bands may be feasible through current-aided manipulation [344] and by the
spin-flop transition [167].

Furthermore, owing to the $\mathcal{PT}$-symmetric but $\mathcal{P}$-violating
symmetry, the symmetry and its characteristic responses can be tuned in a
nontrivial manner. As tabulated in Fig. 2, the $\mathcal{PT}$ symmetry forbids
uniform electric and magnetic polarizations and thereby keeps the spin
degeneracy intact at each crystal momentum. Since the degenerate states show
the intimate coupling between the spin and sublattice degree of freedom, the
spin-momentum-sublattice locking can exert unique transport phenomena like the
layer Hall effect [64] where the anomalous Hall response occurs in the
staggered way between the twined layers. The degeneracy is easily lifted by
the external fields violating the $\mathcal{PT}$ symmetry such as the external
magnetic and electric fields. The available elimination of
$\mathcal{PT}$-ensured degeneracy implies that one can manipulate the physical
phenomena arising from the ferroelectric and ferromagnetic order. For
instance, it has been shown that the electric-field effect gives rise to the
uniform Berry curvature and associated magnetooptical and magnetogalvanic
effects in layered materials such as CrI3 [112, 111, 98] and even-layer
MnBi2Te4 [44].

Similarly, the external magnetic field leads to the physical phenomena
originating from the $\mathcal{PT}$-symmetry violation. For instance, induced
small canting of antiferromagnetic moments leads to sizable anomalous Hall
response which is called chiral Hall effect [157]. This sizable response is in
contrast to the typical case of canted antiferromagnets where the anomalous
Hall response may not be significant when the induced magnetization is small.
Intriguingly, the spin-momentum texture experiences the magnetic-field-induced
correction that is entirely different from the typical case. To corroborate
the induced spin-momentum structure, let us consider the bilayer system
undergoing the layer-dependent spin polarization. The obtained
$\mathcal{PT}$-symmetric antiferromagnetic state can be found in layered
magnetic materials. When the magnetic field is in the out-of-plane direction,
the combination of the $\mathcal{PT}$-symmetric magnetic order and external
field results in lifting the spin-layer-coupled degeneracy at every crystal
momentum and thereby in the Rashba-type spin-momentum locking.

Figure 14:  Sketch for the mechanism of magnetic ASOC in the bilayer
antiferromagnet. (a) Energy spectrum $\varepsilon_{\bm{k}}$ with the zero
interlayer tunneling $t=0$. Energy bands in the antiferromagnetic phase with a
finite molecular field are shown by solid lines while those in the nonmagnetic
phase are by dashed lines. The magnetic structure is in-plane ferromagnetic
and interlayer antiferromagnetic as depicted in the inset. Lower panels
indicate the spin-momentum texture on the Fermi surface for $\bm{L}\neq\bm{0}$
parametrized by the chemical potential $\mu$ (red arrows denote the spin
polarization at each momentum). The Fermi surface, being doubly degenerate due
to the $\mathcal{PT}$ symmetry, is decomposed into those in the upper
($\rho_{z}=+1$) and lower layers ($\rho_{z}=-1$). (b) Energy spectrum obtained
by adding the out-of-plane Zeeman field $\bm{h}$. Solid lines in blue (red)
show the energy of electrons on the upper (lower) layers. Dashed lines are
equivalent to the solid lines in (a) with $\bm{h}=0$. The degenerate Fermi
surface is split into two showing the opposite helical spin texture with
different out-of-plane polarization as depicted in the lower panel, where the
spin-momentum structure on each Fermi surface is illustrated.

The resultant Rashba spin-orbit coupling stems from the purely magnetic
modification of electronic structure and does not require the
noncentrosymmetric crystal structure, being in sharp contrast to the known
cases of spin-momentum locking such as those found in nonmagnetic polar
crystals. Thus, such magnetically-induced antisymmetric spin-orbit coupling is
dubbed with magnetic antisymmetric spin-orbit coupling (magnetic ASOC) [306].
Let us exemplify the magnetic ASOC by taking the model Hamiltonian for the
bilayer system [Eq. (8)]. We further add the antiferromagnetic molecular field
and Zeeman field written by

$H_{\text{mag}}=\sum_{\rho_{z}=\pm}-\left(\bm{h}+\bm{L}\rho_{z}\right)\cdot\bm{\sigma},$
(82)

with $\bm{L}\parallel\hat{z}$. Note that $\bm{\sigma}$ and $\bm{\rho}$ are
Pauli matrices for the spin and layer degrees of freedom, respectively. In the
case of zero tunneling hopping ($t=0$), the momentum-resolved spin
polarization quantifying the magnetic ASOC is analytically obtained as

$\Braket{\bm{\sigma}}({\bm{k}},\rho_{z})=\frac{\bm{G}}{||\bm{G}||},~{}\bm{G}=-\alpha_{\text{R}}\hat{\bm{g}}_{\bm{k}}\rho_{z}+\bm{h}+\bm{L}\rho_{z},$
(83)

for the lower-energy eigenstates of the upper ($\rho_{z}=+1$) and lower
($\rho_{z}=-1$) layers. The hidden Rashba spin-orbit coupling is given with
the vector $\hat{\bm{g}}_{\bm{k}}=\left(k_{y},-k_{x},0\right)$. The two lower
energy spectrums are originally degenerate due to the $\mathcal{PT}$ symmetry,
and their energies are slightly separated by the Zeeman field as much as
$\sim|\bm{h}|$. The resultant difference in Fermi-surface volumes and Fermi
wavelengths ($k_{\text{F}}^{\pm}$) between the layers gives rise to the
uncompensated Rashba-like spin-momentum texture.

Furthermore, the magnetic ASOC may take a form different from that of the
hidden ASOC determined by the crystal structure in the nonmagnetic phase. For
instance, under the in-plane magnetic field such as $\bm{h}\parallel\hat{x}$,
the induced magnetic ASOC has the form of $k_{x}\sigma_{z}$ being irrelevant
to the existing Rashba spin-orbit coupling hidden by the layer degree of
freedom. The interlayer tunneling hopping $t$ plays an essential role in
producing this type of magnetic ASOC. This is in sharp contrast to the fact
that the large inter-subsector hopping weakens the hidden spin polarization
governed by the paramagnetic Hamiltonian.

The magnetic ASOC may offer tunable spin-momentum locking, while the
conventional ASOC is usually determined by the noncentrosymmetric crystal
structure whose parity violation is hard to control externally. Considering
vast $\mathcal{PT}$-symmetric magnetic materials [304], we can find various
types of magnetic ASOC different from the Rashba ASOC, such as the
Dresselhaus-type [306] and the chiral ones. The concept of the magnetic ASOC
can be comprehensively generalized to magnetically-induced electric parity
violations. Then, combining the $\mathcal{PT}$-symmetric magnet with the
external magnetic field, one can obtain the various physical phenomena unique
to the nonmagnetic and noncentrosymmetric crystals; e.g., piezoelectricity,
nonlinear Hall effect arising from the Berry curvature dipole [306],
photogalvanic effects for charge and spin current [172], and so on. The
emergence of the Berry curvature dipole is identified by the discussions
parallel to those of the magnetic ASOC because the spin and momentum-space
Berry curvature have the same space-time symmetry ($\mathcal{P}$-even and
$\mathcal{T}$-odd). The magnetically-tunable responses discussed above can be
systematically identified with the help of magnetic symmetry analysis [52,
306, 326] and are expected to be a key to rich spintronic phenomena based on
antiferromagnetic materials.

##  V Parity-violating Superconducting Responses

In the preceding sections, we overviewed the physical phenomena induced by the
magnetic parity violation. Various emergent responses and itinerant properties
originate from the electronic structure unique to the parity violation.
Similarly, such intriguing electronic structures have a significant influence
on the quantum phases emerging in metals such as superconductivity.
Noncentrosymmetric superconductors, superconductors with the lack of the
$\mathcal{P}$ symmetry, have been intensively studied in light of the strong
spin-orbit coupling as found in heavy fermion and van der Waals materials [18,
101]. Those studies mainly worked on modifications of the basic
superconducting properties such as critical magnetic fields and pairing
states. This section is devoted to discussions of physical phenomena unique to
the parity violation in superconductors.

Let us consider candidates for the noncentrosymmetric superconductors other
than known materials crystalizing in the noncentrosymmetric structure. The
parity violation can be invoked in the following three ways; (1) parity-
violating order concurrently existing with the superconductivity, (2) the
supercurrent injection, and (3) multiple superconducting transitions where the
$\mathcal{P}$ parity is different between those superconducting pairing
potentials [301].

Firstly, the parity violation due to the spontaneous order is ascribed to the
nonmagnetic and magnetic origins. Specifically, for the former the
ferroelectric- or piezoelectric-like order has been identified to coexist with
the superconductivity in doped SrTiO3 [224] and heavy-fermion systems [97],
while for the latter the odd-parity magnetic multipolar order may be involved
in the superconductivity of a locally-noncentrosymmetric superconductor
CeRh2As2 [127, 132]. The order-induced modifications of the electronic
structures may assist the emergence of exotic superconductivity such as the
finite-momentum pairing state in the presence of the magnetic parity violation
[265, 264]. Although similar interplay between the superconductivity and
magnetic parity violation can be realized in the noncentrosymmetric
superconductors under the external magnetic field [18, 254, 126, 293, 43], the
odd-parity magnetic order may offer more significant modification of the
electronic property with the energy scale of the Hund’s coupling, which can be
much larger than the Zeeman coupling, and cause a prominent parity-violating
effect on the superconductivity.

Secondly, the parity violation can be built into the superconductor by the
supercurrent injection. The superfluidity allows the current to flow without
the Joule heating and thereby realizes the parity-violating phase even in
prototypical $s$-wave superconductors. Owing to its space-time symmetry, the
biased electric current gives rise to the magnetic parity violation. The
resulting finite-momentum superconductivity has been verified in experiments
detecting the optical signals unique to the parity violation [322, 191, 287].

Lastly, the superconductivity itself can break the $\mathcal{P}$ symmetry.
Since the $\mathcal{P}$ parity is definite in the conventional pairing states
such as the even-parity $s$-wave and odd-parity for $p$-wave superconductivity
in centrosymmetric materials, the parity violation does not happen at the
single superconducting transition. Note that the $\mathcal{P}$ operation is
effectively retained in the odd-parity superconducting states because the
$\mathcal{P}$ symmetry is preserved in the form of its combination with the
U(1) gauge transformation. This is a significant difference from the odd-
parity multipolar order. On the other hand, spontaneous symmetry breaking
occurs if the system exhibits multiple superconducting transitions. Evidencing
such possibility, Ginzburg-Landau analysis of the multiple superconducting
transitions showed that the $s+ip$-wave pairing state, an exotic pairing state
showing the purely magnetic parity violation, is stable when both of $s$\- and
$p$-wave superconductivity emerge [298]. Interestingly, the multiband nature
in the normal phase allows for richer properties of magnetically parity-
violating superconductivity in the Bogoliubov-quasiparticle spectrum [119,
134]. Following the parallel discussion of the electrically-switchable
$\mathcal{PT}$-symmetric magnets, one can identify the possibility of
supercurrent-induced switching of the magnetically parity-violating
superconductivity, such as between the $s\pm ip$-wave states. When the
superconducting order parameters have the same symmetry as the magnetic
toroidal order, it is called anapole superconductivity [119] and allows
supercurrent-induced switching. The spontaneously parity-violating
superconductivity demands the stringent condition of sizable fluctuations
leading to the multiple Cooper instability and thus material realization
requires further studies, while such fluctuations have been implied in a
heavy-fermion superconductor [107, 7].

The superconductivity-induced symmetry breaking is similarly found in the
chiral superconductivity manifesting the $\mathcal{T}$-symmetry breaking. The
chiral superconductivity, however, differs from the parity-violating
superconductivity as it can occur in the single transition labeled by a multi-
dimensional irreducible representation [248, 246].

Considering the playgrounds for the parity-violating superconductivity, one
may be interested in their characteristic physical responses. Recently, active
experimental and theoretical research has identified various nonreciprocal
responses related to superconductivity. For instance, tremendous interest has
been drawn by the recent discoveries of the nonreciprocal charge transport of
superconductors such as nonreciprocal conductivity, superconducting diode
effect, and nonreciprocal Josephson effect [101, 187, 190]. Furthermore, the
superconducting property shows up in various responses including cross-
correlated and optical responses as follows.

Here, we review the cross-correlated responses in superconductors. Let us
consider the free energy in the superconducting state dependent on external
fields (magnetic field $\bm{H}$, stress $\hat{s}$, and vector potential
$\bm{A}$)

$F=F(\bm{H},\hat{s},\bm{A}).$ (84)

Importantly, the vector potential characterizes the supercurrent injection
through the London equation. Taking the derivative with respect to each field,
we obtain the conjugate quantities as

$\bm{M}=-\frac{\partial
F}{\partial\bm{H}},~{}\hat{\varepsilon}=-\frac{\partial
F}{\partial\hat{s}},~{}\bm{J}^{\text{sc}}=-\frac{\partial F}{\partial\bm{A}},$
(85)

that is, magnetization, strain, and supercurrent. The free energy is
transformed as

$F(\bm{H},\hat{s},\bm{A})\to F(\bm{H},\hat{s},-\bm{A}),$ (86)

under the $\mathcal{P}$ operation and

$F(\bm{H},\hat{s},\bm{A})\to F(-\bm{H},\hat{s},-\bm{A}),$ (87)

under the $\mathcal{T}$ operation. Then, corrections to the free energy
arising from the parity violation is given by

$\delta F_{\text{cc}}=-\alpha_{ab}H_{a}A_{b}-\beta_{ab}s_{a}A_{b},$ (88)

up to the bilinear form. The coefficient $\alpha_{ab}$ is $\mathcal{T}$-even
but $\mathcal{PT}$-odd, while the parity of $\beta_{ab}$ is opposite for each
operation. It follows that the cross-correlation $\alpha_{ab}$ ($\beta_{ab}$)
is unique to the superconductors with the electric (magnetic) parity
violation. One can obtain the cross-correlated response from the former
coupling as

$M_{a}=\alpha_{ab}J^{\text{sc}}_{b},$ (89)

that is the superconducting Edelstein effect [50] being in stark contrast to
the normal Edelstein (inverse magnetogalvanic) effect in which the ohmic
current leads to the magnetization [147, 49] 555 Note that the inverse
response written by $J^{\text{sc}}_{b}=\alpha_{ab}H_{a}$ does not occur in the
DC regime due to the Bloch’s no-go theorem [200].. Notably, the magnetization
response of Eq. (89) can be significant due to nontrivial contribution from
the topological edge states of nodal superconductors [103].

For the latter coupling in Eq. (88), one can identify the strain response to
the supercurrent

$\varepsilon_{a}=\beta_{ab}J^{\text{sc}}_{b},$ (90)

called superconducting piezoelectric effect [32, 31]. As in the case of the
magnetization-supercurrent correlation, the superconducting piezoelectric
effect is clearly distinguished from the conventional (inverse) piezoelectric
and magnetopiezoelectric effects in terms of the fields stimulating the
strain, since the latter two effects are involved in the electric field and
ohmic current, respectively. One can refer to Eqs. (27) and (30) for
comparison. Unlike them, superconducting Edelstein and superconducting
piezoelectric effects are the responses to the supercurrent.

We summarize the cross-correlated responses in Fig. 15. To be of interest, it
has been shown that the superconducting piezoelectric effect displays an
abrupt change during the crossover between the two superconducting phases in
the Rashba superconductor [32], that is, helical and Fulde-Ferrell states [43,
126]. This implies that the cross-correlated response plays a key to
identifying the exotic phase induced by the coupling between superconductivity
and magnetic parity violation. Notably, the supercurrent injection in bulk can
be replaced with the biased Josephson junction [125] because the vector
potential plays a role equivalent to the gradient of the superconducting phase
due to the minimal coupling. For instance, the giant superconducting Edelstein
effect is possibly realized in the biased Josephson junction, which could be
essential for superconductor-based spintronics. Although the research of
superconducting nonreciprocal responses has advanced significantly as reviewed
below and elsewhere [190, 187], cross-correlated responses in superconductors
have not been explored experimentally, pointing to the next issue in
superconducting science.

Figure 15:  Classification of the cross-correlated responses. The cross
correlation is tabulated for the strain (elastic) and magnetization (magnetic)
responses to the electrical stimuli, that is electric field $\bm{E}$, ohmic
electric current $\bm{J}$, and supercurrent $\bm{J}_{\text{sc}}$. The
responses are colored in blue and yellow depending on the type of parity
breaking; the responses shaded in yellow are induced by the electric parity
violation, while those in blue are induced by the magnetic parity violation.

Next, we discuss the nonreciprocal responses in superconductors. The free
energy can be expanded to nonlinear order with respect to the external fields
as well. Considering only the effects of vector-potentials, we obtain the
correction up to the cubic components as

$\delta
F_{\bm{A}}=-\frac{1}{2}\rho_{ab}A_{a}A_{b}-\frac{1}{3}f_{abc}A_{a}A_{b}A_{c}.$
(91)

The quadratic correction results in the well-known Meissner response, while
the cubic coefficient $f_{abc}$ gives $\mathcal{P}$-odd, $\mathcal{T}$-odd,
and $\mathcal{PT}$-even corrections. Then, one arrives at the nonreciprocal
response induced by the magnetic parity violation

$\frac{1}{2}\left[J_{a}^{\text{sc}}(\bm{A})-J_{a}^{\text{sc}}(-\bm{A})\right]=f_{abc}A_{b}A_{c}.$
(92)

This formula represents the nonreciprocal Meissner effect, by which the
supercurrent response differs depending on the direction of the supercurrent
shielding the external magnetic field [300]. The nonreciprocal response is
determined by the nonreciprocal correction to the superfluid weight $f_{abc}$,
namely, nonreciprocal superfluid weight. Its relevance to the superfluid
weight follows from the relation

$f_{abc}=\lim_{\bm{A}\to\bm{0}}\partial_{A_{a}}\rho_{bc}.$ (93)

The nonreciprocal superfluid weight plays a key role in various nonreciprocal
responses of superconductors, not only the nonreciprocal Meissner effect [Eq.
(92)], but also the nonreciprocal conductivity [96] and the nonreciprocal
optical responses [301].

For the nonreciprocal current generation of Eq. (37), let us again consider
the photocurrent response as in Sec. III.3.2. The perturbative calculations
are straightforwardly performed if one works on the molecular-field (BCS)
approximation for the superconductivity [318, 301] where the electronic
excitation is attributed to the Bogoliubov quasiparticles. The nonreciprocal
optics of superconductors have unique features that are not found in the
normal conducting phase. One is the optical excitation associated with the van
Hove singularity in the Bogoliubov spectrum, which gives the peak of the DOS
at the gap edge, and another is the anomalous nonreciprocal optical responses.

To introduce these features, we consider the expression for the photocurrent
conductivity in the clean limit. The formula consists of two contributions

$\sigma_{a;bc}(0;\omega,-\omega)=\sigma_{a;bc}^{\text{reg}}+\sigma_{a;bc}^{\text{ano}}.$
(94)

The first and second terms are conventional and anomalous contributions,
respectively. The conventional contribution ($\sigma^{\text{reg}}$) is given
by the expressions similar to those in the normal state [Eq. (54)]; e.g., it
originates from the Fermi-surface excitation and the resonantly-created
Bogoliubov quasiparticles [318, 301]. The Fermi-surface contribution may be
significant in nodal-gap superconductors such as those manifesting the
Bogoliubov Fermi surface [2], although it disappears in the full gap
superconductors at zero temperatures. Note that the Berry connection
contributing to various nonreciprocal optical responses is defined in the
parameter space described by the vector potential, not by the momentum. This
is because the electrons and holes are treated on equal footing in the
framework of the Bogoliubov-de Gennes Hamiltonian and thereby the minimal
coupling does not imply the equivalence between the derivatives with respect
to the vector potential ($\partial_{\bm{A}}$) and to the crystal momentum
($\partial_{\bm{k}}$).

Let us consider the resonant optical response of superconductors included in
the conventional contribution $\sigma_{a;bc}^{\text{reg}}$. An intriguing
property of the resonant superconducting optical responses is the optical
excitation available after the formation of the superconducting gap. When the
system undergoes the superconducting transition, the superconducting gap gives
rise to the van Hove singularity, which is related to the characteristic
behaviors of physical properties. In terms of the optical response, the energy
spectrum of Bogoliubov quasiparticles indicates the optical excitation $\beta$
bridging the van Hove singularities as $\omega\sim 2|\Delta|$ ($\Delta$: pair
potential) in addition to the high-frequency optical path $\alpha^{\prime}$
[Fig. 16(b)]. We note that the path $\alpha^{\prime}$ gives the optical
response similar to that given by the path $\alpha$ in the normal state [Fig.
16(a)] if the frequency of light is much higher than the superconducting gap
energy. Thus, the optical path $\beta$ is unique to the superconducting state,
however, it was considered to make no contribution to the optical response due
to the selection rule related to the $\mathcal{T}$ symmetry [232].

The absence of optical excitations relevant to the optical path $\beta$ has
been recently revisited in the context of the superconducting fitness [4]
quantifying the multiband nature of the pair potential [56, 217]. The newly-
contributing optical path $\beta$ does not work in prototypical
superconductors such as the conventional $s$-wave superconductors, whereas it
is not the case if one takes into account the strong-coupling effect, multi-
band nature in the normal state, and supercurrent flow [25, 4, 117].
Consistent with this argument, the peak structure stemming from the van Hove
singularity has been identified in the optical spectrum for the linear [4] and
second-order optical responses [275].

Figure 16:  Schematics of resonant optical excitations in the normal and
superconducting states. (a) Energy spectrum $\varepsilon_{\bm{k}}$ in the
normal state of the Rashba model and its optical excitation across the
chemical potential $\mu$ denoted by $\alpha$. (b) Energy spectrum $E_{\bm{k}}$
of Bogoliubov quasiparticles in the superconducting state, optical paths
($\alpha^{\prime},\beta$), and density of states $D(E)$. The energy spectrum
is obtained by introducing the pair potential to the Rashba Hamiltonian in
(a). The high-frequency optical excitation $\alpha^{\prime}$ is almost the
same as that in the normal state ($\alpha$), while the optical path $\beta$
bridging the coherence peaks is unique to the superconducting state.

In contrast to the conventional contributions discussed above, the anomalous
contribution to the photocurrent conductivity Eq. (94) is characteristic of
the parity-violating superconductors and has no counterpart in the normal
phase. The formula is explicitly given by the two terms both of which are
written by the total derivative with respect to the vector potential as

$\sigma_{a;bc}^{\text{ano}}=-\frac{1}{2\omega^{2}}f_{abc}+\frac{1}{4\omega^{2}}\sum_{p\neq
q}\lim_{\bm{A}\to\bm{0}}\partial_{A_{a}}\left[J_{pq}^{b}J_{qp}^{c}\left(\frac{1}{\omega-\varepsilon_{p}+\varepsilon_{q}}+\frac{1}{\varepsilon_{p}-\varepsilon_{q}}\right)\left\\{f(\varepsilon_{p})-f(\varepsilon_{q})\right\\}\right],$
(95)

with the paramagnetic current operator $\hat{J}^{a}$ and indices ($p,q$) for
the energy eigenstates. We note that the gapful superconductivity is assumed
in Eq. (95). It should be noticed that the chain rule between
$\partial_{\bm{A}}$ and $\partial_{\bm{k}}$ holds in the normal state, and
thereby the anomalous contribution vanishes due to the periodicity in the
momentum space [174]. The nonreciprocal superfluid weight covers various
superconducting properties such as the Meissner and photocurrent responses,
similarly to the (linear) superfluid weight [54].

The anomalous contribution is comprised of the nonreciprocal superfluid
density and the second term of Eq. (95). The latter part is proportional to
the vector-potential derivative of the reactive part in the linear optical
conductivity

$\sigma_{ab}^{\text{reg}}(\omega)=\frac{1}{i\omega}\sum_{p\neq
q}J_{pq}^{b}J_{qp}^{c}\left(\frac{1}{\omega-\varepsilon_{p}+\varepsilon_{q}}+\frac{1}{\varepsilon_{p}-\varepsilon_{q}}\right)\left\\{f(\varepsilon_{p})-f(\varepsilon_{q})\right\\}.$
(96)

Given this expression, the second term in Eq. (95) is called the conductivity
derivative. By taking the DC limit, the conductivity derivative becomes the
derivative of the Berry curvature as

$\displaystyle\frac{i}{4\omega}\lim_{\bm{A}\to\bm{0}}\partial_{A_{a}}\sigma_{bc}^{\text{reg}}(\omega)$
$\displaystyle=\frac{1}{4\omega^{2}}\lim_{\bm{A}\to\bm{0}}\partial_{A_{a}}\sum_{p\neq
q}\left[J_{pq}^{b}J_{qp}^{c}\left(\frac{1}{\omega-\varepsilon_{p}+\varepsilon_{q}}+\frac{1}{\varepsilon_{p}-\varepsilon_{q}}\right)\left\\{f(\varepsilon_{p})-f(\varepsilon_{q})\right\\}\right],$
(97)
$\displaystyle=\frac{i}{4\omega}\lim_{\bm{A}\to\bm{0}}\epsilon_{bcd}\,\partial_{A_{a}}\left(\sum_{p}\Omega_{p}^{d}f_{p}\right).$
(98)

Note that the derivative of the Berry curvature is totally different from the
Berry curvature dipole defined in Eq. (45) where the derivative acts only on
the Berry curvature. Being concerned with the anomalous photocurrent response
at the frequency below the superconducting gap, we can attribute the response
to the derivative of the superfluid weight $f_{abc}$ and the total Berry
curvature $\partial_{A_{a}}\left(\sum_{p}\Omega_{p}^{d}f_{p}\right)$. These
terms give rise to the photocurrent responses to the linearly and circularly
polarized light, respectively.

An intriguing property of the anomalous contribution is the diverging behavior
in the low-frequency limit; i.e., the nonreciprocal superfluid weight leads to
the diverging photocurrent conductivity $\sigma_{a;bc}\propto\omega^{-2}$ with
the frequency of irradiating light $\omega$, and the derivative of the Berry
curvature does $\sigma_{a;bc}\propto\omega^{-1}$. The diverging behavior does
not result from the assumption of the clean limit in Eq. (94), while
scattering effects restrict the normal contributions to a finite value in the
DC limit [46, 174]. The robustness of the low-frequency divergence has been
confirmed in the numerical studies by varying the scattering rate [301]. The
low-frequency divergence and disorder-tolerant nature may be advantageous for
applications to optoelectronic devices based on superconductors.

Furthermore, the two anomalous contributions show the contrasting space-time
symmetry and are therefore conveniently classified by the electric and
magnetic parity violations. The nonreciprocal superfluid weight is
$\mathcal{T}$-odd and $\mathcal{PT}$-even as mentioned around Eq. (91), while
the derivative of the Berry curvature has the same space-time symmetry as the
Berry curvature dipole, that is $\mathcal{T}$-even and $\mathcal{PT}$-odd. It
implies that the anomalous nonreciprocal optical response is a convenient tool
for identifying the symmetry of parity violation. Under the irradiation of the
low-frequency light, the $\mathcal{PT}$-symmetric superconductors show
significant photocurrent response to the linearly-polarized light and the
$\mathcal{T}$-symmetric superconductors do that to the circularly-polarized
light.

The anomalous mechanism works not only in the photocurrent response but also
in other nonreciprocal optical responses in superconductors. In the DC limit,
the anomalous terms are determined by several indicators irrespective of the
frequency $(\omega_{1},\omega_{2})$ in the response function
$\sigma_{a;bc}(\omega_{1},\omega_{2})$ [301]. For the gapful superconductors,
the formula is given by

$\sigma_{a;bc}^{\text{ano}}(\omega_{1},\omega_{2})=\frac{1}{2\omega_{1}\omega_{2}}f_{abc}-\frac{i}{8}\lim_{\bm{A}\to\bm{0}}\left[\frac{1}{\omega_{1}}\epsilon_{acd}\,\partial_{A_{b}}\left(\sum_{p}\Omega_{p}^{d}f_{p}\right)+\frac{1}{\omega_{2}}\epsilon_{abd}\,\partial_{A_{c}}\left(\sum_{p}\Omega_{p}^{d}f_{p}\right)\right],$
(99)

to which the nonreciprocity in the superfluid weight and the Berry curvature
universally contribute. Thus, the low-frequency divergence is similarly
predicted in various responses including the second-harmonic generation
($\omega_{1}=\omega_{2}$). It may be relevant to recent experiments where the
supercurrent-induced parity breaking leads to the second-harmonic generation
[191].

Finally, it should be noted that the nonreciprocal and superconducting optical
responses may be informative for quantifying the parity mixing in the
superconducting pairing. The superconducting gap symmetry does not show the
definite parity under the $\mathcal{P}$ violation in noncentrosymmetric
systems, and the resultant parity mixing is expected to give rise to
intriguing superconducting properties [18]. It has been a longstanding issue
whether such parity mixing can be evaluated in the experiments. In this
regard, the nonreciprocal optical response was found to be enhanced in the
presence of moderate parity mixing [301, 275], similar to the nonreciprocal
paraconductivity [292]. The quantitative estimates of the parity mixing can be
obtained by future optical measurements of tunable materials, such as
Li2(Pt3-xPdx)B [13] where the parity mixing presumably changes by chemical
substitution.

##  VI Summary and Outlook

The role of magnetic parity violation in solid state electron systems has been
reviewed with the comparison to that of the electric parity violation, a known
type of parity violation found in the materials crystalizing in a
noncentrosymmetric structure. Since the magnetic parity violation is
accompanied by the time-reversal-symmetry breaking, it may be considered less
fundamental than the electric parity violation. The preserved space-time
symmetry, however, allows us to make clear distinctions between physical
phenomena induced by either the electric or magnetic parity violation; i.e.,
the $\mathcal{T}$-symmetric system manifests the physical properties arising
from the electric parity violation, while the $\mathcal{PT}$-symmetric system
shows phenomena unique to the magnetic parity violation.

The physical responses originating purely from the magnetic parity violation
have been intensively discussed in the field of multiferroics in terms of the
magnetoelectric phenomena. In addition, recent studies have been devoted to
the investigation of magnetically parity-violating physical phenomena, which
have advanced the earlier studies of the electric parity violation, such as
for the photocurrent response. The identified responses are complementary to
those induced by the electric parity violation. The complementary roles of
magnetic and electric parity violation have been highlighted and grasped by
their contrasting electronic structures.

The magnetic parity violation may lead to nontrivial physical phenomena due to
its combination with other quantum nature of materials, for example, giant
optoelectronic responses enhanced by the quantum-geometrical effect of
topological electrons, nonreciprocity in the electromagnetic responses in
superconductors, and so on. Despite the rapid growth in theoretical
understandings of magnetic parity violation and the ubiquity of
$\mathcal{PT}$-symmetric magnetic materials [304] (see also Appendix A), the
material realization and experimental observations of emergent responses
remain elusive. Promissing materials include those having the Dirac electrons
such as quasi-two-dimensional Mn compounds (e.g., even-layer MnBi2Te4 and
EuMnBi2) [336, 227] and massless Dirac systems (e.g., CuMnAs) [276, 344, 151].

The material realization may be feasible in playgrounds other than the bulk
materials. The magnetic parity violation has been realized by metamaterials
where the magnetic islands are arrayed [146] and the ferromagnetic thin films
are patterned [168, 92]. The high tunability of physical responses is
advantageous, and it has been demonstrated, for example, for the photocurrent
generation in the magnetic metamaterials [168]. Furthermore, the magnetic
parity violation may be found in the fictitious fields of the topological
solitons. Recent studies clarified that the emergent magneto-multipolar fields
of the magnetic hopfion give rise to various spin-charge-coupled phenomena
[211, 156].

The systematics based on the preserved space-time symmetry is expected to be
applied to a broad range of physical responses other than those explained in
this review. For instance, the electric and magnetic decomposition is
similarly found in various physical phenomena such as second-harmonic
generation [23], parametric conversion [295, 309] and photo-induced and
current-induced spin current responses [331, 83, 128, 316, 315, 168, 88, 1,
172]. The mechanism for each response has been mainly investigated in the
framework of band electrons’ Hamiltonian, whereas further precise treatments
including disorders and interactions may allow for quantitative estimates and
imply the potential of the responses in probing the quantum nature of matter.
For instance, parity-violating responses may show enhancement due to the
electron correlation effect [213] and collective modes such as phonons [202],
spin fluctuation [327, 182, 105, 102], plasmon [228, 282], and exciton [183,
259, 121].

## Acknowledgement

H.W. expresses a lot of thanks to Naoto Nagaosa, Riki Toshio, and Yoshihiro
Michishita for valuable discussions and comments. This review partly features
our works done with collaborators. We here express our courtesy to them,
particularly to Akito Daido, Atsuo Shitade, Hinako Murayama, Hiroto Tanaka,
Junta Iguchi. Kohei Shinohara, Kousuke Ishida, Liu Yizhou, Masakazu Matsubara,
Michiya Chazono, Yugo Onishi, Yuki Shiomi.

The authors were supported by Grant-in-Aid for Scientific Research on
Innovative Areas “J-Physics” (Grant No. JP15H05884), SPIPITS 2020 of Kyoto
University, and JSPS KAKENHI (Grants No. JP15K05164, No. JP15H05745, No.
JP16H00991, No. JP18H01178, No. JP18H04225, No. JP18H05227, No. JP18J23115,
No. JP20H05159, No. JP21K18145, No. JP21J00453, No. JP22H04933, No.
JP22H01181, No. JP23K17353, No. JP23K13058).

We made use of the useful software vesta for visualization of crystal and
magnetic structures [175].

##  Appendix A Table of magnetic multipolar magnets

Many magnetic materials are characterized by the zero propagation vector
$\bm{q}=\bm{0}$ of order parameter, by which the unit cell does not change at
the magnetic phase transition. Zero propagation vector implies that the
seemingly antiferroic magnetic order does induce uniform fields through the
coupling to the crystal sublattice degree of freedom. From the viewpoint of
symmetry, such a uniform field is classified into the even-parity and odd-
parity magnetic multipolar fields in terms of parity under the space-inversion
($\mathcal{P}$) operation.

Let us consider the point group symmetry of those magnetic multipolar
materials. The even-parity magnetic multipolar order breaks the $\mathcal{T}$
symmetry as well as the $\mathcal{PT}$ symmetry while preserving the
$\mathcal{P}$ symmetry. On the contrary, the odd-parity magnetic multipolar
order, though it similarly breaks the $\mathcal{T}$ symmetry, is
$\mathcal{PT}$-even and $\mathcal{P}$-odd. Then, when we assume the
centrosymmetric symmetry of the paramagnetic state, the point group of the
magnetic state is comprised of the space-inversion operation ($g=I$) but lacks
parity-time-reversal operation ($g=I\theta$) in the case of even-parity
magnetic multipolar systems, while it is comprised of $g=I\theta$ but lacks
$g=I$ for the odd-parity magnetic multipolar systems. Owing to the preserved
$\mathcal{P}$ or $\mathcal{PT}$ symmetry, either even-parity or odd-parity
magnetic multipolar field is allowed.

Letting $\bm{G}$ be the magnetic point group, the coset decomposition is
obtained as

$\bm{G}=\bm{H}_{\text{even}}\cup I\cdot\bm{H}_{\text{even}},$ (100)

for the even-parity magnetic multipolar systems and

$\bm{G}=\bm{H}_{\text{odd}}\cup I\theta\cdot\bm{H}_{\text{odd}},$ (101)

for the odd-parity magnetic multipolar systems. The subgroup $\bm{H}$ of
$\bm{G}$ is convenient to identify the emergent responses induced by each
magnetic multipolar field; e.g. a response unique to the odd-parity magnetic
multipolar systems is subject to symmetry constraints of $\bm{H}_{\text{odd}}$
and thereby may be zero even when it is not forbidden by the $\mathcal{PT}$
symmetry.

In Tables LABEL:Table-even_parity_multipolar, LABEL:Table-
odd_parity_multipolar, we tabulate the pairs of $\bm{G}$ and
$\bm{H}_{\text{even/odd}}$ with some characteristics and candidate materials.
Notably, since we consider the $\mathcal{P}$-symmetric or
$\mathcal{PT}$-symmetric magnetic point groups, the even- and odd-parity
magnetic multipolar fields show up without being admixed with each other. The
even-parity magnetic multipolar symmetry is characterized by whether the
magnetic dipolar field (M) and piezomagnetic effect (PM) are allowed. For
instance, if the magnetic dipole field is active (M: $\checkmark$), the
magnetic materials denoted by the symmetry show the anomalous Hall effect.

Similarly, the odd-parity magnetic multipolar symmetry is characterized by the
magnetic toroidal moment (T), magnetic quadrupole moment (MQ), and
magnetopiezoelectric effect (MPE). If ‘T: $\checkmark$’, the
$\mathcal{PT}$-symmetric magnetic order may be switchable with the electric
current [305] (see also Sec. IV). Magnetic materials with ‘MQ: $\checkmark$’
allow for various odd-parity responses formulated by the rank-2 and
$\mathcal{PT}$-symmetric axial tensor such as the magnetoelectric effect,
photocurrent response under the circularly-polarized lights (see Sec.
III.3.2), and so on. Magnetic materials with ‘MPE: $\checkmark$’ similarly
host odd-parity responses described by the $\mathcal{PT}$-symmetric tensor
sharing the same symmetry as that of the magnetopiezoelectric effect [e.g,
nonreciprocal electric conductivity (Sec. III.3.1) and photocurrent response
under the linearly-polarized and unpolarized lights (Sec. III.3.2)]. Note that
‘T’ is always active if ‘MQ’ is active because the magnetic quadrupole moments
include the magnetic toroidal moment.

Table 4:  Magnetic point groups (MPG) $\bm{G}$ hosting pure even-parity magnetic multipolar fields. Each group is decomposed by its subgroup $\bm{H}_{\text{even}}$ by Eq. (100). Each item is characterized by the magnetic dipole moment (M), piezomagnetic activity (PM), and candidate material. In some cases, candidate materials are missing (N/A) to our knowledge.  |  |  |  |  |
---|---|---|---|---|---
MPG | $\bm{H}_{\text{even}}$ | M | PM | Candidate materials
$m\bar{3}m$ | $432$ |  |  | N/A
$m\bar{3}m^{\prime}$ | $4^{\prime}32^{\prime}$ |  | $\checkmark$ | Cd2Os2O7 |
$m\bar{3}$ | $23$ |  | $\checkmark$ | NiS2 |
$6/mmm$ | $622$ |  | $\checkmark$ | N/A
$6/mm^{\prime}m^{\prime}$ | $62^{\prime}2^{\prime}$ | $\checkmark$ | $\checkmark$ | Mn5Ge3 |
$6^{\prime}/m^{\prime}mm^{\prime}$ | $6^{\prime}22^{\prime}$ |  | $\checkmark$ | CrNb4S8 |
$6/m$ | $6$ | $\checkmark$ | $\checkmark$ | N/A
$6^{\prime}/m^{\prime}$ | $6^{\prime}$ |  | $\checkmark$ | N/A
$\bar{3}m$ | $32$ |  | $\checkmark$ | CoF3 |
$\bar{3}m^{\prime}$ | $32^{\prime}$ | $\checkmark$ | $\checkmark$ | Mn3Ir |
$\bar{3}$ | $3$ | $\checkmark$ | $\checkmark$ | Mn3NiN |
$4/mmm$ | $422$ |  | $\checkmark$ | KMnF3 |
$4/mm^{\prime}m^{\prime}$ | $42^{\prime}2^{\prime}$ | $\checkmark$ | $\checkmark$ | Nd2NiO4 |
$4^{\prime}/mm^{\prime}m$ | $4^{\prime}22^{\prime}$ |  | $\checkmark$ | MnF2 |
$4/m$ | $4$ | $\checkmark$ | $\checkmark$ | MnV2O4 |
$4^{\prime}/m$ | $4^{\prime}$ |  | $\checkmark$ |  |
$mmm$ | $222$ |  | $\checkmark$ | MnTe |
$m^{\prime}m^{\prime}m$ | $2^{\prime}2^{\prime}2$ | $\checkmark$ | $\checkmark$ | Mn3Sn |
$2/m$ | $2$ | $\checkmark$ | $\checkmark$ | MnCO3 |
$2^{\prime}/m^{\prime}$ | $2^{\prime}$ | $\checkmark$ | $\checkmark$ | KMnF4 |
$\bar{1}$ | $1$ | $\checkmark$ | $\checkmark$ | RbMnF4 |
Table 5:  Magnetic point groups $\bm{G}$ hosting pure odd-parity magnetic multipolar fields. Each group is decomposed by its subgroup $\bm{H}_{\text{odd}}$ by Eq. (101). Each item is characterized by the magnetic toroidal moment (T), magnetic quadrupole moment (MQ), magnetopiezoelectric activity (MPE), and candidate material. In some cases, candidate materials are missing to our knowledge.  |  |  |  |  |  |
---|---|---|---|---|---|---
MPG | $\bm{H}_{\text{odd}}$ | T | MQ | MPE | Candidate materials
$m^{\prime}\bar{3}^{\prime}m^{\prime}$ | $432$ |  | $\checkmark$ |  | N/A
$m^{\prime}\bar{3}^{\prime}m$ | $\bar{4}3m$ |  |  | $\checkmark$ | N/A
$m^{\prime}\bar{3}^{\prime}$ | $23$ |  | $\checkmark$ | $\checkmark$ | N/A
$6^{\prime}/mmm^{\prime}$ | $\bar{6}m2$ |  |  | $\checkmark$ | N/A
$6/m^{\prime}mm$ | $6mm$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | N/A
$6/m^{\prime}m^{\prime}m^{\prime}$ | $622$ |  | $\checkmark$ | $\checkmark$ | N/A
$6^{\prime}/m$ | $\bar{6}$ |  |  | $\checkmark$ | U14Au51 |
$6/m^{\prime}$ | $6$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | PbMn2Ni6Te3O18 |
$\bar{3}^{\prime}m^{\prime}$ | $32$ |  | $\checkmark$ | $\checkmark$ | Cr2O3 |
$\bar{3}^{\prime}m$ | $3m$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | Ca2YZr2Fe3O12 |
$\bar{3}^{\prime}$ | $3$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | MnGeO3 |
$4/m^{\prime}m^{\prime}m^{\prime}$ | $422$ |  | $\checkmark$ | $\checkmark$ | GdB4 |
$4/m^{\prime}mm$ | $4mm$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | Co3Al2Si3O12 |
$4^{\prime}/m^{\prime}m^{\prime}m$ | $\bar{4}2m$ |  | $\checkmark$ | $\checkmark$ | BaMn2As2 |
$4/m^{\prime}$ | $4$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | TlFe1.6Se2 |
$4^{\prime}/m^{\prime}$ | $\bar{4}$ |  | $\checkmark$ | $\checkmark$ | KOsO4 |
$m^{\prime}m^{\prime}m^{\prime}$ | $222$ |  | $\checkmark$ | $\checkmark$ | LiMnPO4 |
$mmm^{\prime}$ | $mm2$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | CuMnAs |
$2/m^{\prime}$ | $2$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | Na2RuO4 |
$2^{\prime}/m$ | $m$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | SrMn2As2 |
$\bar{1}^{\prime}$ | $1$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | YbMn2Sb2 |

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