# Magneto-optical Kerr effect in Weyl semimetals with broken inversion and
time-reversal symmetries

Olivier Trépanier  Rémi N. Duchesne  Jérémie J. Boudreault  René Côté
Département de physique and Institut Quantique, Université de Sherbrooke,
Sherbrooke, Québec, Canada J1K 2R1

###### Abstract

The topological nature of the band structure of a Weyl semimetal leads to a
number of unique transport and optical properties. For example, the
description of the propagation of an electromagnetic wave in a Weyl semimetal
with broken time-reversal and inversion symmetry, for example, requires a
modification of the Maxwell equations by the axion field
$\theta\left(\mathbf{r},t\right)=2\mathbf{b}\cdot\mathbf{r}-2b_{0}t,$ where
$2\mathbf{b}$ is the separation in wave vector space between two Weyl nodes of
opposite chiralities and $2\hslash b_{0}$ is their separation in energy. In
this paper, we study theoretically how the axion terms $b_{0}$ and
$\mathbf{b}$ modify the frequency behavior of the Kerr rotation and
ellipticity angles $\theta_{K}\left(\omega\right)$ and
$\psi_{K}\left(\omega\right)$ in a Weyl semimetal. Both the Faraday and Voigt
configurations are considered since they provide different information on the
electronic transitions and plasmon excitation. We derive the Kerr angles
firstly without an external magnetic field where the rotation of the
polarization is only due to the axion terms and secondly in a strong magnetic
field where these terms compete with the gyration effect of the magnetic
field. In this latter case, we concentrate on the ultra-quantum limit where
the Fermi level lies in the chiral Landau level and the Kerr and ellipticity
angles have more complex frequency and magnetic field behaviors.

##  I INTRODUCTION

A Weyl semimetalReview  (WSM) is a three-dimensional topological phase of
matter where pairs of nondegenerate bands cross at isolated points in the
Brillouin zone called ”Weyl nodes”. Near these points, the electronic
dispersion is gapless and linear in wave vector. The electrons in each node
are described by the Weyl equation which is a two-component analog of the
Dirac equation and to each node is associated a chirality index $\chi=\pm 1$
which reflects the topological nature of the band structure. The Nielsen-
Ninomiya theoremNielsen  requires that the number of Weyl points in the
Brillouin zone be even so that Weyl nodes must occur in pairs of opposite
chiralities. Stability of the Weyl nodes requires that time-reversal and/or
inversion symmetry be broken. If both symmetries are broken, the minimal
number of Weyl nodes is two.

The topological nature of the band structure of a WSM leads to a number of
interesting transport and optical properties such as an anomalous Hall
effectAHE , the chiral-magnetic effectCME , Fermi arcsFermiArc , a chiral
anomaly leading to a negative longitudinal negative
magnetoresistanceChiralAnomaly  and to a plasmonSpivak  whose frequency
increases with the magnetic field $B$ as $\sqrt{B}$.

Valuable information on the band structure and topological properties of WSMs
can be obtained by magneto-optical methods such as the magneto-optical Kerr
effect (MOKE)Cheskis  which is a powerful probe of the magnetic and electronic
properties of a material. The MOKE is a noncontact technique that measures the
rotation of the polarization vector and the ellipticity of an electromagnetic
wave reflected from the surface of a material. The rotation is due to the
presence of off-diagonal elements in the dielectric tensor of a material that
are caused by an external magnetic field and, in WSMs, by the axion terms
$b_{0}$ and $\mathbf{b}$ in addition to the magnetic field. The transmission
analog of the MOKE, the Faraday effect, has been shown to be very large in
some WSMsHan2022  and, as we demonstrate in this paper, it is also the case
for the Kerr effect.

In order to study the optical properties of WSMs, one must take into account
the fact that the source terms in Maxwell equations are modified by the axion
field $\theta\left(\mathbf{r},t\right)=2\mathbf{b}\cdot\mathbf{r}-2b_{0}t,$
where $2\hslash\mathbf{b}$ in the separation in momentum space between two
Weyl nodes of opposite chiralities and $2\hslash b_{0}$ is their separation in
energyWilczek ; Wu ; Burkov2012 . The axion terms generate an extra current
$\mathbf{j}=-\frac{e^{2}}{2\pi^{2}\hslash}\mathbf{b}\times\mathbf{E}+\frac{e^{2}}{2\pi^{2}\hslash}b_{0}\mathbf{B}$
in the Ampère-Maxwell equation and an extra density
$\rho=\frac{2\alpha}{\pi}\sqrt{\frac{\varepsilon_{0}}{\mu_{0}}}\mathbf{b}\cdot\mathbf{B}$
in the Gauss law (here, $\alpha=e^{2}/4\pi\varepsilon_{0}\hslash c$ is the
fine-structure constant and $\varepsilon_{0},\mu_{0}$ are the permittivity and
permeability of free space). The first axion term, $\mathbf{b,}$ is present if
time-reversal symmetry is broken and is responsible for the anomalous Hall
effect while the second axion term, $b_{0},$ breaks space-inversion symmetry
and leads to the chiral magnetic effect. Even in the absence of a magnetic
field, a WSM has a gyrotropic property if $\mathbf{b}$ or (to a lesser extent)
$b_{0}$ is non-zero (as we will show, the rotating power of $b_{0}$ is
negligible in comparison with that of $\mathbf{b}$).

The effect of the axion term $\mathbf{b}$ on the Kerr and Faraday rotations
has been studied previouslyKargarian  using a simple, undoped, two-node model
of a WSM with broken time-reversal symmetry and in zero magnetic field. This
work has subsequently been extended to include doping and tilting of the Weyl
cones in Ref. Sonowal2019,  where the authors predicted a giant Kerr angle of
the order of $0.1$ rad for optical frequencies below $10$ THz in both type-I
and type-II thin films of Weyl semimetals. They also found a large Kerr effect
caused by the axion term $\mathbf{b}$ in a bulk WSM. In a previous paper by
one of usParent2020 , it was shown that the MOKE in a Weyl semimetal with
time-reversal symmetry (i.e., $\mathbf{b}=0$) and tilted Weyl cones in a
strong magnetic field is modified in comparison with its behavior in a normal
metal. It was also shown that a valley polarizationBertrand2019  can be
detected in the MOKE signal which also bears a signature of the chiral anomaly
in the form of a large peak at the plasmon frequency in the Kerr angle in the
Voigt configuration. This peak is blueshifted as $\omega\sim\sqrt{B}$ as
expectedSpivak .

In this paper, we study the frequency and magnetic field behaviors of the Kerr
and ellipticity angles $\theta_{K}\left(\omega,B\right)$ and
$\psi_{K}\left(\omega,B\right)$ in a WSM with broken time-reversal and
inversion symmetries with: 1) the axion terms $\mathbf{b}$ and $b_{0}$ alone
(i.e., the natural gyrotropy of the WSM) and 2) in the presence of a strong
magnetic field where these terms compete with the gyrotropy that the magnetic
field induces. We work at $T=0$ K with a two-node model, the simplest possible
in order to get the essential physics. We allow for doping so that, in
conjunction with $b_{0}$, the WSM can be electron or hole doped or at
compensation. The Dirac cones are untilted and the WSM is in equilibrium with
a common Fermi level $e_{F}$ for both nodes.

In a strong magnetic field, the electronic energy is quantized into a set of
negative and positive energy levels $n=0,\pm 1,\pm 2,...$. An important
characteristic of a WSM is the presence of a single linearly dispersive chiral
Landau level (the $n=0$ level) at each node where the electrons disperse in
only one direction which is dictated by the chirality index $\chi$ of the
node. This chiral Landau level modifies the dielectric tensor and so the
optical properties of a WSM. This is especially true in the ultra-quantum
regime where the Fermi level $e_{F}$ lies in this chiral level. The lowest-
energy electronic transitions are then given by the frequencies $\omega_{n,m}$
with $n=-1,m=0$ and $n=0,m=1.$ In this regime, the Kerr angle is particularly
interesting. Indeed, we find a plateau where
$\theta_{K}\left(\omega\right)=\pi/2$ and another one where
$\theta_{K}\left(\omega\right)=0$ when $\omega$ is smaller than the value
$\min\left(\omega_{-1,0},\omega_{0,1}\right)$ i.e., in the Pauli blocked
regime. We also find that for the special case where $b\hslash v_{F}=e_{F}$
for $\mathbf{b}\|\mathbf{B}$ and $b_{0}=0$ ($v_{F}$ is the Dirac-Weyl
velocity), the Kerr angle is exactly zero for
$\omega\in\left[0,\min\left(\omega_{-1,0},\omega_{0,1}\right)\right].$

In a magnetic field, two configurations are usually studiedBalkanski . One is
the Faraday configuration where the incident light propagates along the
direction of the magnetic field $\mathbf{B}$ with its polarization vector
perpendicular to $\mathbf{B}$ and the Voigt configuration where the
propagation is perpendicular to $\mathbf{B}$ but the polarization vector has a
component along $\mathbf{B}.$ The two configurations give different
information on the system capturing, for example, different electronic
transitions. In a WSM, we find that the plasmon mode has a signature in
$\theta_{K}\left(\omega\right)$ in the Voigt configuration only if
$\mathbf{b}\|$ $\mathbf{B}$ while it is signaled in both configurations if
$\mathbf{b}$ is tilted with respect to $\mathbf{B}.$

The remainder of this paper is organized as follows: we derive the dielectric
tensor of a WSM with and without magnetic field in Sections II and III.
Section IV discusses the modification of the Maxwell equations by the axion
terms. We introduce the formalism necessary to compute the Kerr and
ellipticity angles in the Faraday and Voigt configurations in Sec. V. We
briefly review the Kerr rotation for a normal metal in Sec. VI and then
present our numerical results for the frequency dependence of the Kerr angle
in a WSM with and without magnetic field in Secs. VII and VIII. We study the
effect of tilting the vector $\mathbf{b}$ with respect to the magnetic field
in Sec. IX. In Section X, we discuss the magnetic field and $\mathbf{b}$
dependence (for $\mathbf{b}\|\mathbf{B}$) of the Kerr angle in the Faraday
configuration. We conclude in Sec. XI.

##  II DIELECTRIC TENSOR OF A WSM IN ZERO MAGNETIC FIELD

We consider a simple two-node model of a Weyl semimetal (WSM) with broken
space-inversion and time-reversal symmetries so that the two nodes have
opposite chiralities. The nodes are centered at wave vectors
$-\chi\mathbf{b}$, where $\chi=\pm 1$ is the chirality index. The Hamiltonian
of the electron gas in each Weyl node is given by

$h_{\chi}\left(\mathbf{k}\right)=\chi\hslash
v_{F}\mathbf{k}\cdot\mathbf{\sigma}+\chi\hslash b_{0}I_{2},$ (1)

where $v_{F}$ is the Dirac-Weyl velocity, $\mathbf{k}$ is the wave vector
measured from the position of the Weyl node, $\mathbf{\sigma}$ is the vector
of Pauli matrices defined in the basis of the two electronic bands that cross,
$I_{2}$ is the $2\times 2$ unit matrix, $2\hslash b_{0}$ is the separation in
energy of the two Dirac points, and $2\hslash\mathbf{b}$ is their separation
in momentum space. The former term breaks the space-inversion symmetry while
the latter breaks time-reversal symmetry.

The energy spectrum is given by

$E_{\chi,s}\left(\mathbf{k}\right)=s\hslash
v_{F}\left|\mathbf{k}\right|+\chi\hslash b_{0},$ (2)

where $s=\pm 1$ is the band index i.e., the upper ($s=+$) or lower part
($s=-$) of the Dirac cone. The corresponding eigenvectors are of the form
$v_{\chi,\mathbf{k},s}\left(\mathbf{r}\right)=\frac{1}{\sqrt{V}}\eta_{\chi,s}\left(\mathbf{k}\right)e^{i\mathbf{k}\cdot\mathbf{r}}$,
where $V$ is the volume of the WSM and the spinors
$\eta_{\chi,s}\left(\mathbf{k}\right)$ are given by

$\displaystyle\eta_{+,+}\left(\mathbf{k}\right)$ $\displaystyle=$
$\displaystyle\eta_{-,-}\left(\mathbf{k}\right)=\left(\begin{array}[]{c}e^{-i\varphi_{\mathbf{k}}/2}\cos\left(\frac{\theta_{\mathbf{k}}}{2}\right)\\\
e^{i\varphi_{\mathbf{k}}/2}\sin\left(\frac{\theta_{\mathbf{k}}}{2}\right)\end{array}\right),$
(5) $\displaystyle\eta_{+,-}\left(\mathbf{k}\right)$ $\displaystyle=$
$\displaystyle\eta_{-,+}\left(\mathbf{k}\right)=\left(\begin{array}[]{c}-e^{-i\varphi_{\mathbf{k}}/2}\sin\left(\frac{\theta_{\mathbf{k}}}{2}\right)\\\
e^{i\varphi_{\mathbf{k}}/2}\cos\left(\frac{\theta_{\mathbf{k}}}{2}\right)\end{array}\right),$
(8)

where $\theta_{\mathbf{k}},\varphi_{\mathbf{k}}$ are respectively the polar
and azimuthal angles of the wave vector $\mathbf{k}.$

To derive the many-body Hamiltonian, we define the field operators

$\Psi_{\chi}\left(\mathbf{r}\right)=\frac{1}{\sqrt{V}}\sum_{s,\mathbf{k}}e^{i\mathbf{k}\cdot\mathbf{r}}\eta_{\chi,s}\left(\mathbf{k}\right)c_{\chi,s,\mathbf{k}}.$
(9)

The operators $c_{\chi,s,\mathbf{k}},c_{\chi,s,\mathbf{k}}^{{\dagger}}$
annihilate or create an electron in state
$\left(\chi,s,\mathbf{k}\right).$They satisfy the commutation relation
$\left\\{c_{\chi,s,\mathbf{k}},c_{\chi^{\prime},s^{\prime},\mathbf{k}^{\prime}}^{{\dagger}}\right\\}=\delta_{\chi,\chi^{\prime}}\delta_{s,s^{\prime}}\delta_{\mathbf{k},\mathbf{k}^{\prime}}.$
The second-quantized Hamiltonian for each node is then

$\displaystyle H_{\chi}$ $\displaystyle=$ $\displaystyle\int
d^{3}r\Psi_{\chi}^{{\dagger}}\left(\mathbf{r}\right)h_{\chi}\left(\mathbf{k}\right)\Psi_{\chi}\left(\mathbf{r}\right)$
$\displaystyle=$
$\displaystyle\sum_{s,\mathbf{k}}E_{\chi,s}\left(\mathbf{k}\right)c_{\chi,s,\mathbf{k}}^{{\dagger}}c_{\chi,s,\mathbf{k}}.$

The current operator is obtained, after making the Peierls substitution
$h_{\chi}\left(\mathbf{k}\right)\rightarrow
h_{\chi}\left(\mathbf{k}+e\mathbf{A}/\hslash\right)$ in the hamiltonian ($e>0$
for an electron and $\mathbf{A}_{e}$ is the vector potential of an external
electromagnetic field), by taking the derivative

$\mathbf{j}_{\chi}=-\frac{\delta h_{\chi}}{\delta\mathbf{A}_{e}}=-\chi
ev_{F}\mathbf{\sigma,}$ (11)

so that the total current at each node is given by

$\displaystyle\mathbf{J}_{\chi}$ $\displaystyle=$ $\displaystyle\int
d^{3}r\Psi_{\chi}^{{\dagger}}\left(\mathbf{r}\right)\mathbf{j}_{\chi}\Psi_{\chi}\left(\mathbf{r}\right)$
$\displaystyle=$ $\displaystyle-e\chi
v_{F}\sum_{s,s^{\prime},\mathbf{k}}\left[\eta_{\chi,s}^{{\dagger}}\left(\mathbf{k}\right)\mathbf{\sigma}\eta_{\chi,s^{\prime}}\left(\mathbf{k}\right)\right]c_{\chi,s,\mathbf{k}}^{{\dagger}}c_{\chi,s^{\prime},\mathbf{k}}.$

The current operator enters the definition of the retarded current-current
response tensor
$\overleftrightarrow{\mathbf{\chi}}_{\chi}^{R}\left(\omega\right)$ which is
obtained by taking the analytic continuation
$i\Omega_{n}\rightarrow\omega+i\delta$ of the two-particle Matsubara Green’s
function

$\overleftrightarrow{\mathbf{\chi}}_{\chi}\left(i\Omega_{n}\right)=-\frac{1}{V\hslash}\int_{0}^{\beta\hslash}d\tau\left\langle
T\mathbf{J}_{\chi}\left(\tau\right)\mathbf{J}_{\chi}\left(0\right)\right\rangle
e^{i\Omega_{n}\tau},$ (13)

where $\tau$ is the imaginary time, $\Omega_{n}$ is a bosonic Matsubara
frequency and $\beta=1/k_{B}T$ with $k_{B}$ the Boltzmann constant. The
optical conductivity tensor can then be calculated from the expression

$\overleftrightarrow{\mathbf{\sigma}}\left(\omega\right)=\frac{i}{\omega+i\delta}\sum_{\chi}\left[\overleftrightarrow{\mathbf{\chi}}_{\chi}\left(\omega\right)-\overleftrightarrow{\mathbf{\chi}}_{\chi}\left(0\right)\right].$
(14)

In the absence of an external static magnetic field and for the simple
Hamiltonian given in Eq. (1),
$\sigma_{i,j}\left(\omega\right)=\delta_{i,j}\sigma\left(\omega\right)$ so
that we only need to compute the response function

$\displaystyle\chi_{\chi}\left(\omega\right)$ $\displaystyle=$
$\displaystyle-\frac{e^{2}v_{F}^{2}}{\hslash
V}\sum_{s,s^{\prime},\mathbf{k}}\left|\eta_{\chi,s}^{{\dagger}}\left(\mathbf{k}\right)\sigma^{\left(\alpha\right)}\eta_{\chi,s^{\prime}}\left(\mathbf{k}\right)\right|^{2}$
$\displaystyle\times\frac{\left\langle
n_{\chi,s^{\prime}}\left(\mathbf{k}\right)\right\rangle-\left\langle
n_{\chi,s}\left(\mathbf{k}\right)\right\rangle}{\omega+i\delta-\left(E_{\chi,s^{\prime}}\left(\mathbf{k}\right)-E_{\chi,s}\left(\mathbf{k}\right)\right)/\hslash},$

where $\left\langle
n_{\chi,s}\left(\mathbf{k}\right)\right\rangle=\left\langle
c_{\chi,s,\mathbf{k}}^{{\dagger}}c_{\chi,s,\mathbf{k}}\right\rangle$ is the
average occupation of the state $\left(\chi,s,\mathbf{k}\right)$ at zero
temperature and $\alpha$ can be any of $x,y,z.$

The relative dielectric tensor is

$\varepsilon\left(\omega\right)=\varepsilon_{b}+\frac{i\sigma\left(\omega\right)}{\varepsilon_{0}\left(\omega+i\delta\right)},$
(16)

where $\varepsilon_{b}$ is the bound charge contribution. We take
$\varepsilon_{b}=1$ in our calculations, but comment later on the effect of a
larger $\varepsilon_{b}$ on the Kerr rotation. We assume that the two nodes
are at a thermodynamical equilibrium with a Fermi level $e_{F}$ so that the
Fermi wave vector $k_{F,\chi}$ and the Fermi level $e_{F,\chi}$ for each node
are related by

$e_{F,\chi}=\hslash v_{F}k_{F,\chi}=e_{F}-\chi\hslash b_{0}.$ (17)

The optical conductivity has contributions from the interband and intraband
transitions. Both contributions have been calculated in parts in a number of
papersKargarian ; Sonowal2019 ; Carbotte2014 ; Carbotte2016 ; Carbotte2018 ;
Carbotte2021 . For completeness, we derive the intraband contribution in
Appendix A. In the absence of magnetic field and in the continuum model, the
relative dielectric function is

$\displaystyle\varepsilon\left(\omega\right)$ $\displaystyle=$ $\displaystyle
1$
$\displaystyle+\gamma\sum_{\chi}\left[i\theta\left(\omega-2v_{F}k_{F,\chi}\right)+\frac{1}{\pi}\ln\left(\left|\frac{\omega^{2}-4v_{F}^{2}k_{c}^{2}}{\omega^{2}-4v_{F}^{2}k_{F,\chi}^{2}}\right|\right)\right]$
$\displaystyle+\gamma\sum_{\chi}\frac{4}{\pi\hslash^{2}\omega}\frac{i\tau}{1+\omega^{2}\tau^{2}}\left(e_{F,\chi}^{2}+\frac{1}{3}\hslash^{2}\omega^{2}+\frac{\hslash^{2}}{4\tau^{2}}\right)$
$\displaystyle-\gamma\sum_{\chi}\frac{4}{\pi\hslash^{2}}\frac{\tau^{2}}{1+\omega^{2}\tau^{2}}\left(e_{F,\chi}^{2}-\frac{\hslash^{2}}{12\tau^{2}}\right),$

where $\gamma=\frac{\alpha}{6v_{F}/c},k_{c}$ is a cutoff wave vector which is
related to the applicability of the linear dispersion relation and $\tau$ is
the scattering time that enters in the calculation of the intraband
conductivity (see Appendix A). The last two lines in Eq. (II) are intraband
contributions to the dielectric function.

##  III DIELECTRIC TENSOR OF A WSM IN A MAGNETIC FIELD

In a magnetic field
$\mathbf{B}=\mathbf{\nabla}\times\mathbf{A=}B\widehat{\mathbf{z}}$ and in the
Landau gauge $\mathbf{A}=\left(0,Bx,0\right)$, the single-particle Hamiltonian
becomes, after making the Peierls substitution $\mathbf{k\rightarrow
k}+e\mathbf{A}/\hslash,$

$h_{\chi}\left(\mathbf{k}\right)=\chi\hslash b_{0}I_{2}+\chi\hslash
v_{F}\left(\begin{array}[]{cc}k_{z}&\frac{\sqrt{2}}{\ell}a\\\
\frac{\sqrt{2}}{\ell}a^{{\dagger}}&-k_{z}\end{array}\right),$ (19)

where $I_{2}$ is the $2\times 2$ unit matrix and the ladder operators are
defined by

$\displaystyle a$ $\displaystyle=$
$\displaystyle\frac{\ell}{\sqrt{2}\hslash}\left(P_{x}-iP_{y}\right),$ (20)
$\displaystyle a^{{\dagger}}$ $\displaystyle=$
$\displaystyle\frac{\ell}{\sqrt{2}\hslash}\left(P_{x}+iP_{y}\right),$ (21)

where $\mathbf{P}\equiv\hslash\mathbf{k}+e\mathbf{A.}$They obey the
commutation relation $\left[a,a^{{\dagger}}\right]=1.$ Here
$h_{\chi}\left(\mathbf{k}\right)$ is written in the same basis as that used in
Eq. (1).

The energy spectrum of a Weyl node now consists of a chiral Landau level
(index $n=0$) which disperses in one direction only according to

$E_{\chi,0}\left(k\right)=\chi\frac{\hslash
v_{F}}{\ell}\left(-k\ell+\frac{b_{0}\ell}{v_{F}}\right)$ (22)

and of a set of positive and negative energy Landau levels $n=1,2,3,...$ with
dispersion

$E_{\chi,n>0,s}\left(k\right)=\frac{\hslash
v_{F}}{\ell}\left(s\sqrt{k^{2}\ell^{2}+2n}+\chi\frac{b_{0}\ell}{v_{F}}\right).$
(23)

We use the index $s=+1$($-1$) for the Landau levels that originate from the
conduction (valence) band of the Dirac cone and take $\hslash k$ as the
momentum of the electron in the direction of the magnetic field. At some
places in this paper, we use the alternative notation $n<0$ for the Landau
levels of the valence band. The corresponding eigenvectors are

$v_{{}_{\chi,n,X,s}}\left(k,\mathbf{r}\right)=\frac{1}{\sqrt{L_{z}}}e^{ikz}\eta_{\chi,n,X,s}\left(k,\mathbf{r}\right),$
(24)

where $L_{z}$ is the length of the WSM in the $z$ direction and $\mathbf{r}$ a
two-dimensional vector in the plane perpendicular to $z.$ The spinors are
given by

$\eta_{\chi,0,X}\left(k,\mathbf{r}\right)=\left(\begin{array}[]{c}0\\\
h_{0,X}\left(\mathbf{r}\right)\end{array}\right),$ (25)

for the chiral level and by

$\eta_{\chi,n,X,s}\left(k,\mathbf{r}\right)=\left(\begin{array}[]{c}u_{\chi,n,s}\left(k\right)h_{n-1,X}\left(\mathbf{r}\right)\\\
v_{\chi,n,s}\left(k\right)h_{n,X}\left(\mathbf{r}\right)\end{array}\right)$
(26)

for the others. We set $h_{n,X}\left(\mathbf{r}\right)=0$ when $n<0.$ The
$u^{\prime}s$ and $v^{\prime}s$ obey the normalization condition
$\left|u_{\chi,n,s}\left(k\right)\right|^{2}+\left|v_{\chi,n,s}\left(k\right)\right|^{2}=1$
and are given by

$\left(\begin{array}[]{c}u_{\chi,n,s}\left(k\right)\\\
v_{\chi,n,s}\left(k\right)\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}-\chi
si\left(\sqrt{1+\chi s\frac{k\ell}{e_{n}\left(k\right)}}\right)\\\
\sqrt{1-\chi s\frac{k\ell}{e_{n}\left(k\right)}}\end{array}\right),$ (27)

where

$e_{n}\left(k\right)=\sqrt{k^{2}\ell^{2}+2n}.$ (28)

In Eq. (26), the functions
$h_{n,X}\left(\mathbf{r}\right)=\frac{1}{\sqrt{L_{y}}}\varphi_{n}\left(x-X\right)e^{-iXy/\ell^{2}}$
are the Landau level wave functions in the Landau gauge with $X$ the guiding-
center index and the $\varphi_{n}\left(x\right)^{\prime}s$ are the wave
functions of the one-dimensional harmonic oscillator. Each Landau level
$\left(\chi,n,s,k\right)$ has degeneracy $N_{\varphi}=S/2\pi\ell^{2}$ where
$\ell=\sqrt{\hslash/eB}$ is the magnetic length and $S=L_{x}L_{y}$ is the area
of the WSM perpendicular to the magnetic field.

The field operator can now be written as

$\Psi_{\chi}\left(\mathbf{r},z\right)=\frac{1}{\sqrt{L_{z}}}\sum_{n,s,X,k}e^{ikz}\eta_{\chi,n,X,s}\left(k,\mathbf{r}\right)c_{\chi,n,X,s,k}$
(29)

and the second quantized form of the current operator is

$\mathbf{J}_{\chi}=\sum_{n,s,n^{\prime},s^{\prime},k}\sum_{X}\mathbf{\Gamma}_{\chi;ns;n^{\prime}s^{\prime}}\left(k\right)c_{\chi,n,X,s,k}^{{\dagger}}c_{\chi,n^{\prime},X,s^{\prime},k},$
(30)

with the matrix elements defined by

$\displaystyle\Gamma_{\chi;ns;n^{\prime}s^{\prime}}^{\left(x\right)}\left(k\right)$
$\displaystyle=$ $\displaystyle-e\chi
v_{F}\left[u_{\chi,n,s}^{\ast}\left(k\right)v_{\chi,n-1,s^{\prime}}\left(k\right)\delta_{n-1,n^{\prime}}\right.$
$\displaystyle\left.+v_{\chi,n,s}^{\ast}\left(k\right)u_{\chi,n+1,s^{\prime}}\left(k\right)\delta_{n,n^{\prime}-1}\right].$
$\displaystyle\Gamma_{\chi;ns;n^{\prime}s^{\prime}}^{\left(y\right)}\left(k\right)$
$\displaystyle=$ $\displaystyle ie\chi
v_{F}\left[u_{\chi,n,s}^{\ast}\left(k\right)v_{\chi,n-1,s^{\prime}}\left(k\right)\delta_{n-1,n^{\prime}}\right.$
$\displaystyle\left.-v_{\chi,n,s}^{\ast}\left(k\right)u_{\chi,n+1,s^{\prime}}\left(k\right)\delta_{n,n^{\prime}-1}\right],$
$\displaystyle\Gamma_{\chi;ns;n^{\prime}s^{\prime}}^{\left(z\right)}\left(k\right)$
$\displaystyle=$ $\displaystyle-e\chi
v_{F}\left[u_{\chi,n,s}^{\ast}\left(k\right)u_{\chi,n,s^{\prime}}\left(k\right)\right.$
$\displaystyle\left.-v_{\chi,n,s,}^{\ast}\left(k\right)v_{\chi,n,s^{\prime}}\left(k\right)\right]\delta_{n,n^{\prime}}.$

The current-current response tensor defined in Eq. (13) becomes

$\displaystyle\chi_{\chi,\left(\alpha,\beta\right)}\left(\omega\right)$
$\displaystyle=$
$\displaystyle\zeta\sum_{\begin{subarray}{c}k,n,n^{\prime},\\\
s,s^{\prime}\end{subarray}}\Gamma_{\chi;n,s;n^{\prime},s^{\prime}}^{\left(\alpha\right)}\left(k\right)\Gamma_{\chi;n^{\prime},s^{\prime};n,s}^{\left(\beta\right)}\left(k\right)$
$\displaystyle\times\frac{\left\langle
n_{\chi,n^{\prime},s^{\prime}}\left(k\right)\right\rangle-\left\langle
n_{\chi,n,s}\left(k\right)\right\rangle}{\omega+i\delta-\left(E_{\chi,n^{\prime}s^{\prime}}\left(k\right)-E_{\chi,n,s}\left(k\right)\right)/\hslash},$

where $\zeta=-1/2\pi\ell^{2}\hslash L_{z},$ with $L_{z}$ the width of the WSM
in the $z$ direction. The filling factor for each Landau level is given by

$\left\langle
n_{\chi,n,s}\left(k\right)\right\rangle=\frac{1}{N_{\varphi}}\sum_{X}\left\langle
c_{\chi,n,X,s,k}^{{\dagger}}c_{\chi,n,X,s,k}\right\rangle.$ (35)

The conductivity tensor is obtained from Eq. (14) and the relative dielectric
tensor from Eq. (16).

The response functions $\chi_{\chi,\left(\alpha,z\right)}$and
$\chi_{\chi,\left(z,\alpha\right)}$ with $\alpha=x,y$ are zero because of the
Kronecker deltas in the matrix elements
$\mathbf{\Gamma}_{\chi;ns;n^{\prime}s^{\prime}}.$ Moreover, for the interband
part of the response functions (i.e., $s\neq s^{\prime}$), we have the
symmetry relations $\chi_{\chi,\left(x,x\right)}=\chi_{\chi,\left(y,y\right)}$
and $\chi_{\chi,\left(x,y\right)}=-\chi_{\chi,\left(y,x\right)}.$ The optical
selections rules imposed by the matrix elements are such that only the dipolar
transitions $\left|n\right|=\left|n\right|\pm 1$ (here $n=0,\pm 1,\pm 2,...$)
are permitted in $\chi_{\chi,\left(\alpha,\beta\right)}$ with
$\alpha_{,}\beta=x,y$ while only the transitions $-n\rightarrow n$ are
permitted in $\chi_{\chi,\left(z,z\right)}.$ (A tilt of a Weyl cone allows for
a much richer interband spectrumParent2020 ; Goerbig2016 ). We limit our study
to the ultra-quantum regime where the Fermi level lies in the chiral level. In
this case, intraband transitions can only occur in level $n=0$ and they
contribute only to $\chi_{\chi,\left(z,z\right)}.$ The calculation of the
dynamic conductivity tensor including inter- and intra-Landau-level
contributions is summarized in Appendix B.

##  IV MAXWELL EQUATIONS WITH THE AXION TERMS

To compute the optical properties of a WSM, we must take into account the
modification of the Maxwell equations by the topological axion term
$\theta\left(\mathbf{r},t\right)=2\mathbf{b}\cdot\mathbf{r}-2b_{0}t.$ The
modified equations areBurkov2012 :

$\displaystyle\nabla\cdot\mathbf{D}$ $\displaystyle=$
$\displaystyle\rho_{f}+\kappa\sqrt{\frac{\varepsilon_{0}}{\mu_{0}}}\mathbf{b}\cdot\mathbf{B,}$
(36) $\displaystyle\nabla\cdot\mathbf{B}$ $\displaystyle=$ $\displaystyle 0,$
(37) $\displaystyle\nabla\times\mathbf{E}$ $\displaystyle=$
$\displaystyle-\frac{\partial\mathbf{B}}{\partial t},$ (38)
$\displaystyle\nabla\times\mathbf{H}$ $\displaystyle=$
$\displaystyle\frac{\partial\mathbf{D}}{\partial
t}+\mathbf{J}_{f}-\kappa\sqrt{\frac{\varepsilon_{0}}{\mu_{0}}}\left(\mathbf{b}\times\mathbf{E}-gb_{0}\mathbf{B}\right)\mathbf{,}$
(39)

where $\kappa=2\alpha/\pi$ with $\alpha$ the fine-structure constant. In these
equations, $\rho_{f}$ is the free (or induced) charge density and
$\mathbf{J}_{f}=\overleftrightarrow{\mathbf{\sigma}}\cdot\mathbf{E}$ is the
induced current density with $\overleftrightarrow{\mathbf{\sigma}}$ the
conductivity tensor for the two-node model calculated in appendices A and B.
In this paper, we assume that the relative permeability $\mu_{r}=\mu/\mu_{0}$
is unity.

In the context of the chiral magnetic effect which occurs in the
nonequilibrium condition where the chiral chemical potentials of both nodes
are different, the last term in the Ampere-Maxwell equation has $g=1.$ In this
paper, however, we have a different situation since we assume that the two
nodes are at equilibrium and that the two Dirac points are separated in energy
by $2\hslash b_{0}.$ The extra current $gb_{0}\mathbf{B}$ is then related to
the natural optical activity of the WSM and it follows that $g=1/3$ in the
dynamic case and $g=0$ in the static caseMaPesin2015, ; Zhong2016, . The
gyrotropic effect of $b_{0}$ is, as we will show, very small in comparison
with that of axion term $\mathbf{b}$ so that, in the THz frequency range where
we study the Kerr rotation, taking $g=1$ or $g=1/3$ or even $g=0$ does not
make any numerical difference to the results presented in this paper. In the
small frequency limit, however, keeping $g$ finite as $\omega\rightarrow 0$
leads to some strange results as we will point out at some places in this
paper.

The wave equation in the presence of the axion terms can be written in the
matrix form $M_{i,j}E_{j}=0$ where the components of the matrix $M$ are given
by

$M_{ij}=-c^{2}\left(q^{2}\delta_{ij}-q_{i}q_{j}\right)+\omega^{2}\widetilde{\varepsilon}_{ij},$
(40)

where $\delta_{ij}$ is the Kronecker delta. We have defined the tensor
$\widetilde{\varepsilon}_{ij}$ by

$\widetilde{\varepsilon}_{ij}=\varepsilon_{ij}+\frac{i}{\omega}c\kappa\varepsilon_{ijk}\left(b_{k}-\frac{gb_{0}}{\omega}q_{k}\right),$
(41)

with $\varepsilon_{ijk}$ the antisymmetric Levi-Civita tensor and the
components of the dielectric tensor $\varepsilon_{ij}$ are given by

$\varepsilon_{ij}=\varepsilon_{b}\delta_{ij}+\frac{i}{\varepsilon_{0}\omega}\sigma_{ij},$
(42)

where the electrons in each node are described by the hamiltonian of Eq. (1)
or Eq. (19) which contains the axion term $b_{0}$ but not $\mathbf{b}.$ In Eq.
(42), $\varepsilon_{b}$ is the effective background dielectric constant. We
take $\varepsilon_{b}=1,$ but it must be kept in mind that $\varepsilon_{b}$
can be substantially larger in WSMs. For example $\varepsilon_{b}=6.2$ in
type-I WSM TaAsKotov2016 .

The dispersion relations and polarizations of the electromagnetic modes are
obtained from the equations $\det\left[M\right]=0$ and $M_{i,j}E_{j}=0$
respectively. We stress once again that in the continuum model that we use,
the axion term $b_{0}$ is present in the dielectric tensor
$\overleftrightarrow{\mathbf{\varepsilon}}$ and in the Maxwell equations but
the axion term $\mathbf{b}$ appears in the modified Maxwell equations but not
in $\overleftrightarrow{\mathbf{\varepsilon}}$. The induced charge density, if
any, is given by

$\rho_{f}=i\varepsilon_{0}\mathbf{q}\cdot\left(\widetilde{\mathbf{\varepsilon}}\mathbf{\cdot
E}\right).$ (43)

##  V KERR AND ELLIPTICITY ANGLES

In this section, we derive the formulas necessary to compute the Kerr and
ellipticity angles. The external magnetic field, when present, is set along
the $z$ direction and we restrict our analysis to the case where the axion
term $\mathbf{b}=\mathbf{b}_{\|}=b_{z}\widehat{\mathbf{z}}$. We study the
effect of
$\mathbf{b}_{\bot}=b_{x}\widehat{\mathbf{x}}+b_{y}\widehat{\mathbf{y}}$ in
Sec. IX.

We consider a linearly polarized electromagnetic wave with amplitude $E_{0},$
polarization $\mathbf{e}_{0}$ and wave vector $\mathbf{q}_{0}$ arriving at
normal incidence on the surface of a semi-infinite WSM either along the $z$
axis with $\mathbf{e}_{0}\bot\mathbf{B}$ (the Faraday configuration) or along
the $x$ axis with $\mathbf{e}_{0}\cdot\mathbf{B\neq}0$ (the Voigt
configuration)Balkanski . If $\mathbf{B}=0,$ we use the name Faraday
configuration for $\mathbf{e}_{0}\bot\mathbf{b}$ and Voigt configuration for
$\mathbf{e}_{0}\cdot\mathbf{b\neq}0$

###  V.1 Faraday configuration

In the Faraday configuration, the vacuum (medium 1)-WSM (medium 2) interface
is at $z=0.$ The electromagnetic field in medium 1 (with dispersion
$q_{0}=\omega/c$) is given by

$\displaystyle\mathbf{E}_{1}\left(z\mathbf{,}t\right)$ $\displaystyle=$
$\displaystyle E_{0}\mathbf{e}_{0}e^{iq_{0}z}e^{-i\omega
t}+E_{0}r\mathbf{e}_{r}e^{-iq_{0}z}e^{-i\omega t},$ (44)
$\displaystyle\mathbf{B}_{1}\left(z\mathbf{,}t\right)$ $\displaystyle=$
$\displaystyle\frac{q_{0}}{\omega}E_{0}\left(\widehat{\mathbf{z}}\times\mathbf{e}_{0}\right)e^{iq_{0}z}e^{-i\omega
t}$
$\displaystyle-\frac{q_{0}}{\omega}E_{0}r\left(\widehat{\mathbf{z}}\times\mathbf{e}_{r}\right)e^{-iq_{0}z}e^{-i\omega
t},$

while in medium 2, the two transmitted waves with wave vector $q_{1}$ and
$q_{2}$ are given by

$\displaystyle\mathbf{E}_{2}\left(z\mathbf{,}t\right)$ $\displaystyle=$
$\displaystyle E_{0}t_{1}\mathbf{e}_{1}e^{iq_{1}z}e^{-i\omega
t}+E_{0}t_{2}\mathbf{e}_{2}e^{iq_{2}z}e^{-i\omega t},$ (46)
$\displaystyle\mathbf{B}_{2}\left(z\mathbf{,}t\right)$ $\displaystyle=$
$\displaystyle\frac{q_{1}}{\omega}E_{0}t_{1}\left(\widehat{\mathbf{z}}\times\mathbf{e}_{1}\right)e^{iq_{1}z}e^{-i\omega
t}$
$\displaystyle+\frac{q_{2}}{\omega}E_{0}t_{2}\left(\widehat{\mathbf{z}}\times\mathbf{e}_{2}\right)e^{iq_{2}z}e^{-i\omega
t}.$

In these equations,
$\mathbf{e}_{0},\mathbf{e}_{r},\mathbf{e}_{1},\mathbf{e}_{2}$ are the
(complex) polarization vectors for the incoming, reflected and transmitted
waves respectively and $r,t_{1},t_{2}$ are the reflection and transmission
factors.

At the surface of the WSM, the electric and magnetic fields must satisfy the
boundary conditions

$\displaystyle\left(\mathbf{D}_{1}-\mathbf{D}_{2}\right)\cdot\widehat{\mathbf{n}}$
$\displaystyle=$ $\displaystyle\rho_{free},$ (48)
$\displaystyle\mathbf{E}_{1,\|}-\mathbf{E}_{2,\|}$ $\displaystyle=$
$\displaystyle 0,$ (49)
$\displaystyle\left(\mathbf{B}_{1}-\mathbf{B}_{2}\right)\cdot\widehat{\mathbf{n}}$
$\displaystyle=$ $\displaystyle 0,$ (50)
$\displaystyle\mathbf{B}_{1,\|}-\mathbf{B}_{2,\|}$ $\displaystyle=$
$\displaystyle 0,$ (51)

where the unit vector $\widehat{\mathbf{n}}$ points from medium $2$ to medium
$1.$

Defining the complex polarization vectors by

$\mathbf{e}_{j}=\alpha_{j}\widehat{\mathbf{x}}+\beta_{j}\widehat{\mathbf{y}},$
(52)

with $j=0,r,1,2$ (a polarization component along $z$, if any, does not
contribute to the boundary conditions), we get from Eqs. (48)-(51) the system
of equations

$\left(\begin{array}[]{c}r\alpha_{r}\\\ r\beta_{r}\\\ t_{1}\\\
t_{2}\end{array}\right)=\left(\begin{array}[]{cccc}1&0&-\alpha_{1}&-\alpha_{2}\\\
0&1&-\beta_{1}&-\beta_{2}\\\
0&1&\frac{q_{1}}{q_{i}}\beta_{1}&\frac{q_{2}}{q_{i}}\beta_{2}\\\
-1&0&-\frac{q_{1}}{q_{i}}\alpha_{1}&-\frac{q_{2}}{q_{i}}\alpha_{2}\end{array}\right)^{-1}\left(\begin{array}[]{c}-\alpha_{0}\\\
-\beta_{0}\\\ \beta_{0}\\\ -\alpha_{0}\end{array}\right).$ (53)

In the Faraday configuration, the WSM has rotational symmetry along the $z$
axis since both $\mathbf{b}$ and $\mathbf{B}$ are along that axis, and we can
take without loss of generality
$\mathbf{e}_{0}=\widehat{\mathbf{x}}\allowbreak$ for the incident linear
polarization. This choice implies $\alpha_{i}=1,\beta_{i}=0$ and the
reflection and transmission factors are then given by

$r\alpha_{r}=\frac{\left(1-\frac{q_{1}}{q_{0}}\right)\left(1+\frac{q_{2}}{q_{0}}\right)\alpha_{1}\beta_{2}-\left(1+\frac{q_{1}}{q_{0}}\right)\left(1-\frac{q_{2}}{q_{0}}\right)\alpha_{2}\beta_{1}}{\left(\frac{q_{1}}{q_{0}}+1\right)\left(\frac{q_{2}}{q_{0}}+1\right)\left(\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}\right)}$
(54)

and

$r\beta_{r}=-2\frac{\left(\frac{q_{1}}{q_{0}}-\frac{q_{2}}{q_{0}}\right)\beta_{1}\beta_{2}}{\left(\frac{q_{1}}{q_{0}}+1\right)\left(\frac{q_{2}}{q_{0}}+1\right)\left(\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}\right)}.$
(55)

With $b_{0},b_{z}\neq 0$, the matrix $M$ takes the block-diagonal form

$M=\left(\begin{array}[]{ccc}\omega^{2}\widetilde{\varepsilon}_{xx}-c^{2}q^{2}&\omega^{2}\widetilde{\varepsilon}_{xy}&0\\\
-\omega^{2}\widetilde{\varepsilon}_{xy}&\omega^{2}\widetilde{\varepsilon}_{xx}-c^{2}q^{2}&0\\\
0&0&\omega^{2}\widetilde{\varepsilon}_{zz}\end{array}\right).$ (56)

The dispersion relations of the electromagnetic waves are then given by (the
same dispersions are obtained in Refs. Sa2021, ; Deng2021, )

$\displaystyle q_{1,\pm}$ $\displaystyle=$ $\displaystyle-\frac{g\kappa
b_{0}}{2c}\pm\sqrt{\left(\frac{g\kappa
b_{0}}{2c}\right)^{2}+\frac{\omega^{2}\xi_{-}}{c^{2}}},$ (57) $\displaystyle
q_{2,\pm}$ $\displaystyle=$ $\displaystyle\frac{g\kappa
b_{0}}{2c}\pm\sqrt{\left(\frac{g\kappa
b_{0}}{2c}\right)^{2}+\frac{\omega^{2}\xi_{+}}{c^{2}}},$ (58)

where we have defined the functions

$\xi_{\pm}=\varepsilon_{xx}\pm i\varepsilon_{xy}\mp\frac{c\kappa
b_{z}}{\omega}.$ (59)

The corresponding polarizations are given by

$\mathbf{e}_{1,\pm}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}i\\\ 1\\\
0\end{array}\right);\,\mathbf{e}_{2,\pm}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}-i\\\
1\\\ 0\end{array}\right),$ (60)

while for the plasmon mode defined by $\varepsilon_{zz}=0,$ we have

$\mathbf{e}_{3}=\left(\begin{array}[]{c}0\\\ 0\\\ 1\end{array}\right).$ (61)

The polarization vectors show that both the magnetic field (if present) and
the axion terms $b_{0},b_{z}$ lead to circular birefringence.

In deriving Eq. (53), we have implicitly assumed that there are only two
electromagnetic modes propagating in each direction. This is not obvious from
the dispersions given by Eqs. (54) and (55) when $b_{0}\neq 0.$ We did check
numerically, however, that such was the case for the results presented in this
paper.

From $\mathbf{e}_{1,+}$ and $\mathbf{e}_{2,+},$ we get
$\alpha_{1}=-\alpha_{2}=i/\sqrt{2}$ and $\beta_{1}=\beta_{2}=1/\sqrt{2}$ for
the parameters in Eqs. (54) and (55). To compute the Kerr angle, we first
define a function $\eta$ by

$\eta=\frac{r\beta_{r}}{r\alpha_{r}}=\frac{i\left(\frac{q_{1,+}}{q_{0}}-\frac{q_{2,+}}{q_{0}}\right)}{1-\frac{q_{1,+}q_{2,+}}{q_{0}^{2}}}.$
(62)

In all our numerical results, we have checked by plotting the dispersions that
$q_{1,+}$ and $q_{2,+}$ always have positive real and imaginary parts in the
frequency range considered.

If $g=0$ in the $M,$ matrix (but $b_{0}$ is present in the conductivity
tensor), Eq. (62) reduces to the simpler form

$\eta=\frac{r\beta_{r}}{r\alpha_{r}}=\frac{i\left(\sqrt{\xi_{-}}-\sqrt{\xi_{+}}\right)}{1-\sqrt{\xi_{-}\xi_{+}}}.$
(63)

In the general case where the polarization of the reflected wave is
elliptical, the Kerr angle $\theta_{K}$ is defined as the angle that the major
axis of the polarization ellipse makes with the direction of the incident
(linear) polarization i.e., the $x$ axis if
$\mathbf{e}_{0}=\widehat{\mathbf{x}}.$ We use the definitionBornWolf

$\tan
2\theta_{K}=\frac{2\operatorname{Re}\left[\eta\right]}{1-\left|\eta\right|^{2}},$
(64)

where $\theta_{K}\in\left[-\pi/2,\pi/2\right].$

The major and minor axis of the ellipse have length $a$ and $d$ respectively.
The ellipticity angle $\psi_{K}$ is definedBornWolf  as

$\tan\psi_{K}=\pm\frac{d}{a},$ (65)

where the $\pm$ signs indicate the direction of rotation of the electric field
vector along the ellipse and $d/a\in\left[0,1\right]$. Thus, a change in the
sign of the ellipticity corresponds to a change in the direction of the
rotation of the electric field vector on the ellipse. The ellipticity angle is
obtained from the equation

$\sin
2\psi_{K}=\frac{2\operatorname{Im}\left[\eta\right]}{1+\left|\eta\right|^{2}}.$
(66)

A linear polarization has $\psi_{K}=0$ while a purely circular polarization
has $\psi_{K}=\pi/4.$ In the case of a WSM, $\left|\eta\right|$ is generally
not small so that approximate formulas with $\left|\eta\right|=0$ in the
denominator of Eqs. (64) and (66) cannot be used. We remark that, when
$\operatorname{Re}\left[\eta\right]=0,$ $\theta_{K}=0$ if
$1-\left|\eta\right|^{2}>0$ but $\theta_{K}=\pi/2$ if
$1-\left|\eta\right|^{2}<0.$ An abrupt $\pi/2$ rotation of the Kerr angle is
thus possible when $1-\left|\eta\right|^{2}$ changes sign with a small
variation of $\omega$ or $B.$

###  V.2 Voigt configuration

In the Voigt configuration, the incident wave propagates along the $x$ axis
with the magnetic field along the $z$ axis. The analysis proceeds as in the
Faraday case but with $z$ replaced by $x$ in Eqs. (44)-(V.1) and with the
polarization vectors of Eq. (52) now defined by

$\mathbf{e}_{j}=\alpha_{j}\widehat{\mathbf{z}}+\beta_{j}\widehat{\mathbf{y}}.$
(67)

(A polarization component along $x$, if present, does not contribute to the
boundary conditions.) The reflection and transmission factors are still given
by Eq. (53). In this configuration, it is necessary to take the incident
polarization at an angle with respect to the $y$ or $z$ axis in order to get a
Kerr rotationBalkanski . We choose for the incident polarization
$\mathbf{e}_{0}=1/\sqrt{2}\left(\widehat{\mathbf{y}}+\widehat{\mathbf{z}}\right)$
and so we get

$\displaystyle r\alpha_{r}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\frac{\left(1-\frac{q_{1}q_{2}}{q_{0}^{2}}\right)}{\left(\frac{q_{1}}{q_{0}}+1\right)\left(\frac{q_{2}}{q_{0}}+1\right)}$
$\displaystyle-\frac{1}{\sqrt{2}}\frac{\left(\frac{q_{1}}{q_{0}}-\frac{q_{2}}{q_{0}}\right)\left(\alpha_{1}\beta_{2}+\alpha_{2}\beta_{1}-2\alpha_{1}\alpha_{2}\right)}{\left(\frac{q_{1}}{q_{0}}+1\right)\left(\frac{q_{2}}{q_{0}}+1\right)\left(\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}\right)}$

and

$\displaystyle r\beta_{r}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\frac{\left(1-\frac{q_{1}q_{2}}{q_{0}^{2}}\right)}{\left(\frac{q_{1}}{q_{0}}+1\right)\left(\frac{q_{2}}{q_{0}}+1\right)}$
$\displaystyle+\frac{1}{\sqrt{2}}\frac{\left(\frac{q_{1}}{q_{0}}-\frac{q_{2}}{q_{0}}\right)\left(\alpha_{1}\beta_{2}+\alpha_{2}\beta_{1}-2\beta_{1}\beta_{2}\right)}{\left(\frac{q_{1}}{q_{0}}+1\right)\left(\frac{q_{2}}{q_{0}}+1\right)\left(\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}\right)}.$

In this configuration, the $M$ matrix with $b_{0},b_{z}\neq 0$ is given by

$M=\left(\begin{array}[]{ccc}\omega^{2}\varepsilon_{xx}&\omega^{2}\overline{\varepsilon}_{,y}&0\\\
-\omega^{2}\overline{\varepsilon}_{xy}&\omega^{2}\varepsilon_{xx}-c^{2}q^{2}&-ic\kappa
gb_{0}q\\\ 0&ic\kappa
gb_{0}q&\omega^{2}\varepsilon_{zz}-c^{2}q^{2}\end{array}\right),$ (70)

where the function

$\overline{\varepsilon}_{xy}=\varepsilon_{xy}+i\frac{c\kappa b_{z}}{\omega}.$
(71)

The dispersion relations of the electromagnetic waves are given by

$q_{i}=\pm\frac{\omega}{c}\sqrt{\frac{\varepsilon_{W}\pm\sqrt{\varepsilon_{W}^{2}-4\varepsilon_{V}\varepsilon_{zz}}}{2}},$
(72)

where we have defined the dielectric functions

$\displaystyle\varepsilon_{V}$ $\displaystyle=$
$\displaystyle\varepsilon_{xx}+\frac{\overline{\varepsilon}_{xy}^{2}}{\varepsilon_{xx}},$
(73) $\displaystyle\varepsilon_{W}$ $\displaystyle=$
$\displaystyle\varepsilon_{V}+\varepsilon_{zz}+\frac{\kappa^{2}g^{2}b_{0}^{2}\allowbreak}{\omega^{2}}.$
(74)

The function $\varepsilon_{V}$ is the so-called Voigt dielectric
functionBalkanski . The four corresponding eigenvectors are then (when
$\omega^{2}\varepsilon_{zz}-c^{2}q_{i}^{2}\neq 0,\varepsilon_{xx}\neq
0,b_{0}\neq 0$)

$\mathbf{e}_{i}=\frac{1}{\Lambda_{i}\left(q\right)}\left(\begin{array}[]{c}-\frac{\overline{\varepsilon}_{xy}}{\varepsilon_{xx}}\\\
1\\\ \frac{-i\kappa
gb_{0}cq_{i}}{\omega^{2}\varepsilon_{zz}-c^{2}q_{i}^{2}}\end{array}\right),$
(75)

with the normalization condition

$\Lambda\left(q_{i}\right)=\sqrt{1+\left|\frac{\overline{\varepsilon}_{xy}}{\varepsilon_{xx}}\right|^{2}+\left|\frac{\kappa
gb_{0}cq_{i}}{\omega^{2}\varepsilon_{zz}-c^{2}q_{i}^{2}}\right|^{2}}.$ (76)

The wave vector $q_{i}$ in Eqs. (75)-76) is one of the four combinations given
by Eq. (72). A particularity of the Voigt configuration is that there is a
component of the polarization vector along the direction of propagation of the
wave. It is given by

$E_{x}=-\frac{\overline{\varepsilon}_{xy}}{\varepsilon_{xx}}E_{y}.$ (77)

From the polarization vectors, we have for the function $\eta$ the expression

$\eta=\frac{\left(\alpha_{1}-\alpha_{2}\right)\left(1-\frac{q_{1}q_{2}}{q_{0}^{2}}\right)+\left(\alpha_{1}+\alpha_{2}-2\right)\left(\frac{q_{1}}{q_{0}}-\frac{q_{2}}{q_{0}}\right)}{\left(\alpha_{1}-\alpha_{2}\right)\left(1-\frac{q_{1}q_{2}}{q_{0}^{2}}\right)-\left(\alpha_{1}+\alpha_{2}-2\alpha_{1}\alpha_{2}\right)\left(\frac{q_{1}}{q_{0}}-\frac{q_{2}}{q_{0}}\right)},$
(78)

where

$\alpha_{i}=\frac{-i\kappa
gb_{0}cq_{i}}{\omega^{2}\varepsilon_{zz}-c^{2}q_{i}^{2}},$ (79)

where $i=1,2$ and $q_{1,}q_{2}$ are the two wave vectors of Eq. (72)
propagating in the positive $x$ direction.

The $M$ matrix becomes block diagonal and the analysis simplifies considerably
if $g=0$ in the matrix $M.$ The four dispersions are then given by

$\displaystyle q_{1,\pm}$ $\displaystyle=$
$\displaystyle\pm\frac{\omega}{c}\sqrt{\varepsilon_{V}},$ (80) $\displaystyle
q_{2,\pm}$ $\displaystyle=$
$\displaystyle\pm\frac{\omega}{c}\sqrt{\varepsilon_{zz}},$ (81)

with the eigenvectors

$\mathbf{e}_{1,\pm}=\frac{1}{\sqrt{\left|\varepsilon_{xx}\right|^{2}+\left|\overline{\varepsilon}_{xy}\right|^{2}}}\left(\begin{array}[]{c}-\overline{\varepsilon}_{xy}\\\
\varepsilon_{xx}\\\ 0\end{array}\right),$ (82)

and

$\mathbf{e}_{2,\pm}=\left(\begin{array}[]{c}0\\\ 0\\\ 1\end{array}\right).$
(83)

The polarization vectors give
$\beta_{1}=\varepsilon_{xx}/\sqrt{\left|\varepsilon_{xx}\right|^{2}+\left|\overline{\varepsilon}_{xy}\right|^{2}},\beta_{2}=0$
and $\alpha_{1}=0,\alpha_{2}=1.$ Using these values in Eq. (53) with
$\mathbf{e}_{0}=\widehat{\mathbf{y}}$ or $\mathbf{e}_{0}=\widehat{\mathbf{z}}$
gives zero Kerr rotation thus showing that it is necessary for the incident
polarization to make an angle with the $y$ or $z$ axis in order to get a non-
zero result. For the function $\eta,$ we have

$\eta=\frac{\left(1-\sqrt{\varepsilon_{V}}\right)\left(1+\sqrt{\varepsilon_{zz}}\right)}{\left(1+\sqrt{\varepsilon_{V}}\right)\left(1-\sqrt{\varepsilon_{zz}}\right)}.$
(84)

The Kerr and ellipticity angles are given by Eq. (64) and (66). In Eq. (64),
however, the angle is measured with respect to the $z$ axis altough the real
rotation angle is that measured with respect to the incident polarization
vector i.e., $\theta_{K}-\pi/4$.

##  VI KERR ROTATION IN A NON TOPOLOGICAL METAL

For comparison with our results for a WSM, we give in this section the Kerr
angle for an ordinary (non topological) metal. In this case, the conductivity
tensor that enters the $M$ matrix is given by

$\sigma=i\varepsilon_{0}\omega_{p}^{2}\tau\left(\begin{array}[]{ccc}\frac{\omega\tau+i}{\left(\omega\tau+i\right)^{2}-\left(\omega_{c}\tau\right)^{2}}&\frac{-i\omega_{c}\tau}{\left(\omega\tau+i\right)^{2}-\left(\omega_{c}\tau\right)^{2}}&0\\\
\frac{i\omega_{c}\tau}{\left(\omega\tau+i\right)^{2}-\left(\omega_{c}\tau\right)^{2}}&\frac{\omega\tau+i}{\left(\omega\tau+i\right)^{2}-\left(\omega_{c}\tau\right)^{2}}&0\\\
0&0&\frac{1}{\omega\tau+i}\end{array}\right),$ (85)

where the magnetic field $\mathbf{B}=B\widehat{\mathbf{z}},$ $\tau$ is the
transport scattering time, $\omega_{c}=eB/m_{e}$ the cyclotron frequency (with
$m_{e}$ the electron mass) and
$\omega_{p}=\sqrt{n_{e}e^{2}/m_{e}\varepsilon_{0}}$ (with $n_{e}$ the
electronic density) is the plasmon frequency given by
$\varepsilon_{zz}\left(\omega_{p}\right)=0.$ For an ordinary metal, $b_{0}=0$
and $\mathbf{b}=0.$

Figure 1: Frequency behavior of the Kerr and ellipticity angles
$\theta_{K},\psi_{K}$ in the (a) Faraday and (b) Voigt configuration for an
ordinary (non topological) metal in a magnetic field. Parameters:
$\tau=10^{-14}$ s, $B=10$ T, $\omega_{c}\tau=0.0178$ and $\omega_{p}\tau=178$
(if $n_{e}=10^{29}$ m-3), $\omega_{p}\tau=0.178$ (if $n_{e}=10^{23}$ m-3).

In the Faraday configuration, the dispersions, polarizations and function
$\eta$ are given by Eqs. (57) and (58) and Eqs. (60) and (63). With a density
$n_{e}\approx 10^{29}$ m${}^{-3},$ and relaxation time $\tau\approx 10^{-14}$
s, we get $\omega_{p}\tau\approx 1\,\allowbreak 78$ rad/s and
$\omega_{c}\tau\approx 1.8\times 10^{-3}B$ with $B$ in Tesla. Figure 1 shows
that, in the optical frequency range the Kerr and ellipticity angles are
extremely small. (The peak in $\theta_{K}$ occurs at the plasmon frequency in
the ultraviolet frequency domain). If the density is decreased to
$n_{e}\approx 10^{23}$ m-3 (a value closer to what can be found is a WSM), the
Kerr and ellipticity angles are substantially increased. Equation (63) shows
that the Kerr angle changes sign if the direction of the magnetic field is
reversed.

In the Voigt configuration, the dispersions, polarizations and function $\eta$
are given by Eqs. (80) and (83) and Eq. (84) respectively. For $n_{e}=10^{29}$
m${}^{-3},$ the Kerr ($\theta_{K}-\pi/4$) and ellipticity angles are virtually
zero ($<10^{-10}$ rad). For $n_{e}=10^{22}$ m${}^{-3},$ both angles are
substantially bigger but much smaller than in the Faraday configuration for
the same density. Since $\varepsilon_{V}$ is even in $B,$ the Kerr angle does
not change sign if the magnetic field is reversed. A particularity of that
configuration is that in the $q_{1,\pm}$ modes,
$E_{x}=-\frac{\varepsilon_{xy}}{\varepsilon_{xx}}E_{y},$ so that there is a
component of the polarization along the direction of propagation. There is,
however, from Eq. (43), no induced charge.

The frequency profile $\theta_{K}\left(\omega\right)$ depends very much on the
relative value of $\omega_{p}$ and $\omega_{c}.$ Values of these parameters
other than those considered in this section are considered in Ref. Parent2020,
.

##  VII KERR ROTATION IN A WSM IN ZERO $\mathbf{B}$ FIELD

We proceed to the study the frequency behavior of the Kerr and ellipticity
angles in a WSM in the absence of a magnetic field thus complementing earlier
worksKargarian ; Sonowal2019  on this subject. We limit our analysis to the
case $b_{0},b_{z}\neq 0$. We discuss the effect of $\mathbf{b}_{\bot}$ in Sec.
IX.

We take the wave vector $k$ and the axion term $b_{z}$ in units of
$k_{a}=10^{8}~{}$m-1 and choose $\hslash v_{F}k_{a}=19.7$ meV with
$v_{F}=3\times 10^{5}$ m/s as our energy unit ($v_{F}k_{a}=3\times 10^{13}$
s-1). In the presentation of our results, we use the dimensionless variables
$b_{0,r,}b_{z,r},k_{c,r},\omega_{r},k_{F,\chi,r}$ which we define by

$\displaystyle b_{z}$ $\displaystyle=$ $\displaystyle
b_{z,r}k_{a},k_{c}=k_{c,r}k_{a},$ (86) $\displaystyle b_{0}$ $\displaystyle=$
$\displaystyle b_{0,r}v_{F}k_{a},e_{F}=e_{F,r}\hslash v_{F}k_{a},$ (87)
$\displaystyle k_{F,\chi}$ $\displaystyle=$ $\displaystyle
k_{F,\chi,r}k_{a},\,\omega=\omega_{r}v_{F}k_{a}.$ (88)

We take these parameters in the range $\hslash
b_{0},e_{F}\in\left[0,10\right]$ meV and $b_{z}\in\left[-5\times
10^{8},5\times 10^{8}\right]$ m${}^{-1},$ consistent with typical values
reported in the literatureLevy2020 ; Arnold2016  (for example, $b=3.2\times
10^{8}$ m-1 in WTe2Plie2017  and $v_{F}\approx 10^{6}$ m/s). In our
dimensionless units, these values correspond to
$b_{0\,r},e_{F,r}\in\left[0,0.5\right]$ and $b_{z,r}\in\left[-5,5\right]$. We
take $\tau=10$ ps for the intranode scattering time (the internode scattering
time requires a large momentum transfer and is substantially biggerHuang2021 ;
Spivak ). We take for the cutoff wave vector $k_{c,r}\approx 1000.$ The
numerical values of the Kerr angle, the position of its maximum, etc. all
depend on the precise choice of $k_{c},$ but not the qualitative aspects of
frequency profile $\theta_{K}\left(\omega\right)$.

###  VII.1 Faraday configuration

In the Faraday configuration, the dispersions are obtained by setting
$\varepsilon_{ij}=\varepsilon\delta_{ij}$ in Eq. (41). The dispersion
relations, polarization and function $\eta$ are given by Eqs. (57) and (58)
and Eqs. (60) and (62) respectively but with $\xi_{\pm}$ of Eq. (59) replaced
by $\xi_{\pm}\left(\omega\right)=\varepsilon\left(\omega\right)\mp c\kappa
b_{z}/\omega.$

Figure 2: Effect of the parameter $b_{0,r}$ alone on the frequency behavior of
the Kerr and ellipticity angles in the Faraday configuration for a Weyl
semimetal in zero magnetic field. Parameters are $\tau=10$ ps,
$e_{F,r}=b_{z,r}=0,b_{0,r}=0.5,\mathbf{b}_{\bot}=0.$ The maximum of the Kerr
angle (not shown) is $0.06$ rad.

The Kerr and ellipticity angles are shown in Fig. 2 for the case where we
artificially set $b_{z}=e_{F}=0$ in order to see the gyrotropic effect of
$b_{0}$ alone. In this situation, there is an equal number of electrons and
holes but the WSM is at equilibrium. As can be seen from Fig. 2, the
gyrotropic effect of $b_{0}$ is very small with the exception of a narrow peak
at a frequency $\omega$ where $\varepsilon\left(\omega\right)=0$ i.e., at the
plasmon frequencyPlasmonBnul ; Zhou2015 . The ellipticity is also very small
and changes sign at the plasmon frequency. The Kerr angle is positive for
$b_{0}>0,$ odd in $b_{0}$ and increases with $\left|b_{0}\right|.$

Because of the Pauli blocking, the frequency threshold $\omega_{th,r}$ for the
electromagnetic absorption in each node is given by

$\omega_{th,r}=2\left|e_{F,r}-\chi b_{0,r}\right|.$ (89)

Figure 2 shows that the Kerr angle has a step at this threshold which is at
$\omega_{r}=2b_{0,r}=1$.

The dispersion relations in the Faraday configuration (Eqs. (57) and (58),
contain both $b_{0}$ and $b_{z}$ in the ratio

$\frac{\frac{\omega^{2}}{c^{2}}\frac{c\kappa
b_{z}}{\omega}}{\left(\frac{\kappa b_{0}}{2c}\right)^{2}}=\frac{4c\omega
b_{z}}{\kappa b_{0}^{2}}=\frac{4\omega_{r}b_{z,r}}{\kappa
b_{0,r}^{2}v_{F,r}}\approx 10^{6},$ (90)

for $b_{z,r}=b_{0,r}=1,v_{F,r}=0.001$ and $\omega_{r}=1.$ Thus, in the THz
frequency range, the gyrotropic effect of $b_{0}$ is negligible in comparison
with that of $b_{z}$. This is also true in the Voigt configuration although
the dispersion relations are more complex to analyze in this case.

Figure 3 shows the effect of $b_{z}$ on $\theta_{K}$ and $\psi_{K}$ when
$b_{0}=e_{F}=0$. Although there is no carriers in this case, the Kerr angle is
not zero because of the presence of $b_{z}$ in $\xi_{\pm}$. With our sign
convention, $\theta_{K}$ is positive for negative $b_{z}$ and vice-versa i.e.,
the Kerr effect is odd in $b_{z}$ as reported previouslySonowal2019 .
Increasing $\left|b_{z}\right|$ does not change the maximal value
($\theta_{K,\max}\approx 0.5$ rad at $\omega_{\max,r}$) but broadens the
frequency range where the Kerr angle is important and blueshifts the maximum.
The Kerr angle is large in comparison with that due to $b_{0}$ alone or to
that in a normal metal (see Fig. 1). When $b_{0}=0,$ the Kerr angle is even in
$e_{F}$ so that electrons and holes contribute the same way to the Kerr
rotation. Consequently, the Kerr angle is not zero at compensation when
$e_{F}=0$ and $b_{0}\neq 0.$

In panel (a) of Fig. 3, the background dielectric constant is
$\varepsilon_{b}=1.$ Since $\varepsilon_{b}$ can be much larger in a WSM, we
plot in panel (b) the same curves but for $\varepsilon_{b}=10.$ We see that
the maximum value of the Kerr angle is unchanged as well as the overall
profile of the curves which are only compressed and pushed to lower
frequencies. At larger frequencies, the Kerr angle is reduced by a factor
$\approx\varepsilon_{b}.$

Figure 3: Frequency behavior of the Kerr (full lines) and ellipticity angles
(dashed lines) in the Faraday configuration for a Weyl semimetal in zero
magnetic field for several values of the parameters $b_{z,r}$ and with
$b_{0}=e_{F}=0.$ The background dielectric function is $\varepsilon_{b}=1$ in
pannel (a) and $\varepsilon_{b}=10$ in pannel (b).

Figure 4 shows the effect of $b_{z}$ on the Kerr angle for finite $b_{0}$ and
$e_{F}.$ The threshold frequencies (which are independent of $b_{z}$) for the
absorption given by Eq. (89) lead to two spikes in
$\theta_{K}\left(\omega\right).$ Figure 4 shows how
$\theta_{K}\left(\omega\right)$ changes when the two threshold frequencies are
on the left, in the middle, and on the right of the frequency
$\omega_{\max,r}$ where the Kerr angle is maximal. An interesting situation
occurs when the two frequencies are on the right of the maximum in
$\theta_{K}.$ In this case, the Kerr angle takes its maximal value
$\theta_{K,\max}=\pi/2$ and a plateau where $\theta_{K}=0$ appears between the
frequency $\omega_{r,-}$ where
$\operatorname{Re}\left[\xi_{-}\left(\omega_{r,-}\right)\right]=0$ and the
smaller of the two frequencies $\omega_{th,r}.$ The width of this plateau
increases with decreasing $\left|b_{z}\right|$ since $\omega_{r,-}$ is then
pushed to lower frequencies. When this plateau is absent, $\omega_{r,-}$
coincides with the peak in the Kerr angle.

The Kerr angle is zero when $\eta$ is real (and $1-\left|\eta\right|^{2}>0$)
or, equivalently when the wave vectors $q_{1+},q_{2+}$ are real so that the
two electromagnetic waves propagate without attenuation. As we argued above,
the effect of $b_{0}$ is negligible in the frequency range of Fig. 4 so that
we can use Eq. (63) instead of Eq. (62) to compute $\eta.$ The plateau where
$\theta_{K}=0$ thus occurs when the two dielectric functions
$\xi_{\pm}\left(\omega\right)$ are real and positive as can be seen in Fig. 4.

The negative peak at small frequencies in Fig. 4 only occurs when the
intraband transitions are included in $\varepsilon\left(\omega\right)$ and
when $b_{0}$ and/or $e_{F}$ are non-zero so that there is a finite density of
electrons and/or holes. We find numerically, for the range of parameters
considered, that the position of this peak is at the frequency
$\omega_{r,+}\left(\omega_{r,-}\right)$ where
$\operatorname{Re}\left[\xi_{+}\right]=0\left(\operatorname{Re}\left[\xi_{-}\right]=0\right)$
when $b_{z}<0\left(b_{z}>0\right).$ Its frequency is redshifted as
$\left|b_{z}\right|$ increases. In the opposite case, when $b_{z,r}\rightarrow
0,$ we have $\omega_{r,-}-\omega_{r,+}\rightarrow 0$ so that the negative peak
merges with the positive peak and the Kerr angle $\theta_{K}=0.$

Figure 4: Kerr angle as a function of frequency in the Faraday configuration
for a Weyl semimetal in zero magnetic field and for different values of
$b_{z,r}$ and constant $b_{0,r}=0.1$ and $e_{F,r}=0.5.$ Also shown are the
dielectric functions $\varepsilon,\xi_{+},\xi_{-}$ for $b_{z,r}=-1$.

###  VII.2 Voigt configuration

In the Voigt configuration, the propagation vector and magnetic field are
respectively along the $x$ and $z$ axis. Setting $B=0$ and keeping only
$b_{0}$ and $b_{z}$ in the $M$ matrix, the dispersion relations are given by
Eq. (72), the polarizations by Eq. (75) and $\eta$ by Eq. (78) with
$\varepsilon_{xx}=\varepsilon_{yy}=\varepsilon_{zz}=\varepsilon.$ Since the
light is incident on a surface with Fermi arcs in this configuration (the
Fermi arcs are present on surfaces parallels to the Weyl node separation),
their contribution to the conductivity should in principle be taken into
account. This contribution was estimatedKargarian  to be
$\sigma_{xy}^{s}\simeq e^{2}\ln\left(2b_{z}\lambda\right)/\pi h$, where
$\lambda$ is the wavelength of light. It is small in comparison with the bulk
contribution
$\widetilde{\sigma}_{xy}=-i\widetilde{\varepsilon}_{xy}\varepsilon_{0}\omega=e^{2}b_{z}/\pi
h$ in the THz range where $\widetilde{\varepsilon}_{ij}$ is defined in Eq.
(41). We thus neglect this contribution in our analysis.

Figure 5 shows the Kerr and ellipticity angles for a finite $b_{z,r}$ at
$e_{F}=b_{0}=0.$ Contrary to the case of an ordinary metal, the Kerr rotation
angle $\theta_{K}-\pi/4$ is not negligible in the Voigt configuration. The
change of sign of $\theta_{K}-\pi/4$ occurs numerically at the frequency where
$\operatorname{Re}\left[\varepsilon_{V}\left(\omega\right)\right]=0$ which is
also the frequency at which the ellipticity is maximal. The Kerr angle is
positive before the change of sign and is evenSonowal2019  in $b_{z}$. Its
maximal value does not change much with $b_{z}$ but the range of frequencies
where the rotation is important increases with $\left|b_{z}\right|.$ As in the
Faraday case, there is a Kerr effect even in the absence of carriers and even
in the Pauli-blocked regime.

Figure 6 shows a more general result where $b_{z,r},b_{0,r}$ and $e_{F,r}$ are
all finite and consequently the Kerr angle has more structure. The two small
spikes at frequency above $\omega_{r}=0.8$ correspond to the two thresholds
for the interband transitions given in Eq. (89). Both transitions are captured
in the Kerr angle. The large discontinuities in the Kerr angle correspond to
transitions from $\pi/2$ to $-\pi/2.$ These two values give the same angular
position for the major axis of the ellipse. There is in consequence no
discontinuity in the rotating motion of the polarization ellipse at these
frequencies. Between these two peaks, the Kerr angle is almost constant. The
separation between the two peaks increases with $\left|b_{z,r}\right|.$ Note
that, with the exception of the very small spikes at the interband threshold,
the other two discontinuities occur at $\omega_{r,1}$ and $\omega_{r,2}$ which
are defined by
$\operatorname{Re}\left[\varepsilon\left(\omega_{1,r}\right)\right]=0$ and
$\operatorname{Re}\left[\varepsilon_{V}\left(\omega_{2,r}\right)\right]=0.$
The small negative peak at small frequencies is again due to the intraband
transitions and is not present in the absence of carriers. We remark that
increasing the relaxation time above $\tau=10$ ps does not change at all the
frequency behavior of the Kerr angle shown in Fig. 6. Reducing $\tau,$
however, smooths all discontinuities at low $\omega_{r}$ as shown by the green
curve in Fig. 6 for which we took $\tau=0.1$ ps.

The first peak in the Kerr angle in Fig. 6 occurs at the plasmon frequency as
indicated by the red curve. In the limit $\omega\tau>>1$ and for
$\omega<2v_{F}k_{F,\chi},$ Eq. (II) gives
$\operatorname{Im}\left[\varepsilon\left(\omega\right)\right]\approx 0$ and

$\operatorname{Re}\left[\varepsilon\left(\omega\right)\right]\approx\beta\left(\omega\right)-\frac{\alpha}{3v_{F}/c}\frac{4}{\pi\hslash^{2}}\frac{\hslash^{2}b_{0}^{2}+e_{F}^{2}}{\omega^{2}},$
(91)

where

$\beta\left(\omega\right)=1+\frac{\alpha}{6v_{F}/c}\sum_{\chi}\frac{1}{\pi}\ln\left(\left|\frac{\omega^{2}-4v_{F}^{2}k_{c}^{2}}{\omega^{2}-4v_{F}^{2}k_{F,\chi}^{2}}\right|\right).$
(92)

The plasmon frequency is then given by
$\operatorname{Re}\left[\varepsilon\left(\omega\right)\right]=0$ i.e.,

$\omega_{p}=\sqrt{\frac{4\alpha}{3\pi\hslash^{2}v_{F}/c}\frac{\hslash^{2}b_{0}^{2}+e_{F}^{2}}{\beta\left(\omega\right)}},$
(93)

consistent with the result given in Ref. Zhou2015, . The intraband plasmon
frequency is red-shifted by the interband contribution given by
$\beta\left(\omega\right).$ Equation (93) can be solved iteratively but a good
approximation to the numerical value of the plasmon frequency given in Fig. 6
is obtained by simply taking $\beta\left(\omega=0\right)$ in Eq. (93).

Figure 5: Frequency behavior of the Kerr rotation angle $\theta_{K}-\pi/4$
(full lines) and ellipticity angle (dashed lines) $\psi_{K}$ in the Voigt
configuration for a Weyl semimetal in zero magnetic field for two values of
$b_{z,r}$ and with $b_{0}=e_{F}=0.$ Figure 6: Frequency behavior of the Kerr
angle and real part of the dielectric functions
$\varepsilon_{V}\left(\omega\right)$ and $\varepsilon\left(\omega\right)$ for
a Weyl semimetal in zero magnetic field in the Voigt configuration for the
parameters $\varepsilon_{F,r}=0.5,b_{z,r}=2,b_{0,r}=0.1$ and $\tau=10$ ps.The
green curve gives the Kerr angle for $\tau=0.1$ ps.

##  VIII KERR EFFECT IN A WSM IN A FINITE MAGNETIC FIELD

To study the effect of the magnetic field on the Kerr rotation, we define the
dimensionless variables

$\displaystyle v_{F,r}$ $\displaystyle=$
$\displaystyle\frac{v_{F}}{c},b_{z,r}=b_{z}\ell,$ $\displaystyle b_{0}$
$\displaystyle=$ $\displaystyle b_{0,r}v_{F}/\ell,e_{F}=e_{F,r}\hslash
v_{F}/\ell,$ (94) $\displaystyle\omega$ $\displaystyle=$
$\displaystyle\omega_{r}v_{F}/\ell,$

and choose $\hslash v_{F}/\ell=7.\,\allowbreak 69$ $v_{F,r}\sqrt{B}$ eV with
$B$ in Tesla as our unit of energy. The ratio $v_{F}/\ell=1.17\times
10^{16}v_{F,r}\sqrt{B}$ s-1 so that the frequency
$f=\omega/(2\pi)=1.\,\allowbreak 86\times 10^{12}\omega_{r}\sqrt{B}$ Hz for
$v_{F,r}=0.001.$ In terms of these dimensionless parameters, the range of
values considered in Sec. VII corresponds to
$b_{0,r},e_{F,r}\in\left[0,1.\,\allowbreak
3/\sqrt{B}\right],b_{z,r}\in\left[-12.\,\allowbreak
83/\sqrt{B},12.\,\allowbreak 83/\sqrt{B}\right]$ for $v_{F,r}=0.001.$

We restrict our analysis to the ultraquantum limit where the Fermi level
$e_{F}$ lies in the chiral Landau level of both nodes. The Fermi wave vector
in each node, $k_{F,\chi},$ is then given by

$e_{F,r}=\chi\left(b_{0,r}-k_{F,\chi,r}\ell\right)$ (95)

and, in order for the $n\geq 1$ levels to be empty, the condition
$e_{F,r}+b_{0,r}<\sqrt{2}$ must be satisfied. The occupation of the Landau
levels at $T=0$ K is given by Eqs. (121)-(123) of Appendix B. Since the
density of states in one node is $g\left(E\right)=1/4\pi^{2}\ell^{2}\hslash
v_{F},$ the Fermi level is given by

$e_{F,r}=2\pi^{2}\ell^{3}n_{e},$ (96)

where $n_{e}$ is the total density of charge carriers in the WSM.

To describe the inter-Landau-level transitions, it is more convenient to use
negative values of $n$ for the Landau levels in the valence band ($s=-1$). The
threshold energy for the absorptive transitions in the $n=-1,0,1$ sector are
given by

$\displaystyle\Delta_{0,1}^{\left(\chi\right)}$ $\displaystyle=$
$\displaystyle\sqrt{\left(e_{F,r}-\chi
b_{0,r}\right)^{2}+2}-\left(e_{F,r}-\chi b_{0,r}\right),$ (97)
$\displaystyle\Delta_{-1,0}^{\left(\chi\right)}$ $\displaystyle=$
$\displaystyle\sqrt{\left(e_{F,r}-\chi
b_{0,r}\right)^{2}+2}+\left(e_{F,r}-\chi b_{0,r}\right),$ (98)

while for the dipolar transitions (with $n>0$)

$\Delta_{-n,n+1}^{\left(\chi\right)}=\sqrt{2\left|n\right|}+\sqrt{2\left(\left|n\right|+1\right)},$
(99)

and for the transitions captured by $\sigma_{zz}\left(\omega\right)$

$\Delta_{-n,n}^{\left(\chi\right)}=2\sqrt{2\left|n\right|}.$ (100)

The transitions $\Delta_{0,1}^{\left(\chi\right)}$
$\left(\Delta_{-1,0}^{\left(\chi\right)}\right)$ appear as peaks in the real
part of the conductivity $\sigma_{+}\left(\sigma_{-}\right)$ where
$\sigma_{\pm}=\sigma_{xx}\pm i\sigma_{xy},$ while the dipolar transitions
$\Delta_{-n,n+1}^{\left(\chi\right)}$ appear as peaks in both $\sigma_{+}$ and
$\sigma_{-}.$ The transitions $\Delta_{-n,n}^{\left(\chi\right)}$ appears as
peaks in $\sigma_{zz}$ only. The lowest frequency dipolar transition has the
threshold frequency $\omega_{r}=\Delta_{-1,2}^{\left(\chi\right)}=\allowbreak
3.\,\allowbreak 41$ i.e., $f=0.32$ THz for $B=10$ T. As in Sec. VII, we choose
$\tau=10$ ps for the scattering time and limit the summation over Landau
levels to $n_{\max}=10$ unless indicated otherwise. The qualitative aspect of
$\theta_{K}\left(\omega\right)$ and $\chi_{K}\left(\omega\right)$ does not
change significantly if we increase $n_{\max}$ which is, in any case, limited
by the bandwith of the Weyl cones.

###  VIII.1 Faraday configuration

In the Faraday configuration, the dispersion, polarizations and $\eta$ are
given by Eqs. (57) and (58) and Eqs. (60) and (62) with $\xi_{\pm}$ defined in
Eq. (59). When $b_{0}=e_{F}=0,$ it follows that $\sigma_{xy}=0$ (since
$x_{F}=0$ in Eqs. (126) and (127)) and so
$\varepsilon_{\pm}=\varepsilon_{xx}.$ If, in addition, $b_{z}=0,$ then
$\xi_{+}=\xi_{-}$ and $\eta=0.$ As expected the magnetic field alone does not
produce a Kerr rotation in the absence of carriers. There is however a Kerr
rotation when $b_{z}\neq 0$ and $e_{F}=b_{0}=0$ even if there is no carrier in
the WSM. There is also a rotation when $e_{F}=0$ and $b_{0}\neq 0$ i.e., at
compensation when the density of electrons is equal to the density of holes.
Both carriers contribute the same way to the Kerr angle as pointed out in Sec.
VII.

Figure 7 shows a typical result for the Kerr and ellipticity angles as well as
for the corresponding dielectric functions $\xi_{\pm}.$ We choose
$b_{0,r}=0.2,b_{z,r}=0.6,e_{F,r}=1.$ We remark that $\xi_{-}$ has peaks at the
transitions $\Delta_{-1,0}^{\left(\chi\right)}$ while $\xi_{+}$ captures the
transitions $\Delta_{0,1}^{\left(\chi\right)}$ for $\chi=\pm.$ The dipolar
transitions $\Delta_{-n,n+1}^{\left(\chi\right)}$ are present in both
functions. In Fig. 7, the Kerr angle increases until it reaches a plateau at
$\theta_{K}=\pi/2$ in a small range of frequencies. The onset of the plateau
occurs when $\operatorname{Re}\left[\xi_{-}\left(\omega\right)\right]=0.$
After the plateau, the Kerr angle drops abruptly to zero and remains zero
until the transition at $\omega_{r}=\Delta_{0,1}^{\left(-\right)}.$
Discontinuities in $\theta_{K}\left(\omega\right)$ occur at all four
transitions in the $n=-1,0,1$ sector as well as at all dipolar transitions.The
transitions $\Delta_{-n,n}^{\left(\chi\right)}$ are not captured by the Kerr
angle in the Faraday configuration since $\varepsilon_{zz}$ does not enter its
calculation. Other transitions will appear in $\theta_{K}\left(\omega\right)$
if the Weyl cones are tilted since a tilt modifies the matrix elements of the
current operator and so the selection rulesGoerbig2016 ; Parent2020 .

It may seem that there is a discontinuous change in the Kerr angle from
$\pi/2$ to $0$ at $\omega_{r}=0.34$ in Fig. 7. This is however not the case
since both the rotation and the ellipticity of the Kerr polarization change
with frequency. We illustrate this in Fig. 8 where we plot the trajectory of
the electric polarization vector for one cycle of oscillation at different
frequency points and for the parameters used in Fig. 7. The trajectory is an
ellipse with its major axis at $\theta_{K}=\pi/2$ at point 3 and becomes a
quasi circle at points $4$ and $5.$ This circle is then elongated along
$E_{x}$ at points $6$ and $7$ and the major axis is now at $\theta_{K}=0.$
There has been, however, no discontinuous change in the trajectory of the
electric polarization vector in between these points.

Figure 7: Frequency behavior of the Kerr and ellipticity angles and real part
of the dielectric functions $\xi_{-},\xi_{+-}$ for a Weyl semimetal in a
magnetic field and in the Faraday configuration. Parameters
$e_{F,r}=1,b_{0,r}=0.2$ and $b_{z,r}=0.6.$ The
$\Delta_{n,m}^{\left(\pm\right)}$ symbols indicate the inter-Landau-level
transitions. Figure 8: Trajectories of the polarization vector for the
different frequency points identified in the inset. The sharp drop in the Kerr
angle corresponds to a continuous change in the shape of the ellipse. The
parameters are those of Fig. 7.

With $b_{z}=0$ and a finite Fermi level $e_{F,r}=1,$ the magnetic field leads
to a positive Kerr angle as shown by the black line in Fig. 9. With our sign
convention for $b_{z}$ and $b_{0},$ a positive value of $b_{z,r}$ leads to a
negative Kerr angle in the absence of a magnetic field, as shown in Sec. VII.
Thus, at small frequencies increasing $b_{z,r}$ progressively reduces the
region where the Kerr angle induced by the magnetic field is large. (In the
opposite case where $b_{z,r}<0$, increasing $\left|b_{z,r}\right|$ increases
the region where the Kerr angle is large). Further increasing $b_{z,r}$
changes the sign of the Kerr angle and decreases it to zero. The plateaus
where the Kerr angle is precisely zero or $\pi/2$ are in a frequency range
where $\xi_{\pm}$ are both real and positive so that $\eta$ is imaginary and
$\tan\left(2\theta_{K}\right)=0$ or $\pi$ depending on the sign of
$1-\left|\eta\right|^{2}<0$ in Eq. (64). These two plateaus disappear when
$b_{z,r}\leq 0$.

An interesting result of our calculation is that at precisely
$b_{z,r}=e_{F,r},$ with $g=0$ in the matrix $M,$ the Kerr angle is zero in the
range $\omega_{r}\in\left[0,\Delta_{0,1}^{\left(-\right)}\right]$ i.e., up to
the smallest of the four transition frequencies in the $n=-1,0,1$ sector. In
this region, we find numerically that $\operatorname{Re}[\eta]=0$ and
$1-\left|\eta\right|^{2}>0$ in Eq. (64) so that $\theta_{K}=0$. We do not
observe this behavior when $B=0$. Indeed, this result is due to the particular
form of the conductivity tensor in the extreme quantum limit i.e., to the fact
that
$\operatorname{Re}\left[\sigma_{xx}\right]=\operatorname{Im}\left[\sigma_{xy}\right]=0$
for $\omega_{r}<e_{F,r}.$ If we expand the conductivity $\sigma_{xx}$ and
$\sigma_{xy}$ to first order in $\omega_{r},$ we find that

$\xi_{\pm}=1-\frac{2\operatorname{Im}\left[\sigma_{xx}\right]}{\omega\varepsilon_{0}}>0,$
(101)

so that $q_{1,+},q_{2,+}$ in Eqs. (57) and (58) are real and positive and
$\operatorname{Re}[\eta]=0$. The axion term $b_{z}$ in $\xi_{\pm}$ cancels the
gyrotropic contribution from the $\varepsilon_{xy}$ in this limit. Equation
(101) is satisfied not just at small frequencies but for
$\omega_{r}\in\left[0,\Delta_{0,1}^{\left(-\right)}\right],$ something we were
only able to check numerically. This observation can provide a way to measure
the axion term $b_{z}$ if the Fermi level is known. When $g=1$ (or $g=1/3$),
the Kerr angle is zero but only in the range
$\omega_{r}\in\left[\omega_{a,r},\Delta_{0,1}^{\left(-\right)}\right]$ where
$\omega_{a,r}\approx 0.0001$ i.e., a very small frequency. Below
$\omega_{a,r}$ we find that $1-\left|\eta\right|^{2}<0$ and the Kerr angle is
$\theta_{K}=\pi/2.$ This remains true if $b_{z}=e_{F}=0$ so that it would seem
that the axion term $b_{0}$ causes a rotation of the polarization from
$\widehat{\mathbf{x}}$ to $\widehat{\mathbf{y}}$ at very small frequency if
$g$ is not zero, another curious (or spurious) effect of a finite $b_{0}$ at
$\omega\rightarrow 0.$

Another interesting fact when
$\omega_{r}\in\left[0,\Delta_{0,1}^{\left(-\right)}\right]$ and
$b_{z,r}=e_{F,r},g=0$ is that the wave vectors $q_{1,\pm}$ and $q_{2,\pm}$ are
real and positive and their dispersions are linear in $q$ and gapless so that
the two modes propagate without attenuation as shown in Fig. 9(b). However, as
soon as $b_{z,r}\neq e_{F,r},$ only one of the two modes remains gapless with
a $\omega\sim q^{2}$ dispersion (if $b_{0}=0$). In the low-magnetic field
limit where the semi-classical Boltzmann equation is used to compute the
conductivity tensor, this mode would be the helicon mode discussed in Ref.
Helicon2015, .

When $b_{0}$ is finite Eqs. (57) and (58) give that the mode $q_{2,+},$ that
propagates in the positive $z$ direction and $q_{1,-},$ a mode propagating in
the opposite direction have a finite wave vector $q\ell=\frac{\kappa
b_{0}\ell}{c}=\kappa v_{F,r}\approx 10^{-5}$ at $\omega=0$. This sort of
instability in chiral matter and some of its consequences have been discussed
beforeZebin2017 . In this reference however, $b_{0}$ is associated with the
chiral magnetic effect (CME) i.e., with a difference in the chiral potential
of the two nodes. In our case, the Fermi pockets are in chemical equilibrium
with a common Fermi level and $b_{0}$ is related to the difference in energy
between the two Dirac points, a property of the band structure of the WSM.
There is a gyrotropic current in this caseMaPesin2015  but it should vanish in
the static limit. In fact, according to Ref. Zhong2016, , the gyromagnetic
contribution of $b_{0}$ and its associate current in the Maxwell equations
should be suppressed by scattering at low frequency. If this is the case, this
instability would vanish. In any case, our results in the THz range are not
affected by the presence of this possible gap in the wave vector at very low
frequency.

Figure 9: (a) Kerr angle as a function of frequency for a Weyl semimetal in a
magnetic field and in the Faraday configuration for different values of
$b_{z,r}$ and with $e_{F,r}=1,b_{0,r}=0$ and $n_{mzx}=20.$ (b) Real (full
lines) and imaginary part (dashed lines) of the wave vector $q\ell$ for
$e_{F,r}=1,b_{0,r}=0$ and $b_{z,r}=0.4,1.0$.

###  VIII.2 Voigt configuration

We consider an incident wave propagating along the $x$ axis with the external
magnetic field again in the $z$ direction. The dispersion relations are given
by Eq. (72), the polarizations by Eq. (75) and $\eta$ by Eq. (78). As for the
$B=0$ case, the same numerical results are obtained by using the simpler
formulas with $g=0$ ($b_{0}$ must be kept in the dielectric tensor, however).

Figure 10 shows $\theta_{K}\left(\omega\right)$ for the same parameters as
those used in Fig. 9 for the Faraday configuration. Since $\varepsilon_{zz}$
now enters in the calculation of the Kerr angle and because the intra-Landau
level transitions in $n=0$ are included in $\varepsilon_{zz},$ we must also
specify the magnetic field and the relaxation time in addition to the
dimensionless variables $b_{0,r},b_{z,r},e_{F,r}$ and $\omega_{r}.$ The Kerr
angle has a plateau at $\theta_{K}=\pi/4$ (i.e., no effective rotation of the
polarization) at small frequencies before dropping abruptly to
$\theta_{K}=-\pi/4.$ As in the Faraday case, small spikes signal the dipolar
transitions $\Delta_{-1,2}^{\left(\chi\right)}=3.42,$
$\Delta_{0,1}^{\left(\chi\right)}=0.73$ and
$\Delta_{-1,0}^{\left(\chi\right)}=2.73$. A unique feature of the Voigt
configuration is that the transitions $\Delta_{-n,n}^{\left(\chi\right)}$ are
now present in $\theta_{K}\left(\omega\right)$. Increasing $b_{z,r}$ decreases
the width of the plateau where $\theta_{K}=\pi/4$ and so increases the range
of frequencies where the effective rotation angle is important. Contrary to
what we found in the Faraday configuration, $\theta_{K}$ is not zero when
$b_{z,r}=e_{F,r}$ but its frequency profile is distinctively different , at
low frequency, than the profiles for $b_{z,r}\neq e_{F,r}.$

Figure 10: Kerr angle as a function of frequency for a Weyl semimetal in a
magnetic field and in the Voigt configuration for different values of
$b_{z,r}$ and with $e_{F,r}=1,b_{0,r}=0,B=10$ T, $\tau=10$ ps. The rotation
angle is $\theta_{K}-\pi/4.$ The Landau level transitions indicated by the
$\Delta_{n,m}$ symbols have the same value for both nodes since $b_{0}=0.$

Another feature of the Voigt configuration is that $\theta_{K}$ has a change
of curvature and a zero ellipticity at a frequency where
$\varepsilon_{zz}\left(\omega\right)=0$ i.e., at the plasmon frequency
$\omega_{p}$ as indicated in Fig. 11. This feature is also present in Fig. 10
but difficult to see because of the superposition of the different curves. If
we consider only the intraband transitions, we have from
$\varepsilon_{zz}\left(\omega\right)=0$ in the limit $\omega\tau>>1$ that the
plasmon frequency is given by

$\omega_{p}=\sqrt{\frac{e^{3}v_{F}B}{2\pi^{2}\hslash^{2}\varepsilon_{0}},}$
(102)

which is the known resultSpivak  for two nodes. The plasmon frequency is
independent of the relaxation time in this limit. The $\omega_{p}\sim\sqrt{B}$
behavior is an experimental signature of a WSMSa2021  and of the chiral
anomaly. This frequency is corrected by inter-Landau-level transitions and
scattering. The more accurate result is found by solving the self-consistent
equation

$\omega_{p}=\sqrt{\frac{2e^{3}v_{F}B}{4\pi^{2}\varepsilon_{0}\hslash^{2}}\frac{1}{1-\frac{8\alpha
c\Lambda\left(\omega_{p,r}\right)}{\pi v_{F}}}-\frac{1}{\tau^{2}}},$ (103)

where

$\Lambda\left(\omega_{r}\right)=\sum_{n>0}n\int_{0}^{+\infty}dx\frac{1}{\left(x^{2}+2n\right)^{3/2}}\frac{1}{\omega_{r}^{2}-4\left(x^{2}+2n\right)}.$
(104)

A very good approximation of the plasmon frequency found in Fig. 11 is
obtained by taking $\omega_{r}=0$ in $\Lambda\left(\omega_{r}\right).$ The
inter-Landau-level transitions decrease the plasmon frequency because
$\Lambda<0$ when $\omega_{r}<\sqrt{2}$. Figure 11 also shows that the end of
the plateau where $\theta_{K}=\pi/4$ and the negative peak leading to
$\theta_{K}=-\pi/4$ are at a frequency corresponding to
$\operatorname{Re}\left[\varepsilon_{V}\left(\omega\right)\right]=0$ as in the
$B=0.$ The negative peak, however, does not disappear if $e_{F}=b_{0}=0$ and
is not due to the intraband transitions.

Figure 11: Frequency behavior of the Kerr and ellipticity angles and of the
real part of the dielectric functions $\varepsilon_{zz}$ and $\varepsilon_{V}$
for a Weyl semimetal in a magnetic field and in the Voigt configuration. The
actual rotation of the polarization is $\theta_{K}-\pi/4.$ Parameters:
$b_{z,r}=0.2,e_{F,r}=b_{0,r}=0,B=10$ T and $\tau=10$ ps. The kink in the Kerr
angle is at the plasmon frequency.

##  IX EFFECT OF $\mathbf{b}_{\bot}$ ON THE KERR ANGLE

We study in this section the effect of a perpendicular component
$\mathbf{b}_{\bot}$ on the Kerr angle. To obtain analytical results that are
manageable, we set $g=0$ in the matrix $M$ but keep $b_{0}$ in the
conductivity tensor. The magnetic field is again along the $z$ axis and the
light propagates along that axis. The $M$ matrix is

$M=\left(\begin{array}[]{ccc}\omega^{2}\varepsilon_{xx}-c^{2}q_{z}^{2}&\omega^{2}\varepsilon_{xy}+i\omega
c\kappa b_{z}&-i\omega c\kappa b_{y}\\\ -\omega^{2}\varepsilon_{xy}-i\omega
c\kappa b_{z}&\omega^{2}\varepsilon_{xx}-c^{2}q_{z}^{2}&i\omega c\kappa
b_{x}\\\ i\omega c\kappa b_{y}&-i\omega c\kappa
b_{x}&\omega^{2}\varepsilon_{zz}\end{array}\right).$ (105)

With the axion term in spherical coordinates

$\mathbf{b}=b\sin\theta\cos\varphi\widehat{\mathbf{x}}+b\sin\theta\sin\varphi\widehat{\mathbf{y}}+b\cos\theta\widehat{\mathbf{z}},$
(106)

the dispersion relations and corresponding polarizations are given by

$q_{\pm}^{2}=-\frac{\kappa^{2}}{2\varepsilon_{zz}}\Lambda_{\pm}\left(\theta,\varphi=0,b\right),$
(107)

where

$\displaystyle\Lambda_{\pm}\left(\theta,\varphi,b\right)$ $\displaystyle=$
$\displaystyle b^{2}\sin^{2}\theta\cos\left(2\varphi\right)$
$\displaystyle\mp\sqrt{b^{4}\sin^{4}\theta+\frac{4\omega^{2}\varepsilon_{zz}^{2}}{c^{2}\kappa^{2}}\left(b\cos\theta-i\frac{\omega\varepsilon_{xy}}{c\kappa}\right)^{2}}$

and (for $b\neq 0$)

$\mathbf{e}_{\pm}=\frac{1}{\sqrt{\left|E_{x,\pm}\right|^{2}+\left|E_{z,\pm}\right|^{2}+1}}\left(\begin{array}[]{c}E_{x,\pm}\\\
1\\\ E_{z,\pm}\end{array}\right),$ (109)

where we have defined

$E_{x,\pm}=\frac{2i\frac{\omega\varepsilon_{zz}}{c\kappa}\left(b\cos\theta-i\frac{\omega\varepsilon_{xy}}{c\kappa}\right)+b^{2}\sin^{2}\theta\sin
2\varphi}{\Lambda_{\pm}\left(\theta,\varphi,b\right)},$ (110)

and (for $\varepsilon_{zz}\neq 0$)

$E_{z,\pm}=-ib\sin\theta\frac{E_{x,\pm}\sin\varphi-\cos\varphi}{\frac{\omega\varepsilon_{zz}}{c\kappa}}.$
(111)

There is a component of the polarization in the direction of propagation of
the wave when $\theta\neq 0.$

If the polarization of the incident wave is along the $x$ axis, the function
$\eta$ that enters in the calculation of the Kerr angle is given by (with
$E_{y}=1$)

$\eta=\frac{-2\left(\frac{q_{+}}{q_{0}}-\frac{q_{-}}{q_{0}}\right)}{\left(1-\frac{q_{+}}{q_{0}}\right)\left(1+\frac{q_{-}}{q_{0}}\right)E_{x,+}-\left(1+\frac{q_{+}}{q_{0}}\right)\left(1-\frac{q_{-}}{q_{0}}\right)E_{x,-}},$
(112)

where $q_{0}=\omega/c.$ Since, the rotational symmetry in the $x-y$ plane is
broken, the Kerr angle changes with the azimuthal angle $\varphi$ and depends
on the orientation of the polarization vector of the incident wave.

Figure 12 shows the Kerr angle for several orientations of the axion term
$\mathbf{b}$ for $e_{F,r}=1,\left|\mathbf{b}_{r}\right|=0.4$ and $b_{0,r}=0.$
The frequency domain where $\theta_{K}$ is large increases with the polar
angle $\theta$ if $e_{F}$ is not zero. This is consistent with the behavior
shown in Fig. 9 since a finite $\theta\in\left[0,\pi/2\right]$ decreases
$b_{z,r}$ when $\left|\mathbf{b}_{r}\right|$ is fixed. When $\varphi=0,\pi/2$
and $\theta\neq 0,$ a new peak appears on the right of the plateau where
$\theta_{K}=\pi/2$ and increases in amplitude with
$\theta\in\left[0^{+},\pi/2\right].$ The frequency at which $\theta_{K}=0$ on
the left side of this peak is the plasmon frequency. Indeed, when,
$\operatorname{Re}\left[\varepsilon_{zz}\right]=0$,
$\operatorname{Im}\left[\varepsilon_{zz}\right]\approx 0,\theta=\pi/2$ and
$\varphi=0,\pi/2$, Eq. (110) becomes $E_{x,\pm}=0$ so that
$\eta\rightarrow\infty$ and $\theta_{K}=0.$

We found previously that a nonzero $b_{z}$ gives a Kerr angle even in the
absence of carriers. In the present case, if $\theta=\pi/2,\varphi=0$ and
$e_{F}=b_{0}=0$ (no carriers), the Kerr angle is zero at all frequencies (not
shown in Fig. 12).

Figure 12: Kerr angle as a function of frequency for a Weyl semimetal in a
magnetic field and in the Faraday configuration for different orientations of
the vector $\mathbf{b}$ and with $e_{F,r}=1,b_{0,r}=0,b=0.4$ and real part of
the dielectric function $\varepsilon_{zz}$ near $\varepsilon_{zz}=0.$

Figure 9 shows that when $b_{z,r}=e_{F,r}$ and $g=0,$ the Kerr angle is zero
in the frequency range
$\omega_{r}\in\left[0,\Delta_{0,1}^{\left(-\right)}\right].$ This remains true
if $\mathbf{b}$ is tilted with $b_{z}$ kept constant at $b_{z,r}=e_{F,r}$ and
$\varphi=0$.

##  X MAGNETIC FIELD BEHAVIOR OF THE KERR ANGLE

We end our study of the Kerr rotation in a WSM by considering the situation
where the frequency $f$ is kept fixed and the magnetic field is varied. We
restrict our analysis to the Faraday configuration. Figure 13 shows the Kerr
angle for $f=5$ THz, $b=5\times 10^{8}$ m-1, $\varphi=0$, $v_{F}=3\times
10^{5}$ m/s, $n_{e}=10^{21}$ m${}^{-3},\tau=10$ ps and $b_{0}=0.$ Note that
the dimensionless parameters $b_{r},b_{0,r},\omega_{r},e_{F,r}$ that enter the
calculation of the conductivity tensor and the $M$ matrix change with magnetic
field. In addition the Fermi level $e_{F}$ decreases with increasing $B$ when
the density is fixed. To observe any transition in $\theta_{K}\left(B\right)$,
we must ensure that $f$ is chosen such that $\omega_{r}=2\pi f/(v_{F}/\ell)$
goes through the different inter-Landau-level transitions when the magnetic
field is varied. In addition, in order to capture the plasmon mode,
$\omega_{r}$ must intersect the frequency $\omega_{p,r}$ where
$\varepsilon_{zz}\left(\omega_{p,r}\right)=0$ and $\theta$ must be finite.
Finally, the condition $e_{F,r}+b_{0,r}<\sqrt{2}$ must be satisfied at all $B$
in order for the WSM to stay in the ultra-quantum limit where our analysis is
valid. The magnetic field range $B\in\left[0.5,6\right]$ T with the above
parameters satisfy these conditions.

The black curve in Fig. 13 is the function $\theta_{K}\left(B\right)$ for the
special case where $b=0$ thus showing the effect of the magnetic field alone.
We see that $\theta_{K}\left(B\right)=0$ when the magnetic field is such that
$\omega_{r}<\Delta_{-1,0}$ which is the lowest-energy transition when the
Fermi level is above the Dirac point. At lower $B$, the transitions
$\Delta_{-1,2}$ and $\Delta_{0,1}$ are captured by the Kerr angle. The
transition $\Delta_{-1,1}$ is captured only if $\theta\neq 0$ because it is
only then that $\varepsilon_{zz}\left(\omega\right)$, which contains these
transitions, enters in the equation for the Kerr angle.

When $\theta$ is non-zero, a downward turn appears in
$\theta_{K}\left(B\right)$ above the magnetic field $B^{\ast}$ for which
$\varepsilon_{zz}\left(\omega_{p,r}\right)=0$ (note that, in our reduced
units, $\omega_{p,r}$ is independent of $B$ in the clean limit). For
$B>B^{\ast},$ Fig. 13 shows that $\left|\theta_{K}\left(B\right)\right|$
increases with $\theta$ and the shoulder is pushed to higher $B.$ The Kerr
angle is effectively zero from $B^{\ast}$ to the shoulder and tends to a
constant at large $B.$

Figure 13: Behavior of the Kerr angle with magnetic field at fixed frequency
$f=5$ THz in the Faraday configuration for different inclination angles
$\theta$ of the axion term $\mathbf{b}.$ The parameters are $b=5\times 10^{8}$
m-1, $n_{e}=10^{21}$ m${}^{-3},\tau=10$ ps, $v_{F}=3\times 10^{5}$
m/s,$\varphi=0$ and $b_{0}=0.$ The black curve is for $b=0$.

The Fermi level $e_{F,r}\sim 1/B^{3/2}$ and frequency $\omega_{r}\sim
1/B^{1/2}$ so that, in the limit of very large $B,$
$\varepsilon_{xy}\rightarrow 0$ and

$\displaystyle\varepsilon_{xx}^{\ast}$ $\displaystyle\equiv$
$\displaystyle\varepsilon_{xx}\left(B\rightarrow\infty\right)$
$\displaystyle=$ $\displaystyle
1+\frac{\kappa}{2v_{F,r}}\left(1+2\sum_{n>1}\int_{-\infty}^{+\infty}dx\frac{\Lambda_{n}\left(x\right)}{\Delta_{n}^{3}\left(x\right)}\right),$

where $\Lambda_{n}\left(x\right)$ and $\Delta_{n}\left(x\right)$ are defined
in Appendix B. The constant $\varepsilon_{xx}^{\ast}$ is real and positive. We
thus have, from Eqs. (59) and (63,) that

$\xi_{\pm}=\varepsilon_{xx}^{\ast}\mp\frac{c\kappa b_{z}}{\omega}$ (114)

and the function $\eta$ that enters the formula for the Kerr angle is, for
$\theta=0$ and $b_{0}=0,$ given by

$\eta=\frac{i\left(\sqrt{\varepsilon_{xx}^{\ast}+\frac{c\kappa
b_{z}}{\omega}}-\sqrt{\varepsilon_{xx}^{\ast}-\frac{c\kappa
b_{z}}{\omega}}\right)}{1-\sqrt{\varepsilon_{xx}^{\ast 2}-\left(\frac{c\kappa
b_{z}}{\omega}\right)^{2}}}.$ (115)

The Kerr angle in this limit is plotted in Fig. 14 as a function of $b_{z}$
for $f=5$ THz and $B=20$ T. On the positive axis, the Kerr angle is zero at
small $b_{z}$ where $1-\left|\eta\right|^{2}>0$ and changes abruptly to
$\theta_{K}=-\pi/2$ when $1-\left|\eta\right|^{2}<0$ [see Eq. (64)]. As
$b_{z}$ increases further, $\left|\theta_{K}\right|$ decreases. The end of the
plateau where $\theta=-\pi/2$ is at
$b_{z}^{\ast}=\omega\varepsilon_{xx}^{\ast}/c\kappa.$ Note that, since
$\varepsilon_{xy}\rightarrow 0,$ the magnetic field has no effective rotating
power at large $B.$ Figure 14 shows that the Kerr angle is odd in $b_{z}$ as
expected in the Faraday configuration.

Figure 14: Kerr angle as a function of the axion term $b_{z}$ in the limit of
large magnetic field and for $B=20$ T, $f=5$ THz, $\theta=\varphi=0,$
$n_{e}=10^{21}$ m${}^{-3},$ $v_{F}=3\times 10^{5}$ m/s, $b_{0}=0.$

##  XI CONCLUSION

We have studied the effect of the axion terms $b_{0}$ and $\mathbf{b}$ on the
magnetic field and frequency profile of the Kerr angle using a simple two-node
model of a Weyl semimetal with broken time reversal and/or inversion
symmetries. We have considered the effect of the axion terms alone and then in
the presence of an external static magnetic field. In both cases, our study
was made in the Faraday and Voigt configurations. When a magnetic field was
considered, we analyzed the Kerr rotation in the ultra-quantum regime where
the Fermi level lies in the chiral Landau level. For the case $\mathbf{B}=0,$
our study complements earlier works on the subject while the $\mathbf{B}\neq
0$ case, with the axion terms, has not been studied before.

The Kerr angle has a rich frequency behavior which depends sensitively on the
value of the different parameters of the WSM. We find that it can be quite
large, reaching its maximum value $\theta_{K}=\pm\pi/2$ at certain
frequencies. The gyrotropic power of $\mathbf{b}$ is much bigger than that of
$b_{0}$ which, in our calculations, has mainly the effect of unbalancing the
electronic density in both nodes. Depending on its orientation with respect to
the external magnetic field, the axion term $\mathbf{b}$ can increase or
decrease the Kerr rotation caused by the magnetic field. Moreover, the axion
term $\mathbf{b},$ contrary to the magnetic field $\mathbf{B},$ enforces a
Kerr rotation even when the Fermi level $e_{F}=0$ (i.e., no electronic density
in both nodes if $b_{0}=0$ or equal density of electrons and holes if
$b_{0}\neq 0$).

Allowed electronic transitions are captured by the frequency or magnetic field
profile of the Kerr angle. The Faraday and Voigt configurations capture a
different set of transitions. In the Voigt configuration and with
$\mathbf{B}\neq 0,$ the frequency of the plasmon mode is signaled by a small
kink in $\theta_{K}\left(\omega\right).$ The plasmon mode also appears in
$\theta_{K}\left(\mathbf{B}\right)$ if the vector $\mathbf{b}$ is slightly
tilted with respect to the external magnetic field.

In the ultra-quantum limit, the particular form of the dielectric function of
the Weyl semimetal allows the Kerr angle to be zero in a large range of
frequencies in the Faraday configuration. More interestingly, this range can
extend all the way down to $\omega=0$ in the particular case where the
component of the axion term along the magnetic field satisfies the condition
$b\hslash v_{F}=e_{F}$ (for $\mathbf{b}\|\mathbf{B}$) and the axion term
$b_{0}$ is absent from the Maxwell equations.

There are very few measurements of the Kerr angle in WSMs reported in the
literature. Most of them measure the Faraday rotation. In Ref. Okamura2020, ,
however, a large Kerr rotation of $\theta_{K}=0.056$ rad was measured in the
magnetic WSM Co3Sn2S${}_{2},$ which hosts three pairs of Weyl nodesWang2017 ,
in a weak magnetic field at a photon energy of $\hslash\omega=0.1$ eV (in the
infrared spectrum). This angle is large in comparison with that measured in
ferromagnetic metals.

In this paper, we used the simplest hamiltonian possible for a time-reversal
symmetry broken WSM in order to isolate the effect of the axion terms $b_{0}$
and/or $b_{z}$ on the Kerr rotation. To our knowledge, there is a present no
known WSM with this idealized electronic dispersion and a single pair of Weyl
nodes. Most WSMs have several pairs or nodes with a certain amount of tilt,
anisotropy, and nonlinearity in the dispersion. From a previous study by one
of usParent2020  where the axion terms were not considered, we know that the
Kerr rotation, in a magnetic field, is also dependent on the tilt of each
node. It is, of course, possible to extend the approach used in our paper to a
WSM with an arbitrary number of pairs of nodes (possibly tilted) by
calculating the total conductivity tensor if the parameters of each node as
well as their position in energy with respect to the commun Fermi level are
known. It would be difficult, then to isolate the contribution of the axion
terms to the Kerr rotation.

Magnetic Weyl semimetals with a single pair of Weyl nodes are hard to find but
recent $ab$ $initio$ calculationsWang2019 ; Soh2019  and ARPESSoh2019
measurements have shown that EuCd2As2 in a magnetic field greater than
$B_{c}=1.6$ T applied along its $c$ axis is a Weyl semimetal with a single
pair of tilted Weyl nodes. In this WSM, $b_{0}=0$ and $b_{z}=2.6\times 10^{8}$
m-1 and $b_{z}$ is found to be almost constant for $B>B_{c}.$ The magnetic
Weyl semimetal K2Mn3(AsO4)3 is also predicted by first-principles calculations
to host a single pair of nodesNie2022 . In both cases, however, the low-energy
hamiltonian is more complex than the one used in our paper. Moreover, Kerr
rotation measurements on these model Weyl semimetals have not yet been done.

###### Acknowledgements.

R. C. was supported by a grant from the Natural Sciences and Engineering
Research Council of Canada (NSERC).

##  Appendix A CONDUCTIVITY TENSOR OF THE WSM FOR$\ \mathbf{B}=0$

The conductivity of a Weyl node is the sum of the interband and intraband
contributions. The interband term is obtained by calculating the current-
current response function in the clean limit as given by Eq. (II). At $T=0$ K,
using the results of Sec. II, we find

$\displaystyle\sigma_{\chi,inter}\left(\omega\right)$ $\displaystyle=$
$\displaystyle\frac{e^{2}\omega}{12hv_{F}}\left[\theta\left(\omega-2v_{F}\left|k_{F,\chi}\right|\right)\right.$
$\displaystyle\left.-\frac{i}{\pi}\ln\left(\left|\frac{\omega^{2}-4v_{F}^{2}k_{c}^{2}}{\omega^{2}-4v_{F}^{2}k_{F,\chi}^{2}}\right|\right)\right],$

where $v_{F}k_{c}$ is a an ultraviolet cutoff and $k_{F,\chi}$ is the Fermi
wave vector in each node which is related to $b_{0}$ and the Fermi level
$e_{F}$ by Eq. (17).

For the intraband conductivity, it is necessary to include disorder and the
retarded current-current response is then given by

$\displaystyle\chi_{\chi}^{\left(xx\right)}\left(\omega\right)$
$\displaystyle=$
$\displaystyle\frac{e^{2}v_{F}^{2}}{V\hslash}\sum_{s}\sum_{\mathbf{k}}\left|\eta_{\chi,s}^{{\dagger}}\left(\mathbf{k}\right)\sigma^{\left(x\right)}\eta_{\chi,s^{\prime}}\left(\mathbf{k}\right)\right|^{2}$
$\displaystyle\times\int_{-\infty}^{+\infty}d\omega^{\prime}\int_{-\infty}^{+\infty}d\omega^{\prime\prime}A_{\chi,s}\left(\mathbf{k},\omega^{\prime}\right)A_{\chi,s}\left(\mathbf{k},\omega^{\prime\prime}\right)$
$\displaystyle\times\frac{F\left(\omega^{\prime\prime}\right)-F\left(\omega^{\prime}\right)}{\omega+i\delta+\omega^{\prime\prime}-\omega^{\prime}},$

where $F\left(\omega\right)=1/\left(e^{\beta\hslash\omega}+1\right)$ with
$\beta=1/k_{B}T$ and $A_{\chi,s}\left(\mathbf{k},\omega\right)$ is the
spectral weight of the disorder averaged Green’s function $\left\langle
G_{\chi,s}^{R}\left(\mathbf{k},\omega\right)\right\rangle$ which is given by

$\left\langle
G_{\chi,s}^{R}\left(\mathbf{k},\omega\right)\right\rangle=\int_{-\infty}^{+\infty}d\omega\frac{A_{\chi,s}\left(\mathbf{k},\omega\right)}{\omega+i\delta-\omega}.$
(118)

We take for the spectral weight the Lorentzian form

$A_{\chi,s}\left(\mathbf{k},\omega\right)=\frac{\Gamma/\pi}{\left(\omega-\left(E_{\chi,s}\left(\mathbf{k}\right)-\hslash
v_{F}k_{F,\chi}\right)/\hslash\right)^{2}+\Gamma^{2}},$ (119)

where $\Gamma=1/2\tau$ and $\tau$ is the scattering time. After a lengthy
calculation, we recover the result (the real part of this conductivity is
derived in Ref. Carbotte2016, ).

$\displaystyle\sigma_{\chi,intra}\left(\omega\right)$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{6\pi^{2}\hslash^{3}v_{F}}\frac{\tau}{1+\omega^{2}\tau^{2}}\left[e_{F,\chi}^{2}+\frac{1}{3}\hslash^{2}\omega^{2}\right.$
$\displaystyle\left.+\frac{\hslash^{2}}{4\tau^{2}}+i\omega\tau\left(e_{F,\chi}^{2}-e_{C}^{2}\right)\right],$

where $e_{C}$ is an energy cutoff in the valence band. The constant
$e_{C}^{2}$ can be fixed by requiring that $\sigma_{\chi}\left(\omega\right)$
satisfies the Kramers-Kroning relations. We find in this way that
$e_{C}^{2}=\hslash^{2}/12\tau^{2}.$

##  Appendix B CONDUCTIVITY TENSOR OF THE WSM FOR $\mathbf{B}\neq 0$

Formulas for the absorptive parts of the dynamic conductivity tensor due to
interband transitions in a simple two-node model with $b_{0}=0$ are given in
Ref. Carbotte2013, . In this Appendix, we extend these results to compute the
complete dynamic conductivity tensor in the ultra quantum limit, allowing for
different populations for the two nodes. We use the dimensionless variables
defined in Eq. (94) and the current-current response tensor defined in Eq.
(III). In the ultra-quantum limit where the Fermi level is in the chiral level
i.e., we must respect the condition $\left|e_{F,\chi,r}\right|<\sqrt{2}.$ At
$T=0$ K, we have for the filling factors

$\displaystyle\left\langle n_{0,\chi}\left(k\right)\right\rangle$
$\displaystyle=$ $\displaystyle\Theta\left(k_{F,\chi}+\chi k\right),$ (121)
$\displaystyle\left\langle n_{n>0,s=-1,\chi}\left(k\right)\right\rangle$
$\displaystyle=$ $\displaystyle 1,\,$ (122) $\displaystyle\left\langle
n_{n>0,s=1,\chi}\left(k\right)\right\rangle$ $\displaystyle=$ $\displaystyle
0,$ (123)

where $\Theta\left(x\right)$ the step function. We have defined $k_{F,\chi}$
as the Fermi wave vector measured with respect to the Dirac point in the node
considered. Depending on the choice of $b_{0}$ and $e_{F,\chi},$ electrons or
holes can be present in a node. After a lengthy but straightforward
calculation, we obtain the following expressions for the interband
contribution to the conductivity tensor and for $\omega>0$ (the solution for
$\omega<0$ can be obtained by symmetry)

$\displaystyle\operatorname{Re}\left[\sigma_{xx}\left(\omega_{r}\right)\right]$
$\displaystyle=$
$\displaystyle\frac{e^{2}}{\hslash\ell}\frac{1}{8\pi\omega_{r}}\left[\Theta\left(\omega_{r}-\left(x_{F}+\sqrt{x_{F}^{2}+2}\right)\right)+\Theta\left(\omega_{r}-\left(-x_{F}+\sqrt{x_{F}^{2}+2}\right)\right)\right]$
$\displaystyle+\frac{e^{2}}{\hslash\ell}\sum_{n>1}\frac{\omega_{r}^{4}-4}{16\pi\omega_{r}^{3}}\sum_{\pm}\frac{\left(1+\frac{x_{\pm}}{e_{n}\left(x_{\pm}\right)}\right)\sqrt{1+\frac{x_{\pm}}{e_{n+1}\left(x_{\pm}\right)}}\sqrt{1+\frac{x_{\pm}}{e_{n-1}\left(x_{\pm}\right)}}}{\sqrt{\left(\omega_{r}^{2}+2\right)^{2}-8\omega_{r}^{2}n}}\Theta\left(\omega_{r}-\Delta_{n}\left(0\right)\right),$
$\operatorname{Im}\left[\sigma_{xx}\left(\omega_{r}\right)\right]=\frac{e^{2}}{\hslash\ell}\frac{1}{8\pi^{2}\omega_{r}}\left[\log\left[\left|\frac{\omega_{r}^{4}-4\omega_{r}^{2}x_{F}^{2}-4\omega_{r}^{2}+4}{4}\right|\right]+\omega_{r}^{2}\sum_{n>1}2\int_{-\infty}^{+\infty}dx\frac{\Lambda_{n}\left(x\right)}{\Delta_{n}\left(x\right)}\frac{1}{\omega_{r}^{2}-\Delta_{n}\left(x\right)^{2}}\right],$
(125)
$\operatorname{Re}\left[\sigma_{xy}\left(\omega_{r}\right)\right]=-\frac{e^{2}}{\hslash\ell}\frac{1}{8\pi^{2}\omega_{r}}\log\left[\left|\frac{2-\omega_{r}^{2}+2\omega_{r}x_{F}}{2-\omega_{r}^{2}-2\omega_{r}x_{F}}\right|\right],$
(126)
$\operatorname{Im}\left[\sigma_{xy}\left(\omega_{r}\right)\right]=\frac{e^{2}}{\hslash\ell}\frac{1}{8\pi\omega_{r}}\left[\Theta\left(\omega_{r}-\left(x_{F}+\sqrt{x_{F}^{2}+2}\right)\right)-\Theta\left(\omega_{r}-\left(-x_{F}+\sqrt{x_{F}^{2}+2}\right)\right)\right],$
(127)
$\operatorname{Re}\left[\sigma_{zz}\left(\omega_{r}\right)\right]=\frac{e^{2}}{\hslash\ell}\frac{1}{2\pi}\sum_{n>0}\frac{2n}{\omega_{r}^{2}\sqrt{\omega_{r}^{2}-8n}}\theta\left(\omega_{r}-2\sqrt{2n}\right),$
(128)
$\operatorname{Im}\left[\sigma_{zz}\left(\omega_{r}\right)\right]=\frac{e^{2}}{\hslash\ell}\frac{\omega_{r}}{\pi^{2}}\sum_{n>0}n\int_{0}^{+\infty}dx\frac{1}{\left(x^{2}+2n\right)^{3/2}}\frac{1}{\omega_{r}^{2}-4\left(x^{2}+2n\right)},$
(129)

These formulas are for one node and $x_{F}\equiv k_{F}\ell.$ We have also
defined the functions

$e_{n}\left(x\right)=\sqrt{x^{2}+2n},\,$ (130)
$\,\Delta_{n}\left(x\right)=e_{n}\left(x\right)+e_{n-1}\left(x\right),$ (131)
$\,x_{n,\pm}=\pm\sqrt{\left(\frac{\omega_{r}^{2}+2}{2\omega_{r}}\right)^{2}-2n},$
(132) $\displaystyle\Lambda_{n}\left(x\right)$ $\displaystyle=$
$\displaystyle\left(1+\frac{x}{e_{n}\left(x\right)}\right)\sqrt{1+\frac{x}{e_{n+1}\left(x\right)}}$
(134) $\displaystyle\times\sqrt{1+\frac{x}{e_{n-1}\left(x\right)}}.$

Using the definitions of the eigenspinors given in the main text, it is easy
to show that the only difference between the contribution of both nodes to the
conductivity comes from the difference in the local Fermi wave vector
$k_{F,\chi}$ which depends on the Fermi level $e_{F}$ and $b_{0}$ through Eq.
(17).

Defining a dimensionless conductivity tensor
$\overline{\sigma}_{\chi,\left(\alpha\beta\right)}$ by

$\sigma_{\chi,\left(\alpha\beta\right)}\left(\omega\right)=\frac{e^{2}}{\hslash\ell}\overline{\sigma}_{\chi}^{\left(\alpha,\beta\right)}\left(\omega\right),$
(135)

we have for the dielectric tensor of the two nodes

$\varepsilon_{\alpha\beta}=\delta_{\alpha\beta}+\frac{2\pi^{2}\kappa
i}{\omega_{r}v_{F,r}}\sum_{\chi}\overline{\sigma}_{\chi,\left(\alpha\beta\right)}\left(\omega\right).$
(136)

As in Appendix A, the contribution of the intra-Landau-level transitions in
$n=0$ to the conductivity tensor must be calculated by including disorder. In
our case, the intraband transitions are present only in the matrix element
$\Gamma_{\chi;0;0}^{\left(z\right)}\left(k\right)$ so that we need to include
disorder only in $\chi_{\chi}^{\left(z,z\right)}\left(\omega\right).$ We have
for the intra-Landau-level current response function for one node

$\displaystyle\chi_{\chi,\left(zz\right)}\left(\omega\right)$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{2\pi\ell^{2}\hslash^{2}}\int\frac{dk}{2\pi}\left|\Gamma_{\chi;0,0}^{\left(z\right)}\left(k\right)\right|^{2}$
$\displaystyle\times\int_{-\infty}^{+\infty}d\omega^{\prime}\int_{-\infty}^{+\infty}d\omega^{\prime\prime}A_{\chi,0}\left(k,\omega^{\prime}\right)A_{\chi,0}\left(k,\omega^{\prime\prime}\right)$
$\displaystyle\times\frac{F\left(\omega^{\prime\prime}\right)-F\left(\omega^{\prime}\right)}{\omega+i\delta+\omega^{\prime\prime}-\omega^{\prime}}.$

We again choose a Lorentzian for the spectral weight which is given by

$A_{\chi,0}\left(k,\omega\right)=\frac{\Gamma/\pi}{\left(\omega-\left(E_{\chi,0}\left(k\right)-e_{F,\chi}\right)/\hslash\right)^{2}+\Gamma^{2}}$
(138)

and $e_{F,\chi}$ is the local Fermi level measured with respect to the Dirac
point in node $\chi$. At $T=0$ K,

$\displaystyle\chi_{\chi,\left(zz\right)}\left(\omega\right)$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{\hslash^{2}}\frac{1}{2\pi\ell^{2}}\int\frac{dk}{2\pi}\left|\Gamma_{\chi;0,0}^{\left(z\right)}\left(k\right)\right|^{2}$
$\displaystyle\times\left[\int_{-\infty}^{e_{F,\chi}/\hslash}d\omega^{\prime}a_{\chi}\left(k,\omega^{\prime}\right)g_{\chi}\left(k,\omega^{\prime}+\omega\right)\right.$
$\displaystyle+\int_{-\infty}^{e_{F,\chi}/\hslash}d\omega^{\prime}a_{\chi}\left(k,\omega^{\prime}\right)g_{\chi}\left(k,\omega^{\prime}-\omega\right)$
$\displaystyle-i\pi\left.\int_{e_{F}/\hslash-\omega}^{e_{F}/\hslash}d\omega^{\prime}a_{\chi}\left(k,\omega^{\prime}\right)a_{\chi}\left(k,\omega^{\prime}+\omega\right)\right],$

with the functions

$a_{\chi}\left(k,\omega\right)=\frac{\Gamma/\pi}{\left(\omega-
E_{\chi,0}\left(k\right)\left(k\right)/\hslash\right)^{2}+\Gamma^{2}}$ (140)

and

$g_{\chi}\left(k,\omega\right)=\frac{\omega-
E_{\chi,0}\left(k\right)/\hslash}{\left(\omega-
E_{\chi,0}\left(k\right)/\hslash\right)^{2}+\Gamma^{2}}.$ (141)

Evaluating these expressions analytically, we get for the intra-Landau-level
conductivity

$\sigma_{\chi,\left(zz\right)}\left(\omega\right)=\frac{e^{2}}{\hslash\ell}\frac{v_{F}\tau}{4\pi^{2}\ell}\frac{1}{1-i\omega\tau}.$
(142)

This expression is independent of $e_{F}$ and $b_{0}$ i.e., the same for both
nodes.

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