# Kerr effects in tilted multi-Weyl semimetals

Amit Gupta  Department of Physics, M. R. M. College, Lalit Narayan Mithila
University, Darbhanga, Bihar 846004, India

###### Abstract

We calculate the optical conductivity of type-I and type-II multi-Weyl
semimetals(m-WSMs) in time-reversal(TR) broken case at finite doping. It is
found that both longitudinal and Hall conductivities are renormalized by
J(monopole charge) perpendicular to Weyl nodes separation and longitudal
conductivity along nodes has non-trivial dependence on J. The Kerr rotations
in thin films and bulk have different dependence on system parameters. The
Kerr rotation in ultra-thin films is independent of monopole charge J and
depends only on Fermi velocity $v_{F}$ while in bulk it dependent on monopole
charge J, Fermi velocity and material parameters.

††preprint: APS/123-QED

##  I Introduction

The last decade has witnessed Weyl semimetals (WSMs) as a focus of research
attraction due to their exotic properties. It manifest unusual Fermi arc
surface states and chiral anomaly that is intimately related to topology and
symmetry of WSMs armitage2018weyl ; hasan2017discovery . In WSMs, the bulk
band gap closes at an even number of discrete points in the Brillouin
zone(BZ). These special band closing non-generate points are referred as Weyl
nodes or points wan2011topological ; Weyl nodes act as a source or sink of
monopoles of Berry curvature characterized by monopole charge J. Two Weyl
nodes of opposite chirality are located at opposite sign momenta when the
system breaks thet time-reversal(TR symmetry; four Weyl nodes are noticed in
general for systems with broken inversion(I) symmetry only armitage2018weyl ;
mccormick2017minimal . Moreover, I symmetry breaking may also lead to the
energy-separated Weyl points while TR symmetry breaking can result in Weyl
points at the same energy dey2020dynamic ; zyuzin2012weyl . The existence of
Fermi arc surface states, chiral-anomaly induced negative magnetoresistance,
and the quantum anomalous Hall effect is the direct consequence of the nonzero
fluxes of the Berry curvaturezyuzin2012topological ; son2013chiral ;
burkov2015chiral . As compared to the conventional WSMs with J=1, reported in
TaP, TaAs, and NbAs xu2015discovery ; lv2015experimental ; lv2015observation ,
it has been recently shown that J can be generically greater than 1, with the
crystalline environment bounding its maximum value to 3xu2011chern ;
yang2014classification ; fang2012multi . These are called multi-WSMs (m-WSMs).
The m-WSMs can be considered as 3D analog of multilayers graphene
min2008chiral . Close to the Weyl points, m-WSMs host low-energy
quasiparticles which has linear only in one direction and quadratic(cubic) in
other two directions leading to usual properties which differ from linear WSMs
park2017semiclassical ; gorbar2017anomalous ; dantas2018magnetotransport ;
lepori2018axial ; nag2020magneto ; gupta2017floquet ; gupta2019novel . Density
functional theory (DFT)calculations suggest that $HgCr_{2}Se_{4}$and
$SrSi_{2}$can be the candidate materials for double WSM fang2012multi ;
xu2011chern ; huang2016new  whereas $A(MoX)_{3}$(with A=Rb,TI; X=Te) kind of
materials can accommodate triple-Weyl pointsliu2017predicted .

Further, WSMs can be classified as type I, in this it has a conical spectrum
and a point-like Fermi surface at the Weyl point which corresponds to
vanishing density of states at the Weyl nodes, and type II (which occurs when
large tilting of the Weyl cones results in a Lifshitz transition), in which an
electron pocket and a hole pocket meet at a singular point of momentum space
results in finite density of states, allowing for distinct topological
properties armitage2018weyl ; yan2017topological ; xu2015structured ;
soluyanov2015type ; menon2018photoinduced ; nag2020magneto ; fei2017nontrivial
; yu2016predicted ; udagawa2016field ; lv2017experimental . The type-II WSM
has been observed experimentally li2017evidence ; kimura2019optical  while
theoretical study suggests that a type-II WSM can be engineered by applying
strain or chemical doping to the type-I WSM trescher2015quantum . The type-II
WSM phase is characterized by a different class of Weyl fermions manifesting
the violation of Lorentz symmetry. While a lot of progress has been made
experimentally and theoretically in investigating J=1 type-I and type-II WSMs,
the experimental discovery of mWSMs with $J\geq 2$ is yet to be made; however,
using density functional theorycalculations some materials are conjectured to
host Weyl nodes with monopole charges J=2,3 xu2011chern ; fang2012multi ;
huang2016new ; liu2017predicted .

For thin films as well as bulk conventional WSMs, the giant Kerr rotation,
polarization rotation have been demonstrated recently in the electrodynamics
reponse of reflected beam, originates from the optical Hall conductivity
including the quantum anomalous Hall(QAH) effect sonowal2019giant . We extend
this idea for higher monopole charge J in tilted mWSMs. We calculate
analytically as well as numerically all optical conductivity tensor elements
$\sigma_{ij}(\omega)$ for tilted type-I and type-II m-WSMs. The longitudinal
conductivity along node separation has differently as a function of optical
frequency than perpendicular components. This leads to different variation of
Kerr rotations in ultra-thin film and bulk of the m-WSMs. The Kerr angle has
sizeable value($\approx$ 0.1 radians for $\omega<10^{14}$ Hz) compared to the
usually observed values in topological insulators kargarian2015theory .

This paper is organized as follows: In Sec. II, We write the continuum model
Hamiltonian for m-WSMs. In Sec. III, we obtain the full optical conductivity
matrix for tilted type-I and type-II mWSMs, including the QAH contribution. In
Sec. IV we discuss the electromagnetic response in thin films of tilted mWSM,
and demonstrate the existence of giant polarization rotation is independent of
monopole charge J. This is followed by a discussion of axion electrodynamics
induced polarization rotation in bulk mWSMs in Sec. V and show that this
rotation is dependent on monopole charge J as well as material parameters.
Finally, we conclude our main findings in Sec. VI.

##  II Model Hamiltonian

The non-interacting low energy effective Hamiltonian for tilted multi-Weyl
semimetals is given by mukherjee2018doping ; nag2020thermoelectric ;
nag2020magneto ; menon2020anomalous ,

$\mathcal{H}_{J}=\xi\hbar v_{t}k_{z}\mathbb{1}_{2\times
2}+\alpha_{J}\hbar[(\hat{k}_{-})^{J}\sigma_{+}+(\hat{k}_{+})^{J}\sigma_{-}]+\xi\hbar
v_{z}\hat{k}_{z}\sigma_{z},$ (1)

where $\sigma_{\pm}=\frac{1}{2}(\sigma_{x}\pm i\sigma_{y})$ and
$\hat{k}_{\pm}=\hat{k}_{x}\pm i\hat{k}_{y}$, J represents monopole charge,
$v_{z}$ is the effective velocity along $\hat{z}$ direction and $\alpha_{J}$
is the material dependent parameter, e.g. $\alpha_{1}$ and $\alpha_{2}$ are
the Fermi velocity and inverse of the mass respectively for the isotropic and
double WSMs.
To avoid difficulties with anisotropic dispersions, we consider the following
coordinate transformation ahn2017optical

$\displaystyle
k_{x}\rightarrow\rho^{1/J}\cos\phi,k_{y}\rightarrow\rho^{1/J}\sin\phi,k_{z}\rightarrow
k_{z}$ (2)

whose Jacobian is given by

$\mathcal{J}=\frac{\rho^{\frac{2}{J}-1}}{J}\equiv\mathcal{J}(\rho)$ (3)

In the transformed coordinates, the energy dispersion is given by
$E_{s}(\mathbf{k})=\hbar
v_{t}k_{z}+s\hbar\sqrt{\alpha_{n}^{2}(\rho_{J})^{2}+\left(k_{z}v_{z}\right)^{2}}$
with $\rho_{J}=(k_{x}^{2}+k_{y}^{2})^{J/2}$.

##  III Optical Conductivity

The interband optical conductivity in the non-interacting limit
$\gamma\rightarrow 0$ can be expressed as

$\displaystyle\sigma_{\alpha\beta}(\omega)=-\lim_{\gamma\to 0}\sum_{\bf
k}\frac{in_{k}}{\hbar\omega_{k}}\frac{M_{i}^{ss^{\prime}}(k)M_{j}^{s^{\prime}s}(k)}{\hbar\omega+\hbar\omega_{k}+i\gamma}$
(4)

where $i,j=x,y,z,n_{k}^{eq}=\Theta(\mu-\hbar
v_{t}k_{z}-\epsilon_{J})-\Theta(\mu-\hbar v_{t}k_{z}+\epsilon_{J})$ and
$\hbar\omega_{k}=2\hbar\epsilon_{j}(k)$ is the energy difference between the
conduction and the valence band, which is independent of the tilt velocity.
The real and the imaginary parts can be separated in the limit
$\gamma\rightarrow 0$, using the Sokhotski-Plemelj Formula

$\lim_{\gamma\to 0}\frac{1}{x\pm i\gamma}=P\frac{1}{x}\mp i\pi\delta(x)$ (5)

Here P denotes the principal value.

The optical matrix element is given by

$\displaystyle M_{x}$ $\displaystyle=$ $\displaystyle
e\alpha_{J}(i\sin{J\phi}-\xi\frac{v_{F}k_{z}}{\epsilon_{J}}\cos{J\phi}),$
$\displaystyle M_{y}$ $\displaystyle=$ $\displaystyle
e\alpha_{J}(-i\cos{J\phi}-\xi\frac{v_{F}k_{z}}{\epsilon_{J}}\sin{J\phi})$
$\displaystyle M_{z}$ $\displaystyle=$ $\displaystyle\xi
ev_{F}\frac{\alpha_{J}\rho}{\epsilon_{J}}$ (6)

Here we have defined
$\epsilon_{J}=\hbar\sqrt{\alpha_{J}^{2}\rho^{2}+v_{F}^{2}k_{z}^{2}}$. Note
that $\bf{M_{k}}$does not depend on the tilt velocity $v_{t}$ at all.

###  III.1 Re[$\sigma_{xx}(\omega)$]

The real part of longitudinal optical conductivity is given by

$\displaystyle
Re[\sigma_{ii}]=-\frac{1}{(2\pi)^{3}}\int_{0}^{\infty}\mathcal{J}d\rho\int_{k_{c}}^{k_{c}}dk_{z}\int_{0}^{2\pi}\frac{n_{k}^{eq}}{\hbar\omega_{k}}$
$\displaystyle{\mid Mi\mid}^{2}\pi\delta(\omega-\omega_{k})$ (7)

The matrix element of $M_{i}^{ss^{\prime}}(\rho,k_{z})=\langle
s,\rho,k_{z}|\frac{\partial\hat{H}}{\partial
k_{i}}|s^{\prime},\rho,k_{z}\rangle$ are given by

$\displaystyle M_{x}^{+-}(\rho,k_{z})$ $\displaystyle=$ $\displaystyle
Je\alpha_{J}\rho^{(J-1)/J}(i\sin{J\phi}-\xi\frac{v_{f}k_{z}}{\epsilon_{J}}\cos{J\phi}),$
$\displaystyle M_{y}^{+-}(\rho,k_{z})$ $\displaystyle=$ $\displaystyle
Je\alpha_{J}\rho^{(J-1)/J}\alpha_{J}(-i\cos{J\phi}-\xi\frac{v_{f}k_{z}}{\epsilon_{J}}\sin{J\phi}),$
$\displaystyle M_{z}^{+-}(\rho,k_{z})$ $\displaystyle=$ $\displaystyle\xi
ev_{F}\frac{\alpha_{J}\rho}{\epsilon_{J}}$ (8)

Here, the optical matrix is dependent on $\phi_{k}$ and the angular
integration leads to

$\displaystyle\int_{0}^{2\pi}{\mid Mx\mid}^{2}d\phi_{k}=J^{2}\pi
e^{2}\alpha_{J}^{2}\rho^{\frac{2(J-1)}{J}}\Bigl{(}1+\frac{v_{f}^{2}k_{z}^{2}}{\epsilon_{J}^{2}}\Bigl{)}$
(9)

Figure 1: (a),(b) show the real part of the $\sigma_{xx}$ , in a type-I
($\alpha_{t}=0.2$) and a type-II ($\alpha_{t}=2.0$) mWSM,
respectively.(c),(d)show the imaginary parts of the $\sigma_{xx}$ , in a
type-I ($\alpha_{t}=0.2$) and a type-II ($\alpha_{t}=2.0$) mWSMs,
respectively.

Thus we obtain

$\displaystyle Re[\sigma_{xx}(\omega)]$ $\displaystyle=$
$\displaystyle-\frac{Je^{2}\alpha_{J}^{2}}{(8\pi}\int_{0}^{\infty}d\rho\rho\int_{-k_{c}}^{k_{c}}dk_{z}$
(10)
$\displaystyle\frac{\delta(\omega-\omega_{k})}{\hbar\epsilon_{J}}\Bigl{(}1+\frac{v_{f}^{2}k_{z}^{2}}{\epsilon_{J}^{2}}\Bigl{)}[\Theta(\mu-\hbar
v_{t}k_{z}-\hbar\epsilon_{J})$ $\displaystyle-\Theta(\mu-\hbar
v_{t}k_{z}+\hbar\epsilon_{J})]$

Now, the integration for $\rho$ can easily be carried out using the root
($\rho_{i}$) of the argument in the Dirac delta function

$\delta(f(\rho))=\sum_{x_{i}}\frac{\delta(\rho-\rho_{i})}{\mid
f^{\prime}(\rho_{i})\mid},\ where\ f(\rho_{i})=0$ (11)

Substituting the roots in the delta function leads to the following one-
dimensional integral over $k_{z}$

$\displaystyle Re[\sigma_{xx}(\omega)]$ $\displaystyle=$
$\displaystyle\frac{Je^{2}}{32\pi\hbar}\int_{k_{c}}^{k_{c}}dk_{z}\Bigl{[}1+\Bigl{(}\frac{2v_{f}k_{z}}{\omega}\Bigl{)}^{2}\Bigl{]}$
(12) $\displaystyle[\Theta(\mu-\hbar v_{t}k_{z}+\hbar\omega/2)$
$\displaystyle-\Theta(\mu-\hbar v_{t}k_{z}-\hbar\omega/2)]$

Equation 12 can be expressed in a dimesionless form:

$Re[\sigma_{xx}(\omega)]=\sigma_{\omega}^{J}F_{xx},\ where\
\sigma_{\omega}^{J}\equiv\frac{Je^{2}\omega}{6hv_{F}},$ (13)

and

$F_{xx}=\frac{3}{8}\int_{-1}^{1}dx(1+x^{2})(\Theta_{+}-\Theta_{-})$ (14)

where we have defined
$\Theta_{\pm}=\Theta[\frac{2\mu}{\hbar\omega}-\alpha_{t}x\pm 1]$, and
$x=2v_{F}k_{z}/\omega$. Evaluating the integral in Eq.(24) for the two cases
of $\mid\alpha_{t}\mid<1$ and $\mid\alpha_{t}\mid>1$ leads to.

For a type-I WSM hosting a pair of oppositely tilted Weyl points with
$\mid\alpha_{t}\mid<1$, we obtain

$Re[\sigma_{xx}(\omega)]=\begin{cases}0&I:\omega<\omega_{l},\\\
\sigma_{\omega}^{J}(1/2-\eta_{l})&II:\omega_{l}<\omega<\omega_{u}\\\
\sigma_{\omega}^{J},&III:\omega>\omega_{u}\end{cases}$ (15)

where we have defined conductivity via optical
frequency:$\sigma_{\omega}=e^{2}\omega/(6hv_{f})$. The photon energy bounds
are
$\hbar\omega_{l}=2\mu/(1+\mid\alpha_{t}\mid)$,$\hbar\omega_{u}=2\mu/(1-\mid\alpha_{t}\mid)$
and

$\eta_{l}=\frac{3}{8\mid\alpha_{t}\mid}\Bigl{(}\frac{2\mu}{\hbar\omega}-1\Bigl{)}\Bigl{[}1+\frac{1}{3\alpha_{t}^{2}}(\frac{2\mu}{\hbar\omega}-1)^{2}\Bigl{]}$
(16)

In region I, for $\omega<\omega_{l}$, the optical transitions is absent due to
Pauli blocking. In region II, $\omega\in(\omega_{l},\omega_{u})$, optical
transitions happen resulting in finite $Re[\sigma_{xx}(\omega)]$. In the
$\alpha_{t}\rightarrow 0$ limit we have
$\omega_{l}\rightarrow\omega_{u}=2\mu/\hbar$ and we have
$Re[\sigma_{xx}(\omega)]=\sigma_{\omega}$ for $\hbar\omega\geq 2\mu$. In the
other limit $\alpha_{t}\rightarrow 1$, we have $\omega_{u}\rightarrow\infty$
with region II extending to higher energies.
For a type-II WSM with $\mid\alpha_{t}\mid>1$, we obtain

$Re[\sigma_{xx}(\omega)]=\begin{cases}0&I:\omega<\omega_{l},\\\
\sigma_{\omega}^{J}(1/2-\eta_{l})&II:\omega_{l}<\omega<\omega_{u}\\\
\sigma_{\omega}^{J}\eta_{2},&III:\omega>\omega_{u}\end{cases}$ (17)

Here we have defined $\hbar\omega_{u}=2\mu/(\mid\alpha_{t}\mid-1)$ and

$\eta_{2}=\frac{3}{4\mid\alpha_{t}\mid}\Bigl{[}1+\frac{1}{3\alpha_{t}^{2}}+\Bigl{(}\frac{2\mu}{\alpha_{t}\hbar\omega}\Bigl{)}^{2}\Bigl{]}$
(18)

The real and imaginary parts of conductivity are related by Kramers-kronig
relations

$\displaystyle
Im[\sigma_{ij}(\omega)]=-\frac{2\omega}{\pi}\mathcal{P}\int_{0}^{\Lambda}d\omega^{\prime}\frac{Re[\sigma_{ij}(\omega^{\prime})}{\omega^{\prime
2}-\omega^{2}}$ $\displaystyle
Re[\sigma_{ij}(\omega)]=\frac{2}{\pi}\mathcal{P}\int_{0}^{\Lambda}d\omega^{\prime}\frac{\omega^{\prime}Im[\sigma_{ij}(\omega^{\prime})}{\omega^{\prime
2}-\omega^{2}}$ (19)

where $\mathcal{P}$ refers to the principle value of the integral and
$\Lambda(>>\omega)$ is the ultravoilet cutoff.
The imaginary part of the longitudinal optical conductivity
$\sigma_{xx}(\omega)$ can be calculated analytically sonowal2019giant  using
above Eq.(19) and plotted in Fig.(1).

Figure 2: (a),(b),(c) show the real(blue) and the imaginary(grey) parts of the
$\sigma_{zz}$ , in a type-I ($\alpha_{t}=0.2$) mWSM respectively. Here we have
chosen $\mu=.125$ eV, $v_{F}=10^{6}m/s$, $Q=3.25\times 10^{8}m^{-1}$,
$\alpha_{2}=.009m^{2}/s$, $\alpha_{3}=4.5\times 10^{-11}m^{3}/s$

###  III.2 Re[$\sigma_{zz}(\omega)$]

Following the same procedure a s above, the angular integration leads to

$\displaystyle\int_{0}^{2\pi}{\mid Mz\mid}^{2}d\phi_{k}=2\pi
e^{2}v_{f}^{2}\Bigl{(}\frac{\alpha_{J}^{2}\rho^{2}}{\alpha_{J}^{2}\rho^{2}+v_{F}^{2}k_{z}^{2}}\Bigl{)}$
(20)

Thus the longitudinal optical conductivity in this case is given by

$\displaystyle Re[\sigma_{zz}(\omega)]$ $\displaystyle=$
$\displaystyle-\frac{e^{2}v_{f}^{2}\alpha_{J}^{2}}{4\pi
J}\int_{0}^{\infty}d\rho\int_{-k_{c}}^{k_{c}}dk_{z}\frac{\rho^{\frac{2}{J}+1}}{\alpha_{J}^{2}\rho^{2}+v_{F}^{2}k_{z}^{2}}$
(21)
$\displaystyle\frac{\delta(\omega-\omega_{k})}{\hbar\epsilon_{J}}[\Theta(\mu-\hbar
v_{t}k_{z}-\hbar\epsilon_{J})$ $\displaystyle-\Theta(\mu-\hbar
v_{t}k_{z}+\hbar\epsilon_{J})]$

Performing the $\rho$ integral using the roots of the $\delta$ function, we
have

$\displaystyle Re[\sigma_{zz}(\omega)]$ $\displaystyle=$
$\displaystyle\frac{e^{2}v_{F}^{2}}{16\pi\hbar
J\alpha_{J}^{2/J}}\int_{-\omega/2v_{F}}^{\omega/2v_{F}}dk_{z}\Bigl{[}1-\Bigl{(}\frac{2v_{F}k_{z}}{\omega}\Bigl{)}^{2}\Bigl{]}$
(22) $\displaystyle[\Theta(\mu-\hbar v_{t}k_{z}+\hbar\omega/2)$
$\displaystyle-\Theta(\mu-\hbar v_{t}k_{z}-\hbar\omega/2)]$

Eq.(22) can be expressed in a dimesionless form:

$Re[\sigma_{zz}(\omega)]=\sigma_{\omega,z}^{J}F_{zz},\ where\
\sigma_{\omega,z}^{J}\equiv\frac{e^{2}(\frac{\omega}{2})^{\frac{2}{J}-1}v_{f}}{3hJ\alpha_{J}^{2/J}},$
(23)

and

$F_{zz}=\frac{3}{8}\int_{-1}^{1}dx(1-x^{2})^{1/J}(\Theta_{+}-\Theta_{-})$ (24)

where $\Theta_{\pm}\equiv\Theta(\mu-\alpha_{t}x\pm\hbar\omega/2)$. The
imaginary part has analytical only in case of J=1sonowal2019giant . For higher
J, the imaginary parts are computed numerically. The real and imaginary parts
are plotted in Fig.(2).

Figure 3: (a),(b) and (c) show real part of $\sigma_{xy}$ , in a type-I
($\alpha_{t}=0.2,0.5$) mWSM, respectively.

###  III.3 Im[$\sigma_{xy}(\omega)$]

The transverse optical Hall conductivity is given by

$\displaystyle\sigma_{xy}(\omega)]=\frac{1}{(2\pi)^{3}}\int_{0}^{2\pi}d\phi_{k}\int_{0}^{\infty}d\rho\int_{k_{c}}^{k_{c}}dk_{z}\sigma_{k}^{ij}(\omega)$
(25)

where the conductivity kernel is given by

$\sigma_{k}^{ij}=\frac{-n_{k}^{eq}}{\hbar\omega_{k}}\Bigl{(}\frac{M_{j}M_{i}^{*}}{\omega+\omega_{k}+i\gamma}+\frac{M_{j}^{*}M_{i}}{\omega-\omega_{k}+i\gamma}\Bigl{)}$
(26)

Figure 4: (a),(b) show the imaginary part of $\sigma_{xy}$ , in a type-I
($\alpha_{t}=-0.2,0.2$) and a type-II ($\alpha_{t}=-2.0,2.0$) mWSM,
respectively as a function of optical frequency $\omega$.(c)show the imaginary
part of $\sigma_{xy}$ as a function of tilt parameter $\alpha_{t}$.

$\displaystyle M_{y}M_{x}^{*}$ $\displaystyle=$
$\displaystyle-1+\Bigl{(}\frac{v_{f}k_{z}}{\epsilon_{j}}\Bigl{)}^{2}\sin{J\phi_{k}}\cos{\phi_{k}}+i\frac{v_{f}k_{z}}{\epsilon_{J}}$
(27) $\displaystyle=M_{y}^{*}M_{x}$

Performing the angular integration over $\phi_{k}$

$\int_{0}^{2\pi}\Bigl{(}M_{y}M_{x}^{*}+M_{y}^{*}M_{x}\Bigl{)}=i\xi
2\pi(\alpha_{J}J)^{2}\rho^{2(J-1)/J}\frac{v_{f}k_{z}^{\xi}}{\epsilon_{J}}$
(28)

The imaginary part of $\sigma_{xy}(\omega)$ can be expressed as

$\displaystyle Im[\sigma_{xy}(\omega)]$ $\displaystyle=$
$\displaystyle\frac{Je^{2}\alpha_{J}^{2}}{4\pi}\sum_{\xi=\pm
1}\xi\int_{0}^{\infty}\rho
d\rho\int_{k_{c}}^{k_{c}}dk_{z}^{\xi}\frac{v_{F}k_{z}^{\xi}}{\epsilon_{J}}$
(29)
$\displaystyle\frac{n_{k}^{eq}}{\hbar\omega_{k}}\delta(\omega-\omega_{k})$

Performing the $\rho$ integral and the $\delta$ function lead to the following
one-dimensional integral

$\displaystyle Im[\sigma_{xy}(\omega)]$ $\displaystyle=$
$\displaystyle\frac{Je^{2}v_{F}}{8\pi\hbar\omega}\sum_{\xi=\pm
1}\xi\int_{-\omega/2v_{F}}^{\omega/2v_{F}}k_{z}^{\xi}dk_{z}^{\xi}$ (30)
$\displaystyle\Bigl{[}\Theta\Bigl{(}1-\frac{\hbar\omega}{2\mu}-\frac{\hbar\xi
v_{t}k_{z}^{\xi}}{\mu}\Bigl{)}$
$\displaystyle-\Theta\Bigl{(}1+\frac{\hbar\omega}{2\mu}-\frac{\hbar\xi
v_{t}k_{z}^{\xi}}{\mu}\Bigl{)}\Bigl{]}$

Figure 5: (a),(b) and (c) show the Kerr angle as a function of optical
frequency for type-1($\alpha_{t}=0.2$) and a type-II ($\alpha_{t}=2.0$)for
single, double and triple WSM, respectively.

Figure 6: (a),(b) show the real and imaginary parts of effective dielectric
term $\varepsilon_{xx}$ for single WSM at ($\alpha_{t}=0.2,2.0$). (c),(d) show
the real and imaginary parts of effective dielectric term $\varepsilon_{xx}$
for double WSM at ($\alpha_{t}=0.2,2.0$).(e),(f) show the real and imaginary
parts of effective dielectric term $\varepsilon_{xx}$ for triple WSM at
($\alpha_{t}=0.2,2.0$).

Eq.(30) can be expressed as

$Im[\sigma_{xy}(\omega)]=J\frac{3\sigma_{\omega}^{J}}{2}\frac{\mu^{2}}{\hbar^{2}\omega^{2}}F_{xy}$
(31)

where we have defined

$F_{xy}=\sum_{\xi=\pm
1}\xi\int_{-\hbar\omega/2\mu}^{\hbar\omega/2\mu}xdx\sum_{p=\pm
1}\Theta\Bigl{(}1-\xi\alpha_{t}x-p\frac{\hbar\omega}{2\mu}\Bigl{)}$ (32)

The real part of the Hall conductivity can be computed analytically using
Eq.(19)sonowal2019giant . The real part has also contribution due to Weyl
nodes separation $\frac{J\sigma_{Q}}{\mid\alpha_{t}\mid}$
nag2020thermoelectric . The real and imaginary parts of Hall conductivity are
plotted in Figs.(3) and(4).

##  IV Kerr rotation in thin films of m-WSMS

In the ultra-thin limit, the thickness of the film(d) $\ll$ the
wavelenth($\lambda$) of light $\lambda$, even though d$\gg$a, the lattice
constant. Under these conditions the m-WSM film can be considered as a 2D
surface. Thus the polarization angle rotation can be obtained simply by
matching the electromagnetic boundary conditions (at z=0) on either side of
the thin film.

The polarization angle $\Phi_{kerr}$ and the azimuth of the major axes of the
polarization ellipse $\Psi_{Kerr}$. For the case of a reflected beam, the
corresponding dimensionless complex quantities are defined as

Figure 7: (a),(b) show the real and imaginary parts of effective dielectric
term $\varepsilon_{zz}$ for single WSM at ($\alpha_{t}=0.2,2.0$). (c),(d) show
the real and imaginary parts of effective dielectric term $\varepsilon_{zz}$
for double WSM at ($\alpha_{t}=0.2,2.0$).(e),(f) show the real and imaginary
parts of effective dielectric term $\varepsilon_{zz}$ for triple WSM at
($\alpha_{t}=0.2,2.0$).

$\chi_{Kerr}^{s}=\frac{r_{ps}}{r_{ss}}\ and\ \chi=-\frac{r_{sp}}{r_{pp}}$ (33)

where $r_{ij}$ = $(s,p)$ are reflection coefficients. In TR broken m- WSM, the
surface conductivity has finite off-diagonal terms for the surface without
Fermi arc states(surfaces $\perp\textbf{Q}$) while it is diagonal for the
surface hosting Fermi arcs states(surfaces $\parallel\textbf{Q}$). We consider
these two cases in the following subsections.

###  IV.1 Incidence on surface without Fermi arc states($\perp Q$)

Consider a linearly polarized incident beam propagates(say in direction
$\hat{\textbf{z}})$ parallel to Q and the electric polarization is in the x-y
plane. Let us assume that the interface of air and m-WSM thin film is located
at the $z=0$ plane. The wave vectors for the incident, reflected and refracted
beam are

Figure 8: (a),(b) show the real and imaginary parts of effective dielectric
term $\varepsilon_{xy}$ for single WSM at ($\alpha_{t}=0.2,2.0$). (c),(d) show
the real and imaginary parts of effective dielectric term $\varepsilon_{xy}$
for double WSM at ($\alpha_{t}=0.2,2.0$).(e),(f) show the real and imaginary
parts of effective dielectric term $\varepsilon_{xy}$ for triple WSM at
($\alpha_{t}=0.2,2.0$).

$\displaystyle\textbf{k}_{i}=(0,k_{i}\sin\theta_{i},k_{i}\cos\theta_{i})$
$\displaystyle\textbf{k}_{r}=(0,k_{r}\sin\theta_{r},-k_{r}\cos\theta_{r})$
$\displaystyle\textbf{k}_{t}=(0,k_{t}\sin\theta_{t},k_{t}\cos\theta_{t})$

Consider an incident plane wave which is a superposition of the linear s- and
p- polarized light,$\textbf{ E}_{i}$ = ( $E_{i}^{s},E_{i}^{p})^{T}$. The
components of electric field E are

$\displaystyle\textbf{E}_{i}=(E_{i}^{s},E_{i}^{p}\cos\theta_{i},-E_{i}^{p}\sin\theta_{i})e^{i(\textbf{k}_{i}.\cdot\textbf{r}-\omega_{i}t)}$
$\displaystyle\textbf{E}_{r}=(E_{r}^{s},E_{r}^{p}\cos\theta_{r},E_{r}^{p}\sin\theta_{r})e^{i(\textbf{k}_{r}.\cdot\textbf{r}-\omega_{i}t)}$
$\displaystyle\textbf{E}_{i}=(E_{t}^{s},E_{t}^{p}\cos\theta_{t},-E_{t}^{p}\sin\theta_{t})e^{i(\textbf{k}_{t}.\cdot\textbf{r}-\omega_{i}t)}$

The magnetic field B components are given by
$\textbf{B}=(n/c)\hat{\textbf{k}}\times\textbf{E}$, with ’n’ denoting the
refractive index of the medium. The fields on the two sides of the WSM thin
film are connected by the boundary conditions:

$\textbf{E}_{1}^{\parallel}=\textbf{E}_{2}^{\parallel},\ \
\hat{\textbf{n}}\times(\frac{\textbf{B}_{1}^{\parallel}}{\mu_{1}}-\frac{\textbf{B}_{2}^{\parallel}}{\mu_{2}})=\textbf{J}_{s}$
(34)

where $\textbf{E}_{1}=\textbf{E}_{i}+\textbf{E}_{r}$,
$\textbf{E}_{2}=\textbf{E}_{t}$,
$\textbf{B}_{1}=\textbf{B}_{i}+\textbf{B}_{r}$,
$\textbf{B}_{2}=\textbf{B}_{t}$. The surface current density
$\textbf{J}_{s}=\sigma^{s}\textbf{E}$, where the surface conductivity tensor
$\sigma_{ij}^{s}=d\sigma_{ij}$ with the bulk $\sigma_{ij}$ for the m-WSM.

In general, the reflected electric field can be expressed as

$\displaystyle\begin{pmatrix}E_{r}^{p}\\\
E_{r}^{s}\end{pmatrix}=\begin{pmatrix}r_{pp}&r_{ps}\\\
r_{sp}&r_{ss}\end{pmatrix}\begin{pmatrix}E_{i}^{p}\\\ E_{i}^{s}\end{pmatrix}$
(35)

Using this Eq.(35)) in Eq.(34), we obtain the reflection coefficients as

$\displaystyle
r_{sp}=\frac{E_{r}^{s}}{E_{i}^{p}}|_{E_{i}^{s}=0}=\frac{2n_{i}}{c\mu_{i}}\frac{\sigma_{xy}^{s}\cos\theta_{i}\cos\theta_{t}}{(\sigma_{xy}^{s}\sigma_{yx}^{s}\cos\theta_{i}\cos\theta_{t}-\sigma_{1}^{s}\sigma_{2}^{s})},$
(36) $\displaystyle r_{ss}$ $\displaystyle=$
$\displaystyle\frac{E_{t}^{s}}{E_{i}^{s}}|_{E_{i}^{p}=0}=-1$ (37)
$\displaystyle+$
$\displaystyle\frac{2n_{i}}{c\mu_{i}}\frac{\sigma_{2}^{s}\cos\theta_{i}}{(\sigma_{1}^{s}\sigma_{2}^{s}-\sigma_{1}^{s}\sigma_{2}^{s}-\sigma_{xy}^{s}\sigma_{yx}^{s}\cos\theta_{i}\cos\theta_{t})},$

along with

$\displaystyle r_{pp}$ $\displaystyle=$
$\displaystyle\frac{E_{t}^{p}}{E_{i}^{p}}|_{E_{i}^{s}=0}=-1$ (38)
$\displaystyle+$
$\displaystyle\frac{2n_{i}}{c\mu_{i}}\frac{\sigma_{1}^{s}\cos\theta_{t}}{(\sigma_{1}^{s}\sigma_{2}^{s}-\sigma_{1}^{s}\sigma_{2}^{s}-\sigma_{xy}^{s}\sigma_{yx}^{s}\cos\theta_{i}\cos\theta_{t})},$

and

$\displaystyle
r_{sp}=\frac{E_{r}^{s}}{E_{i}^{p}}|_{E_{i}^{s}=0}=\frac{2n_{i}}{c\mu_{i}}\frac{\sigma_{xy}^{s}\cos\theta_{i}\cos\theta_{t}}{(\sigma_{xy}^{s}\sigma_{yx}^{s}\cos\theta_{i}\cos\theta_{t}-\sigma_{1}^{s}\sigma_{2}^{s})}$
(39)

Here, we have defined
$\sigma_{1}^{s}\equiv\frac{n_{i}\cos\theta_{i}}{c\mu_{i}}+\frac{n_{t}\cos\theta_{t}}{c\mu_{t}}+\sigma_{xx}^{s}$
and
$\sigma_{2}^{s}\equiv\frac{n_{i}\cos\theta_{t}}{c\mu_{i}}+\frac{n_{t}\cos\theta_{i}}{c\mu_{t}}+\sigma_{yy}^{s}\cos\theta_{i}\cos\theta_{t}$.

For the case of normal incidence($n_{i}=1,n_{t}=n,\mu_{1}=\mu_{2}=\mu_{0}$
szechenyi2016transfer), using Eq.(36) to (39) in Eq.(33), we obtain

$\chi_{kerr}^{p}=-\frac{\sigma_{xy}}{\sigma_{xx}}\Bigl{[}1+\frac{d\sigma_{xx}}{2c\epsilon_{0}}\Bigl{(}1+\frac{\sigma_{xy}^{2}}{\sigma_{xx}^{2}}\Bigl{)}\Bigl{]}^{-1}$
(40)

This shows that that $\chi_{Kerr}^{p}\propto Q$, to lowest order in Q, as
$\sigma_{xy}\propto Q$ and $\sigma_{xx}$ is independent of Q. In the limit of
$d<<\lambda$, the d dependence of $\chi_{kerr}^{p}$ in thin film mWSMs is
insignificant, therefore we can drop out the d dependence in Eq.(40) and the
Kerr angle can be approximated as sonowal2019giant

$\tan(2\Theta_{Kerr})=-2\frac{Re[\sigma_{xy}]Re[\sigma_{xx}]+Im[\sigma_{xy}]Im[\sigma_{xy}]}{\mid\sigma_{xx}\mid^{2}-\mid\sigma_{xy}\mid^{2}}$
(41)

The dependence of the Kerr angle of rotation for normal incidence on a
freestanding tilted thin film m-WSM ($n_{i}=n_{t}\approx 1$ and
$\theta_{i}=\theta_{t}$) is shown in Fig.(33). The Kerr angle is of the order
of a radian but indepndent of monopole charge J. Therefore, Kerr angle can’t
distinguish the m-WSMs in thin film approximation.

###  IV.2 Incidence on surface with Fermi arc states ($\parallel Q$)

In the case, light propagates($\hat{\textbf{n}}$) perpendicular to the z axis
and the polarization of the incident electric field in the y-z(or x-z) plane.
Consequently, the polarization rotating off diagonal reflection coefficients
is

$r_{sp}\propto\sigma_{zy}=0\ or\ r_{sp}\propto\sigma_{zx}=0$ (42)

Consequently, there is no Kerr rotation. This offers an optical probe to
distinguish the surfaces of a m-WSM which host Fermi arc surface states.

##  V Kerr Rotation in semi-infinite multi-WSMs

We now focus our study on the polarization rotation due to a semi-infinite
slab of a m-WSMs. The Maxwell equations are modified by an axoinic term inside
m-WSMs. In TR broken m-WSms, the axionic field
$\theta(\textbf{r},t)=2\textbf{Q}\cdot\textbf{r}$ leads to

$\displaystyle\textbf{D}=\epsilon\textbf{E}+c\epsilon_{0}\alpha_{F}\theta\textbf{B}/\pi$
(43)
$\displaystyle\textbf{H}=\textbf{B}/\mu_{P}-c\epsilon_{0}\alpha_{F}\theta\textbf{E}/\pi$
(44)

Here $\epsilon$ and $\mu_{P}$ are dielectric and permeability tensors
respectively.

Accordingly, wave equation for the electric field propagation is modified to
be

$\displaystyle\nabla^{2}\textbf{E}-\nabla(\nabla\cdot\textbf{E})$
$\displaystyle=$
$\displaystyle\frac{1}{c^{2}\epsilon_{0}}\frac{\partial(\bf{\sigma}\cdot\bf
E)}{\partial t}+\frac{\epsilon_{b}}{c^{2}}\frac{\partial^{2}\bf{E}}{\partial
t^{2}}$ (45) $\displaystyle+$ $\displaystyle\frac{2\alpha_{F}}{\pi
c}\textbf{Q}\times\frac{\partial\bf E}{\partial t}$

where we have taken $\mu_{P}=\mu_{0}$ and $\epsilon_{b}$ is the static
relative permittivity arising from bound charge polarization. The dispersion
relation of electromagnetic wave is

$k^{2}\textbf{E}-(\bf
q.\textbf{E})\textbf{q}=\frac{\omega^{2}}{c^{2}}\tensor{\varepsilon^{\prime}}\bf
E$ (46)

Again, we need to consider two cases:(1) When the light is incident on the WSM
surface hosting Fermi arc states($\hat{\bf n}\perp\bf Q$) and (2)When the
light is incident on the WSM surface without Fermi arc
states($\hat{n}\parallel\bf Q$). These two cases correspond to the Faraday and
Voigt geometries discussed in the context of magnetic-optic effects in
magnetic systems. In a TR broken m-WSMs, $\bf Q$ acts anologously to
magnetization in a ferromagnet.

###  V.1 Incidence on surface without Fermi arc states(magnetooptic(MO) polar
Kerr effect)

Let us consider a normal incidence of light on a surface without Fermi arcs,
i.e., $\hat{n}\parallel\textbf{ Q}=Q\hat{z}$. Intrduce $k=n\omega/c$, the wave
propagation has following matrix equation visnovsky2018optics

$\displaystyle n^{2}\begin{pmatrix}E_{x}\\\ E_{y}\\\
0\end{pmatrix}=\begin{pmatrix}\epsilon^{\prime}_{xx}&\epsilon^{\prime}_{xy}&0\\\
-\epsilon^{\prime}_{xy}&\epsilon^{\prime}_{yy}&0\\\
0&0&\epsilon^{\prime}_{zz}\end{pmatrix}\begin{pmatrix}E_{x}\\\ E_{y}\\\
E_{z}\end{pmatrix}$ (47)

Here, n represent the complex refractive index and $\epsilon^{\prime}_{ij}$ is
the complex dielectric function including the axion terms. The dielectric
tensor $\varepsilon_{ij}$ are given by

$\varepsilon^{\prime}_{ij}=\varepsilon_{b}\delta_{ij}+\frac{i}{\epsilon_{0}\omega}\sigma_{ij}$
(48)

The digonal dielectric elements are

$\displaystyle\epsilon^{\prime}_{ii}=\epsilon_{b}+\frac{i}{\omega\epsilon_{0}}\sigma_{ii},\
i=x,y,z\ $ (49)

The off-diagonal element is

$\epsilon^{\prime}_{xy}=\frac{i}{\omega\epsilon_{0}}\sigma_{xy}+\frac{2i\alpha_{F}c}{\pi}\frac{Q}{\omega}=\frac{i}{\omega\epsilon_{0}}(\sigma_{xy}+\sigma_{Q})$
(50)

Here $\sigma_{ij}$ denotes complex optical conductivities.
The real and imaginary parts of all componens of modified dielectric tensor
are plotted in Fig.(6), Fig.(7), Fig.(8). Eq.(47) have the following
nontrivial solution for the electromagnetic fields

$n^{2}_{+}=\epsilon^{\prime}_{xx}+i\epsilon^{\prime}_{xy}\ and\
n^{2}_{-}=\epsilon^{\prime}_{xx}-i\epsilon^{\prime}_{xy}$ (51)

Here, $n_{+}$ and $n_{-}$ are refractive indices of the left and right
circularly polarized modes in the m-WSMs gorbar2017anomalous .

The Fresnel reflection coefficients corresponding to these modes are

$r_{\pm}=(1-n_{\pm})/(1+n_{\pm})$ (52)

The dimensionless constant $\chi_{PKE}$ is defined as

$\chi_{PKE}=i\frac{r_{+}-r_{-}}{r_{+}+r_{-}}=i\frac{n_{+}-n_{-}}{n_{+}n_{+}-1}$
(53)

$\chi_{PKE}=i\frac{(\sqrt{\epsilon^{\prime}_{xx}+i\epsilon^{\prime}_{xy}}-\sqrt{\epsilon^{\prime}_{xx}-i\epsilon^{\prime}_{xy}})}{\sqrt{\epsilon^{\prime}_{xx}-i\epsilon^{\prime}_{xy}}\sqrt{\epsilon^{\prime}_{xx}+i\epsilon^{\prime}_{xy}}-1}$
(54)

Figure 9: (a),(b) show real(blue) and imaginary(dotted) parts of $\chi_{PKE}$
for single WSM at $\alpha_{t}=0.2$, $\alpha_{t}=2.0$ respectively.(c),(d) show
real(blue) and imaginary(dotted) parts of $\chi_{PKE}$ for double WSM at
$\alpha_{t}=0.2$, $\alpha_{t}=2.0$ respectively.(e),(f) show real(blue) and
imaginary(dotted) parts of $\chi_{PKE}$ for triple WSM at $\alpha_{t}=0.2$,
$\alpha_{t}=2.0$ respectively.

Figure 10: (a),(b) show the Kerr rotation and Kerr ellipticity respectively
for single WSM at $\alpha_{t}=0.2,2.0$. (c),(d) show the Kerr rotation and
Kerr ellipticity respectively for double WSM at $\alpha_{t}=0.2,2.0$. (e),(f)
show the Kerr rotation and Kerr ellipticity respectively for triple WSM at
$\alpha_{t}=0.2,2.0$.

The MO polar Kerr rotation and corresponding MO polar Kerr ellipticity are

$\displaystyle\tan
2\Theta_{PKE}=\frac{2\mathcal{R}(\chi_{PKE})}{1-\mid\chi_{PKE}\mid^{2}},$
$\displaystyle\sin
2\Psi_{PKE}=\frac{2\mathcal{I}(\chi_{PKE})}{1+\mid\chi_{PKE}\mid^{2}}$ (55)

The MO polar Kerr rotation and corresponding MO polar Kerr ellipticity are
plotted in Fig.(9) and Fig.(10). Thus, Kerr rotation in this configuration
depends on monopole charge J.

###  V.2 Incidence on surface with Fermi arc states(Voigt effect)

Let us consider the propagation of the wave in x-direction. In this case, the
wave equation becomes visnovsky2018optics

$\displaystyle n^{2}\begin{pmatrix}E_{x}\\\ E_{y}\\\
0\end{pmatrix}=\begin{pmatrix}\epsilon^{\prime}_{xx}&\epsilon^{\prime}_{xy}&0\\\
-\epsilon^{\prime}_{xy}&\epsilon^{\prime}_{yy}&0\\\
0&0&\epsilon^{\prime}_{zz}\end{pmatrix}\begin{pmatrix}E_{x}\\\ E_{y}\\\
E_{z}\end{pmatrix}$ (56)

The two solutions are

$n_{\parallel}^{2}=\epsilon^{\prime}_{zz}\ and\
n_{\perp}^{2}=\epsilon^{\prime}_{yy}-\frac{\epsilon^{\prime
2}_{xy}}{\epsilon^{\prime}_{xx}}$ (57)

Figure 11: (a) show the real and imaginary parts of $\chi_{Voigt}$ for single-
WSM at $\alpha_{t}=0.2,2.0$. (b) show the real and imaginary parts of
$\chi_{Voigt}$ for double-WSM at $\alpha_{t}=0.2,2.0$. (c) show the real and
imaginary parts of $\chi_{Voigt}$ for triple-WSM at $\alpha_{t}=0.2,2.0$.

Let $\Theta_{0}$ be an arbitrary angle with respect to $\bf Q$, the electric
field components are

$E_{\parallel}=E_{I}\cos(\Theta_{0}),\ \ E_{\perp}=E_{I}\sin(\Theta_{0}),$
(58)

Then the beam becomes elliptically polarized upon reflection from the mWSM
slab. The correponding dimensionless quantity $\chi_{Voigt}$ is defined as

$\chi_{Voigt}=\frac{r_{\perp}E_{\perp}}{r_{\parallel}E_{\parallel}}=\frac{r_{\perp}}{r_{\parallel}}\tan(\Theta_{0})$
(59)

where

$r_{\parallel/\perp}=\frac{1-n_{\parallel/\perp}}{1+n_{\parallel/\perp}}$ (60)

Therefore

$\chi_{Voigt}=\frac{(1+\sqrt{\epsilon^{\prime}_{xx}-\frac{\epsilon^{\prime}_{xy}}{\epsilon^{\prime}_{xx}}})(\sqrt{\epsilon^{\prime}_{zz}}-1)}{(-1+\sqrt{\epsilon^{\prime}_{xx}-\frac{\epsilon^{\prime}_{xy}}{\epsilon^{\prime}_{xx}}})(\sqrt{\epsilon^{\prime}_{zz}}+1)}\tan(\Theta_{0})$
(61)

Therefore, the relative polarization rotation is

$\delta_{Voigt}=\Theta_{Voigt}-\Theta_{0}$ (62)

The rotation($\Theta_{Voigt}$) and corresponding ellipticity($\Psi_{Voigt}$)
are given by

$\displaystyle\tan
2\Theta_{Voigt}=\frac{2\mathcal{R}(\chi_{Voigt})}{1-\mid\chi_{Voigt}\mid^{2}},$
$\displaystyle\sin
2\Psi_{Voigt}=\frac{2\mathcal{I}(\chi_{Voigt})}{1+\mid\chi_{Voigt}\mid^{2}}$
(63)

Figure 12: (a),(b) show the $\delta_{Voigt}$ and $\Psi_{Voigt}$ for single-WSM
at $\alpha_{t}=0.2,2.0$. (c),(d) show the $\delta_{Voigt}$ and $\Psi_{Voigt}$
for double-WSM at $\alpha_{t}=0.2,2.0$. (a),(b) show the $\delta_{Voigt}$ and
$\Psi_{Voigt}$ for triple-WSM at $\alpha_{t}=0.2,2.0$.

The Voigt rotation and corresponding Voigt ellipticity are plotted in Fig.(11)
and Fig.(12). Thus, the Kerr rotaion in this configuartin also depends on
monopole charge and systems parameters $v_{f}$,$\alpha_{2}$,$\alpha_{3}$.

##  VI Conclusions

We have calculated the full optical conductivity tensor for a mWSMs, which in
turn has made it possible to determine the Kerr rotation. We have found that
the tilt direction plays a dominating role in the determination of the
conductivity. The diagonal components $\sigma_{xx/yy}=J\omega$,
$\sigma_{zz}=\omega^{\frac{2}{J}-1}/J$, while the off-diagonal Hall term
$\sigma_{xy}\propto JQ$ , where Q denotes the Weyl node separation (in the
z-direction) and $\sigma_{yz}=\sigma_{zx}=0$. We have shown that a sizeable
Kerr rotation is possible in both ultra-thin film and bulk m-WSMs and it
depends on various system parameters. The Kerr angle is independent of
monopole charge J for light incident ultra-thin fil of m-WSMs without the
Fermi arc states (parallel to the node separation). In contrast, the Kerr
angle depends on monopole charge in bulk m-WSMs in both Polar Kerr and Viogt
configurations.

##  VII Acknowledgements

We thank Debanand Sa for fruitful discussions.

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