<EMAIL_ADDRESS>
# Orbital optical activity in noncentrosymmetric metals and superconductors

Koki Shinada  [  Robert Peters  Department of Physics, Kyoto University, Kyoto
606-8502, Japan

###### Abstract

We present the optical activity induced by the orbital magnetic moment in
metals and superconductors using Green’s function formalization. We show that
an apparent singularity of the optical activity vanishes in the normal state;
however, it remains finite in the superconducting state and is related to the
superconducting Edelstein effect, ensuring the missing area measurement.
Finally, we calculate the optical activity in a model Hamiltonian mimicking
doped transition metal dichalcogenides to investigate its characteristic
spectrum, and we analyze the Kerr effect to discuss a possibility to observe
the optical activity in experiments.

††preprint: APS/123-QED

##  I Introduction

Optical activity is a ubiquitous optical phenomenon in molecular systems and
solids without an inversion center [1, 2]. It originates from the nontrivial
coupling between magnetism and electricity and is related to various
phenomena, such as optical rotation, dichroism, and birefringence. The optical
activity can be roughly divided into two phenomena: natural optical activity
(NOA) and the optical magnetoelectric effect (OMEE). The NOA includes circular
dichroism, which is, e.g., used to discriminate enantiomers of chiral
molecules. The OMEE appears in systems without time-reversal symmetry. Since
earlier observations in noncentrosymmetric molecules and solids under the
magnetic field or magnetism [3, 4, 5], the OMEE has been observed in a variety
of systems [6].

Optical activity is also gaining recognition for its functionality and
significance in condensed matter physics. In multiferroics [7], spin order
accompanying an electric polarization and chirality is spontaneously formed.
The spin order generates correlations between the magnetism and the
electricity, and the OMEE can be finite. In multiferroic magnets, the OMEE is
also induced by spin waves, called electromagnons, and it forms a resonant
peak in the THz regime [8, 9, 10, 11, 12, 13]. In addition, the OMEE is used
for visualizing domains of such magnets, which are often difficult to observe
with conventional methods due to the absence of macroscopic magnetization [14,
15, 16, 17].

Optical activity in solids is not as well developed as in molecular chemistry.
However, various theoretical studies have been conducted in recent years. The
band theory of optical activity is discussed in several works [18, 19, 20, 21,
22, 23, 24, 25, 26, 27], and it has been applied to a variety of systems,
including chiral crystals [28, 29, 30, 31], twisted bilayer graphene [32, 33,
34, 35], and topological antiferromagnets [36]. Recently, the formalization
has been extended to metals [34, 27]. In addition, it is becoming clear that
optical activity also plays an essential role in exotic phases, including
topological materials and superconductors. For example, it is revealed that
the circular dichroism in chiral multifold semimetals is quantized and that it
can be used to probe the topological nature [37, 38, 39]. Furthermore, for
superconductors, it has recently been shown that the optical activity can be
applied to detect the superconducting Edelstein effect, which is unique to
noncentrosymmetric superconductors and has not been observed experimentally,
using the missing area measurement [40]. Despite such progress regarding
optical activity, research in crystals is less advanced than in molecular
chemistry, and further development is required.

In this paper, we discuss the optical activity originating from orbital
magnetic moments in metals and superconductors using a Green’s function
formalism. We discuss a no-go theorem concerning a singularity of the optical
activity in Secs. II and III, and we show that the singularity appearing in
the superconducting state is related to the superconducting Edelstein effect.
Furthermore, we present model calculations investigating two-dimensional
materials such as transition metal dichalcogenides and twisted bilayer
graphene in the normal state and the superconducting state in Secs. IV and V,
respectively. We also analyze the Kerr spectrum. Finally, in Sec. VI, we
conclude this paper and discuss the possibility of an experimental
observation.

##  II orbital optical activity in the normal state

In this section, we derive the orbital optical activity in the normal state
using a Green’s function method. The optical activity originates from the
wavenumber dependence of the optical conductivity $\sigma(\bm{q},\Omega)$,
where $\bm{q}$ and $\Omega$ are the wavenumber and the frequency of the
irradiation light. $\sigma(\bm{q}=0,\Omega)$ represents the uniform
conductivity that changes the refractive index of a material, leading to
observable effects such as the reflectance and the rotation of linearly-
polarized light (Faraday rotation and Kerr rotation) in magnets. In
noncentrosymmetric systems, the first-order term in the wavenumber of the
optical conductivity $\bm{\partial}_{q}\sigma(\bm{q}=0,\Omega)\cdot\bm{q}$ can
be significant. This is the optical activity, which is the main quantity in
this paper, exhibiting optical rotation, dichroism, and birefringence. It
includes two parts: a spin part and an orbital part. In this paper, we only
focus on the orbital part.

###  II.1 Green’s function formalism of the orbital optical activity

We consider noninteracting systems. The Hamiltonian without an electromagnetic
field is assumed to be described by the nonrelativistic quadratic kinetic
energy term with a spin-orbit coupling term as

$H_{0}=\frac{\bm{p}^{2}}{2m}+V(\bm{x})+\frac{1}{4m^{2}}\biggl{(}\frac{\partial
V(\bm{x})}{\partial\bm{x}}\times\bm{p}\biggr{)}\cdot\bm{\sigma}.$ (1)

Here, $m$ is the electron mass, $V(\bm{x})$ is a periodic potential, and
$\bm{\sigma}$ are the Pauli matrices representing the spin degrees of freedom.
This periodic Hamiltonian is diagonalized by the Bloch theorem using the Bloch
wave function $\ket{\psi_{n\bm{k}}}$ as
$H_{0}\ket{\psi_{n\bm{k}}}=\epsilon_{n\bm{k}}\ket{\psi_{n\bm{k}}}$, where $n$
is the band index and $\bm{k}$ is the Bloch wave number. The Bloch wave
function is split into the product of a plane wave and a periodic function
$\ket{u_{n\bm{k}}}$ as
$\ket{\psi_{n\bm{k}}}=e^{i\bm{k}\cdot\bm{x}}\ket{u_{n\bm{k}}}$. We define the
Bloch Hamiltonian
$H_{\bm{k}}=e^{-i\bm{k}\cdot\bm{x}}H_{0}e^{i\bm{k}\cdot\bm{x}}$, which is
diagonalized by $\ket{u_{n\bm{k}}}$ as
$H_{\bm{k}}\ket{u_{n\bm{k}}}=\epsilon_{n\bm{k}}\ket{u_{n\bm{k}}}$.

We introduce the electromagnetic field using the vector potential
$\bm{A}(\bm{x},t)$ by shifting the momentum
$\bm{p}\to\bm{p}+e\bm{A}(\bm{x},t)$ ($-e<0$ is the electron charge). We use
the temporal gauge, where the vector potential alone fully describes the
electromagnetic field without a scalar potential. The vector potential yields
an additional term up to the first order

$H_{A}=\frac{1}{2}(\bm{v}\cdot\bm{A}(\bm{x},t)+\bm{A}(\bm{x},t)\cdot\bm{v}),$
(2)

where $\bm{v}=i[H_{0},\bm{x}]$ is the velocity operator. Here, we ignore the
diamagnetic term, which is the quadratic term, because it only contributes to
the conductivity independent of $\bm{q}$ and, thus, does not contribute to the
optical activity. We also neglect the Zeeman term, which is the coupling
between the magnetic field and the spin moment and contributes to the spin
part of the optical activity [27, 40]. Focusing only on the orbital part of
the optical activity in this paper, we can neglect this term.

The current operator conjugate with the vector potential $\bm{A}(\bm{r},t)$
($\bm{r}$ is just a position coordinate, not an operator) is given by

$\bm{J}(\bm{r})\equiv-\frac{\delta
H_{A}}{\delta\bm{A}(\bm{r},t)}=-\frac{e}{2}\\{\bm{v},\delta(\bm{r}-\bm{x})\\}.$
(3)

Using the dynamical linear response theory, the current-current correlation
function reads

$\displaystyle\Phi_{\mu\nu}(\bm{q},\Omega)$
$\displaystyle=e^{2}\int[d^{4}k]f(\omega)\mathrm{Tr}\Bigl{[}G^{RA}(\bm{k}-,\omega)v^{\mu}_{\bm{k}}G^{R}(\bm{k}+,\omega+\Omega)v^{\nu}_{\bm{k}}$
$\displaystyle\quad+G^{A}(\bm{k}-,\omega-\Omega)v^{\mu}_{\bm{k}}G^{RA}(\bm{k}+,\omega)v^{\nu}_{\bm{k}}\Bigr{]}.$
(4)

Here, $G^{R/A}(\bm{k},\omega)=1/(\omega-H_{\bm{k}}+\mu\pm i\Gamma)$ is the
retarded/advanced Green’s function at a chemical potential $\mu$, and
$\bm{v}_{\bm{k}}=\partial H_{\bm{k}}/\partial\bm{k}$ is the velocity operator
of the Bloch Hamiltonian. $f(\omega)=1/(e^{\beta\omega}+1)$ is the Fermi
distribution function at temperature $1/\beta$. We use the following
abbreviations; $\int[d^{4}k]=\int^{\infty}_{-\infty}d\omega/(2\pi
i)\int_{\mathrm{BZ}}d^{3}k/(2\pi)^{3}$, $G^{RA}=G^{R}-G^{A}$, and
$\bm{k}\pm=\bm{k}\pm\bm{q}/2$. In this paper, we phenomenologically introduce
the dissipation effect by a finite $\Gamma$.

Expanding the correlation function by the wavenumber $\bm{q}$, the first-order
term is given by

$\displaystyle\Phi_{\mu\nu\lambda}(\Omega)$
$\displaystyle=\frac{e^{2}}{2}\int[d^{4}k]f(\omega)$
$\displaystyle\times\mathrm{Tr}\Bigl{[}-\partial_{\lambda}G^{RA}(\bm{k},\omega)v_{\bm{k}}^{\mu}G^{R}(\bm{k},\omega+\Omega)v_{\bm{k}}^{\nu}$
$\displaystyle+G^{RA}(\bm{k},\omega)v_{\bm{k}}^{\mu}\partial_{\lambda}G^{R}(\bm{k},\omega+\Omega)v_{\bm{k}}^{\nu}$
$\displaystyle-\partial_{\lambda}G^{A}(\bm{k},\omega-\Omega)v_{\bm{k}}^{\mu}G^{RA}(\bm{k},\omega)v^{\nu}_{\bm{k}}$
$\displaystyle+G^{A}(\bm{k},\omega-\Omega)v_{\bm{k}}^{\mu}\partial_{\lambda}G^{RA}(\bm{k},\omega)v_{\bm{k}}^{\nu}\Bigr{]}.$
(5)

Here, we use $\partial_{\lambda}=\partial/\partial k_{\lambda}$. We note the
identity $\partial_{\lambda}G^{R/A}=G^{R/A}v_{\bm{k}}^{\lambda}G^{R/A}$, which
is useful for numerical calculations. Finally, we obtain the optical activity
$\sigma_{\mu\nu\lambda}(\Omega)=\Phi_{\mu\nu\lambda}(\Omega)/i(\Omega+i\delta)$,
where $\delta=+0$ is an adiabatic factor. This tensor $\sigma_{\mu\nu\lambda}$
describes all optical activity phenomena induced by the orbital moment,
including the NOA and the OMEE. This formulation has been useful for analyses
in periodic crystals as it avoids difficulties such as origin dependence and
gauge invariance. Some works have derived this tensor with the band
representation and proved explicitly that this formalism avoids these
difficulties, providing a physical interpretation of the obtained gauge-
invariant equations [21, 23, 41, 42, 27].

###  II.2 No-go theorem

The optical activity has a singularity at $\Omega=0$ because of the factor
$1/(\Omega+i\delta)=1/\Omega-i\pi\delta(\Omega)$. This $\delta$-function and
the accompanying anomalous divergence scaling $\Omega^{-1}$ are known as the
false divergences problem when using the velocity gauge from the minimal-
coupling Hamiltonian [34]. This divergence is supposed to vanish in the normal
state, which has been shown using the Bloch band representation [27]. We here
clarify whether it also vanishes using the Green’s function formula. The
coefficient in front of the $\delta$ function in the optical activity reads

$\Phi_{\mu\nu\lambda}(0)=i\varepsilon_{\mu\nu\lambda}\alpha.$ (6)

Here, $\varepsilon_{\mu\nu\lambda}$ is the totally antisymmetric tensor.
Almost all components of $\Phi_{\mu\nu\lambda}(0)$ are zero. Only totally
antisymmetric terms appear due to $\alpha$, which is real (see Appendix A for
a detailed derivation). This term corresponds to the chiral magnetic effect,
where a current density is induced by a magnetic field as
$\bm{J}=\alpha\bm{B}$. However, this phenomenon usually does not occur unless,
for example, the chiral chemical potential $\mu_{5}$ is finite in the non-
equilibrium case [43]. Using the Bloch band representation, this $\alpha$ is
shown to be zero, resulting from its topological nature [23, 27]. However, the
formula using Green’s function can not immediately prove that this term
vanishes. Therefore, in the following discussion, we only consider cases where
the chiral magnetic effect vanishes due to crystal point group symmetry.

##  III orbital optical activity in a superconductor

###  III.1 Green’s function formula for a superconductor

In this paper, we treat the superconducting state using the mean-field
approximation represented by the Bogoliubov-de Gennes (BdG) Hamiltonian as

$\displaystyle H_{\mathrm{BdG}}$
$\displaystyle=\frac{1}{2}\sum_{\bm{k}nm}\psi^{\dagger}_{n\bm{k}}H^{\mathrm{BdG}}_{\bm{k}nm}\psi_{m\bm{k}},$
(7a) $\displaystyle H^{\mathrm{BdG}}_{\bm{k}nm}$
$\displaystyle=\begin{pmatrix}H_{\bm{k}}-\mu&-\Delta_{\bm{k}}\\\
-\Delta^{\dagger}_{\bm{k}}&-H^{\mathrm{T}}_{-\bm{k}}+\mu\end{pmatrix}.$ (7b)

Here, $\bm{\psi}^{\dagger}_{\bm{k}}=(c_{1\bm{k}}^{\dagger},\cdots
c_{N\bm{k}}^{\dagger},c_{1-\bm{k}},\cdots,c_{N-\bm{k}})$ is the Nambu spinor
and $\Delta_{\bm{k}}$ is the pair potential, which is the order parameter of
the superconducting state. We define $M^{\mathrm{T}}$ and $M^{\dagger}$ as the
transpose and the Hermitian conjugate of a matrix $M$. The current-current
correlation function in the superconducting state is given by

$\displaystyle\Phi_{\mu\nu}(\bm{q},\Omega)$
$\displaystyle=\frac{e^{2}}{2}\int[d^{4}k]f(\omega)\mathrm{Tr}\Bigl{[}G^{RA}_{\mathrm{BdG}}(\bm{k}-,\omega)\tilde{v}^{\mu}_{\bm{k}}G^{R}_{\mathrm{BdG}}(\bm{k}+,\omega+\Omega)\tilde{v}^{\nu}_{\bm{k}}$
$\displaystyle\quad+G^{A}_{\mathrm{BdG}}(\bm{k}-,\omega-\Omega)\tilde{v}^{\mu}_{\bm{k}}G^{RA}_{\mathrm{BdG}}(\bm{k}+,\omega)\tilde{v}^{\nu}_{\bm{k}}\Bigr{]}.$
(8)

Here,
$G^{R/A}_{\mathrm{BdG}}(\bm{k},\omega)=1/(\omega-H^{\mathrm{BdG}}_{\bm{k}}+\Sigma^{R/A}(\bm{k},\omega))$
is the retarded/advanced Green’s function for the BdG Hamiltonian, including
the self-energy describing the dissipation effect. A specific equation of this
self-energy will be given in a model calculation below. The velocity operator
is different from the normal state. It is given by

$\tilde{v}^{\mu}_{\bm{k}}=\begin{pmatrix}v^{\mu}_{\bm{k}}&0\\\
0&-(v^{\mu}_{-\bm{k}})^{\mathrm{T}}\end{pmatrix}.$ (9)

We note that this velocity operator is not the usual derivative of the BdG
Hamiltonian $H^{\mathrm{BdG}}_{\bm{k}}$ by the wavenumber $\bm{k}$. Expanding
the correlation function by $\bm{q}$, the first-order term is

$\displaystyle\Phi_{\mu\nu\lambda}(\Omega)$
$\displaystyle=\frac{e^{2}}{4}\int[d^{4}k]f(\omega)$
$\displaystyle\times\mathrm{Tr}\Bigl{[}-\partial_{\lambda}G^{RA}_{\mathrm{BdG}}(\bm{k},\omega)\tilde{v}^{\mu}_{\bm{k}}G^{R}_{\mathrm{BdG}}(\bm{k},\omega+\Omega)\tilde{v}^{\nu}_{\bm{k}}$
$\displaystyle+G^{RA}_{\mathrm{BdG}}(\bm{k},\omega)\tilde{v}^{\mu}_{\bm{k}}\partial_{\lambda}G^{R}_{\mathrm{BdG}}(\bm{k},\omega+\Omega)\tilde{v}^{\nu}_{\bm{k}}$
$\displaystyle-\partial_{\lambda}G^{A}_{\mathrm{BdG}}(\bm{k},\omega-\Omega)\tilde{v}^{\mu}_{\bm{k}}G^{RA}_{\mathrm{BdG}}(\bm{k},\omega)\tilde{v}^{\nu}_{\bm{k}}$
$\displaystyle+G^{A}_{\mathrm{BdG}}(\bm{k},\omega-\Omega)\tilde{v}^{\mu}_{\bm{k}}\partial_{\lambda}G^{RA}_{\mathrm{BdG}}(\bm{k},\omega)\tilde{v}^{\nu}_{\bm{k}}\Bigr{]}.$
(10)

The derivative of the Green’s function by the wavenumber can be transformed
into the useful form for numerical calculations as
$\partial_{\lambda}G^{R/A}_{\mathrm{BdG}}=G^{R/A}_{\mathrm{BdG}}\hat{v}_{\bm{k}}^{\lambda}G^{R/A}_{\mathrm{BdG}}$,
where

$\hat{v}^{\lambda}_{\bm{k}}=\frac{\partial H^{\mathrm{BdG}}_{\bm{k}}}{\partial
k_{\lambda}}=\begin{pmatrix}v^{\lambda}_{\bm{k}}&-\partial_{\lambda}\Delta_{\bm{k}}\\\
-\partial_{\lambda}\Delta_{\bm{k}}^{\dagger}&(v^{\lambda}_{-\bm{k}})^{\mathrm{T}}\end{pmatrix}.$
(11)

This velocity operator $\hat{v}$ is different from $\tilde{v}$ even if the
pair potential is independent of the wavenumber. Finally, we obtain the
optical activity for the superconducting state by
$\sigma_{\mu\nu\lambda}(\Omega)=\Phi_{\mu\nu\lambda}(\Omega)/i(\Omega+i\delta)$.

###  III.2 Singularity and relation to the superconducting Edelstein effect

In the previous section, we have shown that the optical activity appears to
have a $\delta$-function singularity at $\Omega=0$, which, however, vanishes
in the normal state. In the superconducting case, the singularity generally is
finite. The absence of cancellation is attributed to the difference in the
velocity operator from the normal state. The coefficient of the $\delta$
function satisfies (see Appendix A for a detailed derivation)

$\Phi_{\mu\nu\lambda}(0)=\tilde{\Phi}_{\mu\nu\lambda}-(\mu\leftrightarrow\nu).$
(12)

$\tilde{\Phi}_{\mu\nu\lambda}$ is purely imaginary, and its detailed form is
discussed in Appendix A. $\Phi_{\mu\nu\lambda}(0)$ is antisymmetric for the
interchange between $\mu$ and $\nu$. Being purely imaginary is consistent with
the general symmetry argument for a current-current correlation function [40].
Therefore, the $\delta$-function singularity appears in the imaginary part of
the optical activity $\mathrm{Im}\sigma_{\mu\nu\lambda}(\Omega)$, and the
anomalous $1/\Omega$ divergence with the same coefficient
$\Phi_{\mu\nu\lambda}(0)$ as the $\delta$-function singularity appears in the
real part $\mathrm{Re}\sigma_{\mu\nu\lambda}(\Omega)$ to satisfy the Kramers-
Kronig relation. Due to the antisymmetry in the indices $\mu$ and $\nu$, the
anomalous term only appears in the NOA. The OMEE does not exhibit this
singularity because the OMEE originates from the symmetric part of the
interchange between $\mu$ and $\nu$ due to the necessity of the time-reversal
symmetry breaking. Such anomalous frequency dependence is common in
superconductors. The optical conductivity has the same anomaly, and the
coefficient corresponds to the superfluid density [44]. Furthermore, it has
recently been shown that also the nonlinear conductivity has a similar
anomaly, and the coefficient has the meaning of the nonreciprocal superfluid
density [45, 46]. The physical meaning of the coefficient in the case of the
optical activity is also known. It has been studied for the spin contribution,
and the coefficient corresponds to the superconducting Edelstein effect [20,
22, 40]. In this paper, we consider the orbital contribution, and we will
reveal the physical meaning of $\Phi_{\mu\nu\lambda}(0)$ in the following
discussion.

$\Phi_{\mu\nu\lambda}(0)$ has nine components due to its antisymmetric
property. Therefore, it can be rewritten by a rank-2 tensor
$\mathcal{K}_{\mu\nu}$ with a one-to-one correspondence [47, 21] as

$\displaystyle\Phi_{\mu\nu\lambda}(0)=i\varepsilon_{\mu\lambda\theta}\mathcal{K}_{\nu\theta}-i\varepsilon_{\nu\lambda\theta}\mathcal{K}_{\mu\theta},$
(13a)
$\displaystyle\mathcal{K}_{\mu\nu}=-\frac{1}{4i}\varepsilon_{\nu\theta\lambda}\Bigl{(}2\Phi_{\mu\theta\lambda}(0)-\Phi_{\theta\lambda\mu}(0)\Bigr{)}.$
(13b)

Here, $\mathcal{K}_{\mu\nu}$ is real because $\Phi_{\mu\nu\lambda}(0)$ is
purely imaginary. Substituting Eq. (13a) into the original linear response
relation, we obtain

$J_{\mu}=\mathcal{K}_{\mu\theta}B_{\theta}+(i\bm{q}\times\bm{M})_{\mu},\quad
M_{\theta}=\mathcal{K}_{\mu\theta}A_{\mu}.$ (14)

$\mathcal{K}$ represents the superconducting Edelstein effect (SEE), where a
supercurrent induces a magnetization $\bm{M}$ or the magnetic field $\bm{B}$
induces the supercurrent. Thus, $\Phi_{\mu\nu\lambda}(0)$ also has the
physical meaning of the SEE for the orbital contribution, in analogy to the
case of the spin contribution.

Previous works about the SEE [48, 49, 50, 51, 52, 53] have mainly focused on
the induced spin moment in systems with spin-orbit coupling (SOC). Recently,
supercurrent-induced orbital magnetization has been proposed [54] and is
expected to appear in systems where the valley degrees of freedom are of key
importance, such as the transition metal dichalcogenides (TMDs) and twisted
bilayer graphene (TBG), even without SOC. The SEE is a unique response for
noncentrosymmetric superconductors and is expected to provide essential
information about them; however, it has not been observed in experiments for
both spin and orbital parts.

Here, we propose another means of observing this SEE. By measuring the
spectrum of the optical activity, we can obtain $\Phi_{\mu\nu\lambda}(0)$,
which is equivalent to the SEE. $\Phi_{\mu\nu\lambda}(0)$ can be obtained by
measuring the coefficient of the anomalous divergence of
$\mathrm{Re}\sigma_{\mu\nu\lambda}(\Omega)$ scaling as $\Omega^{-1}$.
Furthermore, the spectrum of $\mathrm{Im}\sigma_{\mu\nu\lambda}(\Omega)$ can
also determine the value of $\Phi_{\mu\nu\lambda}(0)$. Reference [40] has
shown that the missing area, the area reduced from the spectrum of the normal
phase created by the superconducting gap, is identical to
$\Phi_{\mu\nu\lambda}(0)$ because of a universal sum rule and the no-go
theorem in the case of the spin contribution. This method is useful for dirty
superconductors, which is a common situation. As discussed in Sec. II.2, the
no-go theorem is also established for the orbital contribution. Thus, the
missing area measurement is also valid for the orbital contribution.

##  IV model calculation in the normal state

In this section, we discuss the orbital optical activity using a model
Hamiltonian to investigate the characteristics of the spectrum.

Figure 1: (Left) Band structure of the used model: We set $t=1\mathrm{eV}$,
$t^{\prime}=0.9t$, and $G=0.2t$. (Middle) Numerical results of the orbital
optical activity $\sigma_{xyx}$ (blue lines and left axis) and the optical
conductivity $\sigma_{xx}$ (red lines and right axis): We set
$t=1\mathrm{eV}$, $t^{\prime}=0.9t$, $G=0.2t$, $\mu=0.5t$, and $\Gamma=0.1t$.
(Right) The magnitude of $\sigma_{xyx}$ and $\sigma_{xx}$ is plotted on a
logarithmic scale. The lines are the same as in the middle panel. We set
$\sigma_{xyx}$ and $\sigma_{xx}$ in units of $e^{2}a/\hbar$ and $e^{2}/\hbar$.

###  IV.1 Optical activity in the normal state

We discuss the orbital optical activity in the normal state. We use a model
Hamiltonian imitating strained $\mathrm{MoS_{2}}$. For simplicity, we ignore
the SOC, which is realized in the conduction band of $\mathrm{MoS_{2}}$ [55,
56]. In this case, the spin part of the optical activity vanishes, and the
orbital part plays the central role.

The Hamiltonian is written as

$\displaystyle H_{\bm{k}}=\begin{pmatrix}G&\epsilon_{\bm{k}}\\\
\epsilon^{*}_{\bm{k}}&-G\end{pmatrix}\otimes\sigma_{0},$ (15a)
$\displaystyle\epsilon_{\bm{k}}=te^{i\bm{k}\cdot\bm{a}_{1}}+t^{\prime}(e^{i\bm{k}\cdot\bm{a}_{2}}+e^{i\bm{k}\cdot\bm{a}_{3}}).$
(15b)

Here, $t$ and $t^{\prime}$ are hopping parameters, and $G$ is the gap
parameter that opens a gap at the K and $\mathrm{K}^{\prime}$ points due to
spatial inversion symmetry breaking. We use $\bm{a}_{1}=(0,1/\sqrt{3})$,
$\bm{a}_{2}=(-1/2,-1/2\sqrt{3})$ and $\bm{a}_{3}=(1/2,-1/2\sqrt{3})$, and we
set the lattice constant $a=1$ ($a=2.4~{}\mathrm{\AA}$ for
$\mathrm{MoS_{2}}$). $\sigma_{0}$ is the identity matrix for the spin degrees
of freedom. In the following, we set $t>t^{\prime}$ to incorporate the effect
of stretching in the $x$ direction. We plot the band structure in the left
panel of Fig. 1. The Hamiltonian belongs to the point group $\mathrm{C_{2v}}$.
Therefore, the Edelstein effect and the orbital optical activity are
restricted to the one component $\chi_{xz}$, where
$J_{\mu}=-\chi_{\mu\nu}\dot{B}_{\nu}$ and $M_{\nu}=\chi_{\mu\nu}E_{\mu}$.
Then, the nonzero component of the optical activity tensor is only
$\sigma_{xyx}=-\sigma_{yxx}$ and is related to $\chi_{xz}=-i\sigma_{xyx}$
[57]. The stretching generates the in-plane polarization and the induced out-
of-plane magnetization. This type of optical activity is different from the
case of the chiral-structured materials, where the induced magnetization is
parallel to the electric field [58, 34].

We show the numerical results of the optical activity $\sigma_{xyx}$ in the
middle and right panels of Fig. 1 (blue lines). We use $t=1~{}\mathrm{eV}$,
$t^{\prime}=0.9t$, $G=0.2t$, $\mu=0.5t$, and $\Gamma=0.1t$. At frequencies
smaller than the dissipation, $\Omega<\Gamma$, intraband transitions are
dominant and show a Drude model-like behavior. In this region, the imaginary
part of the optical activity is larger than the real part, and the real part
is linear in the frequency and becomes comparable with the imaginary part when
the frequency reaches the scale of the dissipation. This behavior is identical
to that of the uniform optical conductivity described by the Drude model. It
is consistent with the equation of the gyrotropic magnetic effect (GME) in a
nonabsorbing regime [23]. This effect originates from the orbital magnetic
moment dipole (OMMD)

$\int_{\mathrm{BZ}}\frac{d^{d}k}{(2\pi)^{d}}\sum_{n}f(\epsilon_{n\bm{k}}-\mu)\partial_{k_{i}}m^{\mathrm{orb}}_{n\bm{k}},$
(16)

where $m^{\mathrm{orb}}_{n\bm{k}}$ is the orbital magnetic moment oriented in
the direction perpendicular to the surface. Without stretching
$\mathrm{MoS_{2}}$, orbital magnetic moments (OMMs) with opposite signs exist
around the $\mathrm{K}$ and $\mathrm{K}^{\prime}$ points [59]; however, the
OMMD vanishes due to $\mathrm{C_{3v}}$ symmetry. When stretching
$\mathrm{MoS_{2}}$, the OMMD in the same direction as the stretching direction
becomes finite and generates the GME. This discussion is similar to the Berry
curvature dipole in the nonlinear Hall effect [60]. The GME in the DC limit
has been experimentally observed as the orbital Edelstein effect in stretched
$\mathrm{MoS_{2}}$ [61, 62]. Above the frequency corresponding to the optical
gap $\sim 2\mu$, interband effects become finite around $\mathrm{K}$ and
$\mathrm{K}^{\prime}$ points.

For the following discussion, we also calculate the optical conductivity. In
Fig. 1 (red lines), we plot the numerical results of the longitudinal optical
conductivity $\sigma_{xx}$. The character of this spectrum is similar to the
optical activity. In the low-frequency regime, $\Omega<\Gamma$, the optical
conductivity follows the Drude theory. The magnitude of the conductivity in
the DC limit is $0.6e^{2}/\hbar$, which is comparable to a dual-gated
monolayer $\mathrm{MoS_{2}}$ [63]. Above the frequency corresponding to the
optical gap ($\sim 2\mu$), interband transitions appear, and the spectrum
forms a Lorentz-peak structure.

However, a clear difference from the optical activity can be identified. We
can see that the interband peak position differs between
$\mathrm{Im}\sigma_{xyx}$ and $\mathrm{Re}\sigma_{xx}$. The peaks are located
at $\Omega=1~{}\mathrm{eV}$ and $\Omega=2~{}\mathrm{eV}$, respectively.

Figure 2: Chemical potential dependence of the optical conductivity (Left) and
the optical activity (Right). We use the same parameters and units as in Fig.
1.

This distinction becomes clearer when looking at the chemical potential
dependence in Fig. 2. While the interband peak of the optical conductivity
$\mathrm{Re}\sigma_{xx}$ does not depend on the chemical potential and is
located at $\Omega=2~{}\mathrm{eV}$, the interband peak in the optical
activity $\mathrm{Im}\sigma_{xyx}$ shifts as the chemical potential is
increased. It is located at the optical gap $2\mu$. The typical optical
transition occurs as a transition from an occupied state to an unoccupied
state. Thus, the spectra should match once the optical gap is exceeded. The
spectra of $\mathrm{Re}\sigma_{xx}$ are indeed identical among different
chemical potentials above the optical gap $2\mu$; however, the spectra of
$\mathrm{Im}\sigma_{xyx}$ behave differently. In the following, we explain the
reason for these distinctions.

Figure 3: (Left) Numerical results of the JDOS, $J(\Omega)$. We use the same
parameters as in Fig. 1. (Middle) Numerical results of the optical rotational
angle $\theta$ and the ellipticity $\varepsilon$. The red and blue lines
correspond to the reflected light (right axis) and the transmitted light (left
axis). The legend is identical to the right panel. We use the same parameters
as in Fig. 1. (Right) The magnitude of $\theta$ and $\varepsilon$ is plotted
on a logarithmic scale.

The interband transition peak is often explained by the joint density of
states (JDOS). The JDOS is defined by

$J(\Omega)=\int_{\mathrm{BZ}}\frac{d^{d}k}{(2\pi)^{d}}\delta(\Omega-\Delta\epsilon_{\bm{k}}).$
(17)

Here, $\Delta\epsilon_{\bm{k}}=\epsilon^{c}_{\bm{k}}-\epsilon^{v}_{\bm{k}}$ is
the energy difference between the conduction band and the valence band. Thus,
the JDOS has a maximum at the frequency where the largest number of states can
be excited. In this paper, to account for the broadening of the DOS due to
dissipation, we substitute the $\delta$ function by the Lorentz function as
$\delta(x)\sim\Gamma/(\pi^{2}x^{2}+\Gamma^{2})$, where $\Gamma$ is the
dissipation used in the calculation of Fig. 1. We show the numerical results
of the JDOS in the left panel of Fig. 3. The JDOS has a maximum around
$\Omega=2~{}\mathrm{eV}$, explaining the optical conductivity’s peak. However,
the JDOS cannot explain the peak shift of the optical activity.

This peak shift is unique to the orbital optical activity in metals. It
becomes clearer when the optical activity is represented by the Bloch band
basis. The full representation is derived in a recent work [27] (see Eq. (26)
in the paper). It shows that there is a Fermi surface effect affecting the
interband transition besides the Fermi sea effect. This is unique to the
orbital contribution. It demonstrates that the effect of optical transitions
in the optical activity cannot be simply described by transitions from
occupied to unoccupied states, which is described by the Fermi sea term. We
note that in this two-band model with particle-hole symmetry, the Fermi sea
term is suppressed. The Fermi surface term is locally enhanced at the optical
gap $2\mu$, which is consistent with the result in Fig. 2.

###  IV.2 Optical Kerr effect

The optical activity $\sigma_{xyx}$ in our model can be observed using the
optical Kerr effect. The Kerr effect is usually caused by the optical Hall
effect in magnets without time-reversal symmetry. Because the present model
obeys time-reversal symmetry, the Kerr effect originating from the optical
Hall effect does not occur. However, the optical activity also generates an
optical Kerr effect by a different mechanism, as described in the following
discussion.

The spectrum of the longitudinal optical conductivity $\sigma_{ii}(\omega)$
can be extracted from the experimentally observed complex reflectance.
Similarly, the spectrum of the optical activity can be extracted from the
experimentally observed rotational angle and the ellipticity caused by the
Kerr effect or the Faraday effect. In two-dimensional materials, transmission
through the material does not exist, and thus, only the Kerr effect is
allowed. In this subsection, we discuss the Kerr effect as a possibility of
observing the optical activity experimentally.

We consider an electromagnetic wave incident in the $xz$ plane at an angle of
incidence of $None$ (see Appendix B for a detailed discussion and
calculation). The two-dimensional material is put on the $z=0$ plane, and the
incident electric field is assumed to be $s$-polarized, i.e., pointing in the
$y$-axis direction. Then, the reflected electric field,
$(E^{\mathrm{r}}_{y},E^{\mathrm{r}}_{xz})$, and the transmitted electric
field, $(E^{\mathrm{t}}_{y},E^{\mathrm{t}}_{xz})$, satisfy the following ratio

$\displaystyle\frac{E^{\mathrm{r}}_{xz}}{E^{\mathrm{r}}_{y}}\sim\frac{\Omega\sigma_{xyx}}{c(\mu_{0}c\sigma_{yy})}\biggl{(}\frac{2\sqrt{2}}{\mu_{0}c}+\sigma_{xx}\biggr{)}^{-1},$
(18)
$\displaystyle\frac{E^{\mathrm{t}}_{xz}}{E^{\mathrm{t}}_{y}}=-\frac{\Omega\sigma_{xyx}}{\sqrt{2}c}\biggl{(}\frac{2\sqrt{2}}{\mu_{0}c}+\sigma_{xx}\biggr{)}^{-1}.$
(19)

Here, we ignore the squared term of $\sigma_{xyx}$ in Eq. (18) because it is
very small. $\mu_{0}$ and $c$ are the permeability of the vacuum and the light
velocity, and the product of them $(\mu_{0}c)^{-1}\sim 10.90e^{2}/\hbar$ is
the inverse of the vacuum impedance. Usually, a two-dimensional material is
attached to a substrate. We assume that the refractive index of the substrate
at the frequency of interest is 1 in this paper. The component
$E^{\mathrm{r},\mathrm{t}}_{xz}$ is the transverse response induced by the
magnetoelectric effect $\chi_{xz}=-i\sigma_{xyx}$, and it is perpendicular to
the usual response $E^{\mathrm{r},\mathrm{t}}_{y}$. Thus, the output electric
field is no longer linearly polarized, but, in general, it becomes an ellipse
whose deflection direction is tilted from the $y$ axis. The ratio relations in
Eqs. (18) and (19) give the ratio of the magnitude and the phase difference
between electric fields perpendicular to each other. They determine the
rotational angle $\theta$ and the ellipticity $\varepsilon$ of the output
electric fields (see Appendix B for the equation of $\theta$ and
$\varepsilon$). We show the rotational angle and the ellipticity in the middle
and right panels in Fig. 3 using the data of Fig. 1. The Kerr effect is
enhanced at $\Omega=1~{}\mathrm{eV}$ corresponding to the peak of the optical
activity. Its magnitude reaches about 2 µ$\mathrm{rad}$ for the reflected
light and $20~{}\mathrm{nrad}$ for the transmitted one. We see that in the
low-frequency regime, the ellipticity predominates because the optical
conductivity is dominated by the real part, and the optical activity is
dominated by the imaginary part. In other words, the phases of these two
electric fields exhibit a deviation of nearly $\pi/2$ from each other. Its
magnitude decreases, reaching about $\varepsilon^{\mathrm{r}}\sim
1~{}\mathrm{nrad}$ and $\varepsilon^{\mathrm{t}}\sim 0.05~{}\mathrm{nrad}$ at
$\Omega=1~{}\mathrm{meV}$.

Next, we comment on the order of the ellipticity for real two-dimensional
materials such as doped monolayer TMDs and TBG in the low-frequency regime.
The conductivity of TMDs and TBG is $10e^{2}/\hbar$ at the maximum [63, 64,
65, 66]. Thus, it is smaller or comparable to $(\mu_{0}c)^{-1}$. Therefore,
the ellipticity of the reflection light in this regime is approximately
written as $k\chi^{\prime}/\sigma^{\prime}$, where $k$ is the wavenumber of
the light, $\chi^{\prime}$ is the real part of $\chi_{\mu\nu}$, and
$\sigma^{\prime}$ is the real part of the conductivity. This equation is the
Edelstein coefficient $\alpha=\chi^{\prime}/\sigma^{\prime}$ multiplied by the
wavenumber. For strained $\mathrm{MoS_{2}}$ with $1\%$ strain, the Edelstein
coefficient is observed to be $\alpha=0.1~{}\mathrm{\AA}$ [61]. Thus, the
ellipticity is converted to $\varepsilon^{\mathrm{r}}=8~{}\mathrm{nrad}$ for
the frequency $\Omega=1~{}\mathrm{meV}$ (i.e. $k=8~{}\mathrm{cm^{-1}}$), which
is comparable to our calculation. For TBG [54], the Edelstein coefficient is
expected to be larger, reaching $10~{}\mathrm{\AA}$. Thus, the ellipticity
will be 0.8 µrad at $\Omega=1~{}\mathrm{meV}$.

##  V Model calculation in the superconducting state

In this section, we discuss the optical activity in the superconducting state.
We use the same Hamiltonian $H_{\bm{k}}$ mimicking strained $\mathrm{MoS_{2}}$
as in Sec. IV. We assume that the superconducting state is a uniform singlet.
Thus, we introduce the pair potential as
$\Delta_{\bm{k}}=\Delta\tau_{0}\otimes i\sigma_{y}$, where $\Delta$ is real
and $\tau_{0}$ is the identity matrix for the sublattice degrees of freedom.
When calculating the optical conductivity and the optical activity using
Green’s functions (Eq. 10), we introduce the dissipation effect
phenomenologically. For this purpose, we introduce the renormalization factor

$\eta_{\omega}=1+\Gamma\biggr{(}\frac{\theta(|\Delta|-|\omega|)}{\sqrt{\Delta^{2}-\omega^{2}}}+\frac{i\mathrm{sign}(\omega)\theta(|\omega|-|\Delta|)}{\sqrt{\omega^{2}-\Delta^{2}}}\biggr{)},$
(20)

where $\theta(x)$ is the step function and $\mathrm{sign}(x)$ is the sign
function returning $+1$ if $x>0$ and $-1$ if $x<0$. Then, we treat the
dissipation effect by impurities as replacing $\omega$ and $\Delta$ in the
Green’s function $G^{R/A}_{\mathrm{BdG}}(\bm{k},\omega)$ by
$\eta_{\omega}\omega$ and $\eta_{\omega}\Delta$. This renormalization is
derived from the Born approximation [67]. In the limit $\Delta\to 0$, it is
consistent with a constant dissipation $i\Gamma$ used in the calculations in
the normal state.

Figure 4: Numerical results of the optical activity $\sigma_{xyx}$ (Left) and
the optical conductivity $\sigma_{xx}$ (Right) in the superconducting state on
a logarithmic scale. We denote the superconducting state and the normal state
as sc and n, respectively. We use the same parameters and units as in Fig. 1.
We set $\Delta=0.005$.

We show the numerical results of the optical activity $\sigma_{xyx}$ and the
optical conductivity $\sigma_{xx}$ of the superconducting state in Fig. 4
(solid lines). We also include the results of the normal state for comparison
(dashed lines). The spectra differ from the normal state below
$\Omega=2\Delta=0.01$. $\mathrm{Im}\sigma_{xyx}$ and $\mathrm{Re}\sigma_{xx}$
decay rapidly with decreasing frequency around $2\Delta$, and the weights
become zero in the gap. This difference in the spectra from the normal state
is called the missing area. $\mathrm{Re}\sigma_{xyx}$ and
$\mathrm{Im}\sigma_{xx}$ are enhanced in the gap and follow a $\Omega^{-1}$
dependence. The missing area and the anomalous divergence indicate the
existence of a condensation. In particular, the missing area of the optical
conductivity is known as the Ferrell-Glover-Tinkham sum rule for measuring the
superfluid density [68, 69]. Similarly, they can be used to measure the SEE by
the optical activity as discussed in Sec. III.2.

Figure 5: Numerical results of the Kerr rotation $\theta$ and ellipticity
$\varepsilon$ for the reflected light (Left) and the transmitted light (Right)
on a logarithmic scale. We denote the superconducting state and the normal
state as sc and n, respectively. We use the same parameters and units as in
Fig. 1. We set $\Delta=0.005$.

The Kerr rotation and ellipticity are also changed because of the spectral
modification by the superconducting state. We show the Kerr rotation $\theta$
and ellipticity $\varepsilon$ in Fig. 5 (blue lines). We also show the results
of the normal state for comparison (black lines). As discussed in Sec. IV.2,
in the normal state, Eqs. (18) and (19) are almost purely imaginary in the
Drude regime. Thus, the phase difference of the electric fields is almost
$\pi/2$, and the ellipticity is dominant. In the superconducting state, for
frequencies inside the superconducting gap, the dominant components of
$\sigma_{xx/yy}$ and $\sigma_{xyx}$ are changed to the imaginary part and the
real part, respectively. Therefore, Eq. (18) is no longer purely imaginary,
and the Kerr rotation is enhanced. At low frequencies,
$\mathrm{Im}\sigma_{xx}$ increases by $\Omega^{-1}$ and becomes comparable to
the inverse vacuum impedance $(\mu_{0}c)^{-1}$. Then, the real part and
imaginary part of Eq. (18) are comparable, and the rotational angle and
ellipticity are in the same order of magnitude for the reflected light. For
transmitted light, Eq. (19) is also no longer purely imaginary and becomes
rather real when $(\mu_{0}c)^{-1}>\mathrm{Im}\sigma_{xx}$, thus the rotational
angle is dominant. As the frequency decreases further,
$\mathrm{Im}\sigma_{xx}$ increases by $\Omega^{-1}$ and becomes comparable to
$(\mu_{0}c)^{-1}$. Then, the rotational angle and the ellipticity also become
comparable in magnitude.

##  VI Conclusion and discussion

In this paper, we have discussed the optical activity originating from the
orbital magnetic moment in metals and superconductors using a Green’s function
formalism. We have shown that the no-go theorem in the normal state, stating
that a singular contribution to the optical activity is absent, is valid
except for the chiral magnetic effect and that this theorem is broken for the
superconducting state in Secs. II and III. Then, this singularity contributes
to the $\delta$ function in $\mathrm{Im}\sigma_{\mu\nu\lambda}$ in the DC-
limit and the anomalous $\Omega^{-1}$-divergence in
$\mathrm{Re}\sigma_{\mu\nu\lambda}$ for superconductors. In the
superconducting state, the coefficient of the singular part is confirmed to be
exactly the superconducting Edelstein effect (SEE), which implies the validity
of the missing area measurement for the SEE. An identical discussion was
recently presented for the spin-induced optical activity. In this paper, it
has been shown that the same discussion holds for the orbital-induced optical
activity. In Sec. IV, we have shown calculations using a model for monolayer
TMDs. We have used an effective two-band model to elucidate the typical
behavior of the optical activity. We have revealed that the low-frequency
spectrum is described by a Drude-like model and that the interband transition
shows an atypical chemical potential dependence originating from the interband
Fermi surface term, unique to the orbital part in metals. In addition, we have
evaluated the Kerr spectrum. In Sec. V, we have calculated the spectrum of the
optical activity in the superconducting state, and we have demonstrated the
existence of the missing area and the anomalous divergence of the optical
activity at low frequencies. In addition, we have discussed the modification
of the Kerr effect by the superconducting state.

Finally, we discuss the possibility of an experimental realization separated
into the interband regime and the intraband regime. As discussed in Sec. IV,
the interband peak is located at about a few $\mathrm{eV}$. The Kerr rotation
and ellipticity for the reflected light reach a few µrad, which makes them
detectable with the state-of-the-art instruments, having a resolution of 10
nrad [70, 71]. On the other hand, the resolution in the terahertz regime,
corresponding to the gap scale of most noncentrosymmetric superconductors, has
recently been reported to reach 20 µrad and is expected to go down to 2 µrad
[72]. The Kerr effect of TBG is close to this technical observable limit.
Thus, unless we can find other two-dimensional materials with a larger
Edelstein coefficient $\alpha$, the experimental observation is currently
challenging in this regime.

Finally, we shortly discuss the three-dimensional materials. In three-
dimensional systems, the Faraday effect is expected. As light cannot easily
penetrate superconductors, transmission can only occur in thin films [73]. The
Faraday effect originates from the difference of the refractive index for left
and right circular polarized lights $\Delta n$. It is given by
$\Phi_{\mathrm{F}}(=\theta_{\mathrm{F}}+i\varepsilon_{\mathrm{F}})=-\Delta
nk\zeta/2$, where $\theta_{\mathrm{F}}$ and $\varepsilon_{\mathrm{F}}$ are the
rotational angle and the ellipticity for the Faraday effect, $k$ is the
wavenumber of the light, and $\zeta$ is the thickness of materials. For the
natural optical activity, the refractive index difference is given by $\Delta
n=\mu_{0}c\chi$ [74]; thus, we can directly obtain the spectrum of the optical
activity $\chi$ from the Faraday spectrum. In the case of noncentrosymmetric
cubics (point group: O and T), the optical activity in the THz regime is
$\chi\sim(e\mu_{\mathrm{B}}\tau k_{F}^{2}/\hbar)(\gamma k_{F}/E_{F})$ with an
antisymmetric SOC term $\gamma\bm{k}\cdot\bm{\sigma}$. Here,
$\mu_{\mathrm{B}}$ is the Bohr magneton, $\tau$ is the relaxation time, and
$k_{F}$ and $E_{F}$ are the Fermi wavenumber and the Fermi energy. Candidates
for superconductors with this SOC include $\mathrm{Li_{2}Pd_{3}B}$,
$\mathrm{Li_{2}Pt_{3}B}$, and $\mathrm{Mo_{2}Al_{3}C}$ [75]. Then, the Faraday
effect reads

$\Phi_{\mathrm{F}}\sim-\frac{1}{2}\frac{e\mu_{\mathrm{B}}\mu_{0}k_{F}}{\hbar}(\Omega\tau)(k_{F}\zeta)\frac{\gamma
k_{F}}{E_{F}},$ (21)

and the typical order of magnitude is estimated to be 4.8 µrad for
$k_{F}=1~{}\mathrm{\AA}^{-1}$, $\tau=10~{}\mathrm{fs}$, and $\gamma
k_{F}/E_{F}=10^{-2}$ at $\Omega=1~{}\mathrm{THz}$ using the thickness
$\zeta=500~{}\mathrm{\AA}$.

## acknowledgements

K.S. acknowledges support as a JSPS research fellow and is supported by JSPS
KAKENHI, Grant No.22J23393 and No.22KJ2008. R.P. is supported by JSPS KAKENHI
No.23K03300.

##  Appendix A No-go theorem

The singular part of the optical activity is restricted in the normal state.
This part was shown to be forbidden for the spin contribution [40]. In this
appendix, we will show that the part for the orbital part is also restricted.
The coefficient of the singular part is

$\displaystyle\Phi_{\mu\nu\lambda}(0)$
$\displaystyle=\frac{e^{2}}{2}\int[d^{4}k]f(\omega)\mathrm{Tr}\Bigl{[}-\partial_{\lambda}G^{RA}(\bm{k},\omega)v_{\bm{k}}^{\mu}G^{R}(\bm{k},\omega)v_{\bm{k}}^{\nu}+G^{RA}(\bm{k},\omega)v_{\bm{k}}^{\mu}\partial_{\lambda}G^{R}(\bm{k},\omega)v_{\bm{k}}^{\nu}$
$\displaystyle\hskip
90.0pt-\partial_{\lambda}G^{A}(\bm{k},\omega)v_{\bm{k}}^{\mu}G^{RA}(\bm{k},\omega)v^{\nu}_{\bm{k}}+G^{A}(\bm{k},\omega)v_{\bm{k}}^{\mu}\partial_{\lambda}G^{RA}(\bm{k},\omega)v_{\bm{k}}^{\nu}\Bigr{]}$
$\displaystyle=\frac{e^{2}}{2}\int[d^{4}k]f(\omega)\mathrm{Tr}\Bigl{[}-\partial_{\lambda}G^{R}(\bm{k},\omega)v_{\bm{k}}^{\mu}G^{R}(\bm{k},\omega)v_{\bm{k}}^{\nu}+G^{R}(\bm{k},\omega)v_{\bm{k}}^{\mu}\partial_{\lambda}G^{R}(\bm{k},\omega)v_{\bm{k}}^{\nu}$
$\displaystyle\hskip
90.0pt+\partial_{\lambda}G^{A}(\bm{k},\omega)v_{\bm{k}}^{\mu}G^{A}(\bm{k},\omega)v^{\nu}_{\bm{k}}-G^{A}(\bm{k},\omega)v_{\bm{k}}^{\mu}\partial_{\lambda}G^{A}(\bm{k},\omega)v_{\bm{k}}^{\nu}\Bigr{]}$
$\displaystyle=e^{2}\int[d^{4}k]f(\omega)\mathrm{Re}\mathrm{Tr}\Bigl{[}-\partial_{\lambda}G^{R}(\bm{k},\omega)v_{\bm{k}}^{\mu}G^{R}(\bm{k},\omega)v_{\bm{k}}^{\nu}+G^{R}(\bm{k},\omega)v_{\bm{k}}^{\mu}\partial_{\lambda}G^{R}(\bm{k},\omega)v_{\bm{k}}^{\nu}\Bigr{]}.$
(22)

Here, $\int[d^{4}k]=\int^{\infty}_{-\infty}d\omega/(2\pi
i)\int_{\mathrm{BZ}}d^{3}k/(2\pi)^{3}$. Thus, this equation is purely
imaginary. Using the relation
$\partial_{\lambda}G^{R}(\bm{k},\omega)=G^{R}(\bm{k},\omega)v^{\lambda}_{\bm{k}}G^{R}(\bm{k},\omega)$,
we obtain

$\displaystyle\Phi_{\mu\nu\lambda}(0)$
$\displaystyle=e^{2}\int[d^{4}k]f(\omega)\mathrm{Re}\mathrm{Tr}\Bigl{[}-G^{R}(\bm{k},\omega)v^{\lambda}_{\bm{k}}G^{R}(\bm{k},\omega)v_{\bm{k}}^{\mu}G^{R}(\bm{k},\omega)v_{\bm{k}}^{\nu}+G^{R}(\bm{k},\omega)v_{\bm{k}}^{\mu}G^{R}(\bm{k},\omega)v^{\lambda}_{\bm{k}}G^{R}(\bm{k},\omega)v_{\bm{k}}^{\nu}\Bigr{]}$
$\displaystyle=e^{2}\varepsilon_{\mu\nu\lambda}\int[d^{4}k]f(\omega)\mathrm{Re}\mathrm{Tr}\Bigl{[}-G^{R}(\bm{k},\omega)v^{x}_{\bm{k}}G^{R}(\bm{k},\omega)v_{\bm{k}}^{y}G^{R}(\bm{k},\omega)v_{\bm{k}}^{z}+G^{R}(\bm{k},\omega)v_{\bm{k}}^{y}G^{R}(\bm{k},\omega)v^{x}_{\bm{k}}G^{R}(\bm{k},\omega)v_{\bm{k}}^{z}\Bigr{]}.$
(23)

This equation is totally antisymmetric. Then, we show that most components
vanish. Only the totally antisymmetric part remains. This remaining part is
known to be the chiral magnetic effect, as mentioned in the main text. It can
be shown that this part is also zero by its topological nature using the band
representation [23, 27].

In the superconducting case, this no-go theorem changes. Calculating the
singular term in the same way, we obtain

$\displaystyle\Phi_{\mu\nu\lambda}(0)$
$\displaystyle=\frac{e^{2}}{2}\int[d^{4}k]f(\omega)\mathrm{Re}\mathrm{Tr}\Bigl{[}-\partial_{\lambda}G^{R}_{\mathrm{BdG}}(\bm{k},\omega)\tilde{v}^{\mu}_{\bm{k}}G^{R}_{\mathrm{BdG}}(\bm{k},\omega)\tilde{v}^{\nu}_{\bm{k}}+G^{R}_{\mathrm{BdG}}(\bm{k},\omega)\tilde{v}^{\mu}_{\bm{k}}\partial_{\lambda}G^{R}_{\mathrm{BdG}}(\bm{k},\omega)\tilde{v}^{\nu}_{\bm{k}}\Bigr{]}.$
(24)

This equation is purely imaginary and antisymmetric under the interchange
between $\mu$ and $\nu$. However, there is one difference to the case of the
normal state. The velocity operator appearing by differentiating the Green’s
function $\hat{v}^{\mu}_{\bm{k}}$ is different from the velocity operator
$\tilde{v}^{\mu}_{\bm{k}}$. Thus, many components of the singular part can be
finite.

##  Appendix B Optical Kerr effect

Here, we derive the rotational angle in two-dimensional metals with an in-
plane polar axis pointing in the $x$ direction. In this case, the in-plane
current induces the out-of-plane magnetization, which is often discussed in
strained TMDs and twisted bilayer graphene. In the following, we consider that
the two-dimensional metal is put on an insulating substrate, which has a real-
valued refractive index $n_{\mathrm{sub}}$ in the frequency regime of
interest, and we assume $n_{\mathrm{sub}}\sim 1$.

Figure 6: Schematic diagram visualizing the incident, reflected, and
transmitted light beam. The two-dimensional material is put on the $xy$ plane
(dashed line).

We will derive the rotational angle using Maxwell’s electrodynamics. We
consider that the incident light is in the $xz$ plane and travels at an angle
of incidence of $None$. Furthermore, the electric field of the incident light
is the $s$-polarized and, thus, is oriented in the $y$ direction. In this
case, the electric field for the incident light $\bm{E}^{\mathrm{in}}$, the
reflected light $\bm{E}^{\mathrm{r}}$, and the refracted light
$\bm{E}^{\mathrm{t}}$ are given by

$\displaystyle\bm{E}^{\mathrm{in}}=E^{\mathrm{in}}\bm{e}_{y}e^{i\bm{k}\cdot\bm{r}-i\omega
t},$ (25a)
$\displaystyle\bm{E}^{\mathrm{r}}=(E^{\mathrm{r}}_{y}\bm{e}_{y}+E^{\mathrm{r}}_{xz}(\bm{e}_{x}+\bm{e}_{z}))e^{i\bm{k}^{\prime}\cdot\bm{r}-i\omega
t},$ (25b)
$\displaystyle\bm{E}^{\mathrm{t}}=(E^{\mathrm{t}}_{y}\bm{e}_{y}+E^{\mathrm{t}}_{xz}(\bm{e}_{x}-\bm{e}_{z}))e^{i\bm{k}\cdot\bm{r}-i\omega
t}.$ (25c)

Here, the wavenumbers are given by
$\bm{k}=\frac{k}{\sqrt{2}}(\bm{e}_{x}+\bm{e}_{z})$ and
$\bm{k}^{\prime}=\frac{k}{\sqrt{2}}(\bm{e}_{x}-\bm{e}_{z})$ and the magnitude
of the wavenumber $k$ satisfies the dispersion relation $\omega=ck$, where
$\omega$ is the frequency and $c$ is the light velocity. The corresponding
magnetic fields are determined by Faraday’s law
($\bm{\nabla}\times\bm{E}=-\partial\bm{B}/\partial t$). They read

$\displaystyle\bm{B}^{\mathrm{in}}=\frac{E^{\mathrm{in}}}{\sqrt{2}c}(\bm{e}_{z}-\bm{e}_{x})e^{i\bm{k}\cdot\bm{r}-i\omega
t},$ (26a)
$\displaystyle\bm{B}^{\mathrm{r}}=\frac{1}{\sqrt{2}c}(-2E^{\mathrm{r}}_{xz}\bm{e}_{y}+E^{\mathrm{r}}_{y}(\bm{e}_{x}+\bm{e}_{z}))e^{i\bm{k}^{\prime}\cdot\bm{r}-i\omega
t},$ (26b)
$\displaystyle\bm{B}^{\mathrm{t}}=\frac{1}{\sqrt{2}c}(2E^{\mathrm{t}}_{xz}\bm{e}_{y}+E^{\mathrm{t}}_{y}(\bm{e}_{z}-\bm{e}_{x}))e^{i\bm{k}\cdot\bm{r}-i\omega
t}.$ (26c)

Electrons are driven by the electromagnetic wave in the two-dimensional
material, and we treat the responses up to the linear order. In the following,
we consider that the linear responses only include a usual longitudinal
conductivity $\sigma$ and a magnetoelectric effect $\chi$. These effects
appear in a current density as

$\displaystyle j_{x}$
$\displaystyle=\sigma_{xx}(\bm{E}^{\mathrm{t}})_{x}+ik_{x}\chi(\bm{E}^{\mathrm{t}})_{y}$
$\displaystyle=\sigma_{xx}E^{\mathrm{t}}_{xz}+ik_{x}\chi E^{\mathrm{t}}_{y},$
(27) $\displaystyle j_{y}$
$\displaystyle=\sigma_{yy}(\bm{E}^{\mathrm{t}})_{y}-ik_{x}\chi(\bm{E}^{\mathrm{t}})_{x}$
$\displaystyle=\sigma_{yy}E^{\mathrm{t}}_{y}-ik_{x}\chi E^{\mathrm{t}}_{xz}.$
(28)

This current flows in these two-dimensional metals and influences the boundary
conditions of the electromagnetic fields. The boundary conditions coming from
Faraday’s law and Gauss’s law are given by

$E^{\mathrm{in}}+E^{\mathrm{r}}_{y}=E^{\mathrm{t}}_{y},\hskip
10.0ptE^{\mathrm{r}}_{xz}=E^{\mathrm{t}}_{xz}.$ (29)

Furthermore, Ampère’s law gives two boundary conditions as

$\displaystyle\frac{-1}{\sqrt{2}\mu_{0}c}(2E^{\mathrm{r}}_{xz}+2E^{\mathrm{t}}_{xz})=\sigma_{xx}E^{\mathrm{t}}_{xz}+i\frac{\omega}{\sqrt{2}c}\chi
E^{\mathrm{t}}_{y},$ (30a)
$\displaystyle\frac{1}{\sqrt{2}\mu_{0}c}(E^{\mathrm{in}}-E^{\mathrm{r}}_{y}-E^{\mathrm{t}}_{y})=\sigma_{yy}E^{\mathrm{t}}_{y}-i\frac{\omega}{\sqrt{2}c}\chi
E^{\mathrm{t}}_{xz}.$ (30b)

Here, $\mu_{0}$ is the vacuum permeability. Solving these equations
simultaneously, we find

$\displaystyle\frac{E^{\mathrm{t}}_{xz}}{E^{\mathrm{t}}_{y}}$
$\displaystyle=-\frac{i\omega\chi}{\sqrt{2}c}\biggl{(}\frac{2\sqrt{2}}{\mu_{0}c}+\sigma_{xx}\biggr{)}^{-1}\equiv
te^{i\phi_{t}},$ (31)
$\displaystyle\frac{E^{\mathrm{r}}_{xz}}{E^{\mathrm{r}}_{y}}$
$\displaystyle=\frac{i\omega\chi}{c(\mu_{0}c\sigma_{yy})}\biggl{(}\frac{2\sqrt{2}}{\mu_{0}c}+\sigma_{xx}-\frac{\omega^{2}\chi^{2}}{2c^{2}\sigma_{yy}}\biggr{)}^{-1}$
$\displaystyle\sim\frac{i\omega\chi}{c(\mu_{0}c\sigma_{yy})}\biggl{(}\frac{2\sqrt{2}}{\mu_{0}c}+\sigma_{xx}\biggr{)}^{-1}\equiv
re^{i\phi_{r}}.$ (32)

Here, we ignore the square of $\chi$ due to its smallness. This ratio gives
the Kerr rotation $\theta$ and ellipticity $\varepsilon$ as

$\displaystyle\theta^{r}$
$\displaystyle=\frac{1}{2}\arctan\biggl{(}\frac{2r\cos\phi_{r}}{1-r^{2}}\biggr{)},$
(33a) $\displaystyle\varepsilon^{r}$
$\displaystyle=\tan\bigg{\\{}\frac{1}{2}\arcsin\biggl{(}\frac{2r\sin\phi_{r}}{1+r^{2}}\biggr{)}\bigg{\\}}.$
(33b)

The Kerr rotation and ellipticity of the transmitted light are obtained by
simply replacing $r$ with $t$.

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