# Light control with Weyl semimetals

CGCheng Guo  VsAViktar S. Asadchy  BZBo Zhao  SFShanhui Fan  Department of
Applied Physics, Stanford University, 94305 Stanford, California, USA  Ginzton
Laboratory and Department of Electrical Engineering, Stanford University,
94305 Stanford, California, USA  Department of Electronics and
Nanoengineering, Aalto University, 02150 Espoo, Finland  Department of
Mechanical Engineering, University of Houston, 77204 Houston, Texas, USA

###### Abstract

Weyl semimetals are topological materials whose electron quasiparticles obey
the Weyl equation. They possess many unusual properties that may lead to new
applications. This is a tutorial review of the optical properties and
applications of Weyl semimetals. We review the basic concepts and optical
responses of Weyl semimetals, and survey their applications in optics and
thermal photonics. We hope this pedagogical text will motivate further
research on this emerging topic.

Weyl semimetal,

topological materials,

axion electrodynamics,

nanophotonics,

thermal photonics,

optical nonreciprocity,

###### keywords:

Tutorial

[id=n1]Equal contributor

[id=n2]Correspondence<EMAIL_ADDRESS>

[id=n3]Correspondence<EMAIL_ADDRESS>
## Introduction

Weyl semimetals are topological materials whose low-energy excitations obey
the Weyl equation [1, 2]. In a Weyl semimetal, the conduction and valence
bands touch at discrete points in momentum space, called Weyl nodes. Weyl
nodes are monopoles of the Berry curvature and are robust under generic
perturbations. The quasiparticles near the Weyl nodes are analogous to Weyl
fermions in high-energy physics [3]; they exhibit linear dispersion and well-
defined chirality.

The nontrivial topology of Weyl semimetals leads to many unusual electronic,
magnetic, thermal, and optical properties [2, 4, 5]. These intriguing features
have been extensively studied in the literature. Besides these fundamental
interests, Weyl semimetals may also enable new opportunities in practical
applications. For example, photonic applications include compact optical
isolators and circulators [6, 7, 8], orbital angular momentum detectors [9,
10], higher-order harmonic generation [11, 12, 13], and nonreciprocal thermal
emitters [14, 15, 16] among many others. However, such an application-oriented
exploration is still at an early stage, which requires more joint efforts from
scientists and engineers.

This text is a tutorial review of Weyl semimetals that should be of interest
to researchers working in photonics, applied physics, and optical engineering.
We start with the basic concepts of semimetals and Weyl semimetals (Sec. 1).
Then, we review the formalism of axion electrodynamics and derive the optical
responses of Weyl semimetals (Sec. 2). Next, we survey the broad potential
applications of Weyl semimetals in optics (Sec. 3). Finally, we discuss the
applications of Weyl semimetals in thermal photonics (Sec. 4). We hope that
our survey will motivate further exploration of photonic applications with
Weyl semimetals.

##  1 Semimetals and Weyl semimetals

Weyl semimetals are a special class of semimetals. They exhibit common
properties of semimetals as well as some unique characteristics. This section
provides a brief introduction to semimetals and Weyl semimetals. In Sec. 1.1,
we review the basic concept and common properties of semimetals. We also
discuss typical behaviors and material examples of conventional semimetals. In
Sec. 1.2, we review the basic concept and physical properties of Weyl
semimetals.

###  1.1 Semimetals

According to the band theory [17, 18], solids can be classified as insulators,
semiconductors, semimetals, and metals (Fig. 1). An insulator or a
semiconductor has a band gap between the valence and conduction bands; the
band gap is larger for an insulator ($>4\text{\,}\mathrm{e}\mathrm{V}$) than
for a semiconductor ($<4\text{\,}\mathrm{e}\mathrm{V}$). A _semimetal_ has a
very small overlap between the conduction and valence bands and a negligible
density of states at the Fermi level. A metal has a partially filled
conduction band and an appreciable density of states at the Fermi level.

Figure 1: Classification of solids. Schematic density of states (DOS) of
insulators, semiconductors, semimetals, and metals. $E_{F}$ denotes the Fermi
level.

Different band structures lead to different physical properties. For example,
the carrier concentration is $>{10}^{22}\text{\,}\mathrm{c}\mathrm{m}^{-3}$
for normal metals, $\sim{10}^{17}{10}^{20}\text{\,}\mathrm{c}\mathrm{m}^{-3}$
for semimetals, and $\sim{10}^{6}{10}^{13}\text{\,}\mathrm{c}\mathrm{m}^{-3}$
for intrinsic semiconductors such as GaAs, Si, and Ge. Consequently, the
electrical conductivities are
$\sim{10}^{5}{10}^{6}\text{\,}{\mathrm{\SIUnitSymbolOhm}}^{-1}\text{\,}{\mathrm{cm}}^{-1}$
for normal metals,
$\sim{10}^{4}\text{\,}{\mathrm{\SIUnitSymbolOhm}}^{-1}\text{\,}{\mathrm{cm}}^{-1}$
for semimetals, and
$\sim{10}^{-8}{10}^{-1}\text{\,}{\mathrm{\SIUnitSymbolOhm}}^{-1}\text{\,}{\mathrm{cm}}^{-1}$
for intrinsic semiconductors [18].

Semimetals, the focus of our study, are probably the least known among the
four types. An intrinsic semimetal has an equal number of electrons and holes.
Like a normal metal, its conductivity increases as the temperature is lowered.
Like a semiconductor, it can be doped with proper impurities to vary the
number of electrons and holes. Its electronic properties are also sensitive to
pressure since pressure changes the internuclear distances, which sensitively
changes the amount of band overlap and causes large changes in the carrier
concentrations. Besides low carrier concentrations, semimetals typically have
small effective masses and high dielectric constants. Nonmagnetic semimetals
typically exhibit high diamagnetic susceptibilities and huge electron
$g$-values [18].

Figure 2: Conventional versus Weyl semimetals. Schematic band structure for
(a) conventional semimetals, (b) Weyl semimetals.

A conventional semimetal is a “semiconductor” with a negative indirect bandgap
(Fig. 2a). In an intrinsic semimetal, the bottom of the conduction band is
slightly lower than the top of the valence band, and the Fermi level sits in
between. Hence, the semimetal has charge carries of both types (holes and
electrons). Typically, the pockets of electrons and holes are located at
different positions in the wavevector space [18].

Classic examples of semimetals include the group 5A elements: arsenic,
antimony, and bismuth [18]. They have five valence electrons per atom, so if
there were one atom per primitive cell the material would be a metal. However,
there are two atoms per primitive cell. With the ten valence electrons, the
crystal could be an insulator but there is a small overlap in energy between
the conduction and valence bands, resulting in semimetal behavior [19].

Another well-known semimetal is graphite. It consists of stacked layers of
graphene. Since the interlayer bonding is very weak, the band overlap is tiny
[20]. Hence, graphite is a semimetal, and the Fermi surface consists primarily
of tiny pockets of electrons and holes at different wavevectors, with carrier
densities of around
$n_{e}=n_{h}=$3\text{\times}{10}^{18}\text{\,}\mathrm{c}\mathrm{m}^{-3}$$
[17].

###  1.2 Weyl semimetals

Weyl semimetals are a unique class of semimetals. Unlike conventional
semimetals, in Weyl semimetals, the valence and conduction bands touch at
discrete points in the wavevector space. Near the touching point, the band
dispersion is linear (Fig. 2b).

Weyl semimetals are named after Hermann Weyl (1885-1955). In 1929, Weyl
proposed the Weyl equation [21], which simplifies the Dirac equation for a
_massless_ relativistic spin $\frac{1}{2}$ particle:

$i\hbar\frac{\partial\Psi}{\partial t}=\hat{H}\Psi=\pm
c\bm{p}\cdot\bm{\sigma}\Psi.$ (1)

Here $\hbar$ is the reduced Planck constant, $c$ is the speed of light,
$\bm{p}=(p_{x},p_{y},p_{z})$ is the momentum operator,
$\bm{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$ is a vector whose components
are the Pauli matrices, and $\Psi$ is a two-component field called a Weyl
spinor [3]. The Weyl equation describes a massless particle with linear
dispersion, called a Weyl fermion. The $\pm$ sign reflects the existence of
two types of Weyl fermions: right-handed ($+$) and left-handed ($-$). Hence,
unlike the Dirac equation, the Weyl equation violates parity.

Physicists have been searching for Weyl fermions in nature. However, none of
the observed elementary particles are Weyl fermions [2]. It was thought that
neutrinos could be Weyl fermions. However, the discovery of neutrino
oscillation in 1998 [22] shows that neutrinos are massive and thus cannot be
Weyl fermions.

Despite their absence in high-energy physics, Weyl fermions may be realized in
condensed-matter systems. In 1937, Conyers Herring proposed the concept of
Weyl semimetals [1]. He realized that when two electronic bands cross
accidentally, the generic band dispersions near a touching point are linear in
all directions. In the simplest scenario, the effective Hamiltonian reads
[23]:

$\hat{H}=\pm v_{F}\bm{p}\cdot\bm{\sigma}$ (2)

where $\bm{p}$ is the momentum deviation from the touching point, and $v_{F}$
is the Fermi velocity. The resemblance between Eqs. (1) and (2) is evident. As
a result, these touching points are called Weyl nodes, and the quasiparticles
near them are reminiscent of Weyl fermions. Each Weyl point has a definite
chirality, either right-handed ($+$) or left-handed ($-$). For crystals where
the wavevector space is the Brillouin zone and hence is compact, the Nielsen-
Ninomiya no-go theorem asserts that left- and right-handed Weyl points always
appear in pairs [24, 25, 26, 27]. Hence, it is impossible that only a single
Weyl point exists in the momentum space for solid state systems.

Weyl points are robust to generic perturbations: The Hamiltonian remains
gapless while Weyl nodes move around. This is because for a Hermitian matrix,
having a pair of coalescing eigenvalues is an effect with codimension three;
generically, one needs to tune three real parameters to realize such an
effect. This fact, known as the von Neumann-Wigner theorem [28], explains why
generically the accidental degeneracies of two bands only occur at isolated
points in the three-dimensional momentum space and why such degeneracies
persist under perturbation. From a topological perspective, the robustness of
Weyl points can be understood from Berry curvature [29]. Berry curvature
arises from the variation of the periodic part of the Bloch wave function with
respect to wavevetors, and is mathematically analogous to a magnetic field in
momentum space. Weyl points are monopoles of Berry curvature with quantized
charges, which are analogous to quantized magnetic monopoles for magnetic
fields. Weyl points can only be destroyed when two Weyl nodes of opposite
chiralities are moved together and annihilated with each other.

For a Weyl semimetal, the existence of Weyl nodes in the bulk band structure
leads to the presence of Fermi arc surface states when the Weyl semimetal is
truncated. Such surface states correspond to an open Fermi arc that connects
the projection in the surface Brillouin zone of two Weyl nodes of opposite
chiralities. The existence of such surface Fermi arcs is protected by the
nontrivial chiral charge of the Weyl nodes [30, 31]. This is an example of the
general principle of bulk-boundary correspondence in topological materials.

Although the Weyl semimetal was theoretically proposed many decades ago, its
experimental demonstration was quite recent. A material must satisfy some
necessary conditions to be a potential candidate for Weyl semimetals [32].
First, it must break either time-reversal or spatial inversion symmetry [2].
In a system that satisfies both time-reversal and spatial inversion symmetry,
all the bands must be doubly degenerate in the whole wavevector space due to
the Kramers’ degeneracy theorem [33]; this excludes the existence of Weyl
nodes, which only appear as the accidental degeneracy of two non-degenerate
bands. The minimum numbers of Weyl nodes are $2$ and $4$ for Weyl semimetals
that break time-reversal symmetry and inversion symmetry, respectively.
Second, the Weyl nodes must be located near the Fermi level, so that Weyl
fermions may emerge as low-energy excitations.

Weyl semimetals were first discovered in 2015 [34, 35] in non-centrosymmetric
crystals, TaAs family (TaAs [34, 35], TaP [36, 37], NbAs [38], and NbP [39,
40, 41]). Later, magnetic Weyl semimetals that break time-reversal symmetry
have also been discovered [42, 43, 44]. A number of other Weyl semimetals have
since been discovered, as reviewed in Ref. [45].

Weyl semimetals exhibit a rich variety of novel phenomena. Besides surface
Fermi arc states, they also exhibit chiral anomaly [46, 47, 48],
unconventional charge and heat transport [49, 41], strain-induced axial gauge
fields [50, 51, 52], novel collective modes [53], and so on. The properties of
Weyl semimetals have been summarized in numerous reviews and monographs: For
general overviews, see Refs. [54, 2, 55]; for electronic properties, see Ref.
[4]; for magnetic properties, see Refs. [56, 57, 58]; for transport
properties, see Refs. [59, 60, 61, 62, 63, 64]; for topological properties,
see Refs. [65, 66]; for _ab initio_ calculation, see Ref. [67]; for
experimental studies, see Ref. [68, 69, 70, 71]; for material properties, see
Ref. [45]. Complementary to these works, the present tutorial review will
focus on the optical properties of Weyl semimetals.

##  2 Optical properties of Weyl semimetals

In this section, we derive the linear optical properties of Weyl semimetals.
We consider the simplest Weyl semimetal with two nodes separated by $2\bm{b}$
in their wavevectors and $2\hbar b_{0}$ in their energy (Fig. 3a). The
constitutive relations for such an ideal Weyl semimetal are:

$\bm{D}=\varepsilon_{D}(\omega)\bm{E}+\frac{ie^{2}}{2\pi^{2}\hbar\omega}\left(-b_{0}\bm{B}+\bm{b}\times\bm{E}\right),\qquad\bm{H}=\frac{1}{\mu_{0}}\bm{B}.$
(3)

The derivation of Eq. (3) is the main aim of this section. The expression of
$\bm{D}$ has two terms; the first term is referred to as the Dirac term and
the second the axion term. The Dirac term describes the permittivity of a
corresponding Dirac semimetal (Fig. 3b), i.e., a Weyl semimetal with two
overlapping Weyl nodes ($\bm{b}=\bm{0},b_{0}=0$). The axion term captures the
effects of Weyl node separation. Its name is taken from axion electrodynamics.
Its first term represents the chiral magnetic effect, while the second term
represents the anomalous Hall effect [72, 2].

In the following subsections, we provide a step-by-step derivation of Eq. (3)
and a closed-form expression of $\varepsilon_{D}(\omega)$. Then we discuss
their physical consequences. We use SI units and $e^{-i\omega t}$ convention.

###  2.1 A minimal low-energy model of Weyl semimetals

Figure 3: A minimal low-energy model. Schematic band dispersion for (a) a Weyl
semimetal, (b) a Dirac semimetal. In (a), the blue and red lines correspond to
right- and left-handed quasiparticles; the blue and red dots denote the right-
and left-handed Weyl nodes. In (b), these lines/dots coalesce.

According to the Nielsen-Ninomiya theorem [24, 25, 26], a Weyl semimetal
always include an even number of Weyl nodes and the total chirality of all
nodes must vanish. Hence, the simplest Weyl semimetal has a single pair of
Weyl nodes. It can be described by the following minimal low-energy
Hamiltonian:

$H=\begin{pmatrix}\hbar v_{F}\bm{\sigma}\cdot\left(\bm{k}+\bm{b}\right)-\hbar
b_{0}I&0\\\ 0&-\hbar v_{F}\bm{\sigma}\cdot\left(\bm{k}-\bm{b}\right)+\hbar
b_{0}I\end{pmatrix},$ (4)

which characterizes two Weyl nodes of opposite chirality that are separated by
$2\bm{b}$ in wavevector and $2\hbar b_{0}$ in energy. Here $\bm{b}$ is also
known as the chiral shift [73, 4], $\bm{k}$ denotes the wavevector, $v_{F}$
denotes the Fermi velocity, $\bm{\sigma}$ is the vector of Pauli matrices, and
$I$ is the $2\times 2$ identity matrix.

From Eq. (4), the band dispersion is determined as

$E_{\lambda}(\bm{k})=-\lambda\hbar b_{0}\pm\hbar
v_{F}\left|\bm{k}+\lambda\bm{b}\right|,$ (5)

where $\lambda=\pm 1$ is the node’s chirality. Fig. 3 shows the scheme of band
dispersion in the cases (a) $\bm{b}\neq\bm{0}$ and $b_{0}\neq 0$ and (b)
$\bm{b}=\bm{0}$ and $b_{0}=0$. The latter case, where the two Weyl nodes
coalesce, is referred to as a Dirac semimetal.

We make a few remarks. First, the model is paradigmatic and highlights the two
hallmark properties of Weyl semimetals: linear dispersion and chiral Weyl
nodes. Second, the model suffices to describe the universal low-energy
phenomena in Weyl semimetals. It can be shown by explicit calculation that the
general properties do not change if one uses a more realistic periodic two-
band model [74, 4]. Third, nonzero $\bm{b}$ and $b_{0}$ require the breaking
of time-reversal and parity-inversion symmetries, respectively. Fourth, we
note that the opposite sign convention of $\bm{b}$ and $b_{0}$ has been used
in some works [4]. Finally, we note that the ideal magnetic Weyl semimetal
with the minimum number of Weyl points may be realized in real materials such
as K2Mn3(AsO4)3 [75], or EuCd2As2 subjected to an external magnetic field
[76].

###  2.2 Axion electrodynamics

To characterize the electromagnetic properties of the Weyl semimetal as
described by Eq. (4), we first briefly review the Lagrangian approach to
Maxwell’s equations [77]. We start with the conventional Maxwell Lagrangian
density:

$\mathcal{L}_{0}(\bm{r},t)=\frac{\epsilon_{0}}{2}\bm{E}^{2}-\frac{1}{2\mu_{0}}\bm{B}^{2}-\rho\phi+\bm{J}\cdot\bm{A}.$
(6)

Here, the potentials $\phi(\bm{r},t)$ and $\bm{A}(\bm{r},t)$ are the
independent variables; the electric field $\bm{E}(\bm{r},t)$ and the magnetic
field $\bm{B}(\bm{r},t)$ are expressed in terms of potentials:

$\bm{B}=\nabla\times\bm{A},\qquad\bm{E}=-\nabla\phi-\frac{\partial{\bm{A}}}{\partial
t}.$ (7)

Equations (7) already imply two of the four Maxwell’s equations:

$\nabla\cdot\bm{B}=0,\qquad\nabla\times\bm{E}+\frac{\partial\bm{B}}{\partial
t}=0.$ (8)

We define the action

$S_{0}=\int\mathrm{d}t\,\mathrm{d}^{3}\bm{r}\,\mathcal{L}_{0}(\bm{r},t),$ (9)

and require $S_{0}$ to be stationary with respect to the variations of
$\bm{\phi}(\bm{r},t)$ and $\bm{A}(\bm{r},t)$. Then we obtain the other two of
the four Maxwell’s equations:

$\nabla\cdot\bm{E}=\frac{\rho}{\epsilon_{0}},\qquad\nabla\times\bm{B}=\mu_{0}\bm{J}+\frac{1}{c^{2}}\frac{\partial
E}{\partial t}.$ (10)

See Ref. [77, §24] for detailed derivation of Eq. (10).

Now, we introduce the formalism of axion electrodynamics [78]. Axion
electrodynamics was first proposed in high-energy physics to solve the strong
CP problem in quantum chromodynamics [79]. In condensed matter physics, axion
electrodynamics is used to understand the properties of 3He [80], topological
insulators [81, 82, 83, 68], and Weyl semimetals [74, 84, 85, 86]. For Weyl
semimetals, axion electrodynamics describes the topological effects of Weyl
node separation. The usage of axion electrodynamics in Weyl semimetals will be
discussed in the next subsection.

Axion electrodynamics is generated by adding to Eq. (6) an additional term
(the $\theta$ term) [78, 87]:

$\mathcal{L}_{\theta}=2\alpha\sqrt{\frac{\epsilon_{0}}{\mu_{0}}}\frac{\theta}{2\pi}\bm{E}\cdot\bm{B},$
(11)

where $\alpha=\frac{e^{2}}{4\pi\epsilon_{0}\hbar c}$ is the fine structure
constant, and $\theta(\bm{r},t)$ is a pseudoscalar field. Now

$S=\int\mathrm{d}t\,\mathrm{d}^{3}\bm{r}\,\left[\mathcal{L}_{0}(\bm{r},t)+\mathcal{L}_{\theta}(\bm{r},t)\right].$
(12)

This does not modify Eqs. (8), but changes Eqs. (10) into (see Ref. [88] for
detailed derivation):

$\displaystyle\nabla\cdot\bm{E}$
$\displaystyle=\frac{\rho}{\epsilon_{0}}-2c\alpha\nabla\left(\frac{\theta}{2\pi}\right)\cdot\bm{B},$
(13) $\displaystyle\nabla\times\bm{B}$
$\displaystyle=\mu_{0}\bm{J}+\frac{1}{c^{2}}\frac{\partial\bm{E}}{\partial
t}+\frac{2\alpha}{c}\left[\frac{\partial}{\partial
t}\left(\frac{\theta}{2\pi}\right)\bm{B}+\nabla\left(\frac{\theta}{2\pi}\right)\times\bm{E}\right].$
(14)

These modified Maxwell’s equations contain most (although not all) of the new
physics of axion electrodynamics [89]. Here we note that the $\theta$ term
enters the equations only through derivatives. This is as expected since if
$\mathcal{\theta}$ is constant [90], the $\theta$ term is a total derivative
and irrelevant for the equation of motion.

There is a different yet equivalent way to describe axion electrodynamics
[90], where one keeps the macroscopic Maxwell’s equations in their original
form:

$\displaystyle\nabla\cdot\bm{B}=0,\qquad\nabla\times\bm{E}=-\frac{\partial{\bm{B}}}{\partial
t},$ (15)
$\displaystyle\nabla\cdot\bm{D}=\rho,\qquad\nabla\times\bm{H}=\bm{J}+\frac{\partial\bm{D}}{\partial
t},$ (16)

with the modified constitutive relations:

$\bm{D}=\epsilon_{0}\bm{E}+2\alpha
c\epsilon_{0}\frac{\theta}{2\pi}\bm{B},\qquad\bm{H}=\frac{1}{\mu_{0}}\bm{B}-2\alpha\frac{1}{c\mu_{0}}\frac{\theta}{2\pi}\bm{E}.$
(17)

There is yet another way to describe axion electrodynamics if
$\theta(\bm{r},t)$ satisfies the additional condition:

$\nabla\left[\frac{\partial\theta}{\partial t}(\bm{r},t)\right]=\bm{0}.$ (18)

Then one can express Maxwell’s macroscopic equations in the frequency domain:

$\displaystyle\nabla\cdot\bm{B}=0,\qquad\nabla\times\bm{E}=i\omega\bm{B},$
(19)
$\displaystyle\nabla\cdot\bm{D}=\rho,\qquad\nabla\times\bm{H}=\bm{J}-i\omega\bm{D}.$
(20)

with the constitutive relation [91]:

$\bm{D}=\varepsilon_{0}\bm{E}+2\alpha
c\varepsilon_{0}\frac{1}{-i\omega}\left[\frac{\partial}{\partial
t}\left(\frac{\theta}{2\pi}\right)\bm{B}+\nabla(\frac{\theta}{2\pi})\times\bm{E}\right],\quad\bm{H}=\frac{1}{\mu_{0}}\bm{B}.$
(21)

Such a description is equivalent to the previous two under the assumption of
Eq. (18). It will be useful later when we discuss Weyl semimetals.

###  2.3 The axion term

It is known that the effects of Weyl node separation are fully described by an
axion term

$\theta(r,t)=2\bm{b}\cdot\bm{r}-2b_{0}t.$ (22)

Substituting Eq. (22) into Eq. (21) and replacing $\varepsilon_{0}$ with
$\varepsilon_{D}(\omega)$, we obtain Eq. (3). (Note that Eq. (22) satisfies
the assumption of Eq. (18).)

The above result is standard; its derivation is beyond the scope of this
tutorial. We only note that the derivation uses Fujikawa’s method in the
fermion path integral formulation; the $\theta$ term corresponds to the chiral
anomaly, a type of quantum anomalies that arises from a nontrivial Jacobian in
the change of path integral variables. We refer readers to Ref. [74] for the
original derivation of Eq. (22), Ref. [92] for fermion path integrals, and
Ref. [93] for quantum anomalies and Fujikawa’s method.

###  2.4 The Dirac term

Now we derive the Dirac term $\varepsilon_{D}(\omega)$ in Eq. (3), i.e., the
permittivity of a Dirac semimetal (Fig. 3b). The Dirac semimetal can be
described by a simple two-band Hamiltonian:

$H_{D}=\hbar v_{F}\bm{k}\cdot\bm{\sigma},$ (23)

while the spin degeneracy is taken into account by a degeneracy factor $g=2$.

From the Hamiltonian Eq. (23), we obtain the low-energy spectra:

$E_{\bm{k},s}=s\hbar v_{F}|\bm{k}|,$ (24)

where $s=\pm 1$ denote the band indices. The eigenstates are:

$\ket{\bm{k},+}=\begin{pmatrix}\cos{\frac{\vartheta}{2}}\\\
e^{i\varphi}\sin{\frac{\vartheta}{2}}\end{pmatrix},\qquad\ket{\bm{k},-}=\begin{pmatrix}-\sin{\frac{\vartheta}{2}}\\\
e^{i\varphi}\cos{\frac{\vartheta}{2}}\end{pmatrix},$ (25)

where $\vartheta$ and $\varphi$ are respectively the polar and azimuthal
angles of the three-dimensional $\bm{k}$.

Then $\varepsilon_{D}(\omega)$ can be determined by the standard linear
response theory [94, 95]:

$\varepsilon_{D}(\omega)=\varepsilon_{b}+i\frac{\sigma(\omega)}{\omega},$ (26)

where $\varepsilon_{b}$ is the background permittivity due to the other bands
and $\sigma(\omega)$ is the dynamic conductivity tensor due to the Dirac cone.
Assuming noninteracting electrons and local response, $\sigma(\omega)$ is
given by Kubo-Greenwood formula [96, 6]:

$\sigma_{\alpha\beta}(\omega)=\frac{-ie^{2}g\hbar}{V}\sum_{\bm{k},s,s^{\prime}}\frac{n(E_{\bm{k},s})-n(E_{\bm{k},s^{\prime}})}{E_{\bm{k},s}-E_{\bm{k},s^{\prime}}}\frac{\bra{\bm{k}s}\hat{v}_{\alpha}\ket{\bm{k}s^{\prime}}\bra{\bm{k}s^{\prime}}\hat{v}_{\beta}\ket{\bm{k}s}}{\hbar(\omega+i0)+E_{\bm{k},s}-E_{\bm{k},s^{\prime}}}.$
(27)

Here $\alpha,\beta=x,y,z$, $\omega$ is the incident light frequency, $V$ is
the volume, $\hat{v}_{\alpha}=\frac{1}{\hbar}\frac{\partial H_{D}}{\partial
k_{\alpha}}=v_{F}\sigma_{\alpha}$ is the velocity operator, $\ket{\bm{k}s}$
and $\ket{\bm{k}s^{\prime}}$ are the initial and final electronic state, and

$n(E)=\frac{1}{e^{(E-E_{F})/k_{B}T}+1}$ (28)

is the Fermi distribution where $E_{F}$ is the Fermi energy and $T$ is the
temperature.

Since the Hamiltonian Eq. (23) is isotropic in $\bm{k}$, the conductivity
tensor $\sigma(\omega)$ is also isotropic and can be treated as a scalar.
Detailed calculation shows [6]:

$\sigma(\omega)=\frac{e^{2}}{\hbar}\frac{gk_{F}}{24\pi}\Omega\tilde{G}(\Omega/2)+i\frac{e^{2}}{\hbar}\frac{gk_{F}}{24\pi^{2}}\left\\{\frac{4}{\Omega}\left[1+\frac{\pi^{2}}{3}(\frac{k_{B}T}{E_{F}})^{2}\right]+8\Omega\int_{0}^{\varepsilon_{c}}\frac{\tilde{G}(\varepsilon)-\tilde{G}(\Omega/2)}{\Omega^{2}-4\varepsilon^{2}}\varepsilon\mathrm{d}\varepsilon\right\\},$
(29)

where $k_{F}=E_{F}/\hbar v_{F}$, $\Omega=\hbar(\omega+i\tau^{-1})/E_{F}$,
$\tau^{-1}$ is the Drude damping rate, $\varepsilon_{c}=E_{c}/E_{F}$, and
$E_{c}$ is the cutoff energy beyond which the band dispersion is no longer
linear. Moreover, the function $\tilde{G}(x)$ in Eq. (29) reads:

$\tilde{G}(x)\equiv
n(-xE_{F})-n(xE_{F})=\frac{\sinh{(xE_{F}/k_{B}T)}}{\cosh{(E_{F}/k_{B}T)}+\cosh{(xE_{F}/k_{B}T)}}.$
(30)

The physical meaning of Eq. (29) is more transparent in the low-temperature
limit when $k_{B}T\ll E_{F}$. Then
$\tilde{G}(\Omega/2)\rightarrow\Theta(\Omega-2)$ where $\Theta(\cdot)$ is the
Heaviside step function, and thus:

$\displaystyle\sigma(\omega)=\frac{e^{2}}{\hbar}\frac{gk_{F}}{24\pi}\Omega\,\Theta(\Omega-2)+i\frac{e^{2}}{\hbar}\frac{gk_{F}}{24\pi^{2}}\left[\frac{4}{\Omega}-\Omega\ln{\frac{4\varepsilon_{c}^{2}}{|\Omega^{2}-4|}}\right]$
(31)

For the real part of $\sigma(\omega)$, the step function captures the
interband absorption when $E>2E_{F}$. For its imaginary part, the first term
is the Drude term due to the intraband transition, while the second term is
the correction due to the interband transitions.

###  2.5 Giant optical nonreciprocity

Now we can discuss one unique property of magnetic Weyl semimetals: giant
optical nonreciprocity without an external magnetic field. Reciprocity is a
fundamental internal symmetry of Maxwell’s equations [97, 98, 99, 100, 101].
It imposes direct constraints on basic optical phenomena including
transmission [102], reflection [103], absorption, and emission [104, 105].
Conversely, breaking reciprocity enables significant new opportunities in
photonic applications such as isolation [106, 107], circulation [108], robust
topological transport [109, 110], and violation of Kirchhoff’s law of thermal
radiation [111, 112].

Optical materials with significant nonreciprocal responses are rare. The most
used ones are magneto-optical materials [113]. Under an external magnetic
field, this type of material has an asymmetric dielectric tensor
$\varepsilon\neq\varepsilon^{T}$ that breaks reciprocity ($\varepsilon^{T}$
denotes the transpose of $\varepsilon$). The strength of nonreciprocal effects
depends on the degree of asymmetry of $\varepsilon$ [113]:

$\gamma=\frac{\left\|\varepsilon-\varepsilon^{T}\right\|}{\left\|\varepsilon+\varepsilon^{T}\right\|},$
(32)

where $\|\cdot\|$ denotes the matrix norm. For magneto-optical materials,

$\gamma\sim\frac{\omega_{c}}{\omega}\,,$ (33)

where $\omega$ is the light frequency, $\omega_{c}=eB/m^{*}$ is the cyclotron
frequency, $m^{*}$ is the effective electron mass, $e$ is the electron charge,
and $B$ is the external magnetic field [113, 111]. For the typical magnetic
field $B\sim$1\text{\,}\mathrm{T}$$,
$\omega_{c}\sim$1\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$$, thus $\gamma\sim
0.001$-$0.01$ at optical frequencies. Therefore, the nonreciprocal effect is
weak in magneto-optical materials.

Figure 4: Permittivity of a magnetic Weyl semimetal. (a) Band structure of a
Weyl semimetal with a chiral shift $2\bm{b}$. (b) $\varepsilon_{a}$ and the
real part of $\varepsilon_{D}$ in the permittivity tensor of Eq. (34).
$\varepsilon_{0}$ denotes the vacuum permittivity. (c) The real part of the
Voigt permittivity $\varepsilon_{V}$ in Eq. (38). The regions where
$\operatorname{Re}(\varepsilon_{V})>0$ are shaded by color. Figures (a, b) are
reproduced with permission from Ref. [14]. Copyright 2020 American Chemical
Society.

In contrast, magnetic Weyl semimetals can exhibit extremely large
nonreciprocal responses, with $\gamma\sim 1$ at optical frequencies. As an
illustration, we consider a Weyl semimetal with $\bm{b}$ along the
$z$-direction and $b_{0}=0$ (Fig. 4a). It has a permittivity tensor:

$\varepsilon(\omega)=\begin{pmatrix}\varepsilon_{D}&i\varepsilon_{a}&0\\\
-i\varepsilon_{a}&\varepsilon_{D}&0\\\ 0&0&\varepsilon_{D}\end{pmatrix},$ (34)

where $\varepsilon_{D}(\omega)$ is given by Eqs. (26) and (29), and

$\varepsilon_{a}(\omega)=\frac{be^{2}}{2\pi^{2}\hbar\omega}.$ (35)

The permittivity tensor $\varepsilon(\omega)$ has the typical form of a
gyrotropic medium [113]. Following Ref. [6], we use the parameters
$\varepsilon_{b}/\varepsilon_{0}=6.2$, $\xi_{c}=3$, $g=2$,
$\tau=$1000\text{\,}\mathrm{f}\mathrm{s}$$,
$b=$2\text{\times}{10}^{9}\text{\,}\mathrm{/}\mathrm{m}$$,
$v_{F}=$0.83\text{\times}{10}^{5}\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}$$,
$T=$300\text{\,}\mathrm{K}$$, and $E_{F}=$0.15\text{\,}\mathrm{e}\mathrm{V}$$.
Fig. 4b plots the calculated $\varepsilon_{D}$ and $\varepsilon_{a}$. The
magnitude of $\varepsilon_{a}$ is comparable to $\varepsilon_{D}$ over a broad
frequency range. Hence $\gamma\sim|\varepsilon_{a}/\varepsilon_{D}|\sim 1$.
Such a strong nonreciprocity originates from the anomalous Hall effect induced
by the Weyl node separation [2, 4, 114]. This mechanism is fundamentally
different from the cyclotron mechanism for magneto-optical materials. In
particular, it requires no external magnetic field.

###  2.6 Electromagnetic waves in the bulk medium

The strong gyrotropy significantly affects the behavior of light in magnetic
Weyl semimetals. As the first example, we study the electromagnetic waves in
the bulk medium as described by $\varepsilon(\omega)$ in Eq. (34). The results
will be useful later in discussing many applications.

We consider the Voigt configuration, i.e., the wavevector $\bm{k}$ is in the
plane perpendicular to the axis of gyration $\bm{b}$. Such a configuration is
mirror symmetric with respect to that plane, thus waves can be decoupled into
TE and TM modes [115]. TE/TM modes have the electric/magnetic fields parallel
to $\bm{b}$; their dispersion relations can be expressed as:

$\displaystyle\text{TE:}\qquad
k^{2}=\omega^{2}\mu_{0}\varepsilon_{D}(\omega),$ (36)
$\displaystyle\text{TM:}\qquad
k^{2}=\omega^{2}\mu_{0}\varepsilon_{V}(\omega).$ (37)

where we have introduced the Voigt permittivity for TM modes as:

$\varepsilon_{V}(\omega)=\varepsilon_{D}-\frac{\varepsilon_{a}^{2}}{\varepsilon_{D}}.$
(38)

Fig. 4c plots the calculated $\varepsilon_{V}(\omega)$ with the parameters as
described in Sec. 2.5.

We note that the gyrotropic medium behaves differently for the two modes due
to the different permittivity components: $\varepsilon_{D}$ for TE modes in
Eq. (36) and $\varepsilon_{V}$ for TM modes in Eq. (37). Moreover, the two
permittivities have different frequency dependence. For TE modes,
$\varepsilon_{D}$ is not affected by the gyrotropy (Fig. 4b). It monotonically
increases with the frequency and has a plasma frequency $\omega_{p}$. For TM
modes, $\varepsilon_{V}$ is affected by the gyrotropy (Fig. 4c). The existence
of the chiral shift $2\bm{b}$ splits $\varepsilon_{V}$ into two branches which
are separated by $\omega_{p}$. In each branch, $\varepsilon_{V}$ monotonically
increases with the frequency and has an effective plasma frequency
$\omega_{p-}$ and $\omega_{p+}$, respectively. Consequently, when
$\omega_{p-}<\omega<\omega_{p}$, $\operatorname{Re}(\varepsilon_{D})$ is
negative while $\operatorname{Re}(\varepsilon_{V})$ is positive; when
$\omega_{p}<\omega<\omega_{p+}$, $\operatorname{Re}(\varepsilon_{D})$ is
positive while $\operatorname{Re}(\varepsilon_{V})$ is negative.

###  2.7 Nonreciprocal surface plasmon polaritons

As the second example, we study the surface plasmon polaritons of a Weyl
semimetal [91]. The results will also be useful for later applications.

Figure 5: Nonreciprocal surface plasmon polaritons. (a) An interface between
air and a magnetic semimetal. (b) Dispersion relation of the surface plasmon
polaritons. The gray region shows the light cone of the vacuum. The blue
region shows the continuum of the propagating modes in the bulk.
$k_{F}=E_{F}/\hbar v_{F}$ is the Fermi wavevector. (c) The temperature
dependence of the chemical potential $E_{F}(T)$. Figures (b, c) are reproduced
with permission from Ref. [14]. Copyright 2020 American Chemical Society.

We consider a planar interface between air and a semi-infinite magnetic Weyl
semimetal with a chiral shift $2\bm{b}$ along the $z$ direction parallel to
the interface (Fig. 5a). Such an interface supports nonreciprocal surface
plasmon polaritons, whose band dispersion along the $x$ direction is
determined by: [6, 116]

$k_{y}+\varepsilon_{V}k_{y0}-k_{x}\dfrac{\varepsilon_{a}}{\varepsilon_{D}}=0\,,$
(39)

where $k_{0}(\omega)=\omega/c$ is the wavenumber in vacuum, $k_{x}$ is the
$x$-component of the wavevector, $k_{y0}=\sqrt{k_{x}^{2}-k_{0}^{2}}$ and
$k_{y}=\sqrt{k_{x}^{2}-\epsilon_{V}k_{0}^{2}}$ are the $y$-component of the
wavevector in air and in the Weyl semimetal, respectively. The last term in
Eq. (39) shows that when the sign of $k_{x}$ flips, the frequency of the
surface plasmon polaritons will be different, i.e.,
$\omega(k_{x})\neq\omega(-k_{x})$. The asymmetry of the dispersion is induced
by the gyrotropy; it occurs only when $\varepsilon_{a}\neq 0$.

Fig. 5b shows the calculated band dispersion of the surface plasmon
polaritons, together with the continuum region of bulk modes (in light blue)
and the light cone of vacuum (in light gray). We note that the surface plasmon
polaritons approximately occupy the frequency ranges $\omega<\omega_{p-}$ and
$\omega_{p}<\omega<\omega_{p+}$; the bulk modes occupy the frequency ranges
$\omega_{p-}<\omega<\omega_{p}$ and $\omega>\omega_{p+}$. This is as expected
from the frequency dependence of $\varepsilon_{V}$ in Fig. 4c. The dispersion
of the surface plasmon polaritons is clearly asymmetric near the effective
plasma frequencies $\omega_{p-}$ and $\omega_{p+}$. Away from the effective
plasma frequencies, the dispersion becomes approximately symmetric. The
frequency range where the dispersion is evidently asymmetric measures the
strength of nonreciprocity. The plots confirm the strong nonreciprocity of a
magnetic Weyl semimetal without an external magnetic field. As a comparison,
to obtain a similar strength of nonreciprocity, conventional magneto-optical
materials, such as InSb, need an unrealistically large external magnetic field
of about $100\text{\,}\mathrm{T}$ [14, 7]. This is consistent with our order-
of-magnitude comparison of $\gamma$ in Sec. 2.5.

###  2.8 Tunable Fermi level

Another unique property of Weyl semimetals is the large tunability of the
Fermi level $E_{F}$ (also referred to as the chemical potential). The
variation of the Fermi level can significantly affect the optical conductivity
according to Eq. (29). This leads to greatly tunable optical phenomena. The
Fermi level of a Weyl semimetal can be adjusted in different ways. Below we
discuss two approaches.

First, the Fermi level can be tuned by electric gating. This is because an
undoped Weyl semimetal features a vanishing density of states at the energy of
Weyl points. Therefore, the carrier concentrations are much lower than metals
and thus are more easily depleted. Second, the Fermi level can be tuned by
thermal tuning. This is because a Weyl semimetal has a small and nonconstant
density of states and a linear dispersion. Consequently, its Fermi level is
strongly temperature dependent. The Fermi level as a function of temperature
can be calculated from the requirement of charge conservation [117]:

$E_{F}(T)=\frac{2^{1/3}\left(9E_{F}(0)^{3}+\sqrt{81E_{F}(0)^{6}+12\pi^{6}k_{B}^{6}T^{6}}\right)^{2/3}-2\pi^{2}3^{1/3}k_{B}^{2}T^{2}}{6^{2/3}\left(9E_{F}(0)^{3}+\sqrt{81E_{F}(0)^{6}+12\pi^{6}k_{B}^{6}T^{6}}\right)^{1/3}}.$
(40)

Fig. 5c shows $E_{F}(T)$, where we set $E_{F}(0)=0.163$ eV such that
$E_{F}(300$K$)=0.150$ eV. Note that $E_{F}(T)$ decreases as $T$ increases.
Consequently, the dielectric tensor and optical properties of the Weyl
semimetal are also temperature dependent.

###  2.9 Effects of Fermi arcs

As we mentioned in Sec. 1.2, Weyl semimetals feature not only Weyl nodes in
the bulk band structure but also Fermi arc surface states. The existence of
Fermi arc surface states can modify the boundary conditions of Maxwell’s
equations, which can affect the optical properties of Weyl semimetals.
Depending on the problems, the modification may be negligible in some cases
but may be significant in other cases.

So far, we have focused on the optical effects that arise from the nontrivial
bulk electronic states in Weyl semimetals. Now we point to some works that
highlight the optical effects of Fermi arcs. Ref. [118] provides a
semiclassical approach to understanding surface Fermi arcs in Weyl semimetals.
Ref. [119] provides a detailed calculation of optical properties of Weyl
semimetals including the effects of both bulk states and Fermi arc surface
states, and compares their contributions to bulk and surface conductivity
tensors. Refs. [120, 121] study “Fermi arc plasmons” that originate from the
hybridization of collective modes associated with Fermi arc carriers and bulk
carriers. Such an unusual surface plasmon mode exhibits a hyperbolic band
dispersion [122]. Ref. [123] systematically studies both the bulk plasmons and
the Fermi-arc plasmons over opposite surfaces of a Weyl semimetal slab.

##  3 Photonic applications and devices

In this section, we survey various photonic applications of Weyl semimetals.
These applications utilize nontrivial linear and nonlinear optical effects of
Weyl semimetals. For each application, we discuss both the physical mechanism
and the device configuration.

###  3.1 Linear optical effects

####  3.1.1 Nonreciprocal optical components

Nonreciprocal optical components, such as isolators [107], circulators [124],
and nonreciprocal waveguides, are crucial in many photonic applications
including optical circuits [125, 126], and lasers [127]. As we discussed in
Sec. 2.5, magnetic Weyl semimetals can exhibit giant optical nonreciprocity
without an external magnetic field [91, 6] thanks to the anomalous Hall effect
[72, 128, 76, 43, 42, 44]. Hence, magnetic Weyl semimetals can be used to
construct efficient and compact nonreciprocal optical components. Here, we
review three examples of nonreciprocal optical components based on magnetic
Weyl semimetals.

In Ref. [6], Kotov et al. design a nonreciprocal waveguide using magnetic Weyl
semimetals. As shown in Fig. 6a, the structure consists of a Weyl semimetal
thin film sandwiched by two dielectrics. The chiral shift $2\bm{b}$ is along
the $x$ direction. Fig. 6b shows the dispersion diagram of the TM-polarized
light propagating in the $y$ direction. The structure supports two different
types of guided modes depending on the light frequency $\omega$: When
$\omega<\omega_{p-}$ or $\omega_{p}<\omega<\omega_{p+}$, the structure
supports nonreciprocal surface plasmon polaritons; when
$\omega_{p-}<\omega<\omega_{p}$ or $\omega>\omega_{p+}$, the structure
supports nonreciprocal waveguide modes. Such behavior is similar to that shown
in Fig. 5b and is as expected from the frequency dependency of
$\varepsilon_{V}$ in Fig. 4c. The dispersions of these guided modes can be
tuned via gating between the THz and mid-IR ranges.

Figure 6: Nonreciprocal optical components. (a, b) A nonreciprocal waveguide
based on magnetic Weyl semimetals. (a) The geometry consists of a Weyl
semimetal film surrounded by two semi-infinite dielectrics. (b) The light
dispersion at the waveguide. The red lines (${\rm SPP}_{>}$) and blue lines
(${\rm SPP}_{<}$) denote the forward and backward nonreciprocal surface
plasmon polaritons, respectively. The cyan regions denote the nonreciprocal
waveguide modes. (c, d) A Voigt isolator based on magnetic Weyl semimetals.
(c) The geometry consists of three unit cells. Each unit cell comprises three
material layers. The dark and bright grey layers are made of Weyl semimetals
with opposite chiral shifts as indicated by the yellow arrows. The yellow
layer is made of crystalline silicon (c-Si). The blue and red arrows denote
forward and backward illuminations, respectively, at an incident angle
$\theta=45.6^{\circ}$. (d) The transmittance spectra in the forward (blue) and
backward (red) directions. Also shown is the isolation ratio (green). (e, f)
An isolator based on a photonic crystal slab made of Weyl semimetal and
silicon. (e) The geometry consists of a photonic crystal slab made of Weyl
semimetals (green) and silicon (gray) and a uniform silicon slab separated by
an air gap. (f) The reflectance and transmittance spectra in the forward (red)
and backward (blue) directions. Figures are reproduced with permission from
(a, b) Ref. [6], Copyright 2018 American Physical Society; (c, d) Ref. [7],
Copyright 2020 John Wiley and Sons; (e, f) Ref. [8], Copyright 2021 American
Chemical Society.

In Ref. [7], Asadchy et al. design optical isolators in the mid-IR range using
magnetic Weyl semimetals, with both Faraday and Voigt geometries. The Faraday
isolator consists of a Weyl semimetal slab sandwiched by two twisted
polarization filters and operates at the normal incidence. The chiral shift
$2\bm{b}$ is along the direction perpendicular to the slab. The Voigt isolator
(Fig. 6c) is a finite one-dimensional photonic crystal with a three-layer unit
cell consisting of Weyl semimetal, dielectric, and Weyl semimetal; the chiral
shifts of the two Weyl semimetal layers are both parallel to the slab but in
the opposite directions. Such a structure is designed for optical isolation at
oblique incidence. It breaks all the symmetries that preclude isolation [129].
Numerical results show that three unit cells suffice to achieve high isolation
and low insertion loss. Remarkably, the total thickness of the device is less
than $1.4$ times the operation wavelengths, and is three orders of magnitude
smaller than that of conventional isolators based on magnetooptical materials.
Fig. 6d shows the calculated power transmittance spectra in the forward and
backward directions at the incident angle of
$\theta=$45.6\text{\,}\mathrm{\SIUnitSymbolDegree}$$. The two spectra are
significantly different. Near the resonance frequency, the device achieves
greater than $30\text{\,}\mathrm{d}\mathrm{B}$ isolation and less than
$1.2\text{\,}\mathrm{d}\mathrm{B}$ insertion loss. In Ref. [130], Li et al.
demonstrate a similar isolation performance using a one-dimensional photonic
crystal containing a defect layer made of magnetic Weyl semimetals.

In Ref. [8], Park et al. design a photonic crystal slab structure made of
magnetic Weyl semimetal and silicon, which can achieve optical isolation at
normal incidence without the need for polarizers. As shown in Fig. 6e, the
structure consists of a photonic crystal slab and a uniform dielectric slab
separated by an air gap. The total thickness of the device is less than the
operational wavelength. Such a structure hosts guided resonances [131] that
can greatly enhance the nonreciprocal effects. Fig. 6f shows the calculated
power transmission and reflection spectra for the TM polarized light in the
forward and backward directions. The spectra exhibit high contrast near the
guided resonance.

####  3.1.2 Polarization filters

Due to the giant gyrotropy of magnetic Weyl semimetals, the left and right
circularly polarized light propagating along the chiral shift direction will
acquire different phases and attenuation [132]. Such effects can be used to
construct polarization filters. In Ref. [133], Chtchelkatchev et al.
demonstate that a single magnetic Weyl semimetal slab can selectively
transmit/reflect circularly polarized light in the Faraday configuration, and
linearly polarized light in the Voigt configuration (Fig. 7a). In Ref. [134],
Yang et al. design a circular polarizer using two layers of magnetic Weyl
semimetals separated by an air gap. The chiral shifts of the two Weyl
semimetals are parallel and perpendicular to the slab. The proposed device
exhibits a high circular polarization efficiency and high average
transmittance in the wavelength region from
$9\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ to
$15\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ at incidence angles up to
$50\text{\,}\mathrm{\SIUnitSymbolDegree}$. In Ref. [135], Mukherjee et al.
study the effect of a tilt of the Weyl cones on the absorption of left and
right circular polarized light. They show that the difference in absorption
depends strongly on the degree of tilt.

Figure 7: Polarization filters and negative refraction. (a) A circular
polarization filter made of a magnetic Weyl semimetal slab in the Faraday
configuration. The device can selectively transmit one circular polarized
light and reflect the opposite circular polarized light. (b) The negative
refraction of TM-polarized light in a Weyl semimetal with the Voigt
configuration. $\bm{S}_{t}$ is the transmitted Poynting vector. $\bm{k}_{i}$,
$\bm{k}_{r}$, and $\bm{k}_{t}$ are the incident, reflected, and transmitted
wavevectors, respectively. (c) A scheme of a scanning tunneling microscope
(STM) with a probing tip built from three Weyl semimetal layers. These three
layers act as an electronic Veselago lens. The electron flow from the top
layer is focused on a small spot in the bottom layer. The tight focus may
increase the spatial and temporal resolution of the STM. Figures are
reproduced with permission from (a) Ref. [133], Copyright 2021 Elsevier; (b)
Ref. [136], Copyright 2017 the Physical Society of Japan; (c) Ref. [137],
Copyright 2017 American Physical Society.

####  3.1.3 Negative refraction

Negative refraction refers to the counter-intuitive phenomenon where light is
refracted at a negative angle with respect to the surface normal [138].
Negative refraction is related to exotic phenomena such as perfect lensing
[139]. There are several known routes to achieving negative refraction. The
original approach proposed by Veselago [138] requires the use of a negative
index material where both the permittivity and the permeability are negative.
Such “double-negative” materials have been experimentally demonstrated by
metamaterials [140]. Negative refraction has also been achieved in photonic
crystals [141, 142, 143]. Another approach utilizes the magnetoelectric effect
in chiral media [144, 145, 146], which has also been experimentally confirmed
using chiral metamaterials [147].

Weyl semimetals can provide a new route to negative refraction using natural
materials. As we discussed in Sec. 2.6, TM-polarized light can propagate in a
magnetic Weyl semimetal with the Voigt configuration when
$\omega_{p-}<\omega<\omega_{p}$ or $\omega>\omega_{p+}$. In the lower
frequency range, the light can propagate in the bulk below the plasma
frequency $\omega_{p}$. This phenomenon is typical for any gyrotropic system
in the Voigt configuration [115, 6] and is related to the modification of
$\varepsilon_{V}$ by the gyrotropy as shown in Fig. 4c. Interestingly, this
phenomenon is also accompanied by negative refraction. In Refs. [136, 148], it
is shown that TM-polarized light in the lower frequency range propagates with
opposite signs of phase and group velocities. Hence the light exhibits a
negative refractive index (Fig. 7b). Although such negative refraction can be
observed in any gyrotropic media, magnetic Weyl semimetals can exhibit the
phenomena over a much broader bandwidth without any external magnetic field.
These predictions still await experimental confirmation.

Although not the focus of this paper, we point out that similar negative
refraction for _electrons_ can also occur in a Weyl semimetal [137, 149]. The
phenomenon of negative refraction for electrons is similar to that for
photons, although the physical mechanisms are different. Such an effect can be
used to construct a three-dimensional electronic Veselago lens. In Ref. [137],
Hills et al. design an electronic Veselago lens made of three layers of
different Weyl semimetals. They propose to use such a lens as a probing tip
for scanning tunneling microscope (STM) to significantly improve its spatial
and temporal resolution (Fig. 7c). In Ref. [149], Tchoumakov et al. design a
three-dimensional electronic Veselago lens made of a single Weyl semimetal.
Such a Veselago lens is based on chiral anomaly and can selectively focus
electrons of a given chirality.

###  3.2 Nonlinear optical effects

So far, we have only discussed the linear optical effects of Weyl semimetals.
These phenomena can be understood in the context of linear response theory
[94, 150], as we have demonstrated in Sec. 2. Weyl semimetals can also exhibit
pronounced nonlinear optical effects [151]. These phenomena must be understood
in the context of nonlinear response theory [152]. Nonlinear responses of
topological materials have only been investigated recently [153, 154, 155,
156, 157, 11, 158, 159, 160, 161]; a systematic discussion is beyond the scope
of this paper. Instead, we survey some nonlinear optical effects and
applications of Weyl semimetals.

####  3.2.1 Photogalvanic effect

One of the simplest nonlinear optical effects in a solid is the photogalvanic
effect. It refers to the generation of the direct current (DC) in the crystal
under exposure to light. In the context of nonlinear optics [162], the
photogalvanic effect is a second-order nonlinear optical effect: A direct
current can appear in a _noncentrosymmetric_ solid due to an oscillating
electric field when one analyzes the response up to (at least) second order in
the applied field [163]. At this order of perturbation theory, the DC
photocurrents $\bm{J}$ are the sum of three contributions [164, 165]:

$\bm{J}=\bm{J}_{\text{injection}}+\bm{J}_{\text{shift}}+\bm{J}_{\text{anomalous}},$
(41)

where the three terms are referred to as “injection”, “shift”, and “anomalous”
current contributions in the literature. These three contributions have
different physical origins and symmetry requirements. Moreover, they exhibit
different dependence on scattering time: $\bm{J}_{\text{injection}}$ is
proportional to the scattering time $\tau$, while $\bm{J}_{\text{shift}}$ and
$\bm{J}_{\text{anomalous}}$ have well-defined DC values even in the absence of
scattering [164]. Roughly speaking,

$|\bm{J}_{\text{injection}}|/|\bm{J}_{\text{shift}}|\sim|\bm{J}_{\text{injection}}|/|\bm{J}_{\text{anomalous}}|\sim\omega\tau,$
(42)

where $\omega$ is the light frequency [165]. Hence, the injection current
dominates in the high-frequency or weak-scattering regime, while the shift or
anomalous current dominates in the low-frequency or strong-scattering regime.

From a practical perspective, converting light to electricity is crucial for
clean energy, imaging, communications, and chemical and biological sensing.
For this purpose, the photogalvanic effect is of interest because it can
potentially overcome the extrinsic limitations of conventional approaches. For
example, traditional solar cells make use of the built-in electric fields in
p-n junctions to generate photocurrent; however, their efficiency is bounded
by the Shockley-Queisser limit due to the constraint of detailed balance
[166]. Thermoelectric devices utilize the optically induced thermal gradients
to produce currents via the Seebeck effect; however, they require a careful
balance of the optical, electronic, and thermal material properties. The
photogalvanic effect provides an important alternative: It has an ultrafast
response with fewer limitations on efficiency or maximum open-circuit voltage
[167].

Figure 8: Glass coefficients. Measured (circles) and calculated (lines) Glass
coefficients for various ferroelectric materials. The Weyl semimetal TaAs
(blue square) exhibits a nearly one order of magnitude larger response due to
its anomalous Berry curvature. TaAs works in the technically important mid-
infrared regime due to its gapless nature. Figure is reproduced with
permission from Ref. [168]. Copyright 2019 Springer Nature.

The photogalvanic effects were primarily observed in ferroelectric insulators
and semiconductors [163]. However, in these materials, the photogalvanic
effects are usually too weak to be technically relevant. Moreover, the
photogalvanic effects are typically restricted to a narrow range of light
wavelengths. For the practical usage of photogalvanic effects, it is crucial
to search for materials that overcome these limitations.

Weyl semimetals can be ideal material candidates for this purpose. Recent
studies show that noncentrosymmetric Weyl semimetals can exhibit a much
stronger photogalvanic effect than conventional materials due to the large
Berry curvature [159, 168]. The operation is broadband since Weyl semimetals
exhibit zero bandgap and linear dispersion [169, 170]. Moreover, the carriers
in Weyl semimetals can exhibit high mobility; the fast motion of carriers also
contributes to the ultrafast and giant photocurrent response [171].
Interestingly, Weyl semimetals exhibit topological features in all three
contributions of photocurrents [155].

To compare the strength of photogalvanic effects in different materials, one
calculates the so-called Glass coefficient $G$ defined by [172, 163]:

$j=G\alpha I,$ (43)

where $j$ is the photocurrent density, $\alpha$ is the absorption coefficient,
and $I$ is the incident intensity. Thus $G$ measures the generated
photocurrent per absorbed power with a unit
$[$\mathrm{c}\mathrm{m}\mathrm{/}\mathrm{V}$]$. Fig. 8 shows the measured and
calculated Glass coefficients for various ferroelectric materials, together
with the measured data for the Weyl semimetal TaAs. Notably, TaAs exhibits a
nearly one order of magnitude larger response than any other materials.
Moreover, TaAs enables a broadband response even in the mid-IR wavelengths, a
regime that is important for thermal, chemical, and biological sensing.

Below we discuss the three contributions of photocurrents in Weyl semimetals.
For each contribution, we first provide a qualitative picture of the process
in general solids to explain its physical origin and symmetry requirement.
Then we discuss its manifestation in Weyl semimetals. We point to relevant
references for detailed quantitative analysis.

Figure 9: Injection current. (a) Scheme of injection current via interband
absorption involving different polarization component $\bm{E}_{h}$
(horizontal) and $\bm{E}_{v}$ (vertical). (b) The injection current from a
single Weyl cone depends on its chirality and tilt. (c, d) Injection currents
in Weyl semimetals. Each Weyl node produces an injection current with
chirality-independent ($\bm{J}_{0}$) and chirality-dependent ($\bm{J}_{\chi}$)
components. (c) Inversion symmetry relates two Weyl nodes with opposite tilt
and opposite chirality, leading to the cancellation of photocurrents. IC: the
inversion center. (d) Time-reversal symmetry relates two Weyl nodes with
opposite tilt but the same chirality. Two pairs of Weyl nodes give rise to an
overall injection current $2(\bm{J}_{\chi}-\bm{J}^{\prime}_{\chi})$. TRIM: the
time-reversal invariant momentum. (e-g) Experimental observation of the
injection current in the Weyl semimetal TaAs. (e) Setup scheme. (f) A
photograph of the sample. $\bm{a},\bm{b},\bm{c}$ denote the crystal axes.
Scale bar: $300\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$. (g)
Polarization-dependent photocurrents at $T=$10\text{\,}\mathrm{K}$$ measured
along the $\bm{b}$ direction with the laser focused on the pink, black and
blue dots in (f). Figures are reproduced with permission from (c, d) Ref.
[159], Copyright 2017 American Physical Society; (e-g) Ref. [157], Copyright
2017 Springer Nature.

First, we discuss the injection current. Injection currents arise in a crystal
that lacks certain symmetries, which give rise to phase differences between
transition amplitudes associated with different polarizations of light [173].
When the crystal is excited with elliptically polarized light, the different
excitation pathways for horizontal and vertical linear polarizations lead to
an interference effect. This results in an asymmetric population in the
wavevector space and thus generates a current (Fig. 9a). The current flips
sign when the helicity of light changes sign. Because it is generated only
with elliptically polarized light, the injection current is also called
“circular photocurrent” [174].

The injection current requires the breaking of inversion symmetry. But this is
not sufficient. Detailed symmetry analysis [164] shows that among the $21$
crystal classes that lack inversion symmetry, $18$ classes can support
injection current; the exceptions are $\bar{6}m2$, $\bar{6}$, and
$\bar{4}\bar{3}m$ [164]. For example, GaAs belongs to $\bar{4}\bar{3}m$, hence
cannot support injection current.

In a Weyl semimetal, the injection current can occur via asymmetric interband
excitations near each Weyl cone (Fig. 9b). Each Weyl node produces an
injection current that depends on its chirality and tilt, which can be
decomposed as [159]:

$\bm{J}^{(n)}=\bm{J}^{(n)}_{0}+\bm{J}^{(n)}_{\chi},$ (44)

where $(n)$ denotes the $n$-th Weyl cone, $\bm{J}^{(n)}_{0}$ changes sign only
for opposite tilts, and $\bm{J}^{(n)}_{\chi}$ changes sign only for opposite
chirality. The total injection current is then the sum of contributions from
all the Weyl cones.

Next, we discuss the symmetry constraints. When there is inversion symmetry, a
Weyl node at $\bm{k}$ is related to another one at $-\bm{k}$ with opposite
tilt and opposite chirality. When there is time-reversal symmetry, a Weyl node
at $\bm{k}$ is related to another one at $-\bm{k}$ with opposite tilt but the
same chirality. Combining these symmetry considerations with Eq. (44), we
conclude that in centrosymmetric Weyl semimetals, both $\bm{J}_{0}$ and
$\bm{J}_{\chi}$ are canceled; the total injection current is zero as expected
(Fig. 9c). In noncentrosymmetric Weyl semimetals with time-reversal symmetry,
$\bm{J}_{0}$ are canceled while $\bm{J}_{\chi}$ can survive; a minimal number
of two pairs of Weyl nodes can produce a nonzero response
$2(\bm{J}_{\chi}-\bm{J}^{\prime}_{\chi})$ (Fig. 9d). In Weyl semimetals
without any symmetry, both $\bm{J}_{0}$ and $\bm{J}_{\chi}$ can survive [159].
These discussions confirm the general symmetry analysis above from a different
microscopic perspective.

The injection current in Weyl semimetals has been investigated both
theoretically [156, 159, 158, 161, 175] and experimentally [157, 176, 177]. In
Ref. [157], Ma et al. report the experimental observation of the injection
current in TaAs. They use a mid-infrared scanning photocurrent microscope
(Fig. 9e) equipped with a CO2 laser with wavelength
$\lambda_{\text{CO}_{2}}=$10.6\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$
and energy $\hbar\omega=$117\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$. This
photon energy is chosen to induce the desired interband transitions near the
Weyl nodes. The TaAs sample is purposedly filed down to show a clean surface
with its normal direction along the $\bm{a}$ ($[100]$) axis of the crystal
(Fig. 9f). The sample is kept at a low temperature
$T\approx$10\text{\,}\mathrm{K}$$ to increase the scattering time $\tau$,
which is crucial to observe the shift current [see Eq. (42)]. Light is focused
and normally incident on the sample, i.e., along the $\bm{a}$ axis. Its
polarization is controlled by the rotation angle $\theta$ of the quarter-wave
plate. In Fig. 9g, the pink, black, and blue data points show the photocurrent
along the $\bm{b}$ axis when the laser is focused on the pink, black, and blue
dots in Fig. 9f, respectively. When the laser is focused near the sample’s
center (the black dot), the photocurrent fits well to a cosine function of
$\theta$: It reaches the maximum for right circularly polarized light, the
minimum for left circularly polarized light, and zero for linearly polarized
light. When the laser spot is moved horizontally to the blue and pink dots,
the corresponding photocurrents exhibit the same polarization dependence but
with an additional, polarization-independent shift. These data reveal two
distinct mechanisms for photocurrent generation. The polarization-dependent
component corresponds to the injection current. The polarization-independent
component arises from the photo-thermal effect [178, 179, 180]: Because the
sample and contact have different thermopower, a current is generated by the
laser-induced temperature gradient. Such an interpretation is further
supported by additional experiments [157].

Second, we discuss the shift current. Shift currents arise from a coordinate
shift accompanying the interband photoexcitation of electrons [181]. It occurs
because the real-space center of charge for the valence bands differs from
that for the conduction bands. As light is absorbed and electrons transit from
the valence to conduction bands, there will be a motion of charge. If the
crystal has low enough symmetry and the light polarization is appropriate,
there will be a net current due to the “shift” of the center of charge [173].
The shift is on the order of a bondlength, and occurs on femtosecond time
scales [164]. A more rigorous treatment of the shift current requires the
modern theory of electric polarization [29] and the formalism of localized
Wannier functions [182, 183].

Figure 10: Shift current. (a, b) Electron density in the $[110]$ plane of GaAs
at $\Gamma$ point for (a) the highest valence band, and (b) the lowest
conduction band. (c-f) Experimental observation of the shift current in the
Weyl semimetal TaAs. (c) Setup scheme. The polarization is controlled via
rotation of the quarter-wave plate (QWP). (d) False colored scanning electron
microscopy (SEM) image of a microscopic TaAs (purple) device with gold
(yellow) contacts. (e) Polarization dependence of the thermal
($J^{a}_{\text{thermal}}$) and shift ($J^{a}_{\text{shift}}$) photocurrent
contributions. The radius and angle of the polar plot correspond to the
magnitude of the photocurrents and the angle ($\phi$) the fast axis of QWP
makes with the crystal $a$ axis, respectively. These responses exhibit out-of-
phase minima and maxima due to the different electric field combinations. (f)
Measured power dependence of QWP angle-independent ($D$), fourfold shift
($L_{s}$), and photothermal ($L_{c}$) terms. The dependence is linear as
expected from the generation mechanisms of the shift and photothermal current.
Figures are reproduced with permission from (a, b) Ref. [173], Copyright 2006
American Physical Society; (c-f) Ref. [168], Copyright 2019 Springer Nature.

As an illustration, we discuss the shift current in the intrinsic
semiconductor GaAs [173]. Fig. 10a and 10b show the electron density of GaAs
for the $\Gamma$ electrons in the valence and conduction bands, respectively.
They can be viewed as snapshots before and after the photoexcitation.
Electrons at the top of the valence band ($\Gamma$ point) are localized around
the As atoms (Fig. 10a). During the photoexcitation, the crystal absorbs
photons of energy larger than the bandgap to populate the states near the
bottom of the conduction band (also $\Gamma$ point). Now electrons have
relocated closer to the Ga atoms (Fig. 10b). Depending on the light
polarization, the electron density evolves differently to reach the excited
state. For example, if the electric field is polarized along the $[100]$
direction, an electron from the As atom can move towards any one of its four
nearest neighboring Ga atoms with equal probability, generating no net
current. If the polarization is along the $[111]$ direction, an electron from
the As atom will move primarily towards its closest neighboring Ga atom in the
$[111]$ direction, generating a net current [173].

The shift current requires the breaking of inversion symmetry. But, again,
this is not sufficient. Detailed symmetry analysis [164] shows that among the
$21$ crystal classes that lack inversion symmetry, $20$ classes can support
the shift current; the only exception is $432$. Thus the shift current exists
in GaAs and other zinc-blende ($\bar{4}\bar{3}m$) semiconductors [184].
Different from the injection current, the shift current can be generated with
linearly polarized light.

The shift current in Weyl semimetals has been investigated both theoretically
[155, 175] and experimentally [168, 185]. In Ref. [168], Osterhoudt et al.
report the experimental observation of the shift current in the Weyl semimetal
TaAs. Their setup is similar to that in Fig. 9e: a photocurrent microscope
equipped with a CO2 laser, where the light polarization is controlled by the
rotation angle $\phi$ of the quarter-wave plate (Fig. 10c). Their sample is
different from the bulk crystal in Fig. 9f: The TaAs sample is fabricated into
a microscopic device (Fig. 10d) with a small area ($20$ times smaller than the
laser spot) and a small thickness (only three times the penetration depth
$\sim$250\text{\,}\mathrm{n}\mathrm{m}$$ of the light). This design mitigates
the resistive losses and thermal effects. The sample is kept at room
temperature, hence the scattering time $\tau$ is short, and the shift current
dominates over the injection current [see Eq. (42)]. Despite the careful
design of the device, the photo-thermal effect cannot be eliminated.
Fortunately, one can separate the photo-thermal effect from the shift current
by their different polarization dependence. Detailed symmetry analysis shows
that for the photocurrent measured along the $a$ axis, the thermal
contribution $J^{a}_{\text{thermal}}=D+L_{c}\cos 4\phi$ and the shift
contribution $J^{a}_{\text{shift}}=L_{s}\sin 4\phi$ (Fig. 10e). By fitting the
$\phi$-dependent photocurrent, one can obtain the parameters $D$, $L_{c}$, and
$L_{s}$. Fig. 10f shows their measured values as functions of input power,
which all exhibit linear dependence. This is as expected since both mechanisms
of the shift current and the photo-thermal effect rely on the square of the
electric field.

Third, we briefly discuss the anomalous current. The anomalous current arises
from the anomalous velocity [186, 187] caused by the Berry curvature [29, 153,
188, 189]. Similar to the injection current, the anomalous current is
generated only with elliptically polarized light, and changes sign when the
helicity of light flips [165]. The anomalous current in Weyl semimetals was
theoretically predicted but not experimentally observed yet. We refer readers
to Ref. [165] for detailed theoretical analysis.

Finally, we point out that the Fermi arc states may induce additional
contribution to the photocurrent generation. See Refs. [190, 191] for more
details.

####  3.2.2 Orbital angular momentum detection

Figure 11: Some nonlinear optical effects in Weyl semimetals. (a) Direct
detection of light orbital angular momentum. (b, c) Second harmonic
generation. (b) Setup scheme. (c) Second harmonic signal as a function of
incident polarization angle. (d) Inverse Faraday effect. (e, f) High harmonic
generation. (e) Scheme. (f) High harmonic spectrum. Figures are reproduced
with permission from (a) Ref. [9], Copyright 2020 American Association for the
Advancement of Science; (b, c) Ref. [11], Copyright 2017 Springer Nature; (d)
Ref. [192]; (e, f) Ref. [193].

Light can carry orbital angular momentum (OAM) [194]. The OAM manifests as a
helical wavefront with an azimuthal phase distribution $e^{im\phi}$ where $m$
is the mode number and $\phi$ is the azimuthal angle. The OAM modes of light
can encode information. However, direct detection of OAM by the photocurrent
mesurement is challenging. This is because most types of photocurrents are
sensitive only to optical intensity, not to optical phase.

In Ref. [195], Quienteiro et al. propose a new mechanism for the generation of
photocurrent, called the orbital photogalvanic effect. In this case, the
incident light can transfer its OAM and energy simultaneously to the
electrons. This process is similar to the photon drag effect [196], where the
linear momentum of absorbed photons is transferred to electrons. Because the
optical phase varies in the azimuthal direction, it induces a spatial
imbalance of excited carriers, producing a net current flowing either along or
perpendicular to the helical phase gradient. The generated photocurrent is
proportional to the OAM; when the OAM reverses signs, the photocurrent also
flips direction.

In Ref. [9], Ji et al. utilize the orbital photogalvanic effect to enable
direct on-chip electric readout of OAM. They fabricate electrodes of various
shapes on WTe2 for use as photocurrent detectors (Fig. 11a). WTe2 is a room-
temperature Weyl semimetal with broken inversion symmetry. It is an ideal
material for observing the orbital photocurrent because it has large nonlinear
optical susceptibilities and certain symmetries that forbid the photocurrents
of other types. In the experiment, Ji et al. observe that the photocurrent
displays steplike changes with different OAM. Such a detector can be exploited
in future photonic circuits for optical communications. In Ref. [10], Lai et
al. demonstrate the generation of orbital photocurrents in the mid-infrared
wavelengths with another Weyl semimetal TaIrTe4.

####  3.2.3 Second harmonic generation

Second harmonic generation is another important second-order nonlinear optical
effect. It refers to the generation of photons with twice the frequency of the
incident light. Recent works [11, 197] show that noncentrosymmetric Weyl
semimetals can exhibit strong second harmonic generation.

In Ref. [11], Wu et al. reveal giant second harmonic generation in the Weyl
semimetals TaAs, TaP, and NbAs. These materials belong to the point group
$4mm$, which has a unique polar ($z$) axis along the $[001]$ direction.
Symmetry analysis shows that second harmonic generation is allowed when the
incident electric field has a nonzero $z$-component. Fig. 11b depicts the
schematic setup. The incident pulses of $800\text{\,}\mathrm{n}\mathrm{m}$
wavelength are focused at near-normal incidence on the $(112)$ surface of the
sample. The generated second-harmonic light is detected in the reflection. The
polarization of the incident light and the second-harmonic light are
controlled by two sets of polarizers and waveplates, referred to as the
generator and the analyzer, respectively. Angles $\theta_{1}$ and $\theta_{2}$
denote the angles of the linear polarization plane after the generator and the
analyzer, respectively, with respect to the $[1,1,-1]$ crystal axis. Fig. 11c
shows the normalized second harmonic intensity as a function of $\theta_{1}$
when $\theta_{1}=\theta_{2}$ at $20\text{\,}\mathrm{K}$, while the inset shows
the polar plot. The data is consistent with the symmetry analysis and can be
fitted well with the theoretical prediction. The second harmonic signal is
highly anisotropic and sensitive to the incident polarization: It reaches the
maximum at $\theta_{1}=$0\text{\,}\mathrm{\SIUnitSymbolDegree}$$ or
$180\text{\,}\mathrm{\SIUnitSymbolDegree}$ when the electric field is along
the $[1,1,-1]$ direction and has the largest $z$-component; it reaches the
minimum at $\theta_{1}=$90\text{\,}\mathrm{\SIUnitSymbolDegree}$$ or
$270\text{\,}\mathrm{\SIUnitSymbolDegree}$ when the electric field is along
the $[1,-1,0]$ direction and has zero $z$-component. The second harmonic
signal is giant: The second-order optical susceptibility $\chi^{(2)}$ has the
largest component of $7200\text{\,}\mathrm{p}\mathrm{m}\mathrm{V}^{-1}$, which
is larger by almost one order of magnitude than the value in the archetypal
electro-optic materials GaAs
($700\text{\,}\mathrm{p}\mathrm{m}\mathrm{V}^{-1}$) and ZnTe
($900\text{\,}\mathrm{p}\mathrm{m}\mathrm{V}^{-1}$), even when measured at
wavelengths where their response is the largest. In fact, such a value is
larger than then reported in any crystal. In Ref. [197], Patankar et al.
further investigate the physical origins of such a giant second harmonic
generation in TaAs.

From a practical perspective, Weyl semimetals are not optimal for frequency-
doubling applications in the visible regime because of their strong
absorption. However, they are promising materials for terahertz generation
[198] and optoelectronic devices such as far-infrared detectors because the
loss is lower at these longer wavelengths.

####  3.2.4 Inverse Faraday effect

The inverse Faraday effect is the reverse of the Faraday effect. Recall that
the Faraday effect refers to the polarization rotation of light induced by a
static magnetization along the light propagation direction [113].
Correspondingly, the inverse Faraday effect refers to the generation of static
magnetization induced by circularly polarized light propagating inside the
material (Fig. 11d) [199, 200]. In this case, the angular momentum from an
oscillating electric field $\bm{E}(t)$ is transferred into the magnetic moment
of electrons, leading to a static magnetization
$\bm{M}\propto\bm{E}(\omega)\times\bm{E}^{*}(\omega)$. The inverse Faraday
effect is a nonlinear phenomenon [201]. It does not require absorption but is
rather based on a Raman-like coherent optical scattering process.
Consequently, this effect is non-thermal and takes place on a femtosecond
timescale. The laser-induced magnetization can be used for ultrafast control
of magnetic devices [202, 203]. It can also cause the Faraday rotation for a
subsequent _probe_ light, which induces a matter-mediated light-light
interaction.

Recent works [204, 205, 206, 207, 208] show that Dirac and Weyl semimetals can
exhibit a strong inverse Faraday effect due to their unique carrier transport
properties and strong spin-orbit coupling. For example, in Ref. [206], Tokman
et al. estimate that in a Weyl semimetal with $n$ Weyl nodes and a Fermi
energy of $100\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$, an incident pump light
with an electric field
$10\text{\,}\mathrm{k}\mathrm{V}\mathrm{/}\mathrm{c}\mathrm{m}$ at a frequency
$1\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$ can induce a magnetization that
causes the Faraday rotation parameter $\alpha\approx 6.6n^{3/2}$
$\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{/}\mathrm{c}\mathrm{m}$ for a probe
light. Such a value is two orders of magnitude larger than that obtained in
typical ferrites [203].

In Ref. [192], Liang et al. propose an unconventional mechanism of inverse
Faraday effect in Dirac and Weyl semimetals, referred to as the “axial
magnetoelectric effect”. As we discussed above, in the conventional inverse
Faraday effect, static magnetization is generated through dynamical
electromagnetic fields. In contrast, in the axial magnetoelectric effect,
static magnetization is generated through dynamical axial gauge fields. Such
effective gauge fields can be generated in a Dirac or Weyl semimetal via
dynamical deformations (sound). They interact with electrons similarly to the
usual electromagnetic fields but with different signs for different
chiralities (see Ref. [4] for an introduction to axial gauge fields). In the
axial magnetoelectric effect, the angular momentum is transferred from the
axial or pseudoelectric field, conventionally denoted as $\bm{E}_{5}(t)$, into
the magnetic moment of electrons, leading to a static magnetization
$\bm{M}\propto\bm{E}_{5}(\omega)\times\bm{E}_{5}^{*}(\omega)$. The synthetic
nature of the axial gauge fields means that one can induce magnetization in
Dirac and Weyl semimetals using phonons without any electromagnetic fields.

####  3.2.5 Higher-order nonlinear effects

Weyl semimetals can also exhibit higher-order nonlinear optical effects, such
as four-wave mixing, optical Kerr effect, and high-harmonic generation. These
nonlinear effects are expected to be strong, especially at long wavelengths,
due to the linear dispersion and high mobility of the Weyl fermions. Moreover,
these effects may exhibit nontrivial features due to the anomalous Berry
curvature associated with the Weyl nodes.

In Ref. [12], Almutairi et al. derive the third-order nonlinear optical
conductivity of Weyl semimetals in the long-wavelength limit and calculate the
intensity of the nonlinear four-wave mixing signal. The calculated nonlinear
generation efficiency is surprisingly high for a lossy material, of the order
of several $\mathrm{m}\mathrm{W}$ per $\mathrm{W}^{3}$ of the incident pump
power. This value is many orders of magnitude higher than in conventional
nonlinear materials [162]. Optimal conditions for the four-wave mixing are
realized in the vicinity of bulk plasma resonance. This work indicates that
ultrathin Weyl semimetal films of the order of skin depth in thickness can
find applications in compact nonlinear optoelectronic devices.

In Ref. [13], Cheng et al. derive the analytic expressions for linear and
third-order optical conductivities of Dirac and Weyl semimetals, and compare
the results with those of two-dimensional Dirac materials such as graphene.
The details of the third-order conductivity are discussed for third-harmonic
generation, the Kerr effect and two-photon carrier injection, parametric
frequency conversion, and two-color coherent current injection. In contrast
with two-dimensional materials, the three-dimensional Dirac and Weyl
semimetals allow for adjusting the nonlinear signals by tuning the sample
thickness. Thus, one can envision broad applications of such materials in
nonlinear photonic devices.

In Ref. [193], Lv et al. report the experimental observation of high-harmonic
generation in Weyl semimetal $\beta$-WP2. High-harmonic generation in solids
has only been discovered recently [209, 210, 211, 212]. Recent works [213,
214] report efficient high-harmonic generation from topological materials.
Moreover, Ref. [215] shows that the polarimetry of high-harmonic emission from
solids can be used to directly retrieve the Berry curvature. In Lv’s
experiment, both odd and even orders of high-harmonic emissions are observed
(Fig. 11f). The high-harmonic spectrum extends into the vacuum ultraviolet
region ($190\text{\,}\mathrm{n}\mathrm{m}$, 10th order) even under fairly low
femtosecond laser intensity
($\sim$0.29\text{\,}\mathrm{T}\mathrm{W}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$$).
It is interpreted that odd-order harmonics come from the Bloch oscillation,
while even-order harmonics arise from the effective Lorentz force due to the
Berry curvature (Fig. 11e). By analyzing the crystallographic orientation-
dependent high-harmonic spectra, they further quantitatively retrieve the
electronic band structure and Berry curvature of $\beta$-WP2.

###  3.3 Other effects

Of course, this tutorial review cannot cover all the reported optical
phenomena and applications of Weyl semimetals. Here we list some examples of
other phenomena and applications not covered in this review: gyrotropic
magnetic effect [216], photoinduced anomalous Hall effect [217], nonlinear
magneto-optical effects [156], topological frequency conversion [218],
emergent electromagnetic induction [219, 220], tunable perfect absorption
[221], unidirectional bound states in the continuum [222], transverse Kerker
effect [223], and topological lasers [224].

##  4 Thermal photonic applications and devices

In this section, we survey various thermal photonic applications of Weyl
semimetals. We discuss the nonreciprocal thermal emitters in Sec. 4.1, the
heat flux control in Sec. 4.2, and the control of Casimir force in Sec. 4.3.

###  4.1 Nonreciprocal thermal emitters

Conventional thermal emitters obey Kirchhoff’s law of thermal radiation [104,
225, 226, 227, 228, 229], which states that for a given direction,
polarization, and frequency, the emissivity and the absorptivity are equal.
However, Kirchhoff’s law is not a requirement of thermodynamics but a
consequence of Lorentz reciprocity [230, 111, 231, 112]. One can construct
thermal emitters with nonreciprocal materials featuring an asymmetric
dielectric tensor. A conventional way to achieve the nonreciprocal effect is
to use the nonreciprocal response of magneto-optical materials under an
external magnetic field. The cyclotron motion of electrons breaks the
microscopic reversibility [232, 233], resulting in different properties when
the medium is absorbing and emitting light [234, 235]. Using this strategy,
Zhu and Fan [236] proposed a photonic thermal emitter that can achieve
complete violation of Kirchhoff’s law. However, since the nonreciprocal effect
is quite weak in the thermal radiation wavelength range
($\sim$10\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$ at room
temperature), the nonreciprocal emitters require a very large external
magnetic field of $3\text{\,}\mathrm{T}$ to operate. Later, Zhao et al. [231]
improved the photonic design and reduced the required magnetic field to
$0.3\text{\,}\mathrm{T}$ at the expense of reduced bandwidth of the
nonreciprocal effect.

As we mentioned in Sec. 2.5, magnetic Weyl semimetals can exhibit much
stronger nonreciprocal effects compared to conventional magneto-optical
materials. Such giant nonreciprocity persists at the temperature below the
Curie temperature ($T_{C}$) of the material [237]. Some of the known magnetic
Weyl semimetals possess high Curie temperature; for example, Co2MnGa has
$T_{C}=$694\text{\,}\mathrm{K}$$ [237], and Ti2MnAl has $T_{C}>650K$ [238,
239]. These materials are ideal candidates for nonreciprocal thermal emission.
In Ref. [14], Zhao et al. proposed a broadband thermal emitter based on
magnetic Weyl semimetals that achieves near-complete violation of Kirchhoff’s
law at room temperatures. The structure is shown in Fig. 12a which supports
nonreciprocal surface plasmon polaritons as shown in Fig. 12b. The emissivity
and absorptivity of the emitter show a significant difference as shown in Fig.
12c. Tsurimaki et al. investigated the effect of the Fermi-arc surface state
and the number of Weyl nodes for Weyl semimetal-based nonreciprocal thermal
emitters [15, 16]. Wu et al. construct nonreciprocal thermal emitters using
Weyl semimetal thin films [240, 241, 242]. Since the nonreciprocal effect is
intrinsic to the materials, these thermal emitters do not require any external
magnetic field.

Here we note that for a nonreciprocal thermal emitter, the emissivity and
absorptivity in the same direction can be different; however, the angular
distribution of emissivity and absorptivity can be still constrained by
compound symmetries as revealed in Ref. [112]. The compound symmetry can be
used to design nonreciprocal thermal emitters with correlated patterns of
emissivity and absorptivity [8]. Breaking the compound symmetries can remove
such correlation constraints [243, 112].

The above-mentioned works primarily focus on linear polarized thermal
radiation. It has been shown that unpatterned thermal emitters with
nonreciprocal responses could radiate thermal photons that carry a net spin
angular momentum [244, 245, 246]. Khandekar et al. studied spin-resolved
Kirchhoff’s laws in nonreciprocal systems [247]. Since these thermal photons
can carry nonzero spin angular momentum, the emission of such photons can
result in a back-action torque on the thermal emitter [244, 248]. Maghrebi et
al. discussed this fluctuation-driven torque for a topological insulator thin
film out of thermal equilibrium with a cold environment [244]. Guo and Fan
proposed a single-particle heat engine utilizing this effect [249].

Figure 12: Thermal photonic applications. (a) A nonreciprocal thermal emitter
based on a magnetic Weyl semimetal photonic crystal. (b) Dispersion of the
nonreciprocal surface plasmon polaritons of the Weyl semimetal. The gray
region is the light cone of the vacuum. The blue region denotes the continuum
of bulk modes. The red region denotes the surface plasmons used for
nonreciprocal thermal emission. The gray dashed lines denote the boundaries of
the Brillouin zones. $k_{F}=E_{F}/\hbar v_{F}$ is the Fermi wavevector. (c)
Emissivity and absorptivity spectra in $\theta=80^{\circ}$ direction. (d)
Persistent directional heat current in a many-body system, at the thermal
equilibrium of temperature $T$. The spheres are made of magneto-optical
materials, with a magnetic field applied perpendicular to the plane of the
triangle. Magnetic Weyl semimetals can provide the same effects without a
magnetic field. (e) The photon thermal Hall effect among four particles made
of magnetic Weyl semimetals forming a square with $C_{4}$ symmetry. A
temperature gradient $\Delta T$ between particles 1 and 2 along the $x$-axis
induces a nonzero temperature difference between particles 3 and 4. (f) A
radiative thermal router based on three spheres of Weyl semimetal
nanoparticles: tuning the Weyl node separation of the center particle will
guide the heat to flow in a different direction. Figures are reproduced with
permission from (a-c) Ref. [14], Copyright 2020 American Chemical Society; (d)
Ref. [250], Copyright 2016 American Physical Society; (e) Ref. [251],
Copyright 2020 American Physical Society; (f) Ref. [252], Copyright 2020
American Chemical Society.

###  4.2 Heat flux control

Radiative heat transfer plays an important role in photon-based energy
conversion and thermal management systems [226, 253]. In traditional thermal
systems, the radiative heat flux is driven by a nonzero temperature gradient,
and the radiative heat transfer is reciprocal, indicating that when the
temperatures of two bodies are exchanged, the magnitude of the heat flux
transferred between them is unchanged. However, in systems that involve
nonreciprocal materials, photons can flow under zero temperature gradient, and
radiative heat transfer could be nonreciprocal.

In Ref. [250], Zhu and Fan reported the existence of a persistent photon heat
current in a nonreciprocal multi-body system even without a temperature
gradient. As shown in Fig. 12d, the arrows indicate the net heat flux
direction when the three magneto-optical spheres are at the same temperature.
Later, the persistent heat current feature was also reported within a single
nonreciprocal body or cavity [254, 248, 255]. This phenomenon is closely
related to the thermal Hall effect [256, 257]. In Ref. [258], Ben-Abdallah
reported that with an external magnetic field, nonzero heat flux is induced in
the direction that is perpendicular to the temperature gradient between
magneto-optical spheres, as illustrated in Fig. 12e. In Ref. [251], Ott et al.
reported the same phenomenon in a multi-body system made of magnetic Weyl
semimetals without the need for an external magnetic field.

Optical nonreciprocity can also enable efficient heat rectification [259], as
reported in Refs. [260, 261, 262]. These thermal diodes rely on the
nonreciprocal optical surface modes or bulk modes that induce an asymmetric
thermal resistance. When the temperature gradient alters direction, the
magnitude of the thermal resistance also changes.

Magnetic Weyl semimetals also provide new opportunities for controlling near-
field radiative heat transfer due to their anisotropic optical properties and
flexible tunability. In Ref. [263], Tang et al. reported a twist-induced near-
field heat modulator based on magnetic Weyl semimetals. The radiative
resistance of the system experiences a significant change depending on the
relative alignment of the directions of Weyl nodes separation in the thermal
emitter and receiver. Other properties of Weyl semimetals such as the Fermi
level, the number of Weyl nodes, the Weyl node separations [264], and external
magnetic field [265, 252] can also be used to modulate near-field heat
transfer. Using these effects, in Ref. [252], Guo demonstrated a radiative
thermal router based on Weyl semimetals as illustrated in Fig. 12f. We also
note several recent reports concerning the general properties of these
radiative heat transfer systems involving nonreciprocal bodies [266, 267, 256,
257, 268].

###  4.3 Casimir force control

Besides the exciting opportunities in heat transfer, magnetic Weyl semimetals
also provide an effective way to control Casimir forces [269, 270, 271, 272],
especially to create repulsive Casimir force, a long sought-after effect [273,
274, 275]. In Ref. [269], Wilson et al. reported repulsive Casimir force
between semi-infinite magnetic Weyl semimetals when the separation of the two
semimetals $d\leq 4c/\sigma_{xy}$, where $c$ is the speed of light in the
vacuum, and $\sigma_{xy}$ is the bulk hall conductivity of Weyl semimetals.
Recently, it was shown that the equilibrium Casimir force in nonreciprocal
systems can be used for propulsion [276]. We expect that an enhanced
propulsion effect can be realized in magnetic Weyl semimetals.

## Conclusion

In conclusion, we have provided an introductory review of Weyl semimetals in
photonics. We covered the basic concept and optical properties of Weyl
semimetals, and surveyed their emerging applications in photonic science and
engineering. We discussed how the nontrivial topology of Weyl semimetals leads
to unusual optical properties. Photonics based on Weyl semimetals is an
emerging topic with many open challenges and new opportunities. We believe
that more exciting applications of Weyl semimetals will come up in the near
future, and we wish this pedagogical review will benefit upcoming researchers
exploring this new direction.

## Declarations

### Availability of data and materials

Not applicable.

### Competing interests

The authors declare that they have no competing interests.

### Funding

This work is supported by MURI projects from the U. S. Army Research Office
(Grant No. W911NF-19-1-0279) and the U. S. Air Force Office of Scientific
Research (FA9550-21-1-0244).

### Author’s contributions

CG, VSA, BZ, and SF wrote the manuscript. CG led the collaboration. SF
supervised the work.

### Acknowledgements

Not applicable.

## References

  * [1] Herring, C.: Accidental degeneracy in the energy bands of crystals. Physical Review 52, 365–373 (1937). doi:10.1103/PhysRev.52.365
  * [2] Armitage, N.P., Mele, E.J., Vishwanath, A.: Weyl and Dirac semimetals in three-dimensional solids. Reviews of Modern Physics 90, 015001 (2018). doi:10.1103/RevModPhys.90.015001
  * [3] Srednicki, M.: Quantum Field Theory. Cambridge University Press, Cambridge (2007)
  * [4] Gorbar, E.V., Miranskij, V.A., Shovkovy, I.A., Sukhachov, P.O.: Electronic Properties of Dirac and Weyl Semimetals. World Scientific, Singapore (2021)
  * [5] Nagaosa, N., Morimoto, T., Tokura, Y.: Transport, magnetic and optical properties of Weyl materials. Nature Reviews Materials 5, 621–636 (2020). doi:10.1038/s41578-020-0208-y
  * [6] Kotov, O.V., Lozovik, Y.E.: Giant tunable nonreciprocity of light in Weyl semimetals. Physical Review B 98(19), 195446 (2018). doi:10.1103/PhysRevB.98.195446
  * [7] Asadchy, V.S., Guo, C., Zhao, B., Fan, S.: Sub-wavelength passive optical isolators using photonic structures based on Weyl semimetals. Advanced Optical Materials 8, 2000100 (2020). doi:10.1002/adom.202000100
  * [8] Park, Y., Asadchy, V.S., Zhao, B., Guo, C., Wang, J., Fan, S.: Violating Kirchhoff’s law of thermal radiation in semitransparent structures. ACS Photonics 8, 2417–2424 (2021). doi:10.1021/acsphotonics.1c00612
  * [9] Ji, Z., Liu, W., Krylyuk, S., Fan, X., Zhang, Z., Pan, A., Feng, L., Davydov, A., Agarwal, R.: Photocurrent detection of the orbital angular momentum of light. Science 368(6492), 763–767 (2020). doi:10.1126/science.aba9192
  * [10] Lai, J., Ma, J., Fan, Z., Song, X., Yu, P., Liu, Z., Zhang, P., Shi, Y., Cheng, J., Sun, D.: Direct light orbital angular momentum detection in mid-infrared based on the type-II Weyl semimetal ${\rm TaIrTe_{4}}$. Advanced Materials 34(29), 2201229 (2022). doi:10.1002/adma.202201229
  * [11] Wu, L., Patankar, S., Morimoto, T., Nair, N.L., Thewalt, E., Little, A., Analytis, J.G., Moore, J.E., Orenstein, J.: Giant anisotropic nonlinear optical response in transition metal monopnictide Weyl semimetals. Nature Physics 13(4), 350–355 (2017). doi:10.1038/nphys3969
  * [12] Almutairi, S., Chen, Q., Tokman, M., Belyanin, A.: Four-wave mixing in Weyl semimetals. Physical Review B 101(23), 235156 (2020). doi:10.1103/PhysRevB.101.235156
  * [13] Cheng, J.L., Sipe, J.E., Wu, S.W.: Third-order optical nonlinearity of three-dimensional massless Dirac fermions. ACS Photonics 7, 2515–2526 (2020). doi:10.1021/acsphotonics.0c00836
  * [14] Zhao, B., Guo, C., Garcia, C.A.C., Narang, P., Fan, S.: Axion-field-enabled nonreciprocal thermal radiation in Weyl semimetals. Nano Letters 20, 1923–1927 (2020). doi:10.1021/acs.nanolett.9b05179
  * [15] Tsurimaki, Y., Qian, X., Pajovic, S., Han, F., Li, M., Chen, G.: Large nonreciprocal absorption and emission of radiation in type-I Weyl semimetals with time reversal symmetry breaking. Physical Review B 101, 165426 (2020). doi:10.1103/PhysRevB.101.165426
  * [16] Pajovic, S., Tsurimaki, Y., Qian, X., Chen, G.: Intrinsic nonreciprocal reflection and violation of Kirchhoff’s law of radiation in planar type-I magnetic Weyl semimetal surfaces. Physical Review B 102, 165417 (2020). doi:10.1103/PhysRevB.102.165417
  * [17] Ashcroft, N.W., Mermin, N.D.: Solid State Physics. Holt, Rinehart and Winston, New York (1976)
  * [18] Burns, G.: Solid State Physics. Academic Press, Boston (1990)
  * [19] Golin, S.: Band structure of bismuth: Pseudopotential approach. Physical Review 166, 643–651 (1968). doi:10.1103/PhysRev.166.643
  * [20] Slonczewski, J.C., Weiss, P.R.: Band structure of graphite. Physical Review 109, 272–279 (1958). doi:10.1103/PhysRev.109.272
  * [21] Weyl, H.: Elektron und gravitation. I. Zeitschrift für Physik 56, 330–352 (1929). doi:10.1007/BF01339504
  * [22] Super-Kamiokande Collaboration: Evidence for oscillation of atmospheric neutrinos. Physical Review Letters 81, 1562–1567 (1998). doi:10.1103/PhysRevLett.81.1562
  * [23] Vishwanath, A.: Where the Weyl Things Are. Physics 8, 84 (2015)
  * [24] Nielsen, H.B., Ninomiya, M.: Absence of neutrinos on a lattice: (I). Proof by homotopy theory. Nuclear Physics B 185, 20–40 (1981). doi:10.1016/0550-3213(81)90361-8
  * [25] Nielsen, H.B., Ninomiya, M.: Absence of neutrinos on a lattice: (II). Intuitive topological proof. Nuclear Physics B 193, 173–194 (1981). doi:10.1016/0550-3213(81)90524-1
  * [26] Nielsen, H.B., Ninomiya, M.: A no-go theorem for regularizing chiral fermions. Physics Letters B 105, 219–223 (1981). doi:10.1016/0370-2693(81)91026-1
  * [27] Sun, X.-Q., Xiao, M., Bzdušek, T., Zhang, S.-C., Fan, S.: Three-Dimensional Chiral Lattice Fermion in Floquet Systems. Physical Review Letters 121, 196401 (2018). doi:10.1103/PhysRevLett.121.196401
  * [28] von Neuman, J., Wigner, E.: Uber merkwürdige diskrete Eigenwerte. Uber das Verhalten von Eigenwerten bei adiabatischen Prozessen. Physikalische Zeitschrift 30, 467–470 (1929)
  * [29] Vanderbilt, D.: Berry Phases in Electronic Structure Theory. Cambridge University Press, Cambridge (2018)
  * [30] Belopolski, I., Xu, S.-Y., Sanchez, D.S., Chang, G., Guo, C., Neupane, M., Zheng, H., Lee, C.-C., Huang, S.-M., Bian, G., Alidoust, N., Chang, T.-R., Wang, B., Zhang, X., Bansil, A., Jeng, H.-T., Lin, H., Jia, S., Hasan, M.Z.: Criteria for directly detecting topological Fermi arcs in Weyl semimetals. Physical Review Letters 116, 066802 (2016). doi:10.1103/PhysRevLett.116.066802
  * [31] Xu, S.-Y., Belopolski, I., Sanchez, D.S., Neupane, M., Chang, G., Yaji, K., Yuan, Z., Zhang, C., Kuroda, K., Bian, G., Guo, C., Lu, H., Chang, T.-R., Alidoust, N., Zheng, H., Lee, C.-C., Huang, S.-M., Hsu, C.-H., Jeng, H.-T., Bansil, A., Neupert, T., Komori, F., Kondo, T., Shin, S., Lin, H., Jia, S., Hasan, M.Z.: Spin polarization and texture of the Fermi arcs in the Weyl fermion semimetal TaAs. Physical Review Letters 116, 096801 (2016). doi:10.1103/PhysRevLett.116.096801
  * [32] Yan, B., Felser, C.: Topological materials: Weyl semimetals. Annual Review of Condensed Matter Physics 8(1), 337–354 (2017). doi:10.1146/annurev-conmatphys-031016-025458
  * [33] Kramers, H.A.: Théorie générale de la rotation paramagnétique dans les cristaux. Proceedings Koninklijke Akademie van Wetenschappen 33, 959–972 (1930)
  * [34] Xu, S.-Y., Belopolski, I., Alidoust, N., Neupane, M., Bian, G., Zhang, C., Sankar, R., Chang, G., Yuan, Z., Lee, C.-C., Huang, S.-M., Zheng, H., Ma, J., Sanchez, D.S., Wang, B., Bansil, A., Chou, F., Shibayev, P.P., Lin, H., Jia, S., Hasan, M.Z.: Discovery of a Weyl fermion semimetal and topological Fermi arcs. Science 349, 613–617 (2015). doi:10.1126/science.aaa9297
  * [35] Lv, B.Q., Weng, H.M., Fu, B.B., Wang, X.P., Miao, H., Ma, J., Richard, P., Huang, X.C., Zhao, L.X., Chen, G.F., Fang, Z., Dai, X., Qian, T., Ding, H.: Experimental discovery of Weyl semimetal TaAs. Physical Review X 5, 031013 (2015). doi:10.1103/PhysRevX.5.031013
  * [36] Xu, S.-Y., Belopolski, I., Sanchez, D.S., Zhang, C., Chang, G., Guo, C., Bian, G., Yuan, Z., Lu, H., Chang, T.-R., Shibayev, P.P., Prokopovych, M.L., Alidoust, N., Zheng, H., Lee, C.-C., Huang, S.-M., Sankar, R., Chou, F., Hsu, C.-H., Jeng, H.-T., Bansil, A., Neupert, T., Strocov, V.N., Lin, H., Jia, S., Hasan, M.Z.: Experimental discovery of a topological Weyl semimetal state in TaP. Science Advances 1, 1501092 (2015). Chap. Research Article. doi:10.1126/sciadv.1501092
  * [37] Zhang, C.-L., Xu, S.-Y., Wang, C.M., Lin, Z., Du, Z.Z., Guo, C., Lee, C.-C., Lu, H., Feng, Y., Huang, S.-M., Chang, G., Hsu, C.-H., Liu, H., Lin, H., Li, L., Zhang, C., Zhang, J., Xie, X.-C., Neupert, T., Hasan, M.Z., Lu, H.-Z., Wang, J., Jia, S.: Magnetic-tunnelling-induced Weyl node annihilation in TaP. Nature Physics 13, 979–986 (2017). doi:10.1038/nphys4183
  * [38] Xu, S.-Y., Alidoust, N., Belopolski, I., Yuan, Z., Bian, G., Chang, T.-R., Zheng, H., Strocov, V.N., Sanchez, D.S., Chang, G., Zhang, C., Mou, D., Wu, Y., Huang, L., Lee, C.-C., Huang, S.-M., Wang, B., Bansil, A., Jeng, H.-T., Neupert, T., Kaminski, A., Lin, H., Jia, S., Zahid Hasan, M.: Discovery of a Weyl fermion state with Fermi arcs in niobium arsenide. Nature Physics 11, 748–754 (2015). doi:10.1038/nphys3437
  * [39] Zheng, H., Xu, S.-Y., Bian, G., Guo, C., Chang, G., Sanchez, D.S., Belopolski, I., Lee, C.-C., Huang, S.-M., Zhang, X., Sankar, R., Alidoust, N., Chang, T.-R., Wu, F., Neupert, T., Chou, F., Jeng, H.-T., Yao, N., Bansil, A., Jia, S., Lin, H., Hasan, M.Z.: Atomic-scale visualization of quantum interference on a Weyl semimetal surface by scanning tunneling microscopy. ACS Nano 10, 1378–1385 (2016). doi:10.1021/acsnano.5b06807
  * [40] Zheng, H., Chang, G., Huang, S.-M., Guo, C., Zhang, X., Zhang, S., Yin, J., Xu, S.-Y., Belopolski, I., Alidoust, N., Sanchez, D.S., Bian, G., Chang, T.-R., Neupert, T., Jeng, H.-T., Jia, S., Lin, H., Hasan, M.Z.: Mirror protected Dirac fermions on a Weyl semimetal NbP surface. Physical Review Letters 119, 196403 (2017). doi:10.1103/PhysRevLett.119.196403
  * [41] Gooth, J., Niemann, A.C., Meng, T., Grushin, A.G., Landsteiner, K., Gotsmann, B., Menges, F., Schmidt, M., Shekhar, C., Süß, V., Hühne, R., Rellinghaus, B., Felser, C., Yan, B., Nielsch, K.: Experimental signatures of the mixed axial–gravitational anomaly in the Weyl semimetal NbP. Nature 547, 324–327 (2017). doi:10.1038/nature23005
  * [42] Morali, N., Batabyal, R., Nag, P.K., Liu, E., Xu, Q., Sun, Y., Yan, B., Felser, C., Avraham, N., Beidenkopf, H.: Fermi-arc diversity on surface terminations of the magnetic Weyl semimetal Co3Sn2S2. Science 365(6459), 1286–1291 (2019). doi:10.1126/science.aav2334
  * [43] Belopolski, I., Manna, K., Sanchez, D.S., Chang, G., Ernst, B., Yin, J., Zhang, S.S., Cochran, T., Shumiya, N., Zheng, H., Singh, B., Bian, G., Multer, D., Litskevich, M., Zhou, X., Huang, S.-M., Wang, B., Chang, T.-R., Xu, S.-Y., Bansil, A., Felser, C., Lin, H., Hasan, M.Z.: Discovery of topological Weyl fermion lines and drumhead surface states in a room temperature magnet. Science 365(6459), 1278–1281 (2019). doi:10.1126/science.aav2327
  * [44] Liu, D.F., Liang, A.J., Liu, E.K., Xu, Q.N., Li, Y.W., Chen, C., Pei, D., Shi, W.J., Mo, S.K., Dudin, P., Kim, T., Cacho, C., Li, G., Sun, Y., Yang, L.X., Liu, Z.K., Parkin, S.S.P., Felser, C., Chen, Y.L.: Magnetic Weyl semimetal phase in a Kagomé crystal. Science 365(6459), 1282–1285 (2019). doi:10.1126/science.aav2873
  * [45] Fisher, D.J.: Topological Semimetals. Materials Research Forum LLC, Millersville (2019)
  * [46] Xiong, J., Kushwaha, S.K., Liang, T., Krizan, J.W., Hirschberger, M., Wang, W., Cava, R.J., Ong, N.P.: Evidence for the chiral anomaly in the Dirac semimetal Na3Bi. Science 350, 413–416 (2015). doi:10.1126/science.aac6089
  * [47] Zhang, C.-L., Xu, S.-Y., Belopolski, I., Yuan, Z., Lin, Z., Tong, B., Bian, G., Alidoust, N., Lee, C.-C., Huang, S.-M., Chang, T.-R., Chang, G., Hsu, C.-H., Jeng, H.-T., Neupane, M., Sanchez, D.S., Zheng, H., Wang, J., Lin, H., Zhang, C., Lu, H.-Z., Shen, S.-Q., Neupert, T., Zahid Hasan, M., Jia, S.: Signatures of the Adler–Bell–Jackiw chiral anomaly in a Weyl fermion semimetal. Nature Communications 7, 10735 (2016). doi:10.1038/ncomms10735
  * [48] Ong, N.P., Liang, S.: Experimental signatures of the chiral anomaly in Dirac–Weyl semimetals. Nature Reviews Physics 3, 394–404 (2021). doi:10.1038/s42254-021-00310-9
  * [49] Zhang, C., Guo, C., Lu, H., Zhang, X., Yuan, Z., Lin, Z., Wang, J., Jia, S.: Large magnetoresistance over an extended temperature regime in monophosphides of tantalum and niobium. Physical Review B 92, 041203 (2015). doi:10.1103/PhysRevB.92.041203
  * [50] Westström, A., Ojanen, T.: Designer curved-space geometry for relativistic fermions in Weyl metamaterials. Physical Review X 7, 041026 (2017). doi:10.1103/PhysRevX.7.041026
  * [51] Cortijo, A., Ferreirós, Y., Landsteiner, K., Vozmediano, M.A.H.: Elastic gauge fields in Weyl semimetals. Physical Review Letters 115, 177202 (2015). doi:10.1103/PhysRevLett.115.177202
  * [52] Pikulin, D.I., Chen, A., Franz, M.: Chiral anomaly from strain-induced gauge fields in Dirac and Weyl semimetals. Physical Review X 6, 041021 (2016). doi:10.1103/PhysRevX.6.041021
  * [53] Song, Z., Dai, X.: Hear the sound of Weyl fermions. Physical Review X 9, 021053 (2019). doi:10.1103/PhysRevX.9.021053
  * [54] Vafek, O., Vishwanath, A.: Dirac fermions in solids: From high-Tc cuprates and graphene to topological insulators and Weyl semimetals. Annual Review of Condensed Matter Physics 5, 83–112 (2014). doi:10.1146/annurev-conmatphys-031113-133841
  * [55] Burkov, A.A.: Weyl metals. Annual Review of Condensed Matter Physics 9, 359–378 (2018). doi:10.1146/annurev-conmatphys-033117-054129
  * [56] Witczak-Krempa, W., Chen, G., Kim, Y.B., Balents, L.: Correlated quantum phenomena in the strong spin-orbit regime. Annual Review of Condensed Matter Physics 5, 57–82 (2014). doi:10.1146/annurev-conmatphys-020911-125138
  * [57] Šmejkal, L., Jungwirth, T., Sinova, J.: Route towards Dirac and Weyl antiferromagnetic spintronics. physica status solidi (RRL) – Rapid Research Letters 11, 1700044 (2017). doi:10.1002/pssr.201700044
  * [58] Mikitik, G.P., Sharlai, Y.V.: Magnetic susceptibility of topological semimetals. Journal of Low Temperature Physics 197, 272–309 (2019). doi:10.1007/s10909-019-02225-3
  * [59] Hosur, P., Qi, X.: Recent developments in transport phenomena in Weyl semimetals. Comptes Rendus Physique 14, 857–870 (2013). doi:10.1016/j.crhy.2013.10.010
  * [60] Burkov, A.A.: Chiral anomaly and transport in Weyl metals. Journal of Physics: Condensed Matter 27, 113201 (2015). doi:10.1088/0953-8984/27/11/113201
  * [61] Wang, S., Lin, B.-C., Wang, A.-Q., Yu, D.-P., Liao, Z.-M.: Quantum transport in Dirac and Weyl semimetals: A review. Advances in Physics: X 2, 518–544 (2017). doi:10.1080/23746149.2017.1327329
  * [62] Gorbar, E.V., Miransky, V.A., Shovkovy, I.A., Sukhachov, P.O.: Anomalous transport properties of Dirac and Weyl semimetals (Review Article). Low Temperature Physics 44, 487–505 (2018). doi:10.1063/1.5037551
  * [63] Wang, H., Wang, J.: Electron transport in Dirac and Weyl semimetals. Chinese Physics B 27, 107402 (2018). doi:2020030514010695
  * [64] Hu, J., Xu, S.-Y., Ni, N., Mao, Z.: Transport of topological semimetals. Annual Review of Materials Research 49, 207–252 (2019). doi:10.1146/annurev-matsci-070218-010023
  * [65] Chiu, C.-K., Teo, J.C.Y., Schnyder, A.P., Ryu, S.: Classification of topological quantum matter with symmetries. Reviews of Modern Physics 88, 035005 (2016). doi:10.1103/RevModPhys.88.035005
  * [66] Witten, E.: Three lectures on topological phases of matter. La Rivista del Nuovo Cimento 39, 313–370 (2016). doi:10.1393/ncr/i2016-10125-3
  * [67] Weng, H., Dai, X., Fang, Z.: Topological semimetals predicted from first-principles calculations. Journal of Physics: Condensed Matter 28, 303001 (2016). doi:10.1088/0953-8984/28/30/303001
  * [68] Hasan, M.Z., Xu, S.-Y., Bian, G.: Topological insulators, topological superconductors and Weyl fermion semimetals: Discoveries, perspectives and outlooks. Physica Scripta T164, 014001 (2015). doi:10.1088/0031-8949/2015/T164/014001
  * [69] Hasan, M.Z., Xu, S.-Y., Belopolski, I., Huang, S.-M.: Discovery of Weyl fermion semimetals and topological Fermi arc states. Annual Review of Condensed Matter Physics 8, 289–309 (2017). doi:10.1146/annurev-conmatphys-031016-025225
  * [70] Yan, B., Felser, C.: Topological materials: Weyl semimetals. Annual Review of Condensed Matter Physics 8, 337–354 (2017). doi:10.1146/annurev-conmatphys-031016-025458
  * [71] Zheng, H., Zahid Hasan, M.: Quasiparticle interference on type-I and type-II Weyl semimetal surfaces: A review. Advances in Physics: X 3, 1466661 (2018). doi:10.1080/23746149.2018.1466661
  * [72] Nagaosa, N., Sinova, J., Onoda, S., MacDonald, A.H., Ong, N.P.: Anomalous Hall effect. Reviews of Modern Physics 82, 1539–1592 (2010). doi:10.1103/RevModPhys.82.1539
  * [73] Gorbar, E.V., Miransky, V.A., Shovkovy, I.A.: Chiral asymmetry of the Fermi surface in dense relativistic matter in a magnetic field. Physical Review C, 4 (2009). doi:10.1103/PhysRevC.80.032801
  * [74] Zyuzin, A.A., Burkov, A.A.: Topological response in Weyl semimetals and the chiral anomaly. Physical Review B 86(11), 115133 (2012). doi:10.1103/PhysRevB.86.115133
  * [75] Nie, S., Hashimoto, T., Prinz, F.B.: Magnetic Weyl semimetal in K2Mn3(AsO4)3 with the minimum number of Weyl points. Physical Review Letters 128, 176401 (2022). doi:10.1103/PhysRevLett.128.176401
  * [76] Soh, J.-R., de Juan, F., Vergniory, M.G., Schröter, N.B.M., Rahn, M.C., Yan, D.Y., Jiang, J., Bristow, M., Reiss, P.A., Blandy, J.N., Guo, Y.F., Shi, Y.G., Kim, T.K., McCollam, A., Simon, S.H., Chen, Y., Coldea, A.I., Boothroyd, A.T.: Ideal Weyl semimetal induced by magnetic exchange. Physical Review B 100, 201102 (2019). doi:10.1103/PhysRevB.100.201102
  * [77] Zangwill, A.: Modern Electrodynamics. Cambridge University Press, Cambridge (2013)
  * [78] Wilczek, F.: Two applications of axion electrodynamics. Physical Review Letters 58, 1799–1802 (1987). doi:10.1103/PhysRevLett.58.1799
  * [79] Cheng, T.-P., Li, L.-F.: Gauge Theory of Elementary Particle Physics. Oxford University Press, Oxford (1984)
  * [80] Volovik, G.E.: The Universe in a Helium Droplet. Oxford University Press, Oxford (2009)
  * [81] Qi, X.-L., Hughes, T.L., Zhang, S.-C.: Topological field theory of time-reversal invariant insulators. Physical Review B 78, 195424 (2008). doi:10.1103/PhysRevB.78.195424
  * [82] Qi, X.-L., Zhang, S.-C.: Topological insulators and superconductors. Reviews of Modern Physics 83, 1057–1110 (2011). doi:10.1103/RevModPhys.83.1057
  * [83] Bernevig, B.A., Hughes, T.L.: Topological Insulators and Topological Superconductors. Princeton University Press, Princeton (2013)
  * [84] Grushin, A.G.: Consequences of a condensed matter realization of Lorentz-violating QED in Weyl semi-metals. Physical Review D 86, 045001 (2012). doi:10.1103/PhysRevD.86.045001
  * [85] Goswami, P., Tewari, S.: Axionic field theory of (3+1)-dimensional Weyl semimetals. Physical Review B 88, 245107 (2013). doi:10.1103/PhysRevB.88.245107
  * [86] Vazifeh, M.M., Franz, M.: Electromagnetic response of Weyl semimetals. Physical Review Letters 111, 027201 (2013). doi:10.1103/PhysRevLett.111.027201
  * [87] Deng, K., Van Dyke, J.S., Minic, D., Heremans, J.J., Barnes, E.: Exploring self-consistency of the equations of axion electrodynamics in Weyl semimetals. Physical Review B 104, 075202 (2021). doi:10.1103/PhysRevB.104.075202
  * [88] Armitage, N.P., Wu, L.: On the matter of topological insulators as magnetoelectrics. SciPost Physics 6, 046 (2019). doi:10.21468/SciPostPhys.6.4.046
  * [89] Tong, D.: Gauge theory. Lecture notes, DAMTP Cambridge (2018)
  * [90] Karch, A.: Electric-magnetic duality and topological insulators. Physical Review Letters 103, 171601 (2009). doi:10.1103/PhysRevLett.103.171601
  * [91] Hofmann, J., Das Sarma, S.: Surface plasmon polaritons in topological Weyl semimetals. Physical Review B 93(24), 241402 (2016). doi:10.1103/PhysRevB.93.241402
  * [92] Witten, E.: Fermion path integrals and topological phases. Reviews of Modern Physics 88, 035001 (2016). doi:10.1103/RevModPhys.88.035001
  * [93] Fujikawa, K., Suzuki, H.: Path Integrals and Quantum Anomalies. Clarendon Press, Oxford (2013)
  * [94] Kubo, R.: Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems. Journal of the Physical Society of Japan 12, 570–586 (1957). doi:10.1143/JPSJ.12.570
  * [95] Greenwood, D.A.: The Boltzmann equation in the theory of electrical conduction in metals. Proceedings of the Physical Society 71, 585–596 (1958). doi:10.1088/0370-1328/71/4/306
  * [96] Moseley, L.L., Lukes, T.: A simplified derivation of the Kubo-Greenwood formula. American Journal of Physics 46, 676–677 (1978). doi:10.1119/1.11229
  * [97] Lorentz, H.A.: The theorem of Poynting concerning the energy in the electromagnetic field and two general propositions concerning the propagation of light. Amsterdammer Akademie der Wetenschappen 4, 176 (1896)
  * [98] Potton, R.J.: Reciprocity in optics. Reports on Progress in Physics 67(5), 717–754 (2004). doi:10.1088/0034-4885/67/5/R03
  * [99] Caloz, C., Alù, A., Tretyakov, S., Sounas, D., Achouri, K., Deck-Léger, Z.-L.: Electromagnetic nonreciprocity. Physical Review Applied 10(4), 047001 (2018). doi:10.1103/PhysRevApplied.10.047001
  * [100] Asadchy, V.S., Mirmoosa, M.S., Díaz-Rubio, A., Fan, S., Tretyakov, S.A.: Tutorial on electromagnetic nonreciprocity and its origins. Proceedings of the IEEE 108, 1684–1727 (2020). doi:10.1109/JPROC.2020.3012381
  * [101] Guo, C., Zhao, Z., Fan, S.: Internal transformations and internal symmetries in linear photonic systems. Physical Review A 105, 023509 (2022). doi:10.1103/PhysRevA.105.023509
  * [102] Haus, H.A.: Waves and Fields in Optoelectronics. Prentice-Hall, Englewood Cliffs, New Jersey (1984)
  * [103] Guo, C., Fan, S.: Reciprocity constraints on reflection. Physical Review Letters 128, 256101 (2022). doi:10.1103/PhysRevLett.128.256101
  * [104] Kirchhoff, G.: Ueber das Verhältniss zwischen dem Emissionsvermögen und dem Absorptionsvermögen der Körper für Wärme und Licht. Annalen der Physik 185(2), 275–301 (1860). doi:10.1002/andp.18601850205
  * [105] Greffet, J.-J., Bouchon, P., Brucoli, G., Marquier, F.: Light emission by nonequilibrium bodies: Local Kirchhoff law. Physical Review X 8, 021008 (2018). doi:10.1103/PhysRevX.8.021008
  * [106] Yu, Z., Fan, S.: Complete optical isolation created by indirect interband photonic transitions. Nature Photonics 3(2), 91–94 (2009). doi:10.1038/nphoton.2008.273
  * [107] Jalas, D., Petrov, A., Eich, M., Freude, W., Fan, S., Yu, Z., Baets, R., Popović, M., Melloni, A., Joannopoulos, J.D., Vanwolleghem, M., Doerr, C.R., Renner, H.: What is – and what is not – an optical isolator. Nature Photonics 7, 579–582 (2013). doi:10.1038/nphoton.2013.185
  * [108] Wang, Z., Fan, S.: Optical circulators in two-dimensional magneto-optical photonic crystals. Optics Letters 30, 1989–1991 (2005). doi:10.1364/OL.30.001989
  * [109] Wang, Z., Chong, Y., Joannopoulos, J.D., Soljačić, M.: Observation of unidirectional backscattering-immune topological electromagnetic states. Nature 461, 772–775 (2009). doi:10.1038/nature08293
  * [110] Fang, K., Yu, Z., Fan, S.: Realizing effective magnetic field for photons by controlling the phase of dynamic modulation. Nature Photonics 6(11), 782–787 (2012). doi:10.1038/nphoton.2012.236
  * [111] Zhu, L., Fan, S.: Near-complete violation of detailed balance in thermal radiation. Physical Review B 90(22), 220301 (2014). doi:10.1103/PhysRevB.90.220301
  * [112] Guo, C., Zhao, B., Fan, S.: Adjoint Kirchhoff’s law and general symmetry implications for all thermal emitters. Physical Review X 12, 021023 (2022). doi:10.1103/PhysRevX.12.021023
  * [113] Zvezdin, A.K., Kotov, V.A.: Modern Magnetooptics and Magnetooptical Materials. CRC Press, New York (1997)
  * [114] Han, X., Markou, A., Stensberg, J., Sun, Y., Felser, C., Wu, L.: Giant intrinsic anomalous terahertz Faraday rotation in the magnetic Weyl semimetal Co2MnGa at room temperature. Physical Review B 105, 174406 (2022). doi:10.1103/PhysRevB.105.174406
  * [115] Huang, H., Fan, Y., Wu, B.-I., Kong, a.J.: Tunable TE/TM Wave Splitter Using a Gyrotropic Slab. Progress In Electromagnetics Research 85, 367–380 (2008). doi:10.2528/PIER08080303
  * [116] Abdol, S.O., Abdollahipour, B., Vala, A.S.: Surface plasmon polaritons in a waveguide composed of Weyl semimetals with different symmetries. Journal of Physics D: Applied Physics 53(4), 045105 (2019). doi:10.1088/1361-6463/ab52ee
  * [117] Ashby, P.E.C., Carbotte, J.P.: Chiral anomaly and optical absorption in Weyl semimetals. Physical Review B 89, 245121 (2014). doi:10.1103/PhysRevB.89.245121
  * [118] Huang, J., Wang, L., Yao, D.-X.: A semiclassical approach to surface Fermi arcs in Weyl semimetals. Science China Physics, Mechanics & Astronomy 65, 266811 (2022). doi:10.1007/s11433-021-1884-x
  * [119] Chen, Q., Kutayiah, A.R., Oladyshkin, I., Tokman, M., Belyanin, A.: Optical properties and electromagnetic modes of Weyl semimetals. Physical Review B 99(7), 075137 (2019). doi:10.1103/PhysRevB.99.075137
  * [120] Song, J.C.W., Rudner, M.S.: Fermi arc plasmons in Weyl semimetals. Physical Review B 96, 205443 (2017). doi:10.1103/PhysRevB.96.205443
  * [121] Lošić, Ž.B.: The coupling effects of surface plasmons and Fermi arc plasmons in Weyl semimetals. Journal of Physics: Condensed Matter 31, 285001 (2019). doi:10.1088/1361-648X/ab1734
  * [122] Gomez-Diaz, J.S., Tymchenko, M., Alù, A.: Hyperbolic plasmons and topological transitions over uniaxial metasurfaces. Physical Review Letters 114, 233901 (2015). doi:10.1103/PhysRevLett.114.233901
  * [123] Zhang, F., Gao, Y., Zhang, W.: Three-dimensional topological plasmons in Weyl semimetals. Physical Review B 104, 205141 (2021). doi:10.1103/PhysRevB.104.205141
  * [124] Kurokawa, K.: An Introduction to the Theory of Microwave Circuits. Academic, New York (1969)
  * [125] Dötsch, H., Bahlmann, N., Zhuromskyy, O., Hammer, M., Wilkens, L., Gerhardt, R., Hertel, P., Popkov, A.F.: Applications of magneto-optical waveguides in integrated optics: Review. Journal of the Optical Society of America B 22, 240–253 (2005). doi:10.1364/JOSAB.22.000240
  * [126] Shoji, Y., Mizumoto, T.: Magneto-optical non-reciprocal devices in silicon photonics. Science and Technology of Advanced Materials 15, 014602 (2014). doi:10.1088/1468-6996/15/1/014602
  * [127] Kravtsov, N.V., Kravtsov, N.N.: Nonreciprocal effects in ring lasers. Quantum Electronics 29, 378 (1999). doi:10.1070/QE1999v029n05ABEH001495
  * [128] Liu, E., Sun, Y., Kumar, N., Muechler, L., Sun, A., Jiao, L., Yang, S.-Y., Liu, D., Liang, A., Xu, Q., Kroder, J., Süß, V., Borrmann, H., Shekhar, C., Wang, Z., Xi, C., Wang, W., Schnelle, W., Wirth, S., Chen, Y., Goennenwein, S.T.B., Felser, C.: Giant anomalous Hall effect in a ferromagnetic kagome-lattice semimetal. Nature Physics 14, 1125–1131 (2018). doi:10.1038/s41567-018-0234-5
  * [129] Figotin, A., Vitebsky, I.: Nonreciprocal magnetic photonic crystals. Physical Review E 63(6), 066609 (2001). doi:10.1103/PhysRevE.63.066609
  * [130] Li, T., Yin, C., Wu, F.: Strong optical non-reciprocity in one-dimensional photonic crystal containing a Weyl semimetal-based defect. Optical Materials 121, 111583 (2021). doi:10.1016/j.optmat.2021.111583
  * [131] Fan, S., Joannopoulos, J.D.: Analysis of guided resonances in photonic crystal slabs. Physical Review B 65, 235112 (2002). doi:10.1103/PhysRevB.65.235112
  * [132] Kargarian, M., Randeria, M., Trivedi, N.: Theory of Kerr and Faraday rotations and linear dichroism in topological Weyl semimetals. Scientific Reports 5, 12683 (2015). doi:10.1038/srep12683
  * [133] Chtchelkatchev, N.M., Berman, O.L., Kezerashvili, R.Y., Lozovik, Y.E.: Chiral filtration of light by Weyl-semimetal medium. Physics Letters A 399, 127294 (2021). doi:10.1016/j.physleta.2021.127294
  * [134] Yang, C., Zhao, B., Cai, W., Zhang, Z.M.: Mid-infrared broadband circular polarizer based on Weyl semimetals. Optics Express 30(2), 3035–3046 (2022). doi:10.1364/OE.445803
  * [135] Mukherjee, S.P., Carbotte, J.P.: Absorption of circular polarized light in tilted type-I and type-II Weyl semimetals. Physical Review B 96, 085114 (2017). doi:10.1103/PhysRevB.96.085114
  * [136] Ukhtary, M.S., Nugraha, A.R.T., Saito, R.: Negative refraction in Weyl semimetals. Journal of the Physical Society of Japan 86(10), 104703 (2017). doi:10.7566/JPSJ.86.104703
  * [137] Hills, R.D.Y., Kusmartseva, A., Kusmartsev, F.V.: Current-voltage characteristics of Weyl semimetal semiconducting devices, Veselago lenses, and hyperbolic Dirac phase. Physical Review B 95, 214103 (2017). doi:10.1103/PhysRevB.95.214103
  * [138] Veselago, V.G.: The electrodynamics of substances with simultaneously negative values of $\epsilon$ and $\mu$. Soviet Physics Uspekhi 10(4), 509 (1968). doi:10.1070/PU1968v010n04ABEH003699
  * [139] Pendry, J.B.: Negative refraction makes a perfect lens. Physical Review Letters 85(18), 3966–3969 (2000). doi:10.1103/PhysRevLett.85.3966
  * [140] Shelby, R.A., Smith, D.R., Schultz, S.: Experimental verification of a negative index of refraction. Science 292(5514), 77–79 (2001). doi:10.1126/science.1058847
  * [141] Notomi, M.: Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap. Physical Review B 62, 10696–10705 (2000). doi:10.1103/PhysRevB.62.10696
  * [142] Luo, C., Johnson, S.G., Joannopoulos, J.D., Pendry, J.B.: All-angle negative refraction without negative effective index. Physical Review B 65, 201104 (2002). doi:10.1103/PhysRevB.65.201104
  * [143] Cubukcu, E., Aydin, K., Ozbay, E., Foteinopoulou, S., Soukoulis, C.M.: Negative refraction by photonic crystals. Nature 423, 604–605 (2003). doi:10.1038/423604b
  * [144] Tretyakov, S., Nefedov, I., Sihvola, A., Maslovski, S., Simovski, C.: Waves and energy in chiral nihility. Journal of Electromagnetic Waves and Applications 17(5), 695–706 (2003). doi:10.1163/156939303322226356
  * [145] Pendry, J.B.: A chiral route to negative refraction. Science 306, 1353–1355 (2004). doi:10.1126/science.1104467
  * [146] Monzon, C., Forester, D.W.: Negative refraction and focusing of circularly polarized waves in optically active media. Physical Review Letters 95, 123904 (2005). doi:10.1103/PhysRevLett.95.123904
  * [147] Zhang, S., Park, Y.-S., Li, J., Lu, X., Zhang, W., Zhang, X.: Negative refractive index in chiral metamaterials. Physical Review Letters 102, 023901 (2009). doi:10.1103/PhysRevLett.102.023901
  * [148] Hayata, T.: A new route to negative refractive index from topological metals. Progress of Theoretical and Experimental Physics 2018, 083–01 (2018). doi:10.1093/ptep/pty082
  * [149] Tchoumakov, S., Cayssol, J., Grushin, A.G.: Three-dimensional chiral Veselago lensing. Physical Review B 105(7), 075309 (2022). doi:10.1103/PhysRevB.105.075309
  * [150] Giuliani, G., Vignale, G.: Quantum Theory of the Electron Liquid. Cambridge University Press, Cambridge (2005)
  * [151] Moore, J.E.: Optical properties of Weyl semimetals. National Science Review 6, 206–208 (2019). doi:10.1093/nsr/nwy164
  * [152] Peterson, R.L.: Formal theory of nonlinear response. Reviews of Modern Physics 39, 69–77 (1967). doi:10.1103/RevModPhys.39.69
  * [153] Sodemann, I., Fu, L.: Quantum nonlinear Hall effect induced by Berry curvature dipole in time-reversal invariant materials. Physical Review Letters 115, 216806 (2015). doi:10.1103/PhysRevLett.115.216806
  * [154] Cortijo, A.: Magnetic-field-induced nonlinear optical responses in inversion symmetric Dirac semimetals. Physical Review B 94, 235123 (2016). doi:10.1103/PhysRevB.94.235123
  * [155] Morimoto, T., Nagaosa, N.: Topological nature of nonlinear optical effects in solids. Science Advances 2, 1501524 (2016). doi:10.1126/sciadv.1501524
  * [156] Morimoto, T., Zhong, S., Orenstein, J., Moore, J.E.: Semiclassical theory of nonlinear magneto-optical responses with applications to topological Dirac/Weyl semimetals. Physical Review B 94(24), 245121 (2016). doi:10.1103/PhysRevB.94.245121
  * [157] Ma, Q., Xu, S.-Y., Chan, C.-K., Zhang, C.-L., Chang, G., Lin, Y., Xie, W., Palacios, T., Lin, H., Jia, S., Lee, P.A., Jarillo-Herrero, P., Gedik, N.: Direct optical detection of Weyl fermion chirality in a topological semimetal. Nature Physics 13, 842–847 (2017). doi:10.1038/nphys4146
  * [158] de Juan, F., Grushin, A.G., Morimoto, T., Moore, J.E.: Quantized circular photogalvanic effect in Weyl semimetals. Nature Communications 8, 15995 (2017). doi:10.1038/ncomms15995
  * [159] Chan, C.-K., Lindner, N.H., Refael, G., Lee, P.A.: Photocurrents in Weyl semimetals. Physical Review B 95, 041104 (2017). doi:10.1103/PhysRevB.95.041104
  * [160] König, E.J., Xie, H.-Y., Pesin, D.A., Levchenko, A.: Photogalvanic effect in Weyl semimetals. Physical Review B 96, 075123 (2017). doi:10.1103/PhysRevB.96.075123
  * [161] Golub, L.E., Ivchenko, E.L., Spivak, B.Z.: Photocurrent in gyrotropic Weyl semimetals. JETP Letters 105(12), 782–785 (2017). doi:10.1134/S0021364017120062
  * [162] Boyd, R.W.: Nonlinear Optics, 4th edn. Academic Press, San Diego (2020). doi:10.1016/C2015-0-05510-1
  * [163] Sturman, B.I., Fridkin, V.M.: The Photovoltaic and Photorefractive Effects in Noncentrosymmetric Materials. Gordon and Breach Science Publishers, Philadelphia (1992)
  * [164] Sipe, J.E., Shkrebtii, A.I.: Second-order optical response in semiconductors. Physical Review B 61(8), 5337–5352 (2000). doi:10.1103/PhysRevB.61.5337
  * [165] Rostami, H., Polini, M.: Nonlinear anomalous photocurrents in Weyl semimetals. Physical Review B 97, 195151 (2018). doi:10.1103/PhysRevB.97.195151
  * [166] Shockley, W., Queisser, H.J.: Detailed balance limit of efficiency of p-n junction solar cells. Journal of Applied Physics 32, 510–519 (1961). doi:10.1063/1.1736034
  * [167] Spanier, J.E., Fridkin, V.M., Rappe, A.M., Akbashev, A.R., Polemi, A., Qi, Y., Gu, Z., Young, S.M., Hawley, C.J., Imbrenda, D., Xiao, G., Bennett-Jackson, A.L., Johnson, C.L.: Power conversion efficiency exceeding the Shockley–Queisser limit in a ferroelectric insulator. Nature Photonics 10, 611–616 (2016). doi:10.1038/nphoton.2016.143
  * [168] Osterhoudt, G.B., Diebel, L.K., Gray, M.J., Yang, X., Stanco, J., Huang, X., Shen, B., Ni, N., Moll, P.J.W., Ran, Y., Burch, K.S.: Colossal mid-infrared bulk photovoltaic effect in a type-I Weyl semimetal. Nature Materials 18, 471–475 (2019). doi:10.1038/s41563-019-0297-4
  * [169] Zhang, L., Chen, Z., Zhang, K., Wang, L., Xu, H., Han, L., Guo, W., Yang, Y., Kuo, C.-N., Lue, C.S., Mondal, D., Fuji, J., Vobornik, I., Ghosh, B., Agarwal, A., Xing, H., Chen, X., Politano, A., Lu, W.: High-frequency rectifiers based on type-II Dirac fermions. Nature Communications 12(1), 1584 (2021). doi:10.1038/s41467-021-21906-w
  * [170] Wang, L., Han, L., Guo, W., Zhang, L., Yao, C., Chen, Z., Chen, Y., Guo, C., Zhang, K., Kuo, C.-N., Lue, C.S., Politano, A., Xing, H., Jiang, M., Yu, X., Chen, X., Lu, W.: Hybrid Dirac semimetal-based photodetector with efficient low-energy photon harvesting. Light: Science & Applications 11(1), 53 (2022). doi:10.1038/s41377-022-00741-8
  * [171] Weng, H.: Lighting up Weyl semimetals. Nature Materials 18, 428–429 (2019). doi:10.1038/s41563-019-0330-7
  * [172] Glass, A.M., von der Linde, D., Negran, T.J.: High-voltage bulk photovoltaic effect and the photorefractive process in LiNbO3. Applied Physics Letters 25, 233–235 (1974). doi:10.1063/1.1655453
  * [173] Nastos, F., Sipe, J.E.: Optical rectification and shift currents in GaAs and GaP response: Below and above the band gap. Physical Review B 74, 035201 (2006). doi:10.1103/PhysRevB.74.035201
  * [174] Laman, N., Shkrebtii, A.I., Sipe, J.E., van Driel, H.M.: Quantum interference control of currents in CdSe with a single optical beam. Applied Physics Letters 75, 2581–2583 (1999). doi:10.1063/1.125084
  * [175] Zhang, Y., Ishizuka, H., van den Brink, J., Felser, C., Yan, B., Nagaosa, N.: Photogalvanic effect in Weyl semimetals from first principles. Physical Review B 97, 241118 (2018). doi:10.1103/PhysRevB.97.241118
  * [176] Sun, K., Sun, S.-S., Wei, L.-L., Guo, C., Tian, H.-F., Chen, G.-F., Yang, H.-X., Li, J.-Q.: Circular photogalvanic effect in the Weyl semimetal TaAs. Chinese Physics Letters 34, 117203 (2017). doi:10.1088/0256-307X/34/11/117203
  * [177] Ji, Z., Liu, G., Addison, Z., Liu, W., Yu, P., Gao, H., Liu, Z., Rappe, A.M., Kane, C.L., Mele, E.J., Agarwal, R.: Spatially dispersive circular photogalvanic effect in a Weyl semimetal. Nature Materials 18, 955–962 (2019). doi:10.1038/s41563-019-0421-5
  * [178] Gabor, N.M., Song, J.C.W., Ma, Q., Nair, N.L., Taychatanapat, T., Watanabe, K., Taniguchi, T., Levitov, L.S., Jarillo-Herrero, P.: Hot carrier–assisted intrinsic photoresponse in graphene. Science 334, 648–652 (2011). doi:10.1126/science.1211384
  * [179] McIver, J.W., Hsieh, D., Steinberg, H., Jarillo-Herrero, P., Gedik, N.: Control over topological insulator photocurrents with light polarization. Nature Nanotechnology 7, 96–100 (2012). doi:10.1038/nnano.2011.214
  * [180] Yuan, H., Wang, X., Lian, B., Zhang, H., Fang, X., Shen, B., Xu, G., Xu, Y., Zhang, S.-C., Hwang, H.Y., Cui, Y.: Generation and electric control of spin-valley-coupled circular photogalvanic current in WSe2. Nature Nanotechnology 9, 851–857 (2014). doi:10.1038/nnano.2014.183