# Role of topological charges in the nonlinear-optical response from Weyl
semimetals

Amar Bharti  Department of Physics, Indian Institute of Technology Bombay,
Powai, Mumbai 400076, India  Gopal Dixit<EMAIL_ADDRESS>Department of
Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
Center for Computational Sciences, University of Tsukuba, Tsukuba 305-8577,
Japan

###### Abstract

The successful realization of the topological Weyl semimetals has
revolutionized contemporary physics. In recent years, multi-Weyl semimetals, a
class of topological Weyl semimetals, has attracted broad interest in
condensed-matter physics. Multi-Weyl semimetals are emerging topological
semimetals with nonlinear anisotropic energy dispersion, which is
characterized by higher topological charges. In this study, we investigate how
the topological charge affects the nonlinear optical response from multi-Weyl
semimetals. It has been observed that the laser-driven electronic current is
characteristic of the topological charge, and the laser polarization’s
direction influences the current’s direction and amplitude. In addition, the
anomalous current, perpendicular to the laser’s polarization, carries a
distinct signature of the topological charges and encodes the information
about the parity and amplitude of the nontrivial Berry curvature. We show that
the anomalous current associated with the anomalous Hall effect remains no
longer proportional to the topological charge at higher laser intensity – a
significant deviation from the linear response theory. High-harmonic
spectroscopy is employed to capture the distinct and interesting features of
the currents in multi-Weyl semimetals where the topological charge drastically
impacts the harmonics’ yield and energy cutoff.

##  I Introduction

Recent years have witnessed a proliferation in studies of quantum materials,
such as topological insulators, Dirac and Weyl semimetals, as these materials
are not only crucial for fundamental research but also have immense
applications in upcoming quantum technologies. Owing to the presence of the
topologically protected Weyl points, Weyl semimetals (WSMs) have become a
focal point among the quantum materials Armitage  _et al._ (2018); Yan and
Felser (2017); Vazifeh and Franz (2013). Isolated points in momentum space at
which valance and conduction bands, with linear energy dispersions, touch are
known as Weyl points. These points appear in pairs with opposite chirality and
can be visualized as the source and sink of the Berry curvature. The low-
energy collective excitations in WSMs are massless Weyl fermions, which make
an elegant connection between high-energy and condensed-matter physics.

The three-dimensional Dirac semimetals possess both time-reversal and
inversion protection, giving four fold degenerate Dirac points. If one of
these two symmetries is broken, then these Dirac points transform into Weyl
points. The absence of the time-reversal or inversion symmetry in the WSMs
gives rise to the Berry curvature. The topology of the Berry curvature in WSMs
results in a series of exotic phenomenons, such as the chiral magnetic effect
Vazifeh and Franz (2013); Li  _et al._ (2016); Kaushik  _et al._ (2019), the
circular photogalvanic effect de Juan  _et al._ (2017); Ma  _et al._ (2017);
Rees  _et al._ (2020); Le and Sun (2021), and the anomalous Hall effect
Shekhar  _et al._ (2018); Meng  _et al._ (2019); Yang  _et al._ (2015) to name
but a few Burkov (2014); Trescher  _et al._ (2015); Kim  _et al._ (2013).
Interaction of light with WSMs has been a frontier approach to probe various
exotic phenomena associated with the Berry curvature in WSMs Lv  _et al._
(2021a, b); Bao  _et al._ (2021); Orenstein  _et al._ (2021); Dantas  _et al._
(2021); Tamashevich  _et al._ (2022); Nathan  _et al._ (2022); Matsyshyn  _et
al._ (2021); Ahn  _et al._ (2017); Sirica and Prasankumar (2021). The WSMs
also hold promise for future technologies based on the ultrafast
photodetector, the chiral terahertz laser source, and quantum physics
Osterhoudt  _et al._ (2019); Gao  _et al._ (2020).

Earlier realizations of the WSMs have a unit topological charge, $n=1$ Lv  _et
al._ (2015); Xu  _et al._ (2015); Morali  _et al._ (2019); Liu  _et al._
(2019); Belopolski  _et al._ (2019). The sign of the Berry-curvature monopole
determines the sign of the topological charge of the WSMs. The magnitude of
the topological charge is
$n=\int_{BZ}\nabla_{\mathbf{k}}\cdot\Omega(\mathbf{k})~{}d\mathbf{k}$, with
$\Omega(\mathbf{k})$ being the Berry curvature. A class of WSMs, multi-Weyl
semimetals (m-WSMs), with topological charges higher than one came into
existence in recent years. The m-WSMs with topological charges two and three
are, respectively, manifested by quadratic and cubic energy dispersions along
particular directions, whereas they exhibit linear dispersion along other
directions Fang  _et al._ (2012); Huang  _et al._ (2016); Xu  _et al._ (2011);
Liu and Zunger (2017). This contrasts with “conventional” WSMs with
topological unit charges, which exhibit isotropic linear energy dispersion.
Weyl points in m-WSMs can be formed by annihilating Weyl points of the same
chirality in WSMs with lower topological charges Roy and Narayan (2022). The
upper bound on the topological charge in m-WSMs is limited to three by certain
crystalline symmetries Yang and Nagaosa (2014); Fang  _et al._ (2012); Xu  _et
al._ (2011). A few possible potential candidates for the m-WSMs are SiSr2 and
HgCr2Se4 Fang  _et al._ (2012); Huang  _et al._ (2016); Xu  _et al._ (2011).

The m-WSMs manifest unique quantum response due to the higher topological
charges and resultant anisotropic energy dispersions Nag  _et al._ (2020). It
has been shown that the chirality accumulation is possible without a magnetic
field due to higher topological charges Huang  _et al._ (2017). A distinct
chiral anomaly-induced nonlinear Hall effect, associated with the different
topological charges in m-WSMs, has been discussed Nandy  _et al._ (2021).
Moreover, transport properties are significantly altered in comparison to the
single-WSMs Menon and Basu (2020); Menon  _et al._ (2021); Nag and Kennes
(2022); Chen and Fiete (2016); Gorbar  _et al._ (2017); Park  _et al._ (2017).
It has been found that the anomalous Hall current in the nonperturbative
regime saturates, and the anomalous Hall conductivity scales linearly with the
topological charges Dantas  _et al._ (2021); Nandy  _et al._ (2019). By
employing the effective field theory, the signature of a non-Abelian anomaly
in m-WSMs has been investigated Dantas  _et al._ (2020).

In this work, we investigate how the nonlinear optical responses in m-WSMs are
sensitive to the topological charges for different laser intensities. In the
following, we show that the current, parallel to the laser’s polarization, and
the anomalous current, perpendicular to the laser’s polarization, exhibit
distinct behaviour as a function of the laser’s intensity for different
topological charges. In addition, we demonstrate that the current is
anisotropic in nature, which strongly depends on the direction of the laser
polarization. Moreover, the anomalous current is sensitive to the line
connecting the Weyl points. The parallel current increases linearly below a
critical laser’s intensity. However, above a critical intensity, the parallel
current shows a nonlinear increment for different topological charges, and it
starts saturating after another critical intensity. On the other hand, the
anomalous current displays linear behavior for relatively larger intensity.
The Berry-curvature driven anomalous current approaches similar values for
different topological charges at large intensity limits. We employ high-
harmonic spectroscopy to probe the distinct behaviors of the normal and
anomalous currents in m-WSMs. It has been found that the topological charge
drastically alters the harmonics’ yield and energy cutoff.

##  II Theoretical Framework

Interaction of a m-WSM with an ultrashort intense laser pulse is simulated by
solving density-matrix-based semiconductor Bloch equations in the Houston
basis Mrudul  _et al._ (2021) as

$\displaystyle\partialderivative{\rho_{cv}^{\mathbf{k}}}{t}$
$\displaystyle=\left[-i\epsilon_{cv}^{\mathbf{k}_{t}}-\frac{1}{\textrm{T}_{2}}\right]\rho_{cv}^{\mathbf{k}}+i\mathbf{E}(t)\cdot\mathcal{D}_{cv}^{\mathbf{k}_{t}}\left[\rho_{vv}^{\mathbf{k}}-\rho_{cc}^{\mathbf{k}}\right]$
(1a) $\displaystyle\partialderivative{\rho_{vv}^{\mathbf{k}}}{t}$
$\displaystyle=i\textbf{E}(t)\cdot\mathcal{D}_{vc}^{\textbf{k}_{t}}~{}\rho_{cv}^{\textbf{k}}+\textrm{c.c.}$
(1b)

Here, $\epsilon_{cv}^{\mathbf{k}_{t}}$ and $\mathcal{D}_{cv}^{\mathbf{k}_{t}}$
are, respectively, energy band gap and dipole matrix elements between
conduction and valence bands at $\mathbf{k}_{t}$. In the presence of an
intense laser, the crystal momentum changes from $\mathbf{k}$ to
$\mathbf{k}_{t}$ as $\mathbf{k}_{t}=\mathbf{k}+\mathbf{A}(t)$, where
$\mathbf{A}(t)$ as the vector potential of the laser and is related to the
laser’s electric field $\mathbf{E}(t)$ as
$\mathbf{A}(t)=-\int_{-\infty}^{t}\mathbf{E}(t^{\prime})dt^{\prime}$. A
phenomenological term $\textrm{T}_{2}$ is introduced to account for the
decoherence between the hole and the electron during the light-matter
interaction process. Fourth-order Runge-Kutta method is used to solve Eq. (1)
numerically for $\rho$ at each time step. The completely filled valence band
($\rho_{vv}=1$) and the empty conduction band ($\rho_{cc}=0$) are used as
initial conditions. The dipole matrix element and the energy are obtained from
eigenstates and eigen energies by the diagonalizing Hamiltonian given in Eqs.
(II-II) as discussed in Ref. Mrudul and Dixit (2021a).

The total current is evaluated as
$\mathbf{J}(\mathbf{k},t)=\sum_{m,n\in\\{c,v\\}}\rho_{mn}^{\mathbf{k}}\mathbf{p}_{nm}^{\mathbf{k}_{t}}$
with $\mathbf{p}_{nm}^{\mathbf{k}_{t}}$ being the momentum matrix-element. The
total current is decomposed into two components: intraband and non-intraband
currents. The intraband current is solely originating due to group velocity
and can be expressed as
$\mathbf{J}_{intra}(\mathbf{k},t)=\sum_{n}\nabla_{\mathbf{k}}\epsilon_{n}^{\mathbf{k}_{t}}\rho_{nn}^{\mathbf{k}}$.
On the other hand, the non-intraband current contains the contribution due to
the Berry curvature, among others Yue and Gaarde (2022). The Berry curvature
is expressed in terms of the curl of the Berry connection:
$\mathbf{\Omega}=\nabla\times\mathcal{A}$. The Berry connection between
conduction and valence bands can be evaluated as
$\mathcal{A}_{cv}(\mathbf{k})=-i\innerproduct{\nabla_{\mathbf{k}}u_{c}({\mathbf{k}})}{u_{v}({\mathbf{k}})}=\mathcal{A}_{vc}^{*}(\mathbf{k})$,
which is visualized as the off-diagonal dipole matrix elements Nematollahi
_et al._ (2020); Ngo  _et al._ (2021). Thus, the Berry curvature between
conduction and valence bands is written as
$\mathbf{\Omega}_{cv}(\mathbf{k})=i(\mathcal{A}_{vc}(\mathbf{k})\times\mathcal{A}_{cv}(\mathbf{k}))$.
The current due to the Berry curvature is evaluated as
$\mathbf{J}_{\mathbf{\Omega}}(\mathbf{k},t)=\mathbf{E}(t)\times\sum_{cc,vv}\mathbf{\Omega}_{cv}(\mathbf{k})~{}\rho_{cc,vv}(\mathbf{k},t)$
and is commonly known as the anomalous current.

Inversion-symmetric m-WSM with broken time-reversal symmetry for different
topological charges $n$ can be collectively written as
$\mathcal{H}^{(n)}(\mathbf{k})=\mathbf{d}^{(n)}(\mathbf{k})\cdot\sigma$, with
$\sigma$’s being the Pauli matrices and $\mathbf{d}=[d_{x},d_{y},d_{z}]$
Dantas  _et al._ (2020). The full expressions of the three components of
$\mathbf{d}^{(n)}(\mathbf{k})$ for topological charges $n=1,2,$ and 3 are
written as

$\displaystyle\mathbf{d}^{(1)}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\left[t\sin(k_{x}a),t\sin(k_{y}a)\right.,$
$\displaystyle\left.t\\{\cos(k_{z}a)-\cos(k_{0}a)+2-\cos(k_{x}a)-\cos(k_{y}a)\\}\right],$
$\displaystyle\mathbf{d}^{(2)}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\left[t\\{\cos(k_{x}a)-\cos(k_{y}a)\\},t\sin(k_{x}a)\sin(k_{y}a),\right.$
$\displaystyle\left.t\\{\cos(k_{z}a)-\cos(k_{0}a)+2-\cos(k_{x}a)-\cos(k_{y}a)\\}\right],$

and

$\displaystyle\mathbf{d}^{(3)}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\left[t\sin(k_{x}a)\\{3\cos(k_{y}a)-\cos(k_{x}a)-2\\},\right.$
$\displaystyle\left.t\sin(k_{y}a)\\{3\cos(k_{x}a)-\cos(k_{y}a)-2\\},\right.$
$\displaystyle\left.t\\{\cos(k_{z}a)-\cos(k_{0}a)+2-\cos(k_{x}a)-\cos(k_{y}a)\\}\right].$

Here, $k_{0}$ determines the positions of the Weyl nodes, which twe have taken
to be $k_{0}=\pi/(2a)$ in this article throughout, unless stated otherwise.
Thus, the positions of Weyl points for Eqs. (II-II) are $(0,0,\pm k_{0})$ as
shown in Fig. 2.

##  III Results and Discussion

Let us apply a laser polarized along the $z$ direction to induce electron
dynamics in m-WSMs. Figure 1 presents the total current in m-WSMs with
different topological charges. The current in a WSM with $n=2$ is
approximately one-order higher in magnitude in comparison to the current for
$n=1$. However, the total current is comparable for $n=2$ and 3 as evident
from Fig. 1. The overall shape of the currents in all cases are significantly
distorted with respect to the shape of the laser’s electric field as shown in
the inset. This distortion indicates the involvement of nonlinear optical
processes during electron dynamics. Note that the laser polarized along the
$z$ direction does not yield current along any direction other than the
polarization direction, which will be discussed later.

Figure 1: Total current along the $z$ direction in multi-Weyl semimetals for
different topological charges $n$. The linearly polarized laser pulse along
the $z$ direction is employed to generate the current and is shown in the
inset. The pulse is approximately 100 fs long with a wavelength of
$3.2~{}\mu$m and an intensity of $10^{11}$ W/cm2. The value of the
phenomenologically decoherence time $\textrm{T}_{2}$ is 1.5 fs.

To understand the unusual behavior in the strength of the total current for
different $n$ values, let us first analyze the energy band structures of the
m-WSMs. Owing to a similar structure of $\mathcal{H}^{(n)}(\mathbf{k})$, the
eigenvalues can be succinctly written as
$E_{c,v}=\pm\sqrt{d_{x}^{2}+d_{y}^{2}+d_{z}^{2}}$, where $E_{c,v}$ corresponds
to conduction and valence bands, respectively; and the $d$’s are corresponding
$\mathbf{d}$’s for each $n$ as given in Eqs. (II \- II).

Figure 2: Energy band structure corresponding to multi-Weyl semimetals with
different topological charges along (a) the $k_{z}$ direction, and (b) the
$k_{x}$ direction around a Weyl node. The band structures along the $k_{y}$
and $k_{x}$ directions are identical. The band structure is obtained by
diagonalizing the Hamiltonian $\mathcal{H}^{(n)}$ for different values of $n$
with the hopping parameter $t=1.2$ eV and the lattice parameter $a=6.28$ Å.
The purple line corresponds to the valence band in all cases.

Figure 2(a) shows the energy band structures along the $k_{z}$ direction for
different $n$ values. The band structures with two Weyl nodes of opposite
chirality at $\pm k_{0}$ are identical for $n=1,2$, and 3 as evident from the
figure. The reason behind the identical energy dispersion along $k_{z}$ can be
attributed to the same hopping parameter $t$ and lattice parameter $a$ for
different $n$’s. However, the band structures near one of the Weyl nodes are
significantly different along $k_{x}$ for different $n$’s as documented in
Fig. 2(b). Note that the band structures along $k_{x}$ and $k_{y}$ are
identical. The different energy dispersions along $k_{x}$ are directly related
to the topological charges of the Weyl nodes. The relation between the energy
dispersion and the topological charge is apparent from the low-energy
Hamiltonian near a Weyl node as
$\mathrm{H}^{(n)}(\mathbf{k})=v\left(k_{x}^{n}\sigma_{x}+k_{y}^{n}\sigma_{y}+k_{z}\sigma_{z}\right)$
for $n=1,2$, and 3. It is straightforward to see that the effective low-energy
band structures are linear, quadratic, and cubic for $n=1$, 2, and 3,
respectively. Furthermore, it can be shown explicitly by calculating Berry
curvature that the Chern numbers corresponding to linear, quadratic and cubic
dispersions are one, two, and three, respectively, as calculated elsewhere Nag
and Kennes (2022).

Analysis of Fig. 2 indicates that the current should be comparable for the
m-WSMs with different $n$’s as the energy dispersions are identical along the
direction of laser polarization. However, the energy gap between valance and
conduction bands is smaller for $n=2$ and 3 in comparison to $n=1$ around the
Weyl node as evident from the energy dispersion along $k_{x}$, which
translates into a higher probability of electron excitation for $n=2$ and 3.
Moreover, the energy dispersions for $n=2$ and 3 are similar. The distinct
energy dispersion naturally leads to the distinct gradient of the group
velocity. Thus, the cubic band dispersion offers a larger intraband current
compared to the quadratic one. Similarly, the linear dispersion has the
smallest current flow out of the three. Both these facts explain why the total
current exhibits similar features for $n=2$ and 3 and is higher in magnitude
in comparison to $n=1$. Note that the entire Brillouin zone contributes to the
total current in three-dimensional systems like WSMs Gu and Kolesik (2022).

Figure 3: Total current along (a) the direction of the laser polarization,
i.e., normal current, and (b) along the perpendicular $y$ direction, i.e.,
anomalous current. The laser is polarized along the $x$ direction. Details of
the simulation and the laser parameters are the same as those in Fig. 1.

So far, we have witnessed that the laser polarized along the $z$ direction
does not generate current along any perpendicular direction. Let us see how
this observation is modified when the polarization of the laser is tuned from
the $z$ direction to the $x$ direction. Figure 3(a) presents the current
parallel to the laser polarization for different topological charges. The
strength of the current is drastically different for different $n$’s, which is
expected from the band structures along $k_{x}$ as shown in Fig. 1(b). The
current increases monotonically as the band dispersion goes from linear to
quadratic and cubic with the increase in $n$. This observation is in contrast
with the previous one where the current was comparable for $n=2$ and 3. Before
we delve into the detailed reason for the increase in current as a function of
$n$, let us focus on the perpendicular component of the total current along
the $y$ direction as shown in Fig. 3(b). A similar perpendicular component
appears along the $x$ direction when the laser is polarized along the $y$
direction. In the following, let us use the terminology normal and anomalous
currents for the currents along and perpendicular to the laser polarization,
respectively, as used in a previous study Bharti  _et al._ (2022).

We employ a semiclassical equation of motion for laser-driven electron
dynamics to understand why the $x$ polarized laser generates anomalous
current, whereas $z$ polarization does not. It is known that the WSM has a non
zero Berry curvature, which results an anomalous velocity as
$\mathbf{E}(t)\times\mathbf{\Omega}$. Moreover, the Berry curvature in an
inversion-symmetric WSM follows
$\mathbf{\Omega}(-\mathbf{k})=\mathbf{\Omega}(\mathbf{k})$, which leads to
nonzero current due to the anomalous velocity of
$\int_{\mathbf{k}}\rho(\mathbf{k})\\{\mathbf{E}\times\mathbf{\Omega}(\mathbf{k})\\}~{}d\mathbf{k}$.
The parity of the Berry curvature’s components in the present case [see
Supplemental Material i for details Not ] is such that there is no anomalous
velocity if the electric field is along the same direction as the line
connecting the Weyl nodes as discussed in Ref. Bharti  _et al._ (2022).
Moreover, the anomalous current in the time-reversal broken WSM produces the
anomalous Hall effect Xiao  _et al._ (2010). Thus, the anomalous current is
zero when the laser is polarized along the $z$ direction, the direction along
which the two Weyl nodes are situated. The situation changes drastically as
the polarization direction changes from $z$ to $x$ or $y$.

The anomalous current is proportional to $\mathbf{\Omega}$, which means it is
also proportional to $n$. Thus, the strength of the anomalous current
increases as $n$ increases [see Fig. 3(b)]. However, the anomalous currents
due to $n=2$ and 3 are out of phase with respect to $n=1$. This behavior is
due to the sign of the integral of the Berry curvature’s components as shown
in Ref. Bharti  _et al._ (2022). The sign of
$\int_{\mathbf{k}}\\{\mathbf{E}\times\mathbf{\Omega}(\mathbf{k})\\}~{}d\mathbf{k}$
is positive for $n$ = 1 and 3 and negative for $n$ = 2, which leads the
anomalous current for $n$ = 2 out of phase. Thus, the strength and the phase
of the anomalous current encode the information about the nontrivial topology
of the Berry curvatures in m-WSMs. Note that the anomalous current is one-
order weaker than the normal current as evident from Fig. 3. At this juncture,
it is interesting to wonder how these features in the normal and anomalous
currents alter with the laser’s intensity.

Figure 4: Variations of the maximum amplitude for (a) the normal current along
$x$ and (b) the anomalous current along $y$ as a function of the laser’s
intensity for multi-Weyl semimetals with topological charges $n=1,2,$ and 3.
The other parameters are the same as those in Fig. 3. Figure 5: Temporal
evolution of the conduction band population, $\rho_{cc}$, for different $n$
values during the laser for two intensities: (a) $10^{8}$ W/cm2 and (b)
$10^{11}$ W/cm2. The other parameters are the same as those in Fig. 3.

Figure 4 presents variation in the peak current for different $n$ values as a
function of the intensity ranging from $10^{8}$ to $10^{11}$ W/cm2. In the
beginning, the anomalous current dominates over the normal current for each
$n$ at $10^{8}$ W/cm2. However, the normal current takes over the anomalous
current at some critical intensity. On comparing the normal and anomalous
currents for each $n$, we find that this critical intensity gets lower as $n$
increases. In addition, the normal current starts to grow exponentially at a
much lower intensity for higher $n$ values. The peak current increases
linearly as the intensity increases from $10^{8}$ to $10^{10}$ W/cm2 [see Fig.
4(a)]. Moreover, the rate of increase is much higher for the normal current as
compared to the anomalous current, which starts to saturate at an intensity
much lower than that of the normal current. There is no exponential growth
region in the anomalous current compared to the normal current, which grows
exponentially in the intensity window of $10^{10}$ to $10^{11}$ W/cm2 [see
Fig. 4(b)]. It is expected that the laser drives electrons further away in the
energy band as the intensity increases, and therefore the normal current
increases. However, the comparatively large anomalous current at lower
intensity is interesting. Moreover, the subdued increment in the anomalous
current needs further investigation. To understand these interesting
observations, we analyze the laser-driven electronic population in the
conduction band.

Figure 6: (a) Ratio of the anomalous current’s peak, $\textrm{J}_{\Omega}$, as
a function of the laser’s intensity, and (b) variation in the anomalous
current’s peak, $\textrm{J}_{\Omega}$, as a function of the separation between
two Weyl nodes, $k_{0}$, at intensity $10^{11}$ W/cm2 for different
topological charges $n$.

Electronic population in the conduction band for $n=1$ is one oder in
magnitude weaker than the population for $n=3$ as shown in Fig. 5(a).
Moreover, the conduction band is sparsely populated for $n=1$ at $10^{8}$
W/cm2. As the intensity increases from $10^{8}$ to $10^{11}$ W/cm2, the
overall population increases by two orders in magnitude [see Fig. 5]. The
conduction band population for $n=3$ remains substantially more significant
than that of the lower topological charges even at $10^{11}$ W/cm2 as evident
from Fig. 5(b). Similarly, the overall nature of the populations remains the
same and the difference in populations is also pronounced for different $n$
values at $10^{9}$ W/cm2 (not shown here). There are two key factors governing
the overall behavior of the conduction band population for different $n$
values at two intensities: the first one is the reduction in the energy gap
between valence and conduction bands around a Weyl node as $n$ increases [see
Fig. 2(b)]. This results in higher probability of excitation and thus more
population as $n$ increases. The other important factor is the dipole matrix
amplitude and its relation with $n$ [see Eq. (1)], which manifests higher
electronic population as $n$ increases. Thus, it can be concluded from Fig. 5
that the population increases monotonically with $n$ irrespective of the
laser’s intensity.

Returning back to the considerable anomalous current at the lowest intensity
$10^{8}$ W/cm2, let us explore how the topological charge impacts the
anomalous current. By following the analysis of the linear response from the
WSM, the anomalous current is proportional to $n$ as
$\mathbf{J}_{\mathbf{\Omega}}\propto n(\mathbf{b}\times\mathbf{E})$, where
$\mathbf{b}$ is the vector joining the Weyl nodes Nandy  _et al._ (2019). The
quantity $\mathbf{b}\times\mathbf{E}$ determines the direction of the
anomalous current. Note that the same reasoning we have used earlier to
explain why the laser polarized along the direction of the line connecting the
Weyl nodes results in no anomalous current. If we consider the expression of
$\mathbf{J}_{\mathbf{\Omega}}$ to be true for the considered laser intensities
then, the anomalous current should increase monotonically with intensity as
$\mathbf{J}_{\mathbf{\Omega}}\propto\mathbf{E}$. However, the anomalous
current deviates significantly from this expectation at higher intensity,
signaling a nonlinear optical response from m-WSMs. It is important to
emphasis that the anomalous current along the perpendicular direction is
mainly driven by the Berry curvature as shown in Fig. S1 Not .

To corroborate our claim about the nonlinear optical response, let us analyze
the ratio of the anomalous current’s peak for different $n$ values as a
function of the laser’s intensity. The ratios of the peak current for $n=3$ to
$n=1$, and $n=2$ to $n=1$ are presented in Fig. 6(a). Within the linear
response framework, it is expected that the ratios
$\mathbf{J}_{\mathbf{\Omega}}|_{n=3}/\mathbf{J}_{\mathbf{\Omega}}|_{n=1}$ and
$\mathbf{J}_{\mathbf{\Omega}}|_{n=2}/\mathbf{J}_{\mathbf{\Omega}}|_{n=1}$
should be 3 and 2, respectively as $\mathbf{J}_{\mathbf{\Omega}}\propto n$.
This expectation is true at $10^{8}$ W/cm2 where the ratios are close to 3 and
2 as evident from Fig. 6(a). However, both ratios decrease monotonically as
intensity increases, albeit at different rates. The ratios
$\mathbf{J}_{\mathbf{\Omega}}|_{n=3}/\mathbf{J}_{\mathbf{\Omega}}|_{n=1}$ and
$\mathbf{J}_{\mathbf{\Omega}}|_{n=2}/\mathbf{J}_{\mathbf{\Omega}}|_{n=1}$
reach approximately 2 and 1.5 at $10^{10}$ W/cm2, respectively – a drastic
deviation from the expectation of linear response theory. The ratio
$\mathbf{J}_{\mathbf{\Omega}}|_{n=3}/\mathbf{J}_{\mathbf{\Omega}}|_{n=1}$
decreases with a much faster rate compared to
$\mathbf{J}_{\mathbf{\Omega}}|_{n=2}/\mathbf{J}_{\mathbf{\Omega}}|_{n=1}$.
This implies that the anomalous current due to the Berry curvature decreases
faster with an increase in $n$. Thus, $\mathbf{J}_{\mathbf{\Omega}}$
originating from two different topological charges may be comparable at a
certain intensity. However, it is practically not feasible to keep increasing
the intensity as it can go above the damage threshold of the material.

So far, we have investigated how the anomalous current behaves with respect to
the intensity. Recently, it has been shown that the nonlinear anomalous
current in a Weyl semimetal with $n=1$ decreases with increasing the distance
between the Weyl nodes Bharti  _et al._ (2022); Avetissian  _et al._ (2022).
Let us see how the reported observation changes for m-WSMs. Figure 6(b)
presents how the anomalous current’s peak varies with the distance between the
Weyl nodes for different $n$ values. The current due to the Berry curvature
decreases with an increment in the distance between the nodes, and the
decrease is different for each $n$. Similar observations can be made when the
intensity lies within the linear-response regime as evident from Fig. S2 Not .
The overall behavior of the anomalous current remains same, but the response
due to the change in the distance is nontrivial when we compare for different
$n$ values.

Figure 7: Higher-order harmonic spectrum corresponding to multi-Weyl
semimetals with topological charge (a) $n=1$, (b) $n=2$, and (c) $n=3$. The
driving laser is 100 fs long with wavelength of $3.2~{}\mu$m and intensity of
$1\times 10^{11}$ W/cm2. The laser pulse is linearly polarised along $x$
direction. A phenomenologically decoherence time $\textrm{T}_{2}$ of 1.5 fs is
added in the simulation. The harmonic spectrum is simulated by taking the
modules square of the Fourier transform of the time-derivative of the total
current as
$\left|\mathcal{FT}\left(\frac{d}{dt}\left[\int\textbf{J}(\mathbf{k},t)~{}d{\textbf{k}}\right]\right)\right|^{2}$
Mrudul  _et al._ (2020). The distance between two Weyl node is
$k_{0}=\pi/(2a)$ with $a=6.28$ Å.

The normal and anomalous currents are analyzed as the optical responses in
m-WSMs, which transit from a linear regime to a nonlinear regime as the
laser’s intensity increases, even from a perturbative regime to a
nonperturbative regime. High-harmonic generation (HHG) is a hallmark example
of a nonperturbative nonlinear optical process, which has become a method of
choice to probe various static and dynamical aspects of solids Schubert  _et
al._ (2014); Mrudul  _et al._ (2021); Mrudul and Dixit (2021b); Hohenleutner
_et al._ (2015); Zaks  _et al._ (2012); Pattanayak  _et al._ (2020); Mrudul
_et al._ (2019); Pattanayak  _et al._ (2022); Rana and Dixit (2022); Rana  _et
al._ (2022). In addition, HHG from topological materials has become a center
of attention as it allows one to investigate nonequilibrium topological
aspects of topological insulators, Dirac and Weyl semimetals in recent years
Baykusheva  _et al._ (2021); Kovalev  _et al._ (2020); Cheng  _et al._ (2020);
Bai  _et al._ (2021); Dantas  _et al._ (2021); Lv  _et al._ (2021b). Recently,
it has been shown that high-harmonic spectroscopy can be used to probe the
light-induced nonlinear anomalous Hall effect in a Weyl semimetal Bharti  _et
al._ (2022). Thus, it is interesting to explore how the topological charge
impacts HHG from m-WSMs.

The higher-order harmonic spectra corresponding to m-WSMs are distinct for
different $n$’s as can be seen from Fig. 7. Odd-order harmonics are only
generated as m-WSMs exhibit inversion symmetry in the present case. The
harmonic cutoff not only increases drastically but also the yield of the
harmonics is boosted by several orders as $n$ increases. The energy cutoff
increases from seven to thirteen as $n$ changes from $n=1$ to $n=2$. Moreover,
the yield of the seventh harmonic is two-orders in magnitude boosted as $n$
transits from 1 to 2 as is evident from Figs. 7(a) and 7(b). Similar
observations can be made for the harmonics shown in Fig. 7(c) for $n=3$. The
Berry-curvature-driven anomalous current in m-WSMs results in anomalous odd
harmonics along the $y$ direction. The presence of the anomalous harmonics is
a signature of the light-induced anomalous Hall effect, and the strength of
these harmonics gives the measure of the Hall effect Bharti  _et al._ (2022).

The characteristic dependence of the relative yield of the normal and
anomalous harmonics on the harmonic-order offers a route to tailor the
polarization of the emitted harmonics, which carry information about the
topology of the Berry curvature in m-WSMs. The ellipticity of the first,
third, and fifth harmonics for $n=1$ reads as 0.85, 0.17, and 0.55,
respectively. As the value of the topological charge changes to $n=2(3)$, the
ellipticity of the first, third, and fifth harmonics changes drastically as
0.41(0.2), 0.25(0.11), and 0.72(0.45), respectively. The reduction of the
ellipticity of a given harmonic with an increase in the topological charge can
be attributed to a significant change in the yield of the anomalous harmonic.
Thus, the ellipticity of the emitted harmonic for a given harmonic order is
significantly different for different $n$ values and can be potentially used
as a characterization tool for the topological charge in m-WSMs.

##  IV CONCLUSION

In summary, we have explored the impact of the topological charge on laser-
driven electron dynamics in Weyl semimetals. For this purpose, multi-Weyl
semimetals with topological charges $n=1,2$, and 3 are considered. It has been
found that the laser-driven electronic currents are distinct for different
$n$’s. Moreover, the direction and amplitude of the current strongly depend on
the laser polarization. The parity and amplitude of the Berry curvature
determine the behavior of the anomalous current – current perpendicular to the
laser polarization. As the laser’s intensity increases, the scaling of the
anomalous current with $n$ deviates drastically from the linear response
theory, i.e., $\mathbf{J}_{\mathbf{\Omega}}\propto n$. Owing to the large
linear response from the Berry curvature, the anomalous current dominates at
lower intensity. However, the ratio of the anomalous current from higher to
lower topological charge decreases, which leads the saturation of the
anomalous current relatively at lower intensity. On the other hand, normal
current – current parallel to the laser polarization – scales with the
topological charge, and therefore remains separated in magnitude at relatively
higher intensity. As the intensity increases, the optical responses from
multi-Weyl semimetals transit from a linear regime to a nonlinear regime, and
from a perturbative regime to a nonperturbative regime. High-harmonic
spectroscopy is used to probe the distinct and interesting features of the
currents in multi-Weyl semimetals. It has been observed that the harmonic
yield and the energy cutoff of the higher-order harmonics increase drastically
as $n$ increases. Moreover, the polarization of the emitted harmonics encodes
information about the phase and magnitude of the Berry curvature’s components.
It can be anticipated that the topological charge of the multi-Weyl semimetal
can be characterized by the polarization of the emitted harmonics in an all-
optical way.

## ACKNOWLEDGMENTS

G. D. acknowledges support from the Science and Engineering Research Board
(SERB) India (Project No. MTR/2021/000138).

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