# Nonlinear magneto-optical response across van Hove singularity in a non-
centrosymmetric magnetic Weyl semimetal

Jian Li  School of Physics and Electronic Science, Changsha University of
Science and Technology, Changsha 410076, China  Provincial Key Laboratory of
Informational Service for Rural Area of Southwestern Hunan, Shaoyang
University, Shaoyang 422000, China  Hunan Province Higher Education Key
Laboratory of Modeling and Monitoring on the Near-Earth Electromagnetic
Environments (Changsha University of Science and Technology)  Kai-He Ding
<EMAIL_ADDRESS>School of Physics and Electronic Science, Changsha
University of Science and Technology, Changsha 410076, China  Lijun Tang
<EMAIL_ADDRESS>School of Physics and Electronic Science, Changsha
University of Science and Technology, Changsha 410076, China

###### Abstract

We investigate the nonlinear magneto-optical response in non-centrosymmetric
magnetic Weyl semimetals featuring a quadratic tilt, focusing particularly on
the influence of the van Hove singularity (VHS). In the absence of a magnetic
field, the second-order nonlinear Drude conductivity components exhibit
inflection or dip behavior across the VHS. In contrast, the second-order
nonlinear anomalous Hall conductivity, primarily governed by the Berry
curvature dipole, manifests a subtle plateau-like structure. As the tilt
strength increases, the VHS energy escalates, thereby amplifying the VHS-
induced characteristics within these second-order conductivity components.
However, in the presence of a magnetic field, we show that the resultant
magnetic moment suppresses nonlinear electron transport while enhancing
nonlinear hole transport. This effect serves to mitigate the impact of the
VHS, resulting specifically in an asymmetric peak or a kinked-like structure
in the magnetic field-induced contribution to the second-order nonlinear
conductivity near the Weyl nodes. These findings provide new insights into the
intricate interplay among the VHS, Berry curvature, and magnetic moment in
nonlinear magneto-optical transport through non-centrosymmetric magnetic Weyl
semimetals.

##  I Introduction

The field of condensed matter physics has witnessed a significant surge in
interest and investigation into a unique class of materials known as Weyl
semimetals (WSMs)Burkov2011 ; Shin-Ming2015 . These materials exhibit a
specific electronic structure characterized by the presence of band-crossing
points, termed Weyl nodeslvsci2015 ; Bucciantini2017 . These nodes introduce a
new kind of quasiparticle, Weyl fermions, distinguished by their linear
dispersion relations and fundamental properties, making them pivotal entities
in the realm of topological materials Weng15 ; Belopolski15 .

One of the striking features of WSMs lies in the Berry curvature flux
emanating from these Weyl nodes within the Brillouin zone, behaving like
magnetic fields in momentum space. These nodes serve as sources and sinks of
Berry curvature flux, intimately tied to the chiralities of the Weyl fermions
Bentmann2021 ; Xiao10 . The distinct topological electronic structure of WSMs
has paved the way for the exploration of unconventional phenomena and emergent
behaviors, such as high mobility effects timuskprb2013 ; orlitanp2014 , Fermi
arcs Burkov2011 ; xuprl2011 ; lvsci2015 ; wanprb2011 ; Belopolski15 , and the
intriguing chiral anomaly Xiong15 ; Shiva20 ; Kipp21 ; Yuan20 ; Xiaochun15 .
These phenomena have spurred considerable interest and investigation into the
transport properties of these materials Tabert2016 ; Armitage18 ; Jin-Feng2022
. The simultaneous application of electric and magnetic fields to WSMs has
unveiled underlying transport mechanisms, yielding notable discoveries like
positive longitudinal magnetoconductivityyangnp2015 ; shekharnp2015 ; Xiong15
; linc2015 ; wangnc2016 ; lvprl2017 ; linc2016 ; dengprl2013  and giant planar
Hall effect burkovprb2017 ; nandyprl2017 . Further research has extended into
exploring the nonlinear magneto-optical responses to external electric fields,
revealing intriguing phenomena such as the quantum nonlinear Hall effect,
which arises solely from the dipole moment of the Berry curvature in the
absence of an applied magnetic field Morimoto16 ; Hai-Zhou18 ; FuruZhang20 ;
Sodemann15 .

The foundational understanding of WSMs traces back to the violation of
symmetries. According to the Nielsen-Ninomiya theorem Nielsen81 ; Nielsen83 ,
the existence of WSMs necessitates the breaking of time-reversal or inversion
symmetryZyuzin12 . Investigations into the impact of breaking these symmetries
have unveiled transitions from Dirac semimetals to WSMs, manifesting in
various transport signatures and anomalous effects Zhijun12 ; Zyuzin12 ;
Armitage18 . The manipulation of these symmetries through external means, such
as introducing the magnetic atoms or the magnetically doped multilayer
heterostructure, has led to the realization and study of magnetic
WSMsliusci2019 ; puphalprl2020 ; yinnat2018 ; LiuE. ; morsci2019 ; Guoqing18 ;
Cong Li2023 , showing fascinating effects like exotic drumhead surface
statesyinnat2018 , chiral magnetic effects and the giant anomalous Hall
effectLiuE. ; morsci2019 .

In certain WSMs, such as the TaAs family, the convergence of Weyl nodes gives
rise to a van Hove singularity (VHS) at relatively low energies YONG HU2022 ;
Zhenyu2022 ; Ebad-Allah2023 . While the impact of VHS on conventional linear
transport, including the emergence of negative magnetic resistance induced by
the VHSschumannprb2017 ; dingprb2023 , has been elucidated, its implications
in nonlinear magneto-optical responses have been largely overlooked.

In this work, we investigate the nonlinear magneto-optical response across the
VHS within non-centrosymmetric magnetic Weyl semimetals. We find that
depending on the coupling between the conventional velocity and Berry
curvature dipole, the VHS induces inflection points, dip behaviors, and subtle
plateau-like structures in the second-order nonlinear conductivity components
at zero magnetic field. Furthermore, we demonstrate that the application of a
magnetic field suppresses nonlinear electron transport while enhancing
nonlinear hole transport due to the presence of the magnetic moment. This
duality mitigates the influence of the VHS but also results in the emergence
of asymmetric peaks or kink-like structures in the magnetic field-induced
contribution to the second-order nonlinear conductivities near the Weyl nodes.
Upon evaluating the magnitude of these nonlinear conductivity components, our
findings indicate the potential observability of VHS-related features in non-
centrosymmetric magnetic WSMs subjected to simultaneous electric and magnetic
fields.

This paper is organized as follows: In Sec. II, we introduce a theoretical
model describing non-centrosymmetric magnetic WSMs in the presence of the VHS.
In Sec. III, we establish the magneto-optical transport equations within the
semiclassical approximation, considering the simultaneous impact of electric
and magnetic fields. In Sec. IV, we present analytical formulations for
second-order nonlinear magneto-optical conductivities, followed by a detailed
analysis of their numerical results. Finally, we conclude in Sec. V.

##  II Theoretical model

A non-centrosymmetric magnetic WSM can be effectively characterized through a
low-energy Hamiltonian Rui-Hao21 ; Hai-Zhou10 :

$H=v_{F}[k_{x}\sigma_{x}s_{z}-k_{y}\sigma_{y}-\lambda(k_{z}^{2}-Q_{D}^{2})\sigma_{z}]+H_{B},$
(1)

where $\sigma_{i}$ and $s_{i}$ ($i=x,y,z$) represent the Pauli matrices acting
on the orbital and spin space, $k_{i}$ denotes the wave vector, and $v_{F}$ is
the Fermi velocity. The term $\lambda(k_{z}^{2}-Q_{D}^{2})\sigma_{z}$
introduces two Dirac nodes located at $\mathbf{k}=(0,0,\pm Q_{D})$. The
Hamiltonian $H_{B}$ consists of three components:

$H_{B}=Rk_{z}^{2}+V_{I}k_{z}\sigma_{z}s_{z}+J_{ex}s_{z},$ (2)

where $V_{I}k_{z}\sigma_{z}s_{z}$ denotes the inversion symmetry-breaking
term, leading to the splitting of each Dirac node into two Weyl nodes with
opposite chirality along the $z$-axis. The term $Rk_{z}^{2}$ disrupts the
particle-hole symmetry(PHS), resulting in a tilted energy dispersion around
the Weyl nodes and modifying their energy. It is important to emphasize that
in Weyl semimetals hosting Fermi pockets, particularly when these pockets are
not directly linked to the Weyl nodes, the inclusion of a PHS-breaking term is
crucial for accurately describing this phenomenonRui-Hao21 ; nagprb2022 ;
msprb2020 . The formulation of this PHS-breaking term can be achieved by
adjusting the lattice structure (see, for example, Refs.Zhijun12 ; dipprb2020
). In the presence of the D4h symmetry of the crystalyqprb2020 , the PHS-
breaking term is characterized by its proportionality to $k_{z}^{2}$ at the
lowest order momentumRui-Hao21 . However, at low energy, higher-order terms
such as cubic or quartic terms have minimal influence and can be disregarded.
Furthermore, by expanding the energy dispersion around the Weyl nodes and
retaining terms up to the first order in $k_{z}$, one can derive the energy
dispersion in the vicinity of the Weyl node of chirality. This representation
corresponds to the low-energy excitations observed in materials such as MoTe2
and WTe2, featuring a linear term proportional to $k_{z}$, indicating a tilt
in the Weyl conedasprb2022 . Finally, $J_{ex}s_{z}$ represents the time-
reversal symmetry-breaking term, arising from the exchange interaction between
the Weyl-fermion spin and the magnetization in a magnetic WSM.

Figure 1: (color online) (a) The schematic portrays energy band dispersion
with four curves forming two pairs labeled by $s=\pm 1$. Each pair intersects,
creating four Weyl nodes denoted as $W_{\Lambda,s}(\Lambda=L,R)$. The Fermi
energy $E_{F}$, indicated by the dashed blue line, traverses the band curves,
progressively increasing from the left to the right-hand side of the upper
panel. Lower panel of figure (a): Representation of the corresponding Fermi
surface, exhibiting asymmetric features and undergoing a Lifshitz
transitionlifjetp1960  as the Fermi energy increases. The density of states
for the tilt $R=0eV\cdot nm^{2}$(black line), $R=0.02eV\cdot nm^{2}$(red
line), and $R=0.04eV\cdot nm^{2}$(blue line) at $V_{I}=0.05eV\cdot nm$ (b),
and for the parameter $V_{I}=0eV\cdot nm$(black line), $V_{I}=0.03eV\cdot
nm$(red line), and $V_{I}=0.05eV\cdot nm$(blue line) at $R=0.02eV\cdot nm^{2}$
(c). The other parameters are set as $Q_{D}=2nm^{-1}$, $J_{ex}=0eV$, and
$\lambda=0.06nm$.

Diagonalizing Hamiltonian (1) yields the energy spectrum:

$\varepsilon_{\alpha}^{s}=\alpha
v_{F}\sqrt{k_{\parallel}^{2}+\Delta_{s}^{2}(k_{z})}+Rk_{z}^{2}+sJ_{ex},$ (3)

where $k_{\parallel}^{2}=k_{x}^{2}+k_{y}^{2}$,
$\Delta_{s}(k_{z})=\lambda(k_{z}^{2}-Q_{D}^{2})-sV_{I}k_{z}/v_{F}$, and
$\alpha=\pm$ denote the conduction or valence bands. The corresponding
eigenvectors are:

$|u_{\mathbf{k}+}^{s}\rangle=\left(\begin{array}[]{cc}\cos\frac{\theta_{s}}{2}&e^{i\phi_{s}}\sin\frac{\theta_{s}}{2}\end{array}\right)^{t},$
(4)
$|u_{\mathbf{k}-}^{s}\rangle=\left(\begin{array}[]{cc}e^{i\phi_{s}}\sin\frac{\theta_{s}}{2}&-\cos\frac{\theta_{s}}{2}\end{array}\right)^{t},$
(5)

where
$\cos\theta_{s}=\frac{\Delta_{s}(k_{z})}{\sqrt{k_{\parallel}^{2}+[\Delta_{s}(k_{z})]^{2}}}$,
and $\tan\phi_{s}=-\frac{sk_{y}}{k_{x}}$.

In Eq. (3), $s=\pm 1$ denotes the chirality of the Weyl node. This can be
identified through the Berry curvature $\mathbf{\Omega}_{\alpha}^{s}$,
expressed as:

$\mathbf{\Omega}_{\alpha}^{s}=-\operatorname{Im}\left[\langle\nabla_{\mathbf{k}}u_{\mathbf{k}\alpha}^{s}|\times|\nabla_{\mathbf{k}}u_{\mathbf{k}\alpha}^{s}\rangle\right].$
(6)

By substituting Eqs. (4) and (5) into Eq. (6), the resulting equations are:

$\Omega_{\alpha x/y}^{s}=-\frac{s\alpha
k_{x/y}\Delta_{s}^{\prime}(k_{z})}{{2K^{3}}},\ \Omega_{\alpha
z}^{s}=-\frac{{s\alpha\Delta_{s}(k_{z})}}{{2K^{3}}},$ (7)

where $\Delta_{s}^{\prime}(k_{z})=\partial\Delta_{s}(k_{z})/\partial k_{z}$,
and $K=(k_{x}^{2}+k_{y}^{2}+\Delta_{s}^{2})^{1/2}$.

The dispersion relation (3) depicted in Fig. 1(a) encompasses two pairs of
band curves, denoted by $s=\pm 1$. Each band pair intersects, forming two Weyl
nodes located at $(0,0,\pm\sqrt{Q_{D}^{2}+q^{2}}-sq)$ with $q=V_{I}/(2\lambda
v_{F})$, labeled as $W_{\Lambda,s}$ ($\Lambda=L,R$) (see upper panel of Fig.
1(a)). When the Fermi energy crosses the bands at energy $\varepsilon>0$, an
increase leads to a Lifshitz transitionlifjetp1960 , transforming the Fermi
surface from two disconnected parts into a single one(see lower panel of
Fig.1(a)). Throughout this transition, the Fermi surface consistently remains
asymmetric. These distinct Fermi surface behaviors, coupled with the chiral
anomaly, significantly impact the nonlinear magneto-optical responses in WSMs
(as discussed below). Fig.1(b) illustrates the density of states (DOS) for
varying tilt $R$. A detailed calculation of DOS is provided in the appendix.
Evidently, the DOS displays inflections at VHS, which remain symmetrical with
respect to the zero-energy point at tilt $R=0$. However, with the application
of tilt, the symmetry of DOS is lost due to the PHS breaking. The asymmetry
induced by $R$ can be further heightened by the breaking of inversion symmetry
(see Fig. 1(c)).

##  III Magneto-optical transport equations within Semiclassical approximation

We consider the simultaneous application of the static magnetic field
${\bf{B}}$ and light field ${\bf{E}}$ to the system. The resultant electric
current can be computed using the following integral:

$\displaystyle{\bf{j}}=-e\int{[d{\bf{k}}]}D{\bf{\dot{r}}}f({\bf{k}},{\bf{r}},t),$
(8)

where ${\rm{[d}}{\bf{k}}]={{d{\bf{k}}}/{{{(2\pi)}^{3}}}}$, and the weighting
factor $D$ emerges due to the alteration of the phase volume caused by the
electric and magnetic fields (see below). ${\bf{\dot{r}}}$ represents the
electron’s velocity and is obtained by solving the following equation of
motion Xiao10 :

$\displaystyle{\bf{\dot{r}}}={1\over\hbar}{\nabla_{\bf{k}}}{\epsilon_{\alpha\bf{k}}^{s}}-{\bf{\dot{k}}}\times{{\bf{\Omega}}_{\alpha}^{s}},$
(9)
$\displaystyle\hbar{\bf{\dot{k}}}=-e{\bf{E}}-e{\bf{\dot{r}}}\times{\bf{B}}.$

This equation describes the trajectory of an electron in phase space, viewed
as a wave packet within the semiclassical approximation. The self-rotation of
the wave packet around its center of mass under the magnetic field induces the
existence of an orbital magnetic moment ${{\bf{m}}_{\alpha}^{s}}$, leading to
a modification of the dispersion relation (3):

$\epsilon_{\alpha\bf{k}}^{s}=\varepsilon_{\alpha}^{s}-{{\bf{m}}_{\alpha}^{s}}\cdot{\bf{B}}$
(10)

with

${{\bf{m}}_{\alpha}^{s}}=-{e\over{2\hbar}}{\mathop{\rm
Im}\nolimits}[\left\langle{{\nabla_{\bf{k}}}{u_{\bf{k}\alpha}^{s}}}\right|\times(H-\varepsilon_{\alpha}^{s})\left|{{\nabla_{\bf{k}}}{u_{\bf{k}\alpha}^{s}}}\right\rangle].$
(11)

By substituting the wave function $|u_{\mathbf{k}\alpha}^{s}\rangle$ in Eq.
(11), the orbit magnetic moment are further expressed as

$\displaystyle m_{\alpha x/y}^{s}=-s\alpha
e{v_{F}}{{{k_{x/y}}\Delta_{s}^{\prime}({k_{z}})}\over{2{K^{2}}}},m_{\alpha
z}^{s}=-s\alpha e{v_{F}}{{\Delta_{s}({k_{z}})}\over{2{K^{2}}}}.$ (12)

By solving the coupled equations (9), one can obtain:

${D\bf{\dot{r}}}=\mathbf{v}_{\alpha}^{s}+\frac{e}{\hbar}{\bf{E}}\times{{\bf{\Omega}}_{\alpha}^{s}}+{e\over\hbar}(\mathbf{v}_{\alpha}^{s}\cdot{{\bf{\Omega}}_{\alpha}^{s}}){\bf{B}},$
(13)
${D\bf{\dot{k}}}=-\frac{e}{\hbar}{\bf{E}}-{e\over\hbar}\mathbf{v}_{\alpha}^{s}\times{\bf{B}}-{{{e^{2}}}\over\hbar^{2}}({\bf{E}}\cdot{\bf{B}}){{\bf{\Omega}}_{\alpha}^{s}},$
(14)

where
$\mathbf{v}_{\alpha}^{s}=\frac{1}{\hbar}\nabla_{\bf{k}}\epsilon_{\alpha\bf{k}}^{s}$,
and $D=1+{e\over\hbar}{\bf{B}}\cdot{{\bf{\Omega}}_{\alpha}^{s}}.$ In Eq.(13),
the first term is the usual group velocity, the second term is anomalous
velocity induced by the Berry curvature, and the third term represents their
coupling induced by the magnetic field. In Eq.(14), the first two terms are
the usual Lorentz force, while the last term is associated with chiral anomaly
Nielsen83 .

In Eq.(8), $f({\bf{k}},{\bf{r}},t)$ is a distribution function obeying semi-
classical Boltzmann kinetic equation as follows:

$\displaystyle{{{\rm{d}}f({\bf{k}},{\bf{r}},t)}\over{dt}}={{\partial
f}\over{\partial t}}+{\bf{\dot{k}}}{{\partial
f}\over{\partial{\bf{k}}}}+{\bf{\dot{r}}}{{\partial
f}\over{\partial{\bf{r}}}}={I_{c}}\\{f\\}$ (15)

with ${I_{c}}\\{f\\}$ a collision integral term. We assume that the light
field has the form of ${\bf{E}}(t)={\bf{E}}{e^{-i\omega t}}$, and the magnetic
field is homogeneously applied on WSM. In this situation, the distribution
function $f({\bf{k}},{\bf{r}},t)$ becomes independent of the spatial
coordinate. Within the relaxation time approximation, Eq.(15) reduces to

$\displaystyle{{\partial f}\over{\partial t}}+{\bf{\dot{k}}}{{\partial
f}\over{\partial{\bf{k}}}}={\rm{-}}{{f-{f_{0}}}\over\tau},$ (16)

where $f_{0}$ is the Fermi distribution function. For simplicity, we ignore
internode scattering, thus considering $\tau$ from Eq.(16) as the intranode
scattering time. Detailed discussions on the influence of internode scattering
in the nonlinear transport regime are available in Ref.dasprb2022 . To solve
Eq.(16), the distribution function $f$ is expanded up to second order in the
electric field:

$\displaystyle f={f_{0}}+{f_{1}}{e^{-i\omega t}}+{f_{2}}{e^{-2i\omega t}}.$
(17)

Inserting Eq. (17) into Eq.(16) and equating equal powers of $\mathbf{E}$, we
get the recursion equations. Further solving these equations finally leads to

$\displaystyle{f_{1}}={\tau\over{1-i\omega\tau}}{1\over{\hbar
D}}[e{\bf{E}}+{{{e^{2}}}\over\hbar}({\bf{E}}\cdot{\bf{B}}){{\bf{\Omega}}_{\alpha}^{s}}]\cdot{\nabla_{\bf{k}}}{f_{0}},$
(18)
$\displaystyle{f_{2}}={{{\tau^{2}}}\over{(1-i\omega\tau)(1-2i\omega\tau)}}{({1\over{\hbar
D}})^{2}}$ (19)
$\displaystyle\times[e{\bf{E}}+{{{e^{2}}}\over\hbar}({\bf{E}}\cdot{\bf{B}}){{\bf{\Omega}}_{\alpha}^{s}}]\cdot{\nabla_{\bf{k}}}\\{[e{\bf{E}}+{{{e^{2}}}\over\hbar}({\bf{E}}\cdot{\bf{B}}){{\bf{\Omega}}_{\alpha}^{s}}]\cdot{\nabla_{\bf{k}}}{f_{0}}\\}.$

Substituting Eqs.(18) and (19) in Eq.(8), we will get the analytic expressions
of the electric current in powers of $\mathbf{E}$(see below), which provides a
basis for exploring the nonlinear magneto-optical transport through WSM.

##  IV Second-order nonlinear magneto-optical conductivities

We now explore the second-order nonlinear magneto-optical response of WSMs.
Employing Eq.(8), the expression for the second-order nonlinear current
response at the frequency $2\omega$ is given by

$\mathbf{j}^{s}=-e\int{[d{\bf{k}}]}D{\bf{\dot{r}}}({f_{1}}+{f_{2}}).$ (20)

By substituting Eq.(13) into Eq.(20), the second-order nonlinear current can
be expressed as:

$\displaystyle\mathbf{j}^{s}=-e\int{[d{\bf{k}}]\\{[{{\bf{v}}_{\alpha}^{s}}+{e\over\hbar}({{\bf{v}}_{\alpha}^{s}}\cdot{{\bf{\Omega}}_{\alpha}^{s}}){\bf{B}}]{f_{2}}+{e\over\hbar}{\bf{E}}\times{{\bf{\Omega}}_{\alpha}^{s}}{f_{1}}\\}}.$
(21)

Combining Eqs.(18), (19) with Eq.(21) yields the following expressions:

$\displaystyle\mathbf{j}^{s}=-{{e\tau}\over{(1-2i\omega\tau)}}\int{[d{\bf{k}}]\over{\hbar
D}}[{{\bf{v}}_{\alpha}^{s}}+{e\over\hbar}({{\bf{v}}_{\alpha}^{s}}\cdot{{\bf{\Omega}}_{\alpha}^{s}}){\bf{B}}][e{\bf
E}+{{{e^{2}}}\over\hbar}({\bf{E}}\cdot{\bf{B}}){\bf{\Omega}_{\alpha}^{s}}]\cdot{{\partial{f_{1}}}\over{\partial{\bf{k}}}}$
(22)
$\displaystyle-{{{e^{2}}\tau}\over{\hbar(1-i\omega\tau)}}\int{[d{\bf{k}}]\over{\hbar
D}}{\bf{E}}\times{{\bf{\Omega}}_{\alpha}^{s}}[e{\bf{{\bf
E}}}+{{{e^{2}}}\over\hbar}({\bf{E}}\cdot{\bf{B}}){{\bf{\Omega}}_{\alpha}^{s}}]\cdot{{\partial{f_{0}}}\over{\partial{\bf{k}}}}.$

From Eq. (22), it is apparent that, alongside the
$\mathbf{E}\times\mathbf{\Omega}_{\alpha}^{s}$ term, the chiral anomaly (i.e.,
the $\mathbf{E}\cdot\mathbf{B}$ term) contributes to the Hall current.
Moreover, the existence of a magnetic field induces a magnetic moment (refer
to Eq. (11)), exerting distinct effects on the transport properties in the
electron and hole regions(detailed below).

###  IV.1 Second-order nonlinear conductivity without magnetic field

In the absence of a magnetic field($B=0$), Eq.(22) simplifies as follows:

$\begin{array}[]{cll}\mathbf{j}^{s}&=&-\kappa_{1}\int[d\mathbf{k}]\mathbf{v}_{\alpha}^{s}e\mathbf{E}\cdot\frac{\partial}{\hbar\partial\mathbf{k}}[(e\mathbf{E}\cdot\mathbf{v}_{\alpha}^{s})\frac{\partial
f_{0}^{s}}{\partial\varepsilon_{\alpha}^{s}}]\\\
&&-\kappa_{2}\int[d\mathbf{k}]\mathbf{E}\times\mathbf{\Omega}_{\alpha}^{s}(e\mathbf{E}\cdot\mathbf{v}_{\alpha}^{s})\frac{\partial
f_{0}^{s}}{\partial\varepsilon_{\alpha}^{s}},\end{array}$ (23)

where ${\kappa_{1}}={{e{\tau^{2}}}\over{(1-2i\omega\tau)(1-i\omega\tau)}}$ and
${\kappa_{2}}={{{e^{2}}\tau}\over{\hbar(1-i\omega\tau)}}$. We express Eq.(23)
in the form of $j_{a}^{s}=\sigma_{abc}^{s}E_{b}(\omega)E_{c}(\omega)$, where
$\sigma_{abc}^{s}$ is the second-order nonlinear conductivity:

$\sigma_{abc}^{s}(2\omega)=\sigma_{abc}^{s,0}+\sigma_{abc}^{s,H}.$ (24)

In Eq.(24), the first term denotes the nonlinear Drude conductivity:

$\sigma_{abc}^{s,0}=\frac{e^{2}\kappa_{1}}{\hbar}\int[d\mathbf{k}]\frac{\partial
v_{\alpha a}^{s}}{\partial k_{b}}v_{\alpha c}^{s}\frac{\partial
f_{0}^{s}}{\partial\varepsilon_{\alpha}^{s}},$ (25)

while the second term is the nonlinear anomalous Hall conductivities induced
by the intrinsic Berry curvature dipoles:

${\sigma^{s,H}_{abc}}=-\varepsilon_{adc}e\kappa_{2}\int[d\mathbf{k}]\Omega^{s}_{\alpha
d}v^{s}_{\alpha b}\frac{\partial
f_{0}^{s}}{\partial\varepsilon_{\alpha}^{s}},$ (26)

where $\varepsilon_{adc}$ represents the three-dimensional Levi-Civita
antisymmetric tensor, and the integral denotes the Berry curvature
dipolesSodemann15 .

####  IV.1.1 Second-order nonlinear conductivity $\sigma_{abc}^{s,0}$

Since the velocities $\mathbf{v}_{\alpha}^{s}$ exhibit odd functional
dependencies on the momentum components $\mathbf{k}$, the integrands in
Eq.(25) maintain consistent odd symmetries regarding the momentum $k_{x}$,
$k_{y}$, or $k_{z}$. In the absence of a tilt in the Weyl cone, the Fermi
energy surface preserves its symmetry relative to the origin. In this context,
contributions to the electric current from both positive($W_{R,s}$) and
negative($W_{L,s}$) Weyl nodes possess equal magnitudes but opposite
directions, resulting in the complete nullification of the conductivity
$\sigma_{abc}^{s,0}$. However, when considering the tilt term $Rk_{z}^{2}$,
the parity characteristics of the velocity $\mathbf{v}_{\mathbf{k}}^{s}$
remain unchanged. Nonetheless, the Fermi surface symmetry is exclusively
disrupted along the $z$ direction while remaining preserved in the $x$ and $y$
directions. Thus, non-zero components of the conductivity tensor
$\sigma_{abc}^{s,0}$ necessitate an even number of $x$ or $y$ indices, as
outlined below:

$\displaystyle\sigma_{zxx}^{s,0}=\frac{e^{2}v_{F}\kappa_{1}}{8\pi^{2}\hbar^{2}}\int{d{k_{z}}}\frac{\Delta_{s}(k_{z})\Delta_{s}^{\prime}(k_{z})}{r(k_{z})}[1-\frac{\Delta_{s}^{2}(k_{z})}{r^{2}(k_{z})}],$
(27)
$\displaystyle\sigma_{xzx}^{s,0}={{{e^{2}}{v_{F}}{\kappa_{1}}}\over{8{\pi^{2}}{\hbar^{2}}}}\int{d{k_{z}}}\\{[\frac{\Delta_{s}(k_{z})\Delta_{s}^{\prime}(k_{z})}{r(k_{z})}+{{2R{k_{z}}}\over{\hbar{v_{F}}}}]$
(28)
$\displaystyle\times[-1-{{\Delta_{s}^{2}({k_{z}})}\over{{r^{2}}({k_{z}})}}]\\},$
$\displaystyle\sigma_{zzz}^{s,0}=-\frac{e^{2}v_{F}\kappa_{1}}{4\pi^{2}\hbar^{2}}\int{d{k_{z}}}\\{[\frac{\Delta_{s}(k_{z})\Delta_{s}^{\prime}(k_{z})}{r(k_{z})}+\frac{2Rk_{z}}{\hbar
v_{F}}]$ (29)
$\displaystyle\times[2\lambda\Delta_{s}(k_{z})+[\Delta_{s}^{\prime}(k_{z})]^{2}-\frac{\Delta_{s}^{2}({k_{z}})[\Delta_{s}^{\prime}(k_{z})]^{2}}{r^{2}(k_{z})}]$
$\displaystyle+{{2R}\over\hbar
v_{F}}[\Delta_{s}({k_{z}})\Delta_{s}^{\prime}({k_{z}})+r(k_{z})\frac{2Rk_{z}}{\hbar
v_{F}}]\\},$

where $r({k_{z}})={{\mu-s{J_{ex}}-{\rm{R}}k_{z}^{2}}\over{\hbar{v_{F}}}}$. The
Fermi surface symmetry concerning the $k_{x}$ and $k_{y}$ axes implies that
the other components satisfy the relation
$\sigma_{xxz}^{s,0}=\sigma_{zyy}^{s,0}=\sigma_{yyz}^{s,0}=\sigma_{zxx}^{s,0}$,
and $\sigma_{yzy}^{s,0}=\sigma_{xzx}^{s,0}$. It is noted that $\sigma_{xxz}$
does not precisely align with the component $\sigma_{xzx}$, while it exhibits
an approximate equivalence to $\sigma_{xzx}$ particularly under conditions of
small tilt amplitudes, denoted by $R$. Additionally, Eqs. (27) and (29) can be
expressed as follows:
$\sigma_{zxx}^{s,0}=\frac{e^{3}v_{F}\sigma_{d1}}{8\pi^{2}\hbar^{2}}{{{\tau^{2}}}\over{(1-2i\omega\tau)(1-i\omega\tau)}}$
and
$\sigma_{zzz}^{s,0}=-\frac{e^{3}v_{F}\sigma_{d2}}{4\pi^{2}\hbar^{2}}{{{\tau^{2}}}\over{(1-2i\omega\tau)(1-i\omega\tau)}}$.
Here, $\sigma_{d1}$ and $\sigma_{d2}$ represent the integral terms in Eqs.
(27) and (29), respectively. It is evident that these parameters,
$\sigma_{d1}$ and $\sigma_{d2}$ are not dependent on the frequency $\omega$.
In the transport limit, where $\omega\tau<<1$, the conductivity components
$\sigma_{zxx}^{s,0}$ and $\sigma_{zzz}^{s,0}$ are proportional to $\tau^{2}$
and independent of frequency. Conversely, in the optical or clean limit, where
$\omega\tau>>1$, these conductivity components are proportional to
$1/\omega^{2}$ and independent of $\tau$. These results are consistent with
those of the single-node modelGaoyang22 .

Figure 2: (color online) The second-order nonlinear conductivities
$\sigma_{zxx}^{+1,0}$(a) and $\sigma_{zzz}^{+1,0}$(c) in unit of
$e^{2}v_{F}\kappa_{1}/8\pi^{2}\hbar^{2}$ as a function of the Fermi energy for
the tilts $R=0.02eV\cdot nm^{2}$(black line), $R=0.04eV\cdot nm^{2}$(red
line), and $R=0.05eV\cdot nm^{2}$(blue line). Figure (b) depicts the tilt
dependence of the Weyl node and VHS that are indicated by the energy band
diagram in the inset of Figure(b). The specific positions of conductivity
curve features, such as peaks and inflection points, in (a) and (c) are
precisely identified by the intersecting horizontal and vertical dashed lines.
The other parameters are taken as $V_{I}=0.04eV\cdot nm$, $Q_{D}=2nm^{-1}$,
$J_{ex}=0eV$, and $\lambda=0.06nm$.

We begin our numerical analysis by examining the scenario where the parameter
$J_{ex}$ is set to zero. In Fig. 2(a) and (c), the conductivity components
$\sigma_{zxx}^{+1,0}$ and $\sigma_{zzz}^{+1,0}$ are presented across the VHS
and the Weyl node (indicated by circles in the inset of Fig. 2(b)) within the
$s=+1$ band, considering various tilt values represented by $R$. Near the Weyl
node, $\sigma_{zxx}^{+1,0}$ demonstrates a linear correlation with the Fermi
energy $E_{F}$, resulting in a cusp peak concerning the Fermi energy at the
Weyl node. This finding aligns with the outcomes of the single-node modelRui-
Hao21 ; Gaoyang22 . Conversely, at the VHS, $\sigma_{zxx}^{+1,0}$ shows an
inflection in relation to the Fermi energy, marking a transition from a linear
to a nonlinear relationship. As $R$ increases, both the peak and the
inflection shift towards higher energies. Fig. 2(b) precisely tracks their
positions, depicting the evolution of the Weyl node and VHS influenced by the
tilt. Remarkably, as the tilt strength amplifies, the Weyl node experiences a
more rapid shift, causing the conductivity peak to approach the inflection
point at the VHS (refer to Fig. 2(a)). Upon reaching a sufficient tilt
magnitude, the Weyl node and VHS merge into a single point. In this scenario,
the band structure around the Weyl node tends to flatten, leading to a rapid
increase in the density of states and consequently enhancing electron
nonlinear transport. This enhancement manifests in the increased conductivity
peak in Fig. 2(a).

Moving to Fig. 2(c), we observe that the second-order nonlinear conductivity
$\sigma_{zzz}^{+1,0}$ displays a dip either at the Weyl node or the VHS. Due
to the distortion of the Fermi surface induced by the tilt, these dip
structures exhibit asymmetry. With an increase in tilt, these two dips
diminish in magnitude, and their separation reduces, resulting in the
emergence of a broad peak between them. At the juncture where the Weyl node
and VHS merge, this peak notably disappears, leading to a substantial dip.

Figure 3: (color online) The second-order nonlinear conductivities
$\sigma_{zxx}^{+1,0}$(a) and $\sigma_{zzz}^{+1,0}$(c) in unit of
$e^{2}v_{F}\kappa_{1}/8\pi^{2}\hbar^{2}$ as a function of the Fermi energy for
the parameter $V_{I}=0.01eV\cdot nm$(black line) and $V_{I}=0.05eV\cdot
nm$(red line) at $R=0.02eV\cdot nm^{2}$. Figure (b) shows the variation of the
Weyl node and VHS, marked by the inset of Fig. (b), with the parameter
$V_{I}$. The precise locations of salient features in the conductivity curves,
including peaks and inflection points, in panels (a) and (c) are indicated by
the intersection of horizontal and vertical dashed lines. All other parameters
remain consistent with those of Fig. 2.

By varying the parameter $V_{I}$, we analyze the second-order nonlinear
conductivity components, $\sigma_{zxx}^{+1,0}$ and $\sigma_{zzz}^{+1,0}$ as a
function of the Fermi energy $E_{F}$ in Fig. 3(a) and (c). Notably, the
amplitudes of $\sigma_{zxx}^{+1,0}$ or $\sigma_{zzz}^{+1,0}$ near the Weyl
node and VHS exhibit inconsistent changes with $V_{I}$. This behavior markedly
differs from that observed when adjusting the tilt (as depicted in Fig. 2).
This discrepancy stems from the distinct evolution of the Fermi surface
induced by $R$ and $V_{I}$. Under variations in $V_{I}$, the Weyl node and VHS
move at nearly identical rates, as illustrated in Fig. 3(b). Consequently, the
separation between the Weyl node and the VHS remains constant, preserving the
symmetric features of the Fermi surface. Conversely, when influenced by $R$,
the symmetry of the Fermi surface is disrupted, indicated by the convergence
of the Weyl node($W_{R,+1}$) and the VHS towards each other (refer to
Fig.2(b)).

Indeed, at $J_{ex}=0$, due to opposite chiralities, the contributions of the
$s=+1$ and $s=-1$ bands to the second-order conductivity tend to counteract
each other. However, when ${J_{ex}}\neq 0$, the chirality symmetry breaks, and
the $s=\pm 1$ bands separate by $J_{ex}$. Consequently, the total second-order
nonlinear conductivities display non-zero values and introduce additional
inflection points compared to the case of a single $s=+1$ band (see Fig.4).
These intervals are determined by the relative displacements between the Weyl
nodes and the VHS, showcasing the subtle relationship between band structure
and electron transport.

Figure 4: (color online) The second-order nonlinear conductivities for the
$s=+1$(dashed line) and $s=-1$(dotted line) bands, and their summation (solid
line), measured in unit of $e^{2}v_{F}\kappa_{1}/8\pi^{2}\hbar^{2}$ as a
function of the Fermi energy at $V_{I}=0.05eV\cdot nm$, $R=0.02eV\cdot nm^{2}$
and $J_{ex}=0.02eV$. The other parameters are taken as the same as those of
Fig.2.

####  IV.1.2 Second-order anomalous Hall conductivity $\sigma_{abc}^{s,H}$

Based on the the parity of the velocity $\mathbf{v}_{\alpha}^{s}$ and Berry
curvature $\mathbf{\Omega}_{\alpha}^{s}$, alongside the involvement of the
antisymmetric tensor $\varepsilon_{acd}$, the distinct component indices among
$x$, $y$ and $z$ are requisite for the existence of $\sigma_{abc}^{s,H}$. By a
straightforward calculation, the expressions are derived as follows:

$\displaystyle\sigma_{xyz}^{s,H}=-\frac{se\kappa_{2}}{16\pi^{2}\hbar}\int{d{k_{z}}}\frac{\Delta_{s}^{\prime}(k_{z})}{r(k_{z})}[1-\frac{\Delta_{s}^{2}(k_{z})}{r^{2}(k_{z})}],$
(30)
$\displaystyle\sigma_{xzy}^{s,H}={{{se\kappa_{2}}}\over{8\pi^{2}\hbar}}\int
dk_{z}\frac{\Delta_{s}(k_{z})}{r^{2}(k_{z})}[{{{\Delta_{s}({k_{z}})}\Delta_{s}^{\prime}({k_{z}})}\over{{r({k_{z}})}}}+{{2R{k_{z}}}\over{\hbar
v_{F}}}].$ (31)

Additional nonzero components adhere to the relations:
$\sigma_{zxy}^{s,H}=-\sigma_{zyx}^{s,H}=-\sigma_{yxz}^{s,H}=\sigma_{xyz}^{s,H}$
and $-\sigma_{yzx}^{s,H}=\sigma_{xzy}^{s,H}$. The frequency dependence of
conductivity, as expressed in Eqs. (LABEL:hhd3) and (LABEL:hhd32), is
encapsulated within the factor $\kappa_{2}$. Under the transport limit, where
$\omega\tau<<1$, the conductivity components $\sigma_{xyz}^{s,H}$ and
$\sigma_{xzy}^{s,H}$ exhibit a direct proportionality to the relaxation time
$\tau$ and remain invariant with respect to frequency. Conversely, under the
optical or pristine limit, these conductivities, $\sigma_{xyz}^{s,H}$ and
$\sigma_{xzy}^{s,H}$, become purely imaginary and demonstrate a
proportionality to $1/\omega$, thereby reflecting the characteristics of
second harmonic generation.

Figure 5: (color online) Depiction of second-order nonlinear conductivities
($\sigma_{xyz}^{+1,H}$ in (a) and (b), $\sigma_{xzy}^{+1,H}$ in (c) and (d))
for the $s=+1$ band, varying with $R$ and $V_{I}$. These conductivities,
measured in units of $e\kappa_{2}/8\pi^{2}\hbar$, are presented as a function
of the Fermi energy at $V_{I}=0.03eV\cdot nm$ (a) and (c), $R=0.03eV\cdot
nm^{2}$ (b) and (d). Remaining parameters are consistent with those specified
in Fig.2.

Fig.5 depicts the behavior of the second-order anomalous Hall conductivities
$\sigma_{xyz}^{+1,H}$ and $\sigma_{xzy}^{+1,H}$ as functions of the Fermi
energy for different values of $R$ and $V_{I}$ at $J_{ex}=0$. Both
conductivities exhibit a plateau-like structure. For $\sigma_{xyz}^{+1,H}$,
there’s either an increase or decrease at low or high energies, respectively,
with increasing $R$, while $\sigma_{xzy}^{+1,H}$ exhibits a inverse dependence
on the $R$. However, with an increase in $V_{I}$, $\sigma_{xyz}^{+1,H}$
($\sigma_{xzy}^{+1,H}$) maintains a nearly constant plateau height at low
energy, and slightly decrease(increases) at high energy. These distinct
variations attributed to $R$ and $V_{I}$ are associated with the observed
asymmetry in the Fermi surface (as depicted in Fig. 1). The plateau widths are
determined by the separation between the Weyl nodes and VHS, hence their
changes with increasing $R$ and $V_{I}$ display inconsistent behaviors. Upon
the introduction of $J_{ex}$, the $s=\pm 1$ bands split due to $J_{ex}$,
consequently altering the plateau structure and resulting in the emergence of
new plateaus in the total second-order nonlinear Hall conductivities, as
illustrated in Fig. 6.

Figure 6: (color online) The second-order nonlinear conductivities for the
$s=+1$(dashed line) and $s=-1$(dotted line) bands, and their summation(solid
line), measured in unit of $e^{2}v_{F}\kappa_{1}/8\pi^{2}\hbar^{2}$ as a
function of the Fermi energy at $V_{I}=0.05eV\cdot nm$, $R=0.02eV\cdot nm^{2}$
and $J_{ex}=0.02eV$. The other parameters are taken as the same as those of
Fig.2.

###  IV.2 Magnetic field induced contribution to second-order conductivity

In the presence of a weak magnetic field, incorporating the effect of the
orbital magnetic moment modifies the distribution function, expressed
asShudan2016 ; Gaoyang22

${f_{0}}=f_{0}^{s}-{{\bf{m}}^{s}_{\alpha}}\cdot{\bf{B}}{{\partial
f_{0}^{s}}\over{\partial\varepsilon_{\alpha}^{s}}}.$ (32)

By substituting Eq.(10), Eq.(18) and Eq.(32) into the Eq.(22), we only keep
the linear term in ${\bf{B}}$:

$\begin{array}[]{lll}\mathbf{j}^{(B)s}&=&-{\kappa_{1}}\int{[d{\bf{k}}]}[{\bf{v}}_{\alpha}^{s}-{\nabla_{\bf{k}}}({\bf{m}}_{\alpha}^{s}\cdot{\bf{B}})+e({\bf{v}}_{\alpha}^{s}\cdot{\bf{\Omega}}_{\alpha}^{s}){\bf{B}}]C_{\mathbf{k}}\mathbf{\Lambda}_{\mathbf{k}}\\\
&&\cdot\nabla_{\mathbf{k}}[C_{\mathbf{k}}\mathbf{\Lambda}_{\mathbf{k}}\cdot\nabla_{\mathbf{k}}(f_{0}^{s}-{\bf{m}}_{\alpha}^{s}\cdot{\bf{B}}{{\partial
f_{0}^{s}}\over{\partial\varepsilon_{\alpha}^{s}}})]\\\
&&-\kappa_{2}\int{[d{\bf{k}}]}{\bf{E}}\times{\bf{\Omega}}_{\alpha}^{s}C_{\mathbf{k}}\mathbf{\Lambda}_{\mathbf{k}}\cdot\nabla_{\bf{k}}(f_{0}^{s}-{\bf{m}}_{\alpha}^{s}\cdot{\bf{B}}{{\partial
f_{0}^{s}}\over{\partial\varepsilon_{\alpha}^{s}}}),\end{array}$ (33)

where
$\Lambda_{\mathbf{k}}=e{\bf{E}}+{{{e^{2}}}\over\hbar}({\bf{E}}\cdot{\bf{B}}){\bf{\Omega}}_{\alpha}^{s}$,
and $C_{\mathbf{k}}=1-{e\over\hbar}({\bf{B}}\cdot{\bf{\Omega}}_{\alpha}^{s})$.
Obviously, in Eq.(33), the direction of the current density contributed by the
first integral term is governed by a combination of the wave packet velocity
${\bf{v}}_{\alpha}^{s}$, the Berry curvature ${\bf{\Omega}}_{\alpha}^{s}$, and
the orbital magnetic moment ${\bf{m}}_{\alpha}^{s}$. While the direction of
the current density contributed by the second integral term is solely
determined by the coupling term of the electric field and Berry curvature
${\bf{E}}\times{\bf{\Omega}}_{\alpha}^{s}$, resulting in a nonlinear Hall
current. We also express Eq.(33) in the form of
$j_{a}^{(B)s}=\sigma_{abc}^{(B)s}E_{b}E_{c}$, where $\sigma_{abc}^{(B)s}$
represents the magnetic field induced contribution to the second-order
nonlinear conductivity.

When the applied magnetic field ${\bf{B}}$ aligns with the electric field
${\bf{E}}$ (i.e., ${\bf{E}}\cdot{\bf{B}}\neq 0$), evidently, the second
integral term in Eq.(33) does not contribute to the second-order conductivity.
Substituting Eqs.(7) and (11) into Eq.(33), we derive the following
expressions from the first integral terms of Eq.(33):

$\displaystyle\sigma_{xxx}^{(B)s}={\sigma_{1}}{B_{x}}\int{[d{\bf{k}}}]\\{{{\partial{v}_{\alpha
x}^{s2}}\over{2\partial{k_{x}}}}\Omega_{\alpha x}^{s}+[{{\partial{v}_{\alpha
x}^{s2}}\over{2\partial{k_{y}}}}+{{\partial(v_{\alpha x}^{s}v_{\alpha
y}^{s})}\over{\partial{k_{x}}}}]\Omega_{\alpha y}^{s}$ (34)
$\displaystyle+[{{\partial{v}_{\alpha
x}^{s2}}\over{2\partial{k_{z}}}}+{{\partial(v_{\alpha x}^{s}v_{\alpha
z}^{s})}\over{\partial{k_{x}}}}]\Omega_{\alpha z}^{s}+{v_{\alpha
x}^{s2}}\nabla\cdot\mathbf{\Omega}_{\alpha}^{s}+v_{\alpha x}^{s}v_{\alpha
y}^{s}{{\partial\Omega_{\alpha y}^{s}}\over{\partial{k_{x}}}}$
$\displaystyle+v_{\alpha x}^{s}v_{\alpha z}^{s}{{\partial\Omega_{\alpha
z}^{s}}\over{\partial{k_{x}}}}-{{{\partial^{2}}m_{\alpha
x}^{s}}\over{e\partial k_{x}^{2}}}v_{\alpha x}^{s}+{{{\partial^{2}}v_{\alpha
x}^{s}}\over{e\partial k_{x}^{2}}}m_{\alpha x}^{s}\\}{{\partial
f_{0}^{s}}\over{\partial{\varepsilon_{\alpha}^{s}}}},$
$\displaystyle\sigma_{zzz}^{(B)s}={\sigma_{1}}{B_{z}}\int{[d{\bf{k}}}]\\{{{\partial{v}_{\alpha
z}^{s2}}\over{2\partial{k_{z}}}}\Omega_{\alpha z}^{s}+[{{\partial{v}_{\alpha
z}^{s2}}\over{2\partial{k_{y}}}}+{{\partial(v_{\alpha z}^{s}v_{\alpha
y}^{s})}\over{\partial{k_{z}}}}]\Omega_{\alpha y}^{s}$ (35)
$\displaystyle+[{{\partial{v}_{\alpha
z}^{s2}}\over{2\partial{k_{x}}}}+{{\partial(v_{\alpha x}^{s}v_{\alpha
z}^{s})}\over{\partial{k_{z}}}}]\Omega_{\alpha
x}^{s}+v_{z\alpha}^{s2}\nabla\cdot\mathbf{\Omega}_{\alpha}^{s}+v_{x\alpha}^{s}v_{z\alpha}^{s}{{\partial\Omega_{x\alpha}^{s}}\over{\partial{k_{z}}}}$
$\displaystyle+v_{y\alpha}^{s}v_{z\alpha}^{s}{{\partial\Omega_{y\alpha}^{s}}\over{\partial{k_{z}}}}-{{{\partial^{2}}m_{\alpha
z}^{s}}\over{e\partial k_{z}^{2}}}v_{\alpha z}^{s}+{{{\partial^{2}}v_{\alpha
z}^{s}}\over{e\partial k_{z}^{2}}}m_{\alpha z}^{s}\\}{{\partial
f_{0}^{s}}\over{\partial{\varepsilon_{\alpha}^{s}}}},$

where
${\sigma_{1}}={{e^{4}\tau^{2}v_{F}^{2}}\over{(1-2i\omega\tau)(1-i\omega\tau){h^{2}}}}$.
Notably, due to the symmetry of the $k_{x}$-axis and the $k_{y}$-axis,
$\sigma_{yyy}^{(B)s}({B_{y}})/B_{y}=\sigma_{xxx}^{(B)s}(B_{x})/B_{x}$. All
other components are rendered as zero.

By orienting the magnetic field ${\bf{B}}$ perpendicular to the electric field
${\bf{E}}$ (i.e., ${\bf{E}}\cdot{\bf{B}}=0$), we ascertain the nonlinear Hall
conductivity components in the following manner:

$\displaystyle\sigma_{xyy}^{(B)s}={\sigma_{1}}{B_{x}}\int{[d{\bf{k}}}][{{\partial
v_{\alpha x}^{s}}\over{\partial{k_{y}}}}v_{\alpha y}^{s}\Omega_{\alpha
x}^{s}-{{\partial(v_{\alpha y}^{s}\Omega_{\alpha y}^{s}+v_{\alpha
z}^{s}\Omega_{\alpha z}^{s})}\over{\partial{k_{y}}}}v_{\alpha y}^{s}$ (36)
$\displaystyle-{{\partial^{2}m_{\alpha
x}^{s}}\over{e\partial{k_{x}}\partial{k_{y}}}}v_{\alpha
y}^{s}+{{{\partial^{2}}v_{\alpha x}^{s}}\over{e\partial{k_{y}^{2}}}}m_{\alpha
x}^{s}]{{\partial f_{0}^{s}}\over{\partial{\varepsilon_{\alpha}^{s}}}},$
$\displaystyle\sigma_{xzz}^{(B)s}={\sigma_{1}}{B_{x}}\int{[d{\bf{k}}]}[-{{\partial
v_{\alpha x}^{s}}\over{\partial{k_{z}}}}v_{\alpha z}^{s}\Omega_{\alpha
x}^{s}+{{\partial(v_{\alpha y}^{s}\Omega_{\alpha y}^{s}+v_{\alpha
z}^{s}\Omega_{\alpha z}^{s})}\over{\partial{k_{z}}}}v_{z}^{s}$ (37)
$\displaystyle-{{\partial m_{\alpha
x}^{s}}\over{\partial{k_{x}}\partial{k_{z}}}}v_{z\alpha}^{s}+{{{\partial^{2}}v_{\alpha
x}^{s}}\over{\partial{k_{z}^{2}}}}m_{\alpha x}^{s}]{{\partial
f_{0}^{s}}\over{\partial{\varepsilon_{\alpha}^{s}}}},$
$\displaystyle\sigma_{zyy}^{(B)s}=\sigma_{1}B_{z}\int[d\mathbf{k}][-{{\partial
v_{\alpha z}^{s}}\over{\partial k_{y}}}v_{\alpha y}^{s}\Omega_{\alpha
z}^{s}+{{\partial(v_{\alpha x}^{s}\Omega_{\alpha x}^{s}+v_{\alpha
y}^{s}\Omega_{\alpha y}^{s})}\over{\partial{k_{y}}}}v_{\alpha y}^{s}$ (38)
$\displaystyle-{{{\partial^{2}}m_{\alpha
z}^{s}}\over{e\partial{k_{z}}\partial{k_{y}}}}v_{\alpha
y}^{s}+{{{\partial^{2}}v_{\alpha z}^{s}}\over{e\partial{k_{y}^{2}}}}m_{\alpha
z}^{s}]{{\partial f_{0}^{s}}\over{\partial{\varepsilon_{\alpha}^{s}}}}.$

The additional non-zero conductivity components conform to relationships:
$\sigma_{yxx}^{(B)s}/{B_{y}}=\sigma_{xyy}^{(B)s}/{B_{x}}=-\sigma_{zyy}^{(B)s}/{B_{z}}$,
$\sigma_{zxx}^{(B)s}=\sigma_{zyy}^{(B)s}$, and
$\sigma_{yzz}^{(B)s}/{B_{y}}=\sigma_{xzz}^{(B)s}/{B_{x}}$. Notably, Eqs.(36),
(37) and (38) stem from the contributions of the primary integral terms in
Eq.(33). The contribution arising from the second integral term in Eq.(33)
necessitates distinct component indices ($a,b,c$) for the conductivity
components $\sigma_{abc}^{(B)s}$. Through further computation involving
Eq.(33), we find

$\displaystyle\sigma_{zxy}^{(B)s}=-{\sigma_{2}}{B_{z}}\int{[d{\bf{k}}]}({{\partial\Omega_{\alpha
x}^{s}}\over{\partial{k_{x}}}}m_{\alpha z}^{s}+ev_{\alpha x}^{s}\Omega_{\alpha
x}^{s}\Omega_{\alpha
z}^{s}){{\partial{f_{0}^{s}}}\over{\partial{\varepsilon_{\alpha}^{s}}}},$ (39)

where we define
${\sigma_{2}}={{{e^{4}}\tau}\over{4\pi^{2}{\hbar^{3}}(1-i\omega\tau)}}$. The
other non-zero conductivity components can be related as follows:
$\sigma_{zxy}^{(B)s}=-\sigma_{zyx}^{(B)s}$,
$\sigma_{yzx}^{(B)s}/B_{y}=-\sigma_{xzy}^{(B)s}/B_{x}=2\sigma_{zxy}^{(B)s}/B_{z}$.
We

Figure 7: (color online) The total second-order nonlinear conductivity
components of the $s=\pm 1$ bands, namely $\sigma_{xxx}^{(B)}$ (a),
$\sigma_{zzz}^{(B)}$ (b), $\sigma_{xyy}^{(B)}$ (c), $\sigma_{xzz}^{(B)}$ (d),
$\sigma_{zyy}^{(B)}$ (e), and $\sigma_{zxy}^{(B)}$ (f), as a function of Fermi
energy for two distinct values of $R=0.025eV\cdot nm^{2}$ (solid line) and
$R=0.035eV\cdot nm^{2}$ (dashed line) at $J_{ex}=0.02eV$ and
$V_{I}=0.04eV\cdot nm$. All other parameters remain consistent with those
defined in Fig.2.

In the presence of tilt $R$ and the splitting term $J_{ex}$, the energies of
the four Weyl nodes exhibit discrepancies, as illustrated in Fig. 1. Assuming
a comparative scale between the splitting energy $J_{ex}$ and the VHS energy,
the ordering of Weyl node energies remains consistent:
$E_{W_{R,+1}}>E_{W_{L,-1}}>E_{W_{L,+1}}>E_{W_{R,-1}}$. Fig.7 illustrates the
dependence of the total second-order nonlinear conductivity of the $s=\pm 1$
bands, $\sigma_{abc}^{(B)}=\sigma_{abc}^{(B)+1}+\sigma_{abc}^{(B)-1}$, induced
by the magnetic field on the Fermi energy $E_{F}$ for varying tilts. The
second-order conductivity components $\sigma_{aaa}^{(B)}(a=x,z)$ and
$\sigma_{xyy}^{(B)}$ across the Weyl nodes $W_{R,+1}$ and $W_{L,-1}$ display
distinct sharp peaks. The asymmetry in these peak structures concerning the
Weyl nodes indicates the influence of the magnetic moment, which suppresses
nonlinear electron transport while enhancing nonlinear hole transport, as
detailed in Eqs. (LABEL:xxx)-(37). Terms related to the magnetic moment
demonstrate positivity, while others rely on the band index $\alpha(=\pm 1)$.
For the conductivity components $\sigma_{xzz}^{(B)}$, $\sigma_{zyy}^{(B)}$,
and $\sigma_{zxy}^{(B)}$, the dominance of the magnetic moment becomes
evident, resulting in a kinked structure showing a sudden shift from positive
to negative values near the Weyl nodes $W_{R,+1}$ and $W_{L,-1}$, as depicted
in Fig. 7(d),(e), and (f). Further examination of the second-order
conductivity expressions (Eqs. (LABEL:xxx)-(39)) reveals that the peak
structures follow an asymptotic relationship
$\sigma_{aaa}^{(B)},\sigma_{xyy}^{(B)}\propto\frac{1}{(E_{F}-E_{W_{\alpha,s}})^{2}}$
near the Weyl nodes $W_{\alpha,s}$. Meanwhile, the kinked structure (as shown
in Fig. 7(d),(e), and (f)) adheres to the relation
$\sigma_{xzz}^{(B)},\sigma_{zyy}^{(B)},\sigma_{zxy}^{(B)}\propto\pm\frac{\text{sgn}(E_{F}-E_{W_{\alpha,s}})}{(E_{F}-E_{W_{\alpha,s}})^{2}}$
near the Weyl nodes $W_{\alpha,s}$. As the tilt $R$ increases, these
characteristic structures shift towards higher energies while retaining their
essential peak and kinked features. However, near the VHS, the distinct
feature induced by the VHS is not clearly observed due to the suppressing
effect of the magnetic moment, overriding the influence of the VHS. Even an
enhancement in the VHS energy nearly fails to amplify its influence. In
analyzing the frequency dependence of conductivity, we observe a similarity
between the magnetic field-induced contribution to the second-order
conductivity and the case of zero magnetic field. Consequently, in the
transport regime where $\omega\tau\ll 1$, both $\sigma_{abb}$ and
$\sigma_{aaa}$ vary proportionally to $\tau^{2}$, whereas $\sigma_{abc}$
scales linearly with $\tau$. Conversely, in the optical limit where
$\omega\tau\gg 1$, $\sigma_{abb}$ and $\sigma_{aaa}$ scale inversely with the
square of frequency ($1/\omega^{2}$), and $\sigma_{aaa}$ possesses an
imaginary component, obeying the relation $\sigma_{aaa}\propto 1/\omega$.

Moreover, from the aforementioned results, we can further assess the scale of
the nonlinear magneto-optical susceptibility using the equation
$\chi^{(2\omega)}=j/(i\omega)\epsilon_{0}E^{2}$Morimoto16 , where
$\epsilon_{0}$ represents vacuum permittivity. By adopting the parameters
$E_{F}=8meV$, $v_{F}=3.2\times{10^{5}}m/s$, $V_{I}=0.015eV\cdot nm^{-1}$,
$J_{ex}=0.01eV$, $\lambda=0.5\text{nm}$, $R=0.3\text{eV}\cdot nm^{2}$, and
$Q_{D}=0.8nm^{-1}$, we derive $\chi^{(2\omega)}\approx 5.6\times{10^{3}}B$
$pm/V$ at $\omega=8\pi THz$. Recent observations in WSMs have demonstrated
significantly enhanced nonlinear optical responses, encompassing
photocurrentmanp2017 , second- or third-harmonic generationwunp2016 , and the
optical Kerr effectarxivchoi . Specifically, an extraordinarily high
coefficient for the linear magneto-optic Kerr effect has been reported within
a magnetic WSMhigonp2018 . These observations indicate the potential
realization of our findings concerning nonlinear magneto-optical features
induced by the magnetic moment within magnetic WSMs, particularly within the
infrared regime.

##  V Conclusions

We study the nonlinear magneto-optical transport properties within non-
centrosymmetric magnetic WSMs. We construct an effective low-energy model that
incorporates essential elements: the VHS, a tilted term, and the presence of
broken space inversion and broken time inversion terms. Using this model, we
derive analytical expressions for the second-order nonlinear conductivity
components through the semiclassical Boltzmann equation. We observe that in
the absence of a magnetic field, the second-order nonlinear Drude conductivity
components display inflection or dip behaviors across the VHS. Conversely, the
second-order nonlinear anomalous Hall conductivity, primarily influenced by
the Berry curvature dipole, showcases a subtle plateau-like structure.
Significantly, intensifying the tilt strength amplifies these second-order
conductivity features at the singularity due to the increased VHS energy.
Additionally, our investigation explores the magnetic field-induced impact on
the second-order nonlinear conductivity, revealing that the resulting magnetic
moment-induced suppression and enhancement of nonlinear electron and hole
transport help counteract the influence of the VHS. This effect specifically
generates an asymmetric peak or a kinked-like structure near the Weyl nodes.
Upon evaluating the magnitude of these second-order conductivity components,
our findings suggest the potential observability of these phenomena in
realistic magnetic WSMs.

## ACKNOWLEDGMENTS

We thank Dr. Yang Gao for the valuable discussions. This work was supported by
the Postgraduate Scientific Research Innovation Project of Hunan Province
(Grant No.CX20220958), the Natural Science Foundation of Hunan Province, China
(Grant No. 2023JJ30005), and the Open Research Fund of the Hunan Province
Higher Education Key Laboratory of Modeling and Monitoring on the Near-Earth
Electromagnetic Environments (GrantNo. N201904), Changsha University of
Science and Technology.

##  Appendix A  Density of states

The density of states for the $s=+1$ band can be calculated by

$\displaystyle\rho_{+1}(\varepsilon)=\frac{1}{V}\sum\limits_{\mathbf{k}\alpha}\delta[\varepsilon-\varepsilon_{\alpha}^{+1}(\mathbf{k})]$
(40)

To solve Eq.(40), we need to convert the summation over $\mathbf{k}$ into an
integral in the three dimensional momentum space. After a straightforward
calculation, we get the analytical expression for DOS as follows:

For $\varepsilon>\varepsilon_{VHS}^{c}$,

$\displaystyle\rho_{+1}(\varepsilon)={1\over{4{\pi^{2}}v_{F}^{2}}}[(\varepsilon-{J_{ex}})({k_{z3}}-{k_{z1}})-{R\over
3}{({k_{z3}}-{k_{z1}})^{3}}].$ (41)

For $\varepsilon_{W_{R,+1}}<\varepsilon<\varepsilon_{VHS}^{\rm{c}}$,

$\displaystyle\rho_{+1}(\varepsilon)={1\over{4{\pi^{2}}v_{F}^{2}}}\\{(\varepsilon-{J_{ex}})({k_{z2}}+{k_{z3}}-{k_{z1}}-{k_{z4}})$
(42) $\displaystyle-{R\over
3}[{({k_{z2}}-{k_{z1}})^{3}}+{({k_{z3}}-{k_{z4}})^{3}}]\\}.$

For $\varepsilon_{W_{L,+1}}<\varepsilon<\varepsilon_{W_{R,+1}}$,

$\displaystyle{\rho_{+1}}(\varepsilon)={1\over{4{\pi^{2}}v_{F}^{2}}}\\{(\varepsilon-{J_{ex}})({k_{z2}}+{k_{z4}}-{k_{z1}}-{k_{z3}})$
(43) $\displaystyle-{R\over
3}[{({k_{z2}}-{k_{z1}})^{3}}+{({k_{z4}}-{k_{z3}})^{3}}]\\}.$

For $\varepsilon_{VHS}^{\rm{v}}<\varepsilon<\varepsilon_{W_{{\rm{L}},+1}}$,

$\displaystyle\rho_{+1}(\varepsilon)=-{1\over{4{\pi^{2}}v_{F}^{2}}}\\{(\varepsilon-{J_{ex}})({k_{z1}}+{k_{z4}}-{k_{z2}}-{k_{z3}})$
(44) $\displaystyle-{R\over
3}[{({k_{z1}}-{k_{z2}})^{3}}+{({k_{z4}}-{k_{z3}})^{3}}]\\}.$

For $\varepsilon<\varepsilon_{VHS}^{v}$,

$\displaystyle\rho_{+1}(\varepsilon)=-{1\over{4{\pi^{2}}v_{F}^{2}}}[(\varepsilon-{J_{ex}})({k_{z4}}-{k_{z2}})-{R\over
3}{({k_{z4}}-{k_{z2}})^{3}}].$ (45)

where

$\displaystyle{k_{z1}}={{{v_{F}}\lambda q-\sqrt{{{({v_{F}}\lambda
q)}^{2}}+({v_{F}}\lambda+R)(\varepsilon-{J_{ex}}+{v_{F}}\lambda
Q_{D}^{2})}}\over{R+{v_{F}}\lambda}},$ (46)
$\displaystyle{k_{z2}}={{-{v_{F}}\lambda q+\sqrt{{{({v_{F}}\lambda
q)}^{2}}+(R-{v_{F}}\lambda)(\varepsilon-{J_{ex}}-{v_{F}}\lambda
Q_{D}^{2})}}\over{R-{v_{F}}\lambda}},$ (47)
$\displaystyle{k_{z3}}={{{v_{F}}\lambda q+\sqrt{{{({v_{F}}\lambda
q)}^{2}}+({v_{F}}\lambda+R)(\varepsilon-{J_{ex}}+{v_{F}}\lambda
Q_{D}^{2})}}\over{R+{v_{F}}\lambda}},$ (48)
$\displaystyle{k_{z4}}={{-{v_{F}}\lambda q-\sqrt{{{({v_{F}}\lambda
q)}^{2}}+(R-{v_{F}}\lambda)(\varepsilon-{J_{ex}}-{v_{F}}\lambda
Q_{D}^{2})}}\over{R-{v_{F}}\lambda}}.$ (49)

Additionally, the VHS energies for the conduction band and valence band are
expressed as $\varepsilon_{VHS}^{c}={{{{({v_{F}}\lambda
q)}^{2}}}\over{{v_{F}}\lambda-R}}+{v_{F}}\lambda Q_{D}^{2}+{J_{ex}}$, and
$\varepsilon_{VHS}^{v}=-{{{{({v_{F}}\lambda
q)}^{2}}}\over{{v_{F}}\lambda+R}}-{v_{F}}\lambda Q_{D}^{2}+{J_{ex}}$. In the
presence of the tilt, the energies of the Weyl nodes are modified to
$\varepsilon_{{\rm{W}}_{R,+1}}={(\sqrt{Q_{D}^{2}+{q^{2}}}+q)^{2}}+{J_{ex}}$,
and
$\varepsilon_{{\rm{W}}_{L,+1}}={(\sqrt{Q_{D}^{2}+{q^{2}}}-q)^{2}}+{J_{ex}}$.

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