\in\\{-1,0,1,2\\}$, yielding
a $4^{3}\times 4^{3}=64\times 64$ matrix.

This charge interval is chosen to be symmetric to $1/2$, to respect the
symmetry of Eq. (27) protecting the nodal loop at
$\text{\boldmath{$n$}}_{\text{g}0}=(0.5,0.5,0.5)$. The range of the charge
states was chosen to be large enough to describe the physical system
accurately, and small enough to save computational time.

To check the error due to the truncation of the Hamiltonian, we have compared
the numerically identified nodal loops for smaller and larger matrix
dimensions, see Fig. 3a. Besides the $64\times 64$ truncation size, the figure
shows results with the $8\times 8$ and $216\times 216$ Hamiltonians,
corresponding to the charge ranges $0\leq n_{1,2,3}\leq 1$ and $-2\leq
n_{1,2,3}\leq 3$, respectively. We found that all truncated Hamiltonians have
the same qualitative behaviour, i.e., presence of nodal loop, Weyl point
teleportation, and susceptibility divergence. Moreover, the quantitative
difference between the results from the two larger Hamiltonians is negligible,
e.g., the distance between the points of the nodal loops in the
$\varphi_{z}=0$ plane is 0.036 rad (see Fig. 3a).

Figure 3: Nodal loop in the Weyl Josephson junction. a) Numerical
identification of nodal loops and Weyl points was carried out by truncating
the infinite-dimensional Hilbert space of the Weyl Josephson circuit. Larger
Hamiltonian matrices give more accurate results: the corresponding nodal loops
converge. b) Numerical search for the points (red) of the nodal loop (black).
The search consists of two nested iterations: the inner iteration (purple)
searches for the intersection of the nodal loop with a plane (green). After
the intersection is found, the outer iteration makes a step of a finite
distance in the tangential direction of the nodal loop (grey) and determines
the new searching plane as the perpendicular plane passing through the new
point (blue). The first inner iteration searches in the $\varphi_{z}=0$ plane.

In the following subsections, we present the numerical techniques for
characterizing the nodal loop and searching for the Weyl points. We use a
generic description where $p$ is the 3-component configurational parameter in
the configurational space where the Weyl points and the nodal loop are
present, and $q$ is the $k$-component control parameter which can tune system.
The correspondence between this notation and the notation of gate voltages and
magnetic fluxes In the Weyl Josephson circuit reads:

$p$ $\displaystyle\equiv$
$\displaystyle\text{\boldmath{$\varphi$}}=(\varphi_{x},\varphi_{y},\varphi_{z}),$
(34) $q$ $\displaystyle\equiv$
$\displaystyle\text{\boldmath{$n$}}_{\text{g}}=(n_{\text{g},1},n_{\text{g},2},n_{\text{g},3}).$
(35)

###  D.1 Effective $2\times 2$ Hamiltonian near two-fold degeneracies

Searching for the degeneracies of a large Hamiltonian can be computationally
demanding. To ease the problem, a useful technique is to reduce the
Hamiltonian to a low-dimensional effective model with Schrieffer–Wolff
transformation. Consider the Hamiltonian
$H(\text{\boldmath{$p$}},\text{\boldmath{$q$}})$ depending on the 3
dimensional configurational space $p$ and the $k$ dimensional $q$ control
space. At $(\text{\boldmath{$p$}}_{0}$,$\text{\boldmath{$q$}}_{0})$, the $i$th
and $(i+1)$th energies, to be denoted as
$E_{i}(\text{\boldmath{$p$}}_{0},\text{\boldmath{$q$}}_{0})$ and
$E_{i+1}(\text{\boldmath{$p$}}_{0},\text{\boldmath{$q$}}_{0})$, are close
together and well separated from the other energy levels.

In our method to find a degeneracy point in the vicinity of such a quasi-
degenerate point ($\text{\boldmath{$p$}}_{0},\text{\boldmath{$q$}}_{0}$), a
useful ingredient is to expand the Hamiltonian in linear order as

$\displaystyle H$
${}^{(1)}(\text{\boldmath{$p$}}_{0}+\delta\text{\boldmath{$p$}},\text{\boldmath{$q$}}_{0}+\delta\text{\boldmath{$q$}})=$
(36) $\displaystyle H$
$\displaystyle(\text{\boldmath{$p$}}_{0},\text{\boldmath{$q$}}_{0})+\left.\frac{\partial
H}{\partial\text{\boldmath{$p$}}}\right|_{\text{\boldmath{$p$}}_{0},\text{\boldmath{$q$}}_{0}}\cdot\delta\text{\boldmath{$p$}}+\left.\frac{\partial
H}{\partial\text{\boldmath{$q$}}}\right|_{\text{\boldmath{$p$}}_{0},\text{\boldmath{$q$}}_{0}}\cdot\delta\text{\boldmath{$q$}},$

and project it to the subspace of the quasi-degenerate levels with
$P_{0}=\ket{\Psi_{i}}\bra{\Psi_{i}}+\ket{\Psi_{i+1}}\bra{\Psi_{i+1}}$, where
$\ket{\Psi_{i}}$ and $\ket{\Psi_{i+1}}$ are the $i$th and $(i+1)$th energy
eigenstates of $H(\text{\boldmath{$p$}}_{0},\text{\boldmath{$q$}}_{0})$. This
projection yields the following $2\times 2$ effective Hamiltonian:

$\displaystyle H^{(1)}_{\text{eff}}$ $\displaystyle=$ $\displaystyle
P_{0}H^{(1)}P_{0}$ (37) $\displaystyle\equiv$
$\displaystyle\text{\boldmath{$\sigma$}}\cdot\text{\boldmath{$\Delta$}}^{(1)}=\text{\boldmath{$\sigma$}}\cdot\left[\text{\boldmath{$\Delta$}}_{0}+g_{\text{\boldmath{$p$}}}\delta\text{\boldmath{$p$}}+g_{\text{\boldmath{$q$}}}\delta\text{\boldmath{$q$}}\right].$

Here, $\text{\boldmath{$\sigma$}}=(\sigma_{x},\sigma_{y},\sigma_{z})$ is the
vector of Pauli operators acting on the quasi-degenerate subspace. We
neglected the $\sigma_{0}$ term in the expansion, since because it does not
affect the splitting between the two states.

By the above definition, the only nonzero component of the _polarization
vector_ $\text{\boldmath{$\Delta$}}_{0}$ is its third component. Furthermore,
we call $g_{\text{\boldmath{$p$}}}$ and $g_{\text{\boldmath{$q$}}}$ as the
configurational-space and control-space _effective $g$-tensors_, which are a
real valued $3\times 3$ and $3\times k$ matrices:

$\displaystyle\left[\Delta_{0}(\text{\boldmath{$p$}}_{0},\text{\boldmath{$q$}}_{0})\right]_{3}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left[E_{i}(\text{\boldmath{$p$}}_{0},\text{\boldmath{$q$}}_{0})-E_{i+1}(\text{\boldmath{$p$}}_{0},\text{\boldmath{$q$}}_{0})\right],$
$\displaystyle\left[g_{\text{\boldmath{$p$}}}(\text{\boldmath{$p$}}_{0},\text{\boldmath{$q$}}_{0})\right]_{\alpha,\beta}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\text{Tr}\left(\sigma_{\alpha}P_{0}\left.\frac{\partial
H}{\partial
p_{\beta}}\right|_{\text{\boldmath{$p$}}_{0},\text{\boldmath{$q$}}_{0}}P_{0}\right),$
(38)
$\displaystyle\left[g_{\text{\boldmath{$q$}}}(\text{\boldmath{$p$}}_{0},\text{\boldmath{$q$}}_{0})\right]_{\alpha,\beta}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\text{Tr}\left(\sigma_{\alpha}P_{0}\left.\frac{\partial
H}{\partial
q_{\beta}}\right|_{\text{\boldmath{$p$}}_{0},\text{\boldmath{$q$}}_{0}}P_{0}\right).$

Note that the polarization vector and the effective $g$-tensors are basis
dependent. The SU(2) transformation of the basis multiplies them with a
corresponding SO(3) matrix from the left. If the transformation is only the
change of the phase difference between the states, the corresponding
orthogonal matrix $R_{z}(\varphi)$ describes a rotation around the $z$ axis.
Later, we use these basis dependent quantities to calculate the location and
the susceptibility matrix of the Weyl points which are independent from the
basis choice. This independence can be inferred from the formulas.

When searching for a degeneracy point
$\text{\boldmath{$p$}}_{0}+\delta\text{\boldmath{$p$}}$ in the vicinity of
$p_{0}$, for a fixed value of
$\text{\boldmath{$q$}}_{0}+\delta\text{\boldmath{$q$}}$, an approximate result
can be obtained by requiring that the energy gap of the effective Hamiltonian
of Eq. (37) should vanish:

$\displaystyle\text{\boldmath{$\Delta$}}^{(1)}=\text{\boldmath{$0$}}.$ (39)

The solution of this equation for $\delta\text{\boldmath{$p$}}$ provides the
approximate position $\text{\boldmath{$p$}}_{0}+\delta\text{\boldmath{$p$}}$
of the degeneracy point.

It is worth noting that the second order Schrieffer–Wolff transformation is
more than the projection of a second order expansion of the Hamiltonian to its
near-degenerate subspace.

###  D.2 Finding Weyl points

The Weyl points are the generic point-like degeneracies in the three
dimensional configurational space (containing the points $p$), for a fixed
control parameter vector $\text{\boldmath{$q$}}_{0}$. We neglect
$\text{\boldmath{$q$}}_{0}$ in the arguments for brevity. The equation
providing an approximate position of the degeneracy is that the effective
Hamiltonian should vanish, i.e.,

$\displaystyle
H^{(1)}_{\text{eff}}\left(\text{\boldmath{$p$}}_{0}+\delta\text{\boldmath{$p$}}\right)=\text{\boldmath{$\sigma$}}\cdot\left[\text{\boldmath{$\Delta$}}_{0}\left(\text{\boldmath{$p$}}_{0}\right)+g_{\text{\boldmath{$p$}}}\left(\text{\boldmath{$p$}}_{0}\right)\delta\text{\boldmath{$p$}}\right]=0.$
(40)

Rearranging yields

$\displaystyle
g_{\text{\boldmath{$p$}}}\left(\text{\boldmath{$p$}}_{0}\right)\delta\text{\boldmath{$p$}}=-\text{\boldmath{$\Delta$}}_{0}\left(\text{\boldmath{$p$}}_{0}\right),$
(41)

which is solved using the inverse of the $g$-tensor

$\displaystyle\delta\text{\boldmath{$p$}}=-g_{\text{\boldmath{$p$}}}^{-1}\left(\text{\boldmath{$p$}}_{0}\right)\text{\boldmath{$\Delta$}}_{0}\left(\text{\boldmath{$p$}}_{0}\right).$
(42)

The resulting $\text{\boldmath{$p$}}_{0}+\delta\text{\boldmath{$p$}}$ is not
an exact Weyl point in general, but it is closer to the Weyl point than
$\text{\boldmath{$p$}}_{0}$. Following this scheme, we iterate the process
with the following formula

$\displaystyle\text{\boldmath{$p$}}_{n+1}=\text{\boldmath{$p$}}_{n}-g_{\text{\boldmath{$p$}}}\left(\text{\boldmath{$p$}}_{n}\right)^{-1}\text{\boldmath{$\Delta$}}_{0}(\text{\boldmath{$p$}}_{n}),$
(43)

until the energy splitting is sufficiently small; in our numerics, we used the
threshold of $10^{-12}$ GHz. The starting point $\text{\boldmath{$p$}}_{0}$
needs to be close to the exact Weyl point. When searching for an ordinary Weyl
point with the above iteration, the effective $g$-tensor is non-singular
throughout the iteration, as it is non-singular in the Weyl point. Hence the
inverse used in the iteration of Eq. (43) does exist.

###  D.3 Finding the nodal loop

In the presence of a nodal loop, our goal is to numerically locate a discrete
set of its points that allows us to draw the loop. To achieve that, we
intersect the latter with planes in the configurational space, as shown in
Fig. 3b. From the initial point $p$ (see, e.g., the blue point in the
$\varphi_{z}=0$ plane in Fig. 3b), we restrict the displacement
$\delta\text{\boldmath{$p$}}$ as

$\displaystyle\delta\text{\boldmath{$p$}}=c_{1}\text{\boldmath{$v$}}_{1}+c_{2}\text{\boldmath{$v$}}_{2}=\underbrace{\left(\begin{array}[]{c|c}\text{\boldmath{$v$}}_{1}&\text{\boldmath{$v$}}_{2}\end{array}\right)}_{V}\underbrace{\begin{pmatrix}c_{1}\\\
c_{2}\end{pmatrix}}_{\text{\boldmath{$c$}}},$ (45)

where $\text{\boldmath{$v$}}_{1}$ and $\text{\boldmath{$v$}}_{2}$ are two
arbitrary vectors in the 3D configurational space that span the plane. The
condition for the Weyl point in the linearized effective Hamiltonian in Eq.
(41) changes to

$\displaystyle\underbrace{g_{\text{\boldmath{$p$}}}\left(\text{\boldmath{$p$}}_{0}\right)V}_{g_{V}}\text{\boldmath{$c$}}$
$\displaystyle=$
$\displaystyle-\text{\boldmath{$\Delta$}}_{0}(\text{\boldmath{$p$}}_{0}).$
(46)

At first sight this is an overdetermined linear set for $c_{1}$ and $c_{2}$,
since it contains 3 conditions but only 2 variables. However, because of the
$\mathcal{PT}$ symmetry discussed in App. C, there is a basis where the
effective Hamiltonian is real-valued. In this basis, the second component of
Eq. (46) simplifies to the identity $0=0$, meaning that there is a unique
solution for the equation.

One way to solve Eq. (46) is to look for the vector $c$ where
$|g_{V}\text{\boldmath{$c$}}+\text{\boldmath{$\Delta$}}_{0}|^{2}$ is minimal.
Note that at the minimum point, the minimum value is actually zero. Equating
the derivative of
$|g_{V}\text{\boldmath{$c$}}+\text{\boldmath{$\Delta$}}_{0}|^{2}$ with respect
to $c$ to zero yields

$\displaystyle
g_{V}^{\text{T}}g_{V}\text{\boldmath{$c$}}=-g_{V}^{\text{T}}\text{\boldmath{$\Delta$}}_{0}.$
(47)

Note that Eq. (47) can also be obtained by multiplying Eq. (46) with
$g_{V}^{\text{T}}$ from the left. Now we can solve Eq. (47) using the inverse
of the $2\times 2$ matrix $g_{V}^{\text{T}}g_{V}$ as

$c$ $\displaystyle=$
$\displaystyle-\left(g_{V}^{\text{T}}g_{V}\right)^{-1}g_{V}^{\text{T}}\text{\boldmath{$\Delta$}}_{0}.$
(48)

Following the spirit of the iteration at the end of the previous section, the
analogous iteration to find the intersection of the nodal loop and the
considered plane is defined as

$p$
${}_{n+1}=\text{\boldmath{$p$}}_{n}+\delta\text{\boldmath{$p$}}=\text{\boldmath{$p$}}_{n}+V\text{\boldmath{$c$}}=$
$p$
${}_{n}-V\left[V^{\text{T}}g^{\text{T}}_{\text{\boldmath{$p$}}}\left(\text{\boldmath{$p$}}_{n}\right)g_{\text{\boldmath{$p$}}}\left(\text{\boldmath{$p$}}_{n}\right)V\right]^{-1}V^{\text{T}}g^{\text{T}}_{\text{\boldmath{$p$}}}\left(\text{\boldmath{$p$}}_{n}\right)\text{\boldmath{$\Delta$}}_{0}\left(\text{\boldmath{$p$}}_{n}\right).$

where we have used Eqs. (45), (46), and (48).

The first point of the loop is searched in the $\varphi_{z}=0$ plane. Then
every starting point searching the points of the nodal loop is 0.01 rad
distance in the tangential direction of the nodal loop from the previous
point. The search is restricted to the plane perpendicular to the step (Fig.
3). The tangent vector $v$ of the nodal loop is determined by the $g$-tensor
as $g_{\text{\boldmath{$p$}}}\text{\boldmath{$v$}}=0$, which means that the
splitting is zero in the linear order in that direction.

###  D.4 Susceptibility matrix of ordinary Weyl points

Here, we express the motional susceptibility matrix $\chi$ introduced in Eq.
(4) of the main text with the configurational- and control-space effective
$g$-tensors $g_{\text{\boldmath{$p$}}}$ and $g_{\text{\boldmath{$q$}}}$,
respectively. The susceptibility matrix $\chi$ describes the motion of the
Weyl points in the configurational parameter space $p$ upon changing the
control parameters $q$. The linear effective Hamiltonian expanded at a Weyl
point does not have a constant term:

$\displaystyle\text{\boldmath{$\Delta$}}^{(1)}(\text{\boldmath{$p$}},\text{\boldmath{$q$}})=g_{\text{\boldmath{$p$}}}(\text{\boldmath{$p$}}_{0},\text{\boldmath{$q$}}_{0})\delta\text{\boldmath{$p$}}+g_{\text{\boldmath{$q$}}}(\text{\boldmath{$p$}}_{0},\text{\boldmath{$q$}}_{0})\delta\text{\boldmath{$q$}}.$
(50)

The condition for Weyl points reads $\text{\boldmath{$\Delta$}}^{(1)}=0$.
Rearranging this equation and acting with $g_{\text{\boldmath{$p$}}}^{-1}$
from the left gives

$\displaystyle\delta\text{\boldmath{$p$}}=-\left(g_{\text{\boldmath{$p$}}}^{-1}g_{\text{\boldmath{$q$}}}\right)\delta\text{\boldmath{$q$}}=\chi(\text{\boldmath{$p$}}_{0},\text{\boldmath{$q$}}_{0})\delta\text{\boldmath{$q$}},$
(51)

which determines the linear order displacement $\delta\text{\boldmath{$p$}}$
of the Weyl point for the arbitrary perturbation $\delta\text{\boldmath{$q$}}$
in the control space, which is in fact the definition of the susceptibility
matrix.

Notice that the susceptibility matrix is independent from the choice of basis
in the two-fold degenerate eigenspace at the Weyl point. Changing the basis
multiplies the $g$-tensors $g_{\text{\boldmath{$p$}}}$ and
$g_{\text{\boldmath{$q$}}}$ with the same orthogonal matrix from the left, and
this exactly cancels out in the expression
$g_{\text{\boldmath{$p$}}}^{-1}g_{\text{\boldmath{$q$}}}$.

##  Appendix E Survivor Weyl points close to the transition point

Figure 4: Survivor Weyl points of the broken nodal loop. a) The zeros of the
function $\eta_{3}(s)$ (third component of right hand side of Eq. (55))
determines where the survivor Weyl points are located for an infinitesimal
perturbation. For perturbations in different directions of the control space,
the function $\eta_{3}(s)$ is different, causing the Weyl-point teleportation.
b) The direction of the perturbation also determines the number of the
survivor Weyl points. The pale gray region has no Weyl points, the medium gray
has 4 Weyl points and the dark grey region has 8 Weyl points. The
perturbations used in the main text is in the 4-point region. For both markers
(square, cross) in b), the arc length parameters of the survivor Weyl points
are depicted by the same markers on a).

In this section, we derive the location of the Weyl points that ‘survive’ upon
an infinitesimal perturbation of a nodal loop. Moreover, we show that the
number of survivor Weyl points depends on _how_ the nodal loop is perturbed:
for the example considered, the number of survivor Weyl points is either 0, 4,
or 8.

The nodal loop $\text{\boldmath{$p$}}_{0}(s)$ in the configurational parameter
space can be parametrized by its arc length $s$ and its vicinity can be
parametrized by the $\delta\text{\boldmath{$p$}}_{\perp}$ perpendicular
displacement from the nodal loop. The perturbed effective Hamiltonian reads

$\displaystyle\text{\boldmath{$\Delta$}}^{(1)}=g_{\text{\boldmath{$p$}}}(s)\delta\text{\boldmath{$p$}}_{\perp}+g_{\text{\boldmath{$q$}}}(s)\delta\text{\boldmath{$q$}},$
(52)

where we omitted the $\text{\boldmath{$q$}}_{0}$-dependence in the arguments
of the $g$-tensors for simplicity because the nodal loop corresponds to only
specific discrete $\text{\boldmath{$q$}}_{0}$ values. The survivor Weyl points
are located at the arc length parameter $s$ where the condition

$\text{\boldmath{{\hbox{ \Delta }}}}^{(1)}=0$ (53)

has solution for $\delta\text{\boldmath{$p$}}_{\perp}$. The
$g_{\text{\boldmath{$p$}}}(s)$ configurational-space effective $g$-tensor is a
singular matrix for the points of the nodal loops, hence we can not solve Eq.
(53) using the inverse of it. Instead, we use the singular value decomposition
of $g_{\text{\boldmath{$p$}}}(s)$ to rewrite Eq. (53) as

$\displaystyle U(s)\Sigma(s)V(s)^{-1}\delta\text{\boldmath{$p$}}_{\perp}$
$\displaystyle=$ $\displaystyle-
g_{\text{\boldmath{$q$}}}(s)\delta\text{\boldmath{$q$}}$ (54)
$\displaystyle\Sigma(s)\underbrace{V(s)^{-1}\delta\text{\boldmath{$p$}}_{\perp}}_{\delta\tilde{\text{\boldmath{$p$}}}_{\perp}}$
$\displaystyle=$
$\displaystyle-\underbrace{U(s)^{-1}g_{\text{\boldmath{$q$}}}(s)\delta\text{\boldmath{$q$}}}_{\text{\boldmath{$\eta$}}(s,\delta\text{\boldmath{$q$}})}$
(55)

where $\Sigma=\text{diag}(\Sigma_{1},\Sigma_{2},0)$ is a diagonal matrix and
$U$ and $V$ are $3\times 3$ orthogonal matrices. The third column of $\Sigma$
is zero, therefore the third component of the left hand side is also zero.
Therefore, the third component of Eq. (55), that is,

$\displaystyle\eta_{3}(s_{0},\delta\text{\boldmath{$q$}})=0,$ (56)

can be used to find the arc length parameters $s_{0}$ where Weyl points appear
upon applying the perturbation $\delta\text{\boldmath{$q$}}$. It is worth
noting that $\text{\boldmath{$\eta$}}(s,\delta\text{\boldmath{$q$}})$ is
independent of the choice of the basis of the two-dimensional degenerate
subspace at the Weyl point. The first-order perpendicular-to-nodal-loop of the
survivor Weyl points is given as

$\displaystyle\delta\text{\boldmath{$p$}}_{\perp}=-g_{\text{\boldmath{$p$}}}^{+}\left(s_{0}(\delta\text{\boldmath{$q$}})\right)g_{\text{\boldmath{$q$}}}\left(s_{0}(\delta\text{\boldmath{$q$}})\right)\delta\text{\boldmath{$q$}},$
(57)

where $g_{\text{\boldmath{$p$}}}^{+}=V\Sigma^{+}U^{-1}$ is the Moore–Penrose
inverse of the $g$-tensor with
$\Sigma^{+}=\text{diag}(\Sigma_{1}^{-1},\Sigma_{2}^{-1},0)$. There is also a
first-order parallel-to-nodal-loop motion of the survivor Weyl points, but
determining that requires a second order calculation.

This derivation indicates the Weyl-point teleportation: for infinitesimal
perturbations $\delta\text{\boldmath{$q$}}$ in different directions the
function $\eta_{3}(s,\delta\text{\boldmath{$q$}})$ has roots at different arc
length values (see Fig. 4a), meaning that different points of the nodal loop
survive. For infinitesimally small perturbations
$\delta\text{\boldmath{$q$}}=\text{\boldmath{$n$}}\delta q$ only the direction
$n$ describes the location of the surviving Weyl points. Their number also
depends on the direction of the perturbation which gives a phase diagram on
the unit sphere (see Fig. 4b).