Present address: ] Department of Condensed Matter Physics, Weizmann Institute
of Science, Rehovot, Israel.

# Topological features in the ferromagnetic Weyl semimetal CeAlSi:
Role of domain walls

M. M. Piva<EMAIL_ADDRESS>Max Planck Institute for Chemical Physics
of Solids, Nöthnitzer Str. 40, D-01187 Dresden, Germany  J. C. Souza  [
Instituto de Física “Gleb Wataghin”, UNICAMP, 13083-859, Campinas, SP, Brazil
Max Planck Institute for Chemical Physics of Solids, Nöthnitzer Str. 40,
D-01187 Dresden, Germany  V. Brousseau-Couture  Département de Physique,
Université de Montréal, C.P. 6128, Succursale Centre-Ville, Montréal, Québec,
Canada H3C 3J7  Sopheak Sorn  Institute for Quantum Materials and Technology,
Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany  K. R. Pakuszewski
Instituto de Física “Gleb Wataghin”, UNICAMP, 13083-859, Campinas, SP, Brazil
Janas K. John  Max Planck Institute for Chemical Physics of Solids, Nöthnitzer
Str. 40, D-01187 Dresden, Germany  C. Adriano  Instituto de Física “Gleb
Wataghin”, UNICAMP, 13083-859, Campinas, SP, Brazil  M. Côté  Département de
Physique, Université de Montréal, C.P. 6128, Succursale Centre-Ville,
Montréal, Québec, Canada H3C 3J7  P. G. Pagliuso  Instituto de Física “Gleb
Wataghin”, UNICAMP, 13083-859, Campinas, SP, Brazil  Los Alamos National
Laboratory, Los Alamos, New Mexico 87545, USA  Arun Paramekanti  Department of
Physics, University of Toronto, 60 St. George Street, Toronto, ON, M5S 1A7
Canada  S. N. Bose National Centre for Basic Sciences, Block JD, Sector - III,
Salt lake, Kolkata-700106, India  M. Nicklas<EMAIL_ADDRESS>Max
Planck Institute for Chemical Physics of Solids, Nöthnitzer Str. 40, D-01187
Dresden, Germany

###### Abstract

In the ferromagnetic (FM) Weyl semimetal CeAlSi both space-inversion and time-
reversal symmetries are broken. Our quantum oscillation (QO) data indicate
that the FM ordering modifies the Fermi surface topology and also leads to an
unusual drop in the QO amplitude. In the FM phase, we find a pressure-induced
suppression of the anomalous and the loop Hall effects. This cannot be
explained based on the electronic band structure or magnetic structure, both
of which are nearly pressure independent. Instead, we show that a simplified
model describing the scattering of Weyl fermions off FM domain walls can
potentially explain the observed topological features. Our study highlights
the importance of domain walls for understanding transport in FM Weyl
semimetals.

##  I INTRODUCTION

Topological phases of matter have lately received considerable attention, due
to the experimental realization of exotic types of charge carriers. One
example is the massless Weyl fermions found in Weyl semimetals (WSMs) [1, 2,
3], which are characterized by remarkable electronic properties, such as
surface Fermi arcs, a bulk chiral anomaly, axial–gravitational anomaly, an
extremely large magnetoresistance (MR) and an anomalous Hall effect (AHE) [1,
3, 4, 5, 6]. Weyl fermions can be generated by either breaking space-inversion
(SI) or time-reversal (TR) symmetry of materials with a Dirac or quadratic
band touching points. So far most experimentally studied WSMs break SI
symmetry [7, 8, 9, 10, 11, 12, 13, 14, 15]; fewer examples are known for WSMs
with broken TR symmetry, i.e. magnetic WSMs [16, 17, 18, 19, 20, 21]. Magnetic
WSMs are of fundamental interest since they intertwine topology and strong
correlations [22, 23]. They also offer the potential to manipulate the
topological phase in a desired way, for instance using a magnetic field to
tune the position of Weyl nodes or to control the chirality or geometry of
magnetic domain walls, which is important for next-generation spintronics
applications [24, 25].

The family of $Ln$Al$Pn$ ($Ln$ = lanthanides, $Pn$ = Ge, Si) materials is
ideal to host nontrivial topological properties due to their
noncentrosymmetric crystalline structure ($I4_{1}md$), which is the same as in
the TaAs family of WSMs [7, 26, 27, 9, 28, 29]. Multiple Weyl nodes and a
large spin Hall effect were predicted to exist in LaAlGe and LaAlSi [30]. Weyl
cones were experimentally observed for LaAlGe [31] and a $\pi$ Berry phase was
recently found in LaAlSi [32]. Remarkably, magnetic members of the family host
rare-earth moments which can order and additionally break TR symmetry - many
of them are predicted to feature Weyl nodes near the Fermi level [33, 34, 35,
36]. Experiments have discovered an anomalous Hall effect (AHE) in
PrAlGe1-xSix [33], chiral surface Fermi arcs in PrAlGe [37, 38], and a
topological magnetic phase and singular angular MR in the semimetal CeAlGe
[39, 40, 41]. In addition, Weyl fermions have been found to mediate magnetism
in NdAlSi [42] and a $\pi$ Berry phase was reported for quantum oscillations
(QO) in SmAlSi [34].

In this Article, we focus on the ferromagnetic Weyl semimetal CeAlSi. CeAlSi,
which hosts an in-plane non-collinear ferromagnetic (FM) order below the Curie
temperature $T_{C}\approx 8$ K with a large anisotropy, the $c$-axis being the
magnetically hard axis [36]. Ce3+ spins in adjacent FM planes display an angle
of $\approx 70^{\circ}$ [36]. Recent angle resolved photo emission
spectroscopy experiments in the paramagnetic phase of CeAlSi above $T_{C}$
revealed Fermi arcs and several Weyl nodes lying close to the Fermi energy
which stem from the non-centrosymmetric structure [43]. Going below $T_{C}$,
into the FM state, a magnetic field applied parallel to the $[100]$ direction
reveals an AHE, while a $[001]$ field leads to an unexplained hysteretic loop
Hall effect (LHE) [36]. In addition, CeAlSi may exhibit nontrivial magnetic
domain walls [44]; indeed, chiral domain walls were recently detected in this
system [45]. Furthermore, magnetoelastic couplings give rise to picometer
displacements in the unit cell due to the internal FM field, which can lead to
different domain wall spin textures [46].

The presence of this magnetoelastic effect suggests that external pressure may
lead to a strong tuning of magnetism and to associated large changes in the
AHE and LHE [46]. Hydrostatic pressure has previously been shown to be an
effective tool in tuning the electronic structure without introducing any
additional disorder and was successfully used to tune Weyl points closer to
the Fermi energy in certain topological materials [47, 48, 49, 50].
Furthermore, application of pressure is known to systematically modify the
magnetic properties in Ce-based materials [51]. Here, we use hydrostatic
pressure as a tool to investigate the origin of the features characteristic of
the nontrivial topological behavior in CeAlSi, focusing on longitudinal and
Hall transport experiments and on quantum oscillation measurements. We combine
these with ab initio density functional theory (DFT) calculations and
phenomenological models for scattering of Weyl fermions off magnetic domain
walls to shed light on our unusual observations.

##  II METHODS

Single crystals of CeAlSi and LaAlSi were grown by the Al-flux technique
similar to [52]. High purity elements with starting composition Ce [La]
(99.99%) : Al (99.999+%) : Si (99.999+%), $1:20:1$, were place into an alumina
crucible and sealed in an evacuated quartz tube. The samples were heated to
$1200^{\circ}$C, kept at this temperature for 15 hours and cooled down to
$720^{\circ}$C at $2^{\circ}$C/h. The excess of Al was removed by spinning the
tube upside down in a centrifuge. The crystal structure was confirmed by x-ray
powder diffraction. Energy dispersive x-ray spectroscopy shows, within the
experimental uncertainty, a Ce:Al:Si proportion of $1:1:1$.

Electrical transport experiments were carried out by a four-probe
configuration using a low-frequency AC resistance bridge. Temperatures down to
1.8 K and magnetic fields up to 9 T were achieved in a physical property
measurement system (PPMS, Quantum Design) and in a liquid helium cryostat
(Janis). Magnetization measurements were conducted in a magnetic property
measurement system (MPMS, Quantum Design). Pressures up to 2.7 GPa (electrical
transport) and 1 GPa (magnetization) were generated using self-contained
piston-cylinder-type pressure cells using silicon oil as pressure transmitting
medium. A piece of lead (tin) served as manometer.

Density functional theory (DFT) calculations were performed with the local
density approximation functional (LDA) and projector-augmented wave (PAW)
method as implemented in the Abinit software package [53], using Jollet-
Torrent-Holzwarth (JTH) pseudopotentials [54]. Spin-orbit coupling (SOC) and
non-collinear magnetism are taken into account. An on-site Coulomb interaction
with $U=6$ eV was added for the Ce $f$ electrons within the LDA+U scheme. We
use a $16\times 16\times 16$ Monkhorst-Pack $\mathbf{k}$-point grid and a
plane-wave energy cutoff of 25 hartree. The lattice parameters and relevant
internal atomic coordinates were optimized at respectively $0$ GPa and $3$ GPa
until all forces on the atoms were below $10^{-6}$ hartree/bohr3. At 0 GPa (3
GPa), we obtain $a=7.926$ bohr ($a=7.832$ bohr) and $c=27.397$ bohr
($c=27.192$ bohr).

##  III RESULTS

### Temperature – pressure phase diagram

Figure 1: (a) Magnetization ($M$) (left axis), obtained in an applied field of
50 mT along the $[100]$ crystal axis, and electrical resistivity (right axis)
as a function of temperature for selected pressures. (b) Temperature–pressure
phase diagram. The solid lines are linear fits. (c) Magnetization measurements
for several pressures. Figure 2: Electronic bands and DOS at ambient pressure
(blue) and 3 GPa (red). The hatched region of right panel corresponds to the
partial DOS associated with Ce $f$ states.

At ambient pressure, the FM ordering transition in CeAlSi is marked by a
singular magnetization $M(T)$ and sharp drop in electrical resistivity as a
function of temperature $\rho(T)$ at $T_{C}\approx 8$ K. Figure 1 shows the
effect of external pressure on the magnetic phase (see Appendix A for
additional data). Application of pressure linearly enhances $T_{C}(p)$ with a
slope of 0.62(2) K/GPa, driving $T_{C}$ from 7.8 K at ambient pressure to 9.4
K at 2.7 GPa (values taken from the resistivity data). More important is our
finding that, for different pressures, the in-plane magnetization curves
$M(H)$ at 2 K as a function of the applied magnetic field along the $[100]$
direction lie on top of each other. This result indicates a negligible
pressure effect on the non-collinear planar magnetic structure found at
ambient pressure [36].

### Electronic band structure

Our DFT calculations at ambient pressure and 3 GPa, which incorporate spin-
orbit coupling and non-collinear magnetic order, reveal only a negligible
effect of pressure on the electronic band structure and the electronic density
of states (DOS) at the Fermi energy [see Fig 2] as well as on the ordered
moments and their orientation. The bands contributing to the hole pockets at
the Fermi surface (FS) barely display any variation of their intercepts of the
Fermi energy in $\mathbf{k}$-space, suggesting a negligible variation of the
FS area (see Appendix D for further information). The only noticeable
modification of the electronic structure is a small shift of the bands
associated with Ce $f$-electrons to higher energies with respect to the Fermi
energy. As these bands lie about 2 eV below the Fermi level, they most likely
do not directly contribute to the transport properties.

### Quantum oscillations

Figure 3: (a) Fast Fourier transformation (FFT) amplitude as a function of
temperature at different applied pressures. The solid lines are simulations
considering the best fits using the Lifshitz-Kosevich formula. The inset shows
longitudinal conductivity for $H\parallel[001]$ after subtraction of a third
order polynomial background $\Delta\sigma_{xx}$ as a function of
$1/(\mu_{0}H)$ at 15 K (top) and 2 K (bottom) for selected pressures. The
curves at 2 K were shifted by $-10$ kS/m for clarity. (b) and (c) Landau fan
diagrams for CeAlSi at 15 and 2 K, respectively.

Next we turn to the results of our QO measurements. Longitudinal conductivity
data $\sigma_{xx}$ well above $T_{C}$ at $T=15$ K and in the FM state at $T=2$
K at several pressures are presented in the inset of Fig. 3(a), where we have
subtracted a smooth background yielding $\Delta\sigma_{xx}$. Within the
investigated field range, the $\Delta\sigma_{xx}$ data as well as its fast
Fourier transform (FFT) analysis reveals a single QO frequency $f\approx
20(5)$ T, which is found to be independent of pressure and temperature (see
Appendix B). We notice two main features: i. the amplitude of the oscillations
at 15 K is larger than that at 2 K and ii. the amplitude of the oscillations
is suppressed by increasing pressure [see Fig. 3(a)]. Generally, the thermal
damping of the QO amplitude can be described by the Lifshitz-Kosevich (LK)
formula [55]. However, our FFT signal follows the LK prediction only in the
paramagnetic (PM) region above $T_{C}$ [solid lines in Fig. 3(a)]. In the FM
state we observe an unusual reduction of the QO amplitude upon cooling. This
remarkable response of the oscillation amplitude as a function of temperature
has not been observed in any other members of the $Ln$Al$Pn$ family [36, 32,
34, 42, 56], and was previously reported in just a few materials [57, 58]. In
SmSb, for instance, a sudden decrease of the Shubnikov-de Haas oscillations
takes place once the material becomes antiferromagnetic, which was conjectured
to be due to the presence of a nontrivial Berry phase [58].

To further analyze the QO, Landau fan diagrams are shown in Figs. 3(b) and
3(c). Our analysis indicates a change in the nature of the topological
properties between the PM and FM phase. At 15 K in the PM phase the intercept
is around $-5/8$, which suggests the presence of topologically trivial charge
carriers [59]. In contrast to that, at 2 K the intercept is $-9/8$, which for
3D magnetic WSMs can be associated with linear dispersive charge carriers and
a nontrivial Berry phase [59].

Our QO data suggest that the momentum space separation between nearby Weyl
nodes with opposite topological charges gets enhanced in the FM state leading
to a change in FS topology - from one which encloses both Weyl nodes above
$T_{C}$ to a split FS enclosing isolated well-separated Weyl nodes below
$T_{C}$. An enhanced separation of the Weyl nodes in the FM phase has been
also previously found in band-structure calculations (SI of Ref. [36]). This
can explain the change in the intercept in our Landau fan plot. Such a change
in the FS topology could nonetheless preserve the area of certain extremal
orbits, so that the observed QO frequency can remain nearly unchanged (see
Appendix E for details). If the topological Fermi pockets are only weakly
split below $T_{C}$, the large density of states due to proximity to the
Lifshitz transition [60] can lead to an increased scattering rate for states
on the extremal orbits, thus enhancing the Dingle temperature and suppressing
the QO amplitude for $T<T_{C}$. We note that our results cannot rule out other
possible scenarios for the suppression of the amplitude of the quantum
oscillations upon cooling. However, a Lifshitz transition in the FS of CeAlSi
can explain both, the suppression of the QO and the phase shift upon entering
the FM phase revealed by our measurements.

Figure 4:  (a) Anomalous Hall effect (AHE) at 2 K as a function of magnetic
fields $H\parallel[001]$ for different applied pressures. The top inset shows
the AHE jump as a function of pressure and the bottom inset displays a scheme
of the circuit used in this measurement. s1 and s2 denote samples 1 and 2,
respectively. (b) Loop Hall effect (LHE) at 2 K as a function of magnetic
field for $H\parallel[001]$ for selected applied pressures. The left inset
shows the Hall resistivity measured upon increasing (red) and decreasing
(blue) magnetic field and the difference of both curves (green) at 0.1 GPa.
The right inset displays a schematic drawing of the measurement circuit. Data
of the nonmagnetic reference material LaSiAl at ambient pressure is shown as
gray line in both panels

### Hall effect

For magnetic field along the $[100]$ direction and current along $[010]$ [see
sketch in Fig. 4(a)] we find a large AHE in CeAlSi in its ferromagnetic state.
The AHE signal has been extracted by fitting the Hall resistivity to the form
$\rho_{yz}(H)=R_{0}H+\rho_{\rm AHE}$, where $R_{0}$ is the ordinary Hall
effect coefficient and $\rho_{\rm AHE}=R_{s}M_{x}$ with $R_{s}$ being the
anomalous Hall coefficient and $M_{x}$ being the magnetization along $[100]$
(see Appendix C). We confirm that an AHE is absent in the non-magnetic analog
LaAlSi [Fig. 4(a)]. We have fitted the longitudinal and ordinary Hall
conductivities to a simple two-band model to obtain information on the density
of the electron- and hole-like charge carriers and their mobilities (see
Appendix C). At low temperatures and ambient pressure we find $5.9(1)\times
10^{19}$ holes/cm3 and $2.5(1)\times 10^{19}$ electrons/cm3. The application
of external pressure suppresses the extracted hole density only slightly,
which reaches $4.6(1)\times 10^{19}$ holes/cm3 at 2 K and 2.6 GPa, whereas the
electrons density remains nearly unchanged. Moreover, the corresponding
mobilities at 2 K and ambient pressure are about $1.4(1)\times 10^{3}$ cm2/Vs
($3.2(1)\times 10^{3}$ cm2/Vs) for holes (electrons). These values are nearly
unaffected by application of external pressure and are on the same order of
magnitude compared with other Weyl semimetals [61, 62]. Application of
external pressure suppresses the jump of the AHE (defined as difference in
$\rho_{\rm AHE}$ between positive and negative magnetic fields) up to 1.5 GPa
[top inset of Fig. 4(a)]. Above 1.5 GPa the anomalous Hall jump saturates to
around 0.5(1) $\mu\Omega$cm. As we have shown above, the $M(H)$ curves taken
at different pressures fall on top of each other [see Fig. 1(b)], suggesting
the absence of changes in the magnetic structure as a source for the
suppression of the AHE. Moreover, the electronic bands close to the Fermi
level are only slightly affected by pressure [see Fig. 1(c)], making it
unlikely that this significant decrease results from a pressure-induced change
in the position of the Weyl nodes. We find that while $R_{s}$ scales nearly
linearly with $\rho_{xx}$ for pressure $\lesssim\\!1$ GPa, the scaling
deviates significantly from linear behavior at higher pressures (see Appendix
C). A linear relation between $R_{s}$ and $\rho_{xx}$ suggests that the
observed AHE at ambient pressure has a significant extrinsic skew-scattering
contribution [63], which gets suppressed at high pressures. Given the
robustness of the electronic structure and magnetic order against pressure,
the most plausible explanation for this is a pressure-dependent change in the
nature or distribution of domain walls. Previous work has shown that Weyl
fermions can undergo skew scattering from magnetic domain walls which contain
the axis of the average magnetization, leading to an extrinsic contribution to
the AHE qualitatively consistent with our data [64].

An even more unusual Hall response is observed for field applied along $[001]$
[see sketch in Fig. 4(b)]. We note that in this geometry the magnetic field is
applied perpendicular to the ferromagnetically ordered moments in the
$\left<001\right>$ plane and this Hall response thus cannot arise from the
bulk in-plane magnetization. This so-called loop Hall effect (LHE) is
displayed in Fig. 4(b). It is only observed in the ferromagnetic regime,
displays hysteresis even in the absence of any observable $M_{z}$
magnetization hysteresis, and is absent in the non-magnetic analog LaAlSi.
$\rho_{\rm LHE}(H)$ is obtained by recording the Hall resistivity $\rho_{xy}$
for $H\parallel[001]$ upon increasing and decreasing magnetic field and taking
the difference between both curves, as shown in the left inset of Fig. 4(b)
for 0.1 GPa as an example. Similar to the AHE, the application of external
pressure leads to a decrease in the LHE [see Fig. 4(b)]. While the existence
of the LHE in CeAlSi has been argued to be tied to the presence of the Weyl
nodes near the Fermi energy [36], no clear physical mechanism has been
provided for its origin.

##  IV DISCUSSION

Figure 5: (a) Domain wall between two bulk magnetic domains showing twisted
magnetization configuration with $M^{DW}_{z}(x)=M\sin\theta(x)$. (b) Skew
scattering of Weyl fermions off hysteretic domain wall loops, with red regions
having $M^{DW}_{z}>0$ and blue having $M^{DW}_{z}<0$, provides a mechanism for
the loop Hall effect (LHE). The black arrows are classical representations of
the Weyl fermions trajectories. (c) Calculated LHE shown as a ratio of
Landauer conductances, versus the maximal out of plane tilt angle $\theta$ of
the domain wall magnetization.

In the following we present a simplified model for Weyl fermions in a non-
centrosymmetric FM, and show that a domain wall scattering mechanism, similiar
to that previously explored to understand the AHE [64], can also lead to the
LHE for domain walls which are perpendicular to the average magnetization. Our
key idea here is that the bulk magnetic domains in CeAlSi host an in-plane
magnetization with hard axis along $[001]$, so that the out-of-plane bulk
contribution to the field-induced magnetization $M^{\rm bulk}_{z}$ is not
expected to be hysteretic as we tune the magnetic field $H_{z}$. We instead
argue that the hysteretic LHE must be attributed to the hysteretic domain wall
magnetization $M^{\rm DW}_{z}$ as we tune $H_{z}$, as schematically depicted
in Fig. 4(a) and 4(b). Our calculations show that the intra-node skew
scattering of Weyl fermions as they cross a domain wall with nonzero $M^{\rm
DW}_{z}$ can explain the LHE.

To illustrate this physics, we study a model with $4$ pairs of Weyl nodes in
the $k_{z}=0$ plane (see Appendix E for details). These could be viewed as a
caricature of the $W_{3}^{\prime}$ Weyl nodes found to lie $\sim 46$ meV above
the Fermi level close to the $k_{z}=0$ plane in CeAlSi [43, 36]. For a single
Weyl node, with chirality $+1$, we consider a simple linearized Hamiltonian:

$\displaystyle\mathcal{H}_{+}$ $\displaystyle=$ $\displaystyle
v_{F}\sigma_{i}G^{(+)}_{ij}q_{j}+\sigma_{i}M_{i}$ (1)

where $v_{F}$ is the nodal velocity, $\sigma$ is the spin Pauli matrix, $q$
denotes the momentum relative to the Weyl node position, and the tensor
$G_{ij}$ is chosen to yield an elliptical Fermi surface at fixed $k_{z}$ with
its major axis rotated away from the $x,y$ axes. The Hamiltonian for the other
Weyl nodes can be reconstructed using symmetries. The Weiss field ${\bf M}$ is
nonzero in the magnetically ordered phase and tunes the momentum of the Weyl
node; we assume this leads to topologically nontrivial FS pockets enclosing
single Weyl nodes. For a magnetic field $H_{z}$ applied along the $z$-axis,
the system will support domains of ${\bf M}$, with magnetization aligned along
different in-plane directions, which are separated by domain walls. As shown
in [45], the domain wall magnetization in such cases supports an out-of-plane
component $M^{\rm DW}_{z}$. Fig. 5(c) shows the computed Hall conductance
scaled by the longitudinal conductance showing that it is an odd function of
$M^{\rm DW}_{z}$ (see Appendix F for details). Crudely, this small Landauer
conductance [65, 66] ratio is expected to be related to the ratio of loop Hall
to longitudinal resistivity - our experiments show that $\rho_{xy}^{\rm
LHE}/\rho_{xx}\\!\sim\\!10^{-3}$, in reasonable agreement with the theoretical
estimate in Fig. 5. We thus propose that the mechanism for the aptly named LHE
is the skew scattering of Weyl fermions off hysteretic domain wall loops or
surfaces. Since we expect the domain wall magnetization $M^{\rm
DW}_{z}\\!\ll\\!M^{\rm bulk}_{z}$, the hysteretic behavior of $M^{\rm DW}_{z}$
cannot be resolved in bulk magnetization measurements. Our model might also
help to understand the observation of a similar LHE reported previously in
other compounds, in which Weyl fermions were predict to exist [67, 68, 69].

##  V CONCLUSIONS

In summary, our study emphasizes, through a key tuning parameter (hydrostatic
pressure), the importance of ferromagnetism for the low temperature
topological features in CeAlSi. Our QO data show a difference in the Berry
phase above and below $T_{C}$, indicating that FM ordering shifts oppositely
charged Weyl nodes away from each other in momentum space, leading to a change
in the Fermi-surface topology. We have argued that this also leads to an
increase in the scattering rate below $T_{C}$, and thus to a drop in the
amplitude of Shubnikov – de Haas oscillations in contrast to the conventional
LK formula. This result calls for angular dependent Shubnikov – de Haas and de
Haas – van Alphen experiments in an extended field range. We have also
discovered pressure dependent changes in the AHE and LHE below $T_{C}$. Since
our DFT calculations indicate that the electronic band structure is robust
against pressure, we argue that these changes in the AHE and LHE must arise
from differences in domain wall defects and we have shown how Weyl fermions
scattering off hysteretic domain walls can lead to the LHE.

## DATA AVAILABILITY

Data that underpin the ﬁndings of this study are available at Edmond – the
open research data repository of the Max Planck Society at [70].

###### Acknowledgements.

We acknowledge fruitful discussions with A. P. Mackenzie. We also thank U.
Burkhardt for carrying out energy dispersive x-ray analysis on the samples.
This project has received funding from the European Union’s Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie grant
agreement No 101019024. This work was also supported by the São Paulo Research
Foundation (FAPESP) grants 2017/10581-1, 2018/11364-7, 2020/12283-0, CNPq
grants $\\#$ 304496/2017-0, 310373/2019-0 and CAPES, Brazil. This research was
financially supported by the Natural Sciences and Engineering Research Council
of Canada (NSERC), under the Discovery Grants program grant No.
RGPIN-2016-06666. Computations were made on the supercomputers Beluga and
Narval managed by Calcul Québec and the Digital Research Alliance of Canada.
The operation of these supercomputers is funded by the Canada Foundation for
Innovation, the Ministère de la Science, de l’Économie et de l’Innovation du
Québec, and the Fonds de recherche du Québec – Nature et technologies. V.B.-C.
and M.C are members of the Regroupement québécois sur les matériaux de pointe
(RQMP).

## APPENDICES

##  APPENDIX A: Electrical resistivity

Figure 6: (a) and (b) Electrical resistivity ($\rho$) as a function of
temperature at several pressures for two different samples of CeAlSi. The
insets displays a magnified view of the low-temperature range. The insets show
the temperature derivative of $\rho$ as a function of temperature at several
pressures for two different samples of CeAlSi.

Figures 6(a) and 6(b) present the electrical resistivity ($\rho$) as a
function of temperature at several pressures for two different samples of
CeAlSi. At high temperatures $\rho(T)$ exhibits a metallic behavior for both
samples at all studied pressures. Moreover, a clear kink is observed at low-
temperatures characterizing the ferromagnetic transition, in good agreement
with previous reports at ambient pressure [36, 46, 45]. A broad shoulder is
observed at around 80 K. It shows up as a maximum in the temperature
derivative of the electrical resistivity [see insets of Figs. 6(a) and 6(b)].
The shape and the position of the maximum is nearly unaffected by the
application of external pressure, suggesting that the gap between the ground
state and the first excited crystalline electrical field state does not change
with increasing pressure.

## APPENDIX B: Quantum Oscillations

Figure 7: (Left panels) Longitudinal conductivity after subtraction of a third
order polynomial $\Delta\sigma_{xx}$ as a function of $1/(\mu_{0}H)$ at
several temperatures for selected pressures. (Right panels) Fast Fourier
transformation of $\Delta\sigma_{xx}$ shown in the left.

The left panels of Fig. 7 present the longitudinal conductivity measured with
$H\parallel[001]$ after subtraction of a third order polynomial (fit between 5
and 9 T) $\Delta\sigma_{xx}$ as a function of $1/(\mu_{0}H)$. We note that
quantum oscillations are clearly seen up to 40 K at all studied pressures.
Furthermore, the unusual behavior of the quantum oscillations amplitudes can
be seen by the naked eye. The oscillations in the paramagnetic state at 15 K
are more pronounced than the oscillations in the ferromagnetic state at 2 K.
The panels on the right side of Fig. 7 present the Fast Fourier transformation
(FFT) of $\Delta\sigma_{xx}$, using a Hamming window from 0.2 to 0.11 T-1, as
a function of frequency ($f$) at several temperatures for selected pressures.
Only one oscillation frequency $f\approx 20(5)$ T is present. It is unaffected
by changes in pressure and/or temperature.

Figure 8: Effective mass $m^{*}$ as a function of pressure for magnetic fields
parallel to $[001]$.

The effective mass ($m^{*}$) was estimated in the paramagnetic state of CeAlSi
by fitting the FFT amplitude as a function of temperature by the Lifshitz-
Kosevich (LK) formula [55]:

$R_{T}=\frac{\alpha Tm^{*}}{B\sinh(\alpha Tm^{*}/B)},$ (2)

in which $\alpha=2\pi^{2}k_{B}/e\hbar\approx 14.69$ T/K, $T$ is the
temperature, $B$ is the magnetic field and $m^{*}$ the effective mass. As
shown in Fig. 8, the application of external pressure leads to a decrease in
the value of $m^{*}$.

## APPENDIX C: Hall Effect

### Anomalous Hall Effect

Figure 9: Hall resistivity ($\rho_{yz}$) as a function of magnetic field
applied parallel to [100] at 2 K for several pressures. The solid orange line
is an extrapolation of a linear fit performed in the range 0.2 T $\leqslant
H\leqslant$ 0.6 T, which yields the ordinary background of $\rho_{yz}$.

Figure 9 presents the Hall resistivity ($\rho_{yz}$) as a function of magnetic
field at 2 K for several pressures. A linear background was determined by
performing a linear fit in the range 0.2 T $\leqslant H\leqslant$ 0.6 T. We
obtained the anomalous Hall ($\rho_{\rm AHE}$) effect by subtracting the
linear background using $\rho_{yz}=R_{0}H+\rho_{\rm AHE}$.

Figure 10: Anomalous Hall coefficient ($R_{S}$) as a function of the
longitudinal resistivity ($\rho_{xx}$) at several pressures.

As we can see in Fig. 10, the linear dependence between the anomalous Hall
coefficient ($R_{S}$) and the longitudinal resistivity ($\rho_{xx}$)
characterizes the presence of a skew scattering contribution to the AHE in
CeAlSi at ambient pressure [63]. The observation of this contribution in a
good metal regime can be attributed to domain wall scattering of Weyl fermions
(see Sec. IV), as this contribution should be the dominant one in highly
conducting samples ($\sigma_{xx}\geqslant 0.5\times 10^{6}$ ($\Omega$cm)-1)
[63]. Furthermore, the application of external pressure suppresses the linear
relation between $R_{s}$ and $\rho_{xx}$, which is better seen in the inset of
Fig. 10, where the exponents obtained with allometric fits
($R_{S}=a+b\rho_{xx}^{n}$) are shown as a function of pressure. One can
clearly see the increase of the exponent $n$ as a function of increasing
pressure, reaching 1.21(1) at 1.1 GPa, indicating that the skew scattering
contribution of the AHE from the domain walls is being suppressed by
application of pressure. The domains themselves (bulk) also contribute to the
AHE. It is possible to differentiate both contributions, as analyzed in great
detail in Ref. [64], by considering that the domain wall scattering
contribution to the AHE is limited by the electron mean free path, whereas the
bulk contribution is not. The total Hall resistivity is therefore an average
between the bulk and domain wall contributions. Our results suggests that at
low pressures ($p\leqslant 1.5$ GPa) the AHE is dominated by the skew
scattering contribution coming from the domain walls, while in the high-
pressure range ($p\geqslant 1.5$ GPa), where the AHE is not skew scattering
type, it is dominated by the contribution of the domains.

### Two-band model fits

Figure 11: (a) Longitudinal ($\sigma_{xx}$) and Hall ($\sigma_{xy}$)
conductivities at 2 K and 1.2 GPa. Carrier densities (b) and mobilities (c)
obtained from the two-band fits as a function of temperature at several
pressures.

To accurately estimate the carrier densities and mobilities of CeAlSi, we have
simultaneously fit the longitudinal ($\sigma_{xx}$) and the Hall
($\sigma_{xy}$) conductivities considering a two-band model described by:

$\displaystyle\sigma_{xx}$ $\displaystyle=$ $\displaystyle
e\left(\frac{n_{e}\mu_{e}}{1+\mu_{e}^{2}\left(\mu_{0}H\right)^{2}}+\frac{n_{h}\mu_{h}}{1+\mu_{h}^{2}\left(\mu_{0}H\right)^{2}}\right)$
$\displaystyle\sigma_{xy}$ $\displaystyle=$ $\displaystyle
e\left(\mu_{0}H\right)\left(\frac{n_{e}\mu_{e}^{2}}{1+\mu_{e}^{2}\left(\mu_{0}H\right)^{2}}-\frac{n_{h}\mu_{h}^{2}}{1+\mu_{h}^{2}\left(\mu_{0}H\right)^{2}}\right),$

where $n$ denotes the electron ($e$) and hole ($h$) carrier densities, and
$\mu_{e}$ and $\mu_{h}$ are the electron and hole mobilities, respectively.
Figure 11(a) presents a representative plot of the fits at 2 K and 1.2 GPa, in
which a good agreement between the experimental data and the fits is observed.
Figure 11(b) displays the carrier densities as a function of temperature at
several pressures. Figure 11(c) shows the mobilities as a function of
temperature at several pressures.

## APPENDIX D: Bandstructure calculations

Figure 12: Electronic bands and DOS at ambient pressure (blue) and 3 GPa
(red), zoomed in the vicinity of the Fermi level.

Figure 13: Orbital decomposition of the electronic wavefunction at (a) ambient
pressure and (b) 3 GPa, zoomed in the vicinity of the Fermi level.

Figure 12 shows the electronic bands and DOS zoomed in the vicinity of the
Fermi level, to emphasize the negligible effect of pressure on the bands
contributing to the AHE and LHE. Note, also that no crossing feature nor
electron pocket was found in our ambient pressure calculation along the
$\Gamma-X$ high symmetry path, in contrary to Fig. 3(a) of [36]. This
discrepancy could be attributed to the different exchange-correlation
functional or to our use of theoretically relaxed lattice parameters, while
[36] used experimental values which, in the case of the PBE-GGA functional
used in their paper, will be smaller than the theoretical one. Nevertheless,
from the pressure dependence of the electronic bands relative to the Fermi
level, an electron pocket could likely appear along this path upon further
increasing the pressure.

We further refine the analysis of the electronic structure by calculating the
orbital decomposition of the electronic wavefunction inside the atom-centered
PAW spheres for Ce $5d$ states (left panels, red), as well as for Al (middle
panels, green) and Si (right panels, blue) $2p$ states, in the same energy
range as Fig. 12. The relative weights of the different orbitals at ambient
pressure (top panels) and $3$ GPa (bottom panels) are essentially identical,
thus confirming the negligible effect of pressure on the electronic bands.

The calculated magnetic structure does not display any significant differences
between $0$ and $3$ GPa either, in agreement with the experimental
observations (see Fig. 1(b) of the main text). For $0$ GPa ($3$ GPa), we find
a magnetic moment of 0.880 $\mu_{B}$ (0.878 $\mu_{B}$) inside the PAW spheres
of the 2 inequivalent Ce atoms in the unit cell. Considering that one moment
points mostly in the $\hat{x}$ direction and the other in the $\hat{y}$
direction with an angle of 87.3∘ (89.8∘) between them, we find a total net
magnetic moment of 1.311 $\mu_{B}$ (1.284 $\mu_{B}$) oriented along $[110]$
for the whole unit cell. Note that the net size of the magnetic moment depends
strongly on the choice of $U$.

## APPENDIX E: SIMPLIFIED MODEL FOR WEYL NODES

We choose a simple model for CeAlSi with $4$ pairs of Weyl nodes as shown in
Fig. 14 (top left panel). We have chosen the Weyl nodes and Fermi pockets for
$T>T_{C}$ to be consistent with the $C_{4v}$ and mirror $M_{x},M_{y}$ crystal
symmetries of CeAlSi, as well as time-reversal symmetry. These nodes crudely
mimic the $W_{3}^{\prime}$ nodes found slightly above the Fermi level in
previous ab initio electronic structure calculations. With the onset of
magnetism, the Weyl nodes get displaced with opposite chirality nodes being
displaced in opposite directions. As shown in Fig. 14(top right panel), this
can lead to a topological transition of the Fermi surface where each pocket
now encloses a single Weyl node. At the same time, for the type of Fermi
surface sketched above, certain extremal orbits can remain unchanged in area
(dashed lines), so that the QO frequency will be unaffected as observed. If
the Weyl nodes are not widely separated even after the topological Fermi
surface phase transition, proximity to a Lifshitz transition may lead to an
enhancement of the electron scattering rate (in the presence of weak disorder)
due to a large density of states, which can enhance the Dingle temperature and
explain the strong observed deviation from the Lifshitz-Kosevich formula.

Figure 14: Top: Illustrative example of $4$ topologically trivial Fermi
surface pockets for $T\\!>\\!T_{c}$, each enclosing a pair of Weyl points (WP)
with opposite topological charge, located at momenta $(K_{0},\pm K_{1},0)$,
$(-K_{0},\pm K_{1},0)$, $(\pm K_{1},K_{0},0)$, $(\pm K_{1},-K_{0},0)$. For
$T\\!<\\!T_{c}$, the in-plane magnetization leads to a momentum space
displacement of the Weyl points, leading to transition into $8$ topologically
nontrivial Fermi pockets. The area of certain maximal orbits (dashed lines)
can remain unchanged across this transition. Bottom: Projected view of the
elliptical cross sections of the topologically nontrivial Fermi surfaces for
$T\\!<\\!T_{c}$.

Within the symmetry broken for $T<T_{C}$, it is reasonable to consider the
physics of isolated Weyl nodes. We model a single Weyl node as having an
elliptical pocket with velocity tensor and a simple coupling to the Weiss
field from the magnetization.

$\displaystyle\mathcal{H}_{+}$ $\displaystyle=$
$\displaystyle\sigma_{i}G^{(+)}_{ij}\tilde{q}_{j}+\sigma_{i}M_{i},$ (3)
$\displaystyle G^{(+)}_{ij}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}|a|&&\\\ &\frac{1}{|a|}&\\\
&&1\end{pmatrix}\begin{pmatrix}\cos\beta&\sin\beta&0\\\
-\sin\beta&\cos\beta&0\\\ 0&0&1\end{pmatrix},$ (4)

Here $\tilde{q}_{i}=v_{F}q_{i}$ where $\mathbf{q}$ denotes the momentum
measured from the Weyl node location, and $v_{F}$ is a velocity scale. The
real matrix, $G_{ij}$, defined this way results in an ellipsoidal Fermi
surface, whose xy-plane cross section has an elliptical shape with the major
and minor axis, $|a|$ and $1/|a|$ respectively for $|a|>1$, and the major axis
is rotated from the $q_{y}$-axis by the angle $\beta$. For $|a|<1$, the major
axis is instead rotated from the $q_{x}$ axis by $\beta$.

Position of Weyl points: When $M_{i}=0$, the Weyl point resides at
$\mathbf{q}=0$. When $M_{i}\neq 0$, the Weyl point shifts to the point
satisfying the following equation

$\displaystyle\tilde{q}^{*}_{i}=-\left[G^{(+)}\right]^{-1}_{ij}M_{j}$ (5)

Eigenspectrum: The eigenvalues of $\mathcal{H}_{+}$ are given by

$\displaystyle E$ $\displaystyle=$
$\displaystyle\pm\sqrt{\tilde{q}_{i}[G^{(+)}]^{T}_{ij}[G^{(+)}]_{jl}\tilde{q}_{l}+2M_{i}[G^{(+)}]_{ij}\tilde{q}_{j}+M_{i}M_{i}}$

Written in the form which is useful for numerics:

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\tilde{q}_{x}^{2}\left(|a|^{2}c^{2}+\frac{1}{|a|^{2}}s^{2}\right)$
(7)
$\displaystyle+\tilde{q}_{x}\left[2cs\left(|a|^{2}-\frac{1}{|a|^{2}}\right)\tilde{q}_{y}+2\left(|a|cM_{x}-\frac{1}{|a|}sM_{y}\right)\right]$
$\displaystyle+\left[\tilde{q}_{z}^{2}+2M_{z}\tilde{q}_{z}+\left(|a|^{2}s^{2}+\frac{1}{|a|^{2}}c^{2}\right)\tilde{q}_{y}^{2}\right.$
$\displaystyle+\left.2\left(|a|sM_{x}+\frac{1}{|a|}cM_{y}\right)\tilde{q}_{y}+M^{2}-E^{2}\right],$

where $s\equiv\sin\beta$ and $c\equiv\cos\beta$. The quadratic equation allows
us to determine the mover modes given the Fermi energy and the Weiss field. If
the $\tilde{q}_{x}$ solutions are real-valued, we obtain travelling waves; the
complex-valued solutions correspond to evanescent waves.

The eigenfunctions are merely the eigenfunctions of a usual 2-by-2 Hermitian
matrix, generally expressed in terms of the Pauli matrices as
$d_{i}\sigma_{i}$, where $d_{i}=G^{(+)}_{ij}\tilde{q}_{j}+M_{i}$. The wave
functions are given by

$\displaystyle\psi$ $\displaystyle=$
$\displaystyle\begin{cases}\frac{1}{\sqrt{2d(d+d_{3})}}\begin{pmatrix}d+d_{3}\\\
d_{1}+\text{i}d_{2}\end{pmatrix},\text{ for }E=d>0,\\\
\frac{1}{\sqrt{2d(d+d_{3})}}\begin{pmatrix}\text{i}d_{2}-d_{1}\\\
d_{3}+d\end{pmatrix},\text{ for }E=-d<0.\end{cases}$ (8)

The group velocity for a mover are given by

$\displaystyle v_{i}$ $\displaystyle=$
$\displaystyle\frac{\left[G^{(+)}\right]^{T}_{ij}\left(\left[G^{(+)}\right]_{jl}\tilde{q}_{l}+M_{j}\right)}{E}$
(9)

Negative-chirality node: With ${\bf M}=0$, we can use $C_{4v}$, time-reversal,
and mirror symmetries ${\cal M}_{x}$, ${\cal M}_{y}$ to write out the
Hamiltonian for all $8$ Weyl nodes. For instance a negative chirality node is
obtained under a mirror operation, where we can relate the g-tensor part of
the Hamiltonian $\mathcal{H}^{(+)}$ to the g-tensor part of
$\mathcal{H}^{(-)}$. For the Weyl point related to the original one by a
mirror $\mathcal{M}_{y}$, we have

$\displaystyle\mathcal{H}^{(-)}$ $\displaystyle=$
$\displaystyle\sigma_{i}G^{(-)}_{ij}\tilde{q}_{j}+\sigma_{i}M_{i},$ (10)
$\displaystyle G^{(-)}$ $\displaystyle=$
$\displaystyle-\begin{pmatrix}|a|&&\\\ &\frac{1}{|a|}&\\\
&&1\end{pmatrix}\begin{pmatrix}\cos\beta&-\sin\beta&0\\\
\sin\beta&\cos\beta&0\\\ 0&0&1\end{pmatrix}.$ (11)

The distinctions from $G^{(+)}$ are (i) the prefactor -1 which leads to the
negative determinant and (ii) the $\sin\beta$ which used to be $-\sin\beta$ in
$G^{(+)}$. The latter amounts to a rotation of the Fermi surface about
$q_{z}$-axis by $-\beta$ instead of $\beta$ in $\mathcal{H}^{(+)}$. All the
formulae derived in earlier in this section can be straightforwardly
generalized for $\mathcal{H}^{(-)}$.

Choice of parameters: As an illustrative example, we choose $|a|=0.5$ and
$\beta=\pi/4$. This results in elliptical cross-sections (at any given
$k_{z}$) for the Fermi surfaces near a Weyl point with major : minor axis
ratio of $4:1$. The major axis of the ellipse is rotated by $\pi/4$, so that
it points along the $45^{\circ}$ direction in the $(k_{x},k_{y})$ plane. We
also choose other parameters to be reasonable values in line with the ab
initio calculations, namely $v_{F}=500$ meVÅand chemical potential $\mu=-30$
meV (below the Weyl node). This leads to Fermi pockets with a typical size
$k_{F}\sim 0.06$ Å-1. We fix the Weiss field to have a magnitude
$|M|=|E_{F}|/4$. We note that our results do not change qualitatively if we
choose somewhat different parameters - however, it is important that the
elliptical Fermi pockets are not aligned along the tetragonal $x$ or $y$ axes
(see Fig. 14).

## APPENDIX F: MODELING THE DOMAIN WALL

Figure 15: Evolution of the magnetization across a domain wall. Region I and
region III indicate bulk domains, and region-II is the domain wall region.
Going across the domain wall, the magnetization vector twists, with the
perpendicular domain wall magnetization $M_{z}^{DW}>0$ for the depicted
configuration. We will denote ${\bf
M}(x)=M(\cos\theta(x)\cos\gamma(x),\cos\theta(x)\sin\gamma(x),\sin\theta(x))$.
In region-I, we choose $\theta=0,\gamma=\pi/4$, while we set
$\theta=0,\gamma=-\pi/4$ in region III. In region II, we assume a twisting
magnetization profile, with the maximum out of plane component determined by
$\theta_{\rm max}$ which is achieved at the center of region II. Figure 16:
Illustration of the Fermi surfaces and the domain wall induced intra-node
scattering (dashed arrow). For simplicity, we have not shown the displacements
of the Fermi pockets relative to each other in the two domains or their
difference in spin textures, but this is taken into account in our
calculations as given below.

We assume a domain wall width $N\times w=40$ nm (corresponding to $N=40$.) We
consider a domain wall between a left and a right region (see illustration in
Fig. 15) with the following Weiss fields, respectively,

$\displaystyle\mathbf{M}_{I}$ $\displaystyle=$ $\displaystyle
M\left(\cos\gamma,\sin\gamma,0\right)^{T},$ (12)
$\displaystyle\mathbf{M}_{III}$ $\displaystyle=$ $\displaystyle
M\left(\cos\gamma,-\sin\gamma,0\right)^{T}.$ (13)

The domain wall region, region II, has a width $Nw$, which is partitioned into
$N$ intervals, each with width $w$. The Weiss field in $j$-th interval is
given by

$\displaystyle\mathbf{M}_{j}$ $\displaystyle=$ $\displaystyle
M\left(\cos\theta_{j}\cos\gamma_{j},\cos\theta_{j}\sin\gamma_{j},\sin\theta_{j}\right)^{T},$
(14) $\displaystyle\gamma_{j}$ $\displaystyle=$
$\displaystyle\gamma-2\gamma\frac{j-1/2}{N},$ (15) $\displaystyle\theta_{j}$
$\displaystyle=$
$\displaystyle\left(1-\frac{2|j-\frac{1}{2}-\frac{N}{2}|}{N}\right)\theta,$
(16)

where $\theta$ is the angle at the center of region II. $\theta_{j}$
monotonically decreases away from the center of region II. A large $N$ models
a smooth variation of the Weiss field in region II.

###  V.1 Transmission and reflection coefficients

The domain wall will lead to a scattering between Weyl Fermi surfaces. For
simplicity, we assume a smooth domain wall and only take into account the
intra-node scattering as shown in Fig. 16. We now sketch the computation of
the transmission coefficient (TC) and reflection coefficient (RC) at the
domain wall, defined earlier. For concreteness, we show the calculation for
$\mathcal{H}^{(+)}$. The step-by-step summary is given below

  * •

For a given Fermi energy $E$ and the parallel momenta $(q_{y},q_{z})$, we
compute the x-momenta for all the regions.

  * •

Compute the eigenfunctions

  * •

Wave function in each region is a superposition of a left mover and a right
mover, except in Region III, where the wave function consists of only a right
mover

$\displaystyle\Psi_{I}$ $\displaystyle=$
$\displaystyle\chi_{R}e^{\text{i}\mathbf{q}_{R}\cdot\mathbf{x}}+r\chi_{L}e^{\text{i}\mathbf{q}_{L}\cdot\mathbf{x}}$
(17) $\displaystyle\Psi_{II,j}$ $\displaystyle=$ $\displaystyle
c^{(j)}_{1}\eta_{1}^{(j)}e^{\text{i}\mathbf{p}^{(j)}_{1}\cdot\mathbf{x}}+c^{(j)}_{2}\eta_{2}^{(j)}e^{\text{i}\mathbf{p}^{(j)}_{2}\cdot\mathbf{x}},$
(18) $\displaystyle\Psi_{III}$ $\displaystyle=$ $\displaystyle
t\xi_{R}e^{\text{i}\mathbf{k}_{R}\cdot\mathbf{x}},$ (19)

where $t$ and $r$ are the transmission and reflection amplitude respectively.

  * •

We then match the wave function at each boundary at $x_{j}=jw$ for
$j=0,1,\cdots,N$ to determine $r,t$ and $c^{(j)}_{1,2}$. This can be
formulated in transfer matrix form. This can be seen below.

$\displaystyle\chi_{R}+r\chi_{L}$ $\displaystyle=$ $\displaystyle
c_{1}^{(1)}\eta_{1}^{(1)}+c_{2}^{(1)}\eta_{2}^{(1)},$ (20) $\displaystyle
c_{1}^{(1)}\eta_{1}^{(1)}e^{\text{i}p_{1x}^{(1)}w}+c_{2}^{(1)}\eta_{2}^{(1)}e^{\text{i}p_{2x}^{(1)}w}$
$\displaystyle=$ $\displaystyle
c_{1}^{(2)}\eta_{1}^{(2)}e^{\text{i}p_{1x}^{(2)}w}+c_{2}^{(2)}\eta_{2}^{(2)}e^{\text{i}p_{2x}^{(2)}w}$
(21) $\displaystyle
c_{1}^{(2)}\eta_{1}^{(2)}e^{\text{i}p_{1x}^{(2)}2w}+c_{2}^{(2)}\eta_{2}^{(2)}e^{\text{i}p_{2x}^{(2)}2w}$
$\displaystyle=$ $\displaystyle
c_{1}^{(3)}\eta_{1}^{(3)}e^{\text{i}p_{1x}^{(3)}2w}+c_{2}^{(3)}\eta_{2}^{(3)}e^{\text{i}p_{2x}^{(3)}2w}$
(22) $\displaystyle
c_{1}^{(j)}\eta_{1}^{(j)}e^{\text{i}p_{1x}^{(j)}jw}+c_{2}^{(j)}\eta_{2}^{(j)}e^{\text{i}p_{2x}^{(j)}jw}$
$\displaystyle=$ $\displaystyle
c_{1}^{(j+1)}\eta_{1}^{(j+1)}e^{\text{i}p_{1x}^{(j+1)}jw}+c_{2}^{(j+1)}\eta_{2}^{(j+1)}e^{\text{i}p_{2x}^{(j+1)}jw}$
(23) $\displaystyle
c_{1}^{(N)}\eta_{1}^{(N)}e^{\text{i}p_{1x}^{(N)}Nw}+c_{2}^{(N)}\eta_{2}^{(N)}e^{\text{i}p_{2x}^{(N)}Nw}$
$\displaystyle=$ $\displaystyle t\xi_{R}e^{\text{i}k_{Rx}Nw}.$ (24)

Rewriting in matrix form

$\displaystyle\begin{pmatrix}\chi_{R}&\chi_{L}\end{pmatrix}\begin{pmatrix}1\\\
r\end{pmatrix}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\eta_{1}^{(1)}&\eta_{2}^{(1)}\end{pmatrix}\begin{pmatrix}c_{1}^{(1)}\\\
c_{2}^{(1)}\end{pmatrix},$ (25)
$\displaystyle\begin{pmatrix}\eta_{1}^{(j)}e^{\text{i}p_{1x}^{(j)}jw}&\eta_{2}^{(j)}e^{\text{i}p_{2x}^{(j)}jw}\end{pmatrix}\begin{pmatrix}c^{(j)}_{1}\\\
c^{(j)}_{2}\end{pmatrix}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\eta_{1}^{(j+1)}e^{\text{i}p_{1x}^{(j+1)}jw}&\eta_{2}^{(j+1)}e^{\text{i}p_{2x}^{(j+1)}jw}\end{pmatrix}\begin{pmatrix}c^{(j+1)}_{1}\\\
c^{(j+1)}_{2}\end{pmatrix}$ (26)
$\displaystyle\begin{pmatrix}\eta_{1}^{(N)}e^{\text{i}p_{1x}^{(N)}Nw}&\eta_{2}^{(N)}e^{\text{i}p_{2x}^{(N)}Nw}\end{pmatrix}\begin{pmatrix}c^{(N)}_{1}\\\
c^{(N)}_{2}\end{pmatrix}$ $\displaystyle=$ $\displaystyle
t\xi_{R}e^{\text{i}k_{Rx}Nw}.$ (27)

From above, we can solve for $r$ and $t$ by the transfer matrix $T^{(j)}$:

$\displaystyle\begin{pmatrix}1\\\ r\end{pmatrix}$ $\displaystyle=$
$\displaystyle
te^{\text{i}k_{Rx}Nw}\begin{pmatrix}\chi_{R}&\chi_{L}\end{pmatrix}^{-1}T^{(1)}\cdots
T^{(N)}\xi_{R}\equiv t\begin{pmatrix}u_{1}\\\ u_{2}\end{pmatrix}$
$\displaystyle T^{(j)}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\eta_{1}^{(j)}e^{\text{i}p_{1x}^{(j)}(j-1)w}&\eta_{2}^{(j)}e^{\text{i}p_{2x}^{(j)}(j-1)w}\end{pmatrix}\begin{pmatrix}\eta_{1}^{(j)}e^{\text{i}p_{1x}^{(j)}jw}&\eta_{2}^{(j)}e^{\text{i}p_{2x}^{(j)}jw}\end{pmatrix}^{-1},$
(29)

where, in the definition of the transfer matrix, the two matrices differ from
each other, apart from the inverse operation, by the phase factors: one
involves $(j-1)w$, whereas the other involves $jw$. We finally obtain

$\displaystyle t$ $\displaystyle=$ $\displaystyle 1/u_{1}$ (30) $\displaystyle
r$ $\displaystyle=$ $\displaystyle u_{2}/u_{1}.$ (31)

TC and RC are given by

$\displaystyle TC$ $\displaystyle=$
$\displaystyle\frac{|v_{x,\text{trans}}|}{|v_{x,\text{inc}}|}|t|^{2},$ (32)
$\displaystyle RC$ $\displaystyle=$
$\displaystyle\frac{|v_{x,\text{refl}}|}{|v_{x,\text{inc}}|}|r|^{2}.$ (33)

The longitudinal conductance $g_{xx}$ and the transverse conductance $g_{yx}$
are then computed using TC and RC [64] (see also Refs. 71 and 72.)

###  V.2 Results: case $|a|=0.5,\beta=\pi/4$

####  V.2.1 Parameters

The $xy$-plane cross section of the Fermi surface near a Weyl point is an
ellipse whose major axis is rotated by $\pi/4$.

The results corresponds to the following parameters:

  * •

Fermi velocity $v_{F}=500$ meV.Å

  * •

Fermi wave vector $k_{F}\sim 0.06$ Å-1

  * •

Fermi energy $E_{F}=-30$ meV

  * •

Weiss field $|M|=0.015$ a.u., which corresponds to $|E_{F}|/4$.

  * •

Domain wall width $N\times w=40$ nm for $N=40$.

####  V.2.2 Results

Figure 17: Contributions from the 4 pairs (green-purple pair of adjacent Fermi
surfaces related by $\mathcal{M}_{x}$ or $\mathcal{M}_{y}$ in Fig. S14) of
Fermi surfaces. Figure 18: $\theta$ dependence of AHE stays the same upon
interchanging Region I and Region III.

We will show results of the anomalous Hall contribution obtained by
antisymmetrizing the off-diagonal conductance:
$g^{A}_{yx}(\theta)=\frac{1}{2}(g_{yx}(\theta)-g_{yx}(-\theta))$, namely
antisymmetrizing w.r.t merely reversing the $M_{z}$ component of the Weiss
field. Figure 17 shows $g^{A}_{yx}$ for the 4 pairs of WPs: (i) Fermi surfaces
in each pair are related by either $\mathcal{M}_{x}$ or $\mathcal{M}_{y}$
mirror operation at zero Weiss field (see Fig. 14), and (ii) different pairs
are related by a $C_{4v}$ rotation at zero Weiss field (see Fig. 14.)

A few main results are summarized below:

  * (1)

AHE contributions from the two Fermi surfaces in each pair has opposite signs,
yet they do not cancel each other out, so AHE is still non vanishing.

  * (2)

AHE from the four WPs related by $C_{4z}$ rotations has the same sign (see
Fig. 17.)

  * (3)

Interchanging region I and region III leads to the same $\theta$-dependence of
$g^{A}_{yx}$ (see Fig. 18.) This suggests that as long as the total
$z$-component of the Weiss field does not vanish, AHE contribution from the
domain wall is non-zero.

  * (4)

Ratio $g^{A}_{yx}/g_{xx}$ is of the order $10^{-3}$ at small angle, see Fig.
5(c).

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