Magnetic parity violation and Parity-time-reversal-symmetric magnets

Two types of parity violation are clearly distinguished by the space-time symmetry. One may take account of $\mathcal{P}$- and $\mathcal{T}$-parities to characterize the space-time symmetry. We, however, stress that the combination of $\mathcal{P}$ and $\mathcal{T}$ operations plays a fundamental role as well as $\mathcal{P}$ and $\mathcal{T}$ symmetries. To this end, we raise examples of symmetry breaking arising from the antiferromagnetic order.

The antiferromagnetic order is defined by the antiferroic alignment of magnetic moments in crystals. Owing to the $\mathcal{T}$-odd nature of magnetic moments, the antiferromagnetic order is accompanied by the $\mathcal{T}$-symmetry violation. The symmetry related to the $\mathcal{T}$ operation, however, may be retained; e.g., for the $\bm{q}=(\pi,\pi)$ antiferromagnetic order in the square lattice, there exists $\mathcal{T}$ symmetry coupled to the lattice translation $\bm{\tau}=(a,0),(0,a)$ [Fig. 1(a)]. The preserved symmetry is equivalent to the $\mathcal{T}$ symmetry in terms of the bulk physical properties because the microscopic lattice translation $\bm{\tau}$ does not lead to any symmetry constraint. Such an effective $\mathcal{T}$ symmetry makes it difficult to detect the antiferromagnetic state without microscopic spectroscopy such as neutron diffraction.

The antiferromagnetic order gives rise to various symmetry breaking due to the crystal structure in contrast to the case of checkerboard antiferromagnetic order depicted in Fig. 1(a). Here, we consider the zigzag chain to corroborate the space-time symmetry arising from the magnetic parity violation. The zigzag chain is comprised of two sites (A and B sites) inside the unit cell Yanase (2014) [Fig. 1(b)]. The system holds the $\mathcal{P}$ symmetry due to interchanging the sublattices, while each site is not an inversion center. Let us introduce the antiferromagnetic order with the zero propagation vector ($\bm{q}=\bm{0}$) [Fig. 1(c)]. The ordered state is described by using the magnetic moments localized at sites as

$$\left(\bm{m}_{\text{A}},\bm{m}_{\text{B}}\right)=(+\hat{x},-\hat{x}).$$

(1)

Intriguingly, the antiferromagnetic order breaks both of the $\mathcal{P}$ and $\mathcal{T}$ symmetries in the macroscopic scale. It is readily checked by applying the operations to the magnetic configuration given in Eq. (1). The $\mathcal{P}$ operation interchanging the two sites does not flip the localized magnetic moments due to the axial symmetry of magnetic moments. Then, the magnetic moments at each site are transformed as

$$\left(\bm{m}_{\text{A}},\bm{m}_{\text{B}}\right)=(+\hat{x},-\hat{x})\xrightarrow{\mathcal{P}}(-\hat{x},+\hat{x}).$$

(2)

Similarly, the $\mathcal{T}$ operation reverses the magnetic moments with keeping structural degrees of freedom invariant as

$$\left(\bm{m}_{\text{A}},\bm{m}_{\text{B}}\right)=(+\hat{x},-\hat{x})\xrightarrow{\mathcal{T}}(-\hat{x},+\hat{x}).$$

(3)

As a result, the antiferromagnetic state shows the odd parity under the $\mathcal{P}$ and $\mathcal{T}$ operation, which is a magnetic parity violation. The symmetry breaking is not compensated by any symmetry operation such as a lattice translation in sharp contrast to the antiferromagnetic order shown in Fig. 1(a).

One can notice the symmetry unique to the antiferromagnetic zigzag chain by considering Eqs. (2) and (3). The antiferromagnetic state returns to the original configuration under the combined operation as

$$(+\hat{x},-\hat{x})\xrightarrow{\mathcal{P}}(-\hat{x},+\hat{x})\xrightarrow{\mathcal{T}}(+\hat{x},-\hat{x}),$$

(4)

which indicates the $\mathcal{PT}$ symmetry. In the light of the $\mathcal{PT}$-even nature, the parity-violating magnets may be called $\mathcal{PT}$-symmetric magnets. The $\mathcal{PT}$-symmetric magnets are comprised of intriguing materials such as magnetoelectric materials Fiebig (2005) and electrically-switchable antiferromagnets Wadley et al. (2016).

The intact $\mathcal{PT}$ symmetry is a fundamental property distinguishing the $\mathcal{PT}$-symmetric magnets from systems showing the electric parity violation. For an example of electric parity violation, the macroscopic electric polarization $\bm{P}$ is flipped under the $\mathcal{PT}$ operation while it is invariant under the $\mathcal{T}$ operation ($\mathcal{PT}$-odd, $\mathcal{T}$-even). The electric polarization $\bm{P}$ is therefore absent in the $\mathcal{PT}$-symmetric systems. As a result, parity-violating systems are divided into three classes; the $\mathcal{T}$-symmetric case with only the electric parity violation, $\mathcal{PT}$-symmetric case with only the magnetic parity violation, and the otherwise manifesting both electric and magnetic parity violations. The third class contains an important series of the multiferroic magnets such as $R$MnO₃ ($R$: rare-earth element) Kimura et al. (2003); Tokura et al. (2014), which is not in the scope of this review. As a result, we have obtained the classification of parity violations based on the parity under $\mathcal{P}$, $\mathcal{T}$, and $\mathcal{PT}$ operations.

Adding the case of the ferromagnets, we tabulate the space-time symmetry of ordered states in Fig. 2. Owing to different space-time symmetry, the order is not admixed with each other unless the preserved symmetry is lost.

Keys to the magnetic parity violation are antiferromagnetism and the crystal structure showing the local parity violation at atomic sites as depicted in Fig. 1(b). The $\mathcal{PT}$-symmetric magnets can be found in a broad range of materials as tabulated in Refs. Schmid (1973); Siratori et al. (1992); Gallego et al. (2016); Watanabe and Yanase (2018a). The structural feature, called locally-noncentrosymmetric property, is ascribed to various structural degrees of freedom other than atomic sites; layers, clusters of atoms, chains, and so on (see Sec. II.2). Furthermore, the magnetic parity violation can be realized without the locally-noncentrosymmetric structure by unconventional order such as the loop-current order Zhao et al. (2015); Seyler et al. (2020); Murayama et al. (2021); Watanabe and Yanase (2021b) and exotic superconductivity Wang and Fu (2017); Kanasugi and Yanase (2022); Kitamura et al. (2023). Interestingly, the magnetic parity violation can be implemented by micro-fabrications as demonstrated in Refs. Lehmann et al. (2019, 2020).

The locally-noncentrosymmetric property can be found in various sectors in crystal structure Fischer et al. (2023). The series of locally-noncentrosymmetric crystals consists of the subsector degree of freedom such as atomic site, cluster, chain, and so on (Fig. 3). We here introduce a key ingredient to understand the itinerant property unique to locally-noncentrosymmetric crystals, that is hidden magnetic degrees of freedom such as spin and Berry curvature.

Firstly, we explain spin-charge coupling hidden by the inversion partners in crystals. The concept has been introduced to predict the $\mathcal{T}$-symmetric counterpart of Chern insulator proposed in Ref. Haldane (1988), that is quantum spin Hall insulator Kane and Mele (2005); Fu et al. (2007). Independently, the concept has been proposed in the contexts of superconductivity Yanase (2010); Fischer et al. (2011), spintronic applications Železný et al. (2014), and the first-principles study of spin-momentum coupling Zhang et al. (2014). A prototypical example is the bilayer two-dimensional electron gas which may be found in double quantum wells and layered materials such as the cuprate Liu et al. (2013) [Fig. 3(c)]. Although the inversion center is present between the layers, the $\mathcal{P}$ symmetry does not hold if each layer is taken to be the origin. The locally-noncentrosymmetric symmetry indicates that each layer is under unidirectional potential arising from the other layer. Owing to the global $\mathcal{P}$ parity, the polarity of the unidirectional field should be opposite between the two layers.

The structural property is built into the peculiar spin-orbit interaction as follows. The two-dimensional electron gas manifests the so-called Rashba spin-orbit coupling in the presence of the polar field as written by the one-body Hamiltonian

$$h_{\text{R}}=\alpha_{\text{R}}\left({\bm{k}}\times\bm{\sigma}\right)_{z}=\alpha_{\text{R}}\left(k_{x}\sigma_{y}-k_{y}\sigma_{y}\right),$$

(5)

where the polar field is along the $z$ direction, and the electron’s momentum ${\bm{k}}$ and spin $\bm{\sigma}$. The Rashba spin-orbit coupling gives rise to the vortex-like spin configuration in the momentum space as confirmed in spin-resolved spectroscopy of bulk materials such as BiTeI Ishizaka et al. (2011); Landolt et al. (2012). In the case of the bilayer system, the Rashba spin-orbit coupling has the opposite signs as

$$\alpha_{\text{R}}\left({\bm{k}}\times\bm{\sigma}\right)_{z},$$

(6)

for the upper layer and

$$-\alpha_{\text{R}}\left({\bm{k}}\times\bm{\sigma}\right)_{z},$$

(7)

for the lower layer. It indicates that spins are closely coupled to the layer degree of freedom as well as the momentum and that the spin-momentum locking is completely compensated.

To elucidate the electronic structure, let us write down a Hamiltonian of the bilayer system in the field-quantization representation as

$$H=\sum_{{\bm{k}}}\left({c_{{\bm{k}}+}}^{\dagger},{c_{{\bm{k}}-}}^{\dagger}\right)\begin{pmatrix}h_{{\bm{k}}+}&t\\ t^{\ast}&h_{{\bm{k}}-}\end{pmatrix}\begin{pmatrix}c_{{\bm{k}}+}\\ c_{{\bm{k}}-}\end{pmatrix}.$$

(8)

The creation (annihilation) operators ${c_{{\bm{k}}\rho_{z}}}^{\dagger}$ ($c_{{\bm{k}},\rho_{z}}$) are for the upper ($\rho_{z}=+$) and lower ($\rho_{z}=-$) layers and implicitly include the spin degree of freedom written by the Pauli matrices $\bm{\sigma}$. A constituent Hamiltonian reads as

$$h_{{\bm{k}}\rho_{z}}=\frac{{\bm{k}}^{2}}{2m}+\rho_{z}\,\alpha_{\text{R}}\left(k_{x}\sigma_{y}-k_{y}\sigma_{y}\right),$$

(9)

for the diagonal components and the tunneling parameter $t$ for the off-diagonal components. When the constant tunneling parameter $t$ varies, the energy spectrum of Hamiltonian Eq. (8) changes as schematically depicted in Fig. 4(a,b,c). It is noteworthy that in the case of no tunneling ($t=0$) one can observe the energy spectrum similar to that with the Rashba spin-orbit coupling nevertheless each spectrum shows double degeneracy due to the $\mathcal{PT}$ symmetry [Fig. 4(a)]. This is because the opposite spin-momentum coupling in two layers leads to the vanishing spin polarization at each momentum as

$$\sum_{\alpha}\Braket{\phi_{{\bm{k}}p\alpha}}{\bm{\sigma}}{\phi_{{\bm{k}}p\alpha}}=\bm{0},$$

(10)

where $\ket{\phi_{{\bm{k}}p\alpha}}$ is the eigenstate for energy $\varepsilon_{{\bm{k}}p}$ of Eq. (8) denoted by the momentum ${\bm{k}}$, level index $p$, and Kramers degree of freedom $\alpha$. On the other hand, there exists the spin-momentum locking in a staggered manner as given by

$$\sum_{\alpha}\Braket{\phi_{{\bm{k}}p\alpha}}{\rho_{z}\,\bm{\sigma}}{\phi_{{\bm{k}}p\alpha}}\neq\bm{0}.$$

(11)

The obtained subsector-dependent spin-momentum coupling is called hidden spin polarization Zhang et al. (2014). If the hopping $t$ surpasses the spin-orbit coupling quantified by $\alpha_{\text{R}}$ [Fig. 4 (b,c)], the entanglement of wavefunction localized at each layer weakens the hidden spin-polarization Maruyama et al. (2012).

The concept of hidden spin polarization is generalized to other locally-noncentrosymmetric crystals Kane and Mele (2005); Fu et al. (2007); Yanase (2014); Železný et al. (2014). For instance, given that local inversion asymmetry comes from the sublattice degree of freedom as in the zigzag chain and the honeycomb net [Fig. 3(b)], the hidden spin polarization is given by the sublattice-dependent spin-orbit coupling

$$h_{\text{SOC}}=\sum_{{\bm{k}}}\bm{g}_{{\bm{k}}}\cdot\bm{\sigma}\,\rho_{z}.$$

(12)

The basis for Pauli matrices $\ket{\rho_{z}=\pm}$ is spanned by the two sublattice degrees of freedom, and thus $\rho_{z}$ indicates that the antisymmetric spin-orbit coupling ($\bm{g}_{-{\bm{k}}}=-\bm{g}_{{\bm{k}}}$) appears in a staggered manner for electrons localized at each sublattice. The expression for the vector $\bm{g}_{\bm{k}}$ is determined by the local site symmetry of the sublattice Fischer et al. (2023); Guan et al. (2022) such as

$$\bm{g}_{\bm{k}}=\alpha\sin{k_{z}}\hat{x},$$

(13)

with the coupling constant $\alpha$ given for the next-nearest neighbor hopping path of the zigzag chain [Fig. 3(a)], and

$$\bm{g}_{\bm{k}}=\alpha\sin{\frac{k_{y}}{2}}\left(\cos{\frac{k_{y}}{2}}-\cos{\frac{\sqrt{3}}{2}k_{x}}\right)\hat{z},$$

(14)

for that of the honeycomb lattice Kane and Mele (2005); Hayami et al. (2014b) [Fig. 3(b)]. The coupling is similarly obtained in more complex cases such as locally-noncentrosymmetric crystals consisting of $n(>2)$ sublattice like Cr₂O₃ Sumita et al. (2017); Daido et al. (2019); Hayami et al. (2014b); Niu et al. (2017).

After the discovery of the hidden spin polarization, many studies have been devoted to utilizing and maximizing this peculiar charge-spin coupling mostly for the case of local asymmetry of atomic sites. For instance, since the hidden spin-orbit coupling gives a significant modification of the spin susceptibility, the superconducting state acquires more robustness to the paramagnetic pair breaking Maruyama et al. (2012) as reviewed in Refs. Sigrist et al. (2014); Fischer et al. (2023). In addition to the enriched property of nonmagnetic superconductors, the hidden spin-momentum coupling leads to various physical phenomena if coupled to the antiferroic order (see the following sections).

The hidden spin-charge coupling has been quantitatively evaluated for various materials. The model studies clarified important ingredients for the large sublattice-dependent spin-orbit coupling, that is the atomic spin-orbit coupling and odd-parity hopping allowed by the local parity violation Fischer et al. (2011); Yanase (2014); Hayami et al. (2014b, a). It follows that the hidden spin polarization is significant for the bands consisting of orbitals at heavy atoms Liu et al. (2013); Yao et al. (2017); Goh et al. (2012). On one hand, as implied in Fig. 4(a-c), the inter-sublattice hopping may smear out the hidden spin-charge coupling. In this regard, large hidden spin polarization may occur due to negligible inter-sublattice hopping that can be realized in the layered materials such as 2H-stacking transition metal dichalcogenides (e.g., WSe₂) Gong et al. (2013); Riley et al. (2014); Jones et al. (2014); Gehlmann et al. (2016); Bertoni et al. (2016); Razzoli et al. (2017); Devarakonda et al. (2020); Tu et al. (2020) and layered superconductor Liu et al. (2013, 2015); Nakamura and Yanase (2017); Wu et al. (2017); Gotlieb et al. (2018). The layered materials are promising platforms for manipulating the spin-orbit coupling due to the high controllability offered by gating fields and the epitaxial-growth method. Gong et al. (2013); Goh et al. (2012); Shimozawa et al. (2016). Interestingly, nonsymmorphic space group symmetry may realize the segregation between sublattice degrees of freedom in the high-symmetry subspace of the Brillouin zone Young and Kane (2015) and therefore leads to enhanced hidden spin polarization. The hidden spin polarization protected by the nonsymmorphic symmetry has been pointed out by theories Yanase (2016); Sławińska et al. (2016); Yuan et al. (2019) and demonstrated in experiments Santos-Cottin et al. (2016); Zhang et al. (2021). In the two-sublattice Hamiltonian Eq. (8), the inter-sublattice decoupling is represented by vanishing hopping $t$ at high-symmetry momentum. Such decoupling can also be protected by symmorphic symmetries, depending on the symmetry of the local orbitals Akashi et al. (2017); Nakamura and Yanase (2017).

Since the electronic bands of layered materials are described in a spin- and layer-resolved manner, the spin-resolved ARPES study may allow us to measure the hidden spin polarization if there is no complete compensation in the spin polarization between the photoelectrons emitted by the scattering at each layer Zhang et al. (2014); Riley et al. (2014). For example, the intimate coupling between the spin, valley, and layer degrees of freedom in van der Waals materials has been reported by Ref. Razzoli et al. (2017) (Fig. 5). Such an interplay between various degrees of freedom has been investigated by photoluminescence measurements as well Wu et al. (2013); Zhu et al. (2014); Jones et al. (2014).

In the preceding discussions, we considered the electrical activity of spins hidden by the locally noncentrosymmetric structure. It is noteworthy that other $\mathcal{T}$-odd quantities are coupled to the momentum in a similar manner to spins Lin et al. (2020). For instance, the celebrated Kane-Mele model shows the hidden Berry curvature offering the quantum spin Hall effect Kane and Mele (2005). The hidden magnetic properties have been explored in terms of the orbital magnetic moment and Berry curvature Go et al. (2018); Cho et al. (2018); Beaulieu et al. (2020).

The locally-noncentrosymmetric symmetry can be incorporated into the model studies by using the hidden spin-orbit coupling introduced in Sec. II.2. This peculiar spin-charge coupling seemingly may not dramatically affect the physical property due to the fact that the symmetry-breaking effect is compensated between subsectors. On the other hand, once the antiferroic ordering such as anti-ferroelectricity and antiferromagnetism occurs, staggered order parameters may give rise to macroscopic symmetry breaking. As a result, the emergent responses arise from the coupling between the hidden spin-charge coupling and ‘antiferroic’ order. It is convenient for clarifying the macroscopic physical properties to introduce the multipolar degrees of freedom Watanabe and Yanase (2018a); Hayami et al. (2018); Winkler and Zülicke (2023); Watanabe and Yanase (2017).

Let us again consider the antiferromagnetic state of the zigzag chain where magnetic moments are ordered along the $y$ direction as an example. Note that the Néel vector is taken to be along the $y$ axis different from that depicted in Fig. 1(c). As implied by the form of spin-orbit coupling in Eq. (13), each sublattice is under the polar field along the $x$ direction in a staggered manner. By taking the cross product of the local polar fields $\bm{P}$ and magnetic moments $\bm{m}$, the products are the same between the sublattices A and B

$$\bm{P}_{\text{A}}\times\bm{m}_{\text{A}}=\bm{P}_{\text{B}}\times\bm{m}_{\text{B}}\parallel\hat{z},$$

(15)

which are termed with an atomic toroidal moment [Fig. 6(a)]. Thus, the antiferromagnetic state is translated into the ferroic arrangement of atomic toroidal moments, which is in agreement with the zero propagation vector indicating the uniformly manifesting physical quantity. As depicted in Fig. 6(b,c), the atomic toroidal moments uniformly align along the $z$ direction, and its direction is opposite between the two antiferromagnetic states. Given that the toroidal moment is the polar vector with the $\mathcal{T}$-odd and $\mathcal{PT}$-even parities, one can see that the obtained ferro-toroidal state shows the magnetic parity violation. Similarly, if the zigzag chain undergoes a $\bm{q}=\bm{0}$ antiferroic ordering of nonmagnetic even-parity objects such as commensurate charge density wave and orbital order, the order results in the ferroic alignment of odd-parity and $\mathcal{T}$-preserving quantities such as electric polarization. It indicates that the ordered state shows the electric parity violation Hitomi and Yanase (2014, 2016).

These examples imply that the antiferroic order in the locally noncentrosymmetric crystals is a promising playground for the spontaneous parity violation in solids and that the ordered state may be classified in terms of the electric or magnetic anisotropy showing the $\mathcal{P}$-odd parity, that is the odd-parity electric or magnetic multipolar field Kusunose (2008); Kuramoto et al. (2009). Similar arguments are obtained for the even-parity magnetic multipolar order without uniform magnetization such as collinear magnetic order in a rutile-type magnet, MnF₂, and noncollinear magnetic order in Mn₃X ($X=$ Ga, Ge, Sn, Ir, and Pt) Nakatsuji et al. (2015); Nayak et al. (2016); Šmejkal et al. (2020). In those cases, the seemingly antiferroic order is characterized by $\bm{q}=\bm{0}$ and classified into an even-parity multipolar phase, whereas the local parity violation is not required because the ferroic even-parity multipolar order does not break the $\mathcal{P}$ symmetry. It is noteworthy that the importance of the magnetic octupole moment has been confirmed in Mn₃Sn and Mn₃Ge Suzuki et al. (2017); Nomoto and Arita (2020); Kimata et al. (2021); Higo et al. (2022); Go et al. (2022); Yoon et al. (2023).

Physical consequences of the odd-parity multipolar order can be derived by group-theoretical tools once the symmetry of the phase is identified. Let us raise some examples of odd-parity multipoles and associated physical properties from the viewpoint of the representation analysis below. Notably, the odd-parity multipolar symmetry can emerge not only due to the antiferroic order in the locally-noncentrosymmetric systems but also by other exotic quantum phases such as the loop-current order Zhao et al. (2015); Murayama et al. (2021). We emphasize that the classification result obtained by group theory can be applied to any system showing electric and magnetic parity violations.

The uniformly emerging order parameter characterizing the (second-order) phase transition is classified by irreducible representations of a given crystallographic point group. For instance, when the ordered phase belongs to the $B_{1u}$ representation of the tetragonal point group $4/mmm$ ($D_{4h}$), odd-parity multipole moments appear in the ferroic manner. When the $B_{1u}$-type phase transition is attributed to multipolar degrees of freedom, candidates for the order parameter are the electric octupole moment

$$Q_{31}^{+}=xyz,$$

(16)

for the electric parity violation and the magnetic quadrupole moment

$$M_{22}^{+}=xm_{x}-ym_{y},$$

(17)

for the magnetic parity violation Watanabe and Yanase (2017). These multipole moments indicate the lowest order of electric or magnetic anisotropy in real space. Since these multipole moments show the opposite parity under the $\mathcal{T}$ or $\mathcal{PT}$ operation, the corresponding irreducible representation should be labeled by the $\mathcal{T}$-parity such as $B_{1u}^{+}$ for $Q_{31}^{+}$ and $B_{1u}^{-}$ for $M_{22}^{+}$; i.e., the odd-parity irreducible representation $\Gamma_{u}$ is $\mathcal{T}$-even and $\mathcal{PT}$-odd for $\Gamma_{u}^{+}$, or $\mathcal{T}$-odd and $\mathcal{PT}$-even for $\Gamma_{u}^{-}$ (subscript ‘u’ denotes the odd $\mathcal{P}$ parity). ¹¹1 We implicitly assume that the para phase is $\mathcal{T}$-symmetric and $\mathcal{P}$-symmetric to characterize the multipolar order by the definite parity of each operation. Generalized representation analysis is similarly obtained by making use of the magnetic point group Erb and Hlinka (2020).

As in the classification of order parameters in terms of the real-space basis [Eqs. (16) and (17)], the relevant basis functions in the momentum space (${\bm{k}}$) are identified in the group-theoretical manner Sigrist and Ueda (1991). Referring to the classification presented in Refs. Watanabe and Yanase (2017, 2018a); Hayami et al. (2018), we can construct a basis function formed by the momentum ${\bm{k}}$ and magnetic moment $\bm{m}$ as

$$k_{x}m_{x}-k_{y}m_{y},$$

(18)

for the $B_{1u}^{+}$ representaion and

$$k_{x}k_{y}k_{z},$$

(19)

for the $B_{1u}^{-}$ representation. One may notice that the momentum-space basis for the electric parity violation of Eq. (18) is obtained by replacing ${\bm{r}}$ with ${\bm{k}}$ in the real-space basis for the magnetic parity violation [Eq. (17)] and vice versa. The cross-correlation is a consequence of the intact symmetry, that is $\mathcal{T}$ or $\mathcal{PT}$ symmetry.

The correspondence between the real- and momentum-space basis functions conveniently gives clues to understanding the itinerant property of the parity-violating materials such as changes in the electronic structure and emergent responses. Leaving discussions of the physical responses to the following sections, let us consider the band structure undergoing the $B_{1u}$-type odd-parity multipolar order for an example. In case of the electric parity violation, the odd-parity electric multipolar anisotropy of Eq. (16) corresponds to spontaneous emergence of the spin-momentum locking of Eq. (18) whose form is similar to that of the Rashba spin-orbit coupling of Eq. (5). Contrastingly, in the case of the magnetic parity violation, the magnetic multipolar anisotropy of Eq. (17) gives rise to the spin-independent modulation of electronic bands of Eq. (19). The asymmetric energy spectrum has been observed in experiments of a $\mathcal{PT}$-symmetric magnet Fedchenko et al. (2022); Lytvynenko et al. (2023). These characteristic changes in the band structure are strictly forbidden if the $\mathcal{T}$ or $\mathcal{PT}$ symmetry is present; the asymmetric modulation such as Eq. (19) is forbidden due to the $\mathcal{T}$ symmetry, while the spin-momentum locking as in Eq. (18) is absent in total if the $\mathcal{PT}$ symmetry is retained. The modified band structures arising from electric and magnetic parity violation are summarized in Fig. 7.

The microscopic grounds for the multipole moments in solids remain to be completed, though the macroscopic aspects have been addressed from the viewpoint of point group symmetry. The origin of the multipole moments may be attributed to the atomic orbitals well-localized at rare-earth atoms in some f-electron systems Kuramoto et al. (2009). The quantitative estimates of the multipole moments have been explored by first-principles calculations Cricchio et al. (2010); Bultmark et al. (2009); Suzuki et al. (2018); e.g., the close relationship between the magneto-electric coupling and the odd-parity magnetic multipole moments has been pointed out Spaldin et al. (2013); Thöle et al. (2016, 2020).

Furthermore, recent theoretical studies revisited the connection between the observable physical property and multipolar degrees of freedom in the context of generalized free energy. For instance, the thermodynamic magnetic quadrupole moments are defined to be conjugate to the gradient of magnetic fields, and they are directly related to the magnetoelectric property Gao et al. (2018); Shitade et al. (2018); Bhowal and Spaldin (2022). It is noteworthy that the thermodynamic multipole moments contribute to other cross-correlated responses such as the nonlinear thermoelectric response Gao and Xiao (2018), temperature-gradient-induced magnetization (gravito-magnetoelectric effect) Shitade et al. (2019); Shinada and Peters (2023), and spin accumulations Shitade and Tatara (2022). In the same spirit, the electric quadrupole moments are defined and estimated in the language of thermodynamics and quantum geometry Daido et al. (2020); Kitamura et al. (2021). It is desirable to perform further explorations for higher-order multipolar degrees of freedom in solids Tahir and Chen (2023).

Finally, we comment on material realizations of the $\mathcal{PT}$-symmetric (odd-parity magnetic multipolar) magnets. Although the uniform odd-parity magnetic multipolar fields show up in the presence of special crystal and antiferromagnetic structures, one can notice that they exist in a broad range of materials as we tabulate candidates in Appendix A. Historically, enormous efforts have been made to explore materials undergoing the $\mathcal{PT}$-symmetric magnetic order, with special attention to the oxides motivated by the interests in the magnetoelectric coupling in magnetic insulators such as Cr₂O₃ and LiCoPO₄ Fiebig (2005). On the other hand, as mentioned in the introductory part, recent studies shed light on magnetic metals as well with the developments in the field of antiferromagnetic spintronics Jungwirth et al. (2016); Baltz et al. (2018). The candidates cover a diverse class of materials such as ferro-pnictide (e.g., BaMn₂As₂, EuMnBi₂, CuMnAs, YbMnBi₂) Borisenko et al. (2019); Wadley et al. (2015); Sakai (2022); Tanida et al. (2022) and rare-earth-based magnetic conductors Saito et al. (2018); Arakawa et al. (2023); Ota et al. (2022). It is expected that the magnetic parity violation may show intriguing interplays with the physical properties unique to the metals like topological electrons nearby the Fermi energy Tang et al. (2016); Šmejkal et al. (2018), giant magnetoresistance Aoyama et al. (2022); Sun et al. (2021); Ogasawara et al. (2021), and what is discussed in the following sections (e.g., magnetopiezoelectric effect and nonreciprocal electrical transport induced by the magnetic parity violation).

The contrasting space-time symmetry of the odd-parity electric/magnetic multipolar materials is reflected in the response as well as the electronic structure described in Sec. II.3. In this section, taking the linear response function, we introduce the $\mathcal{T}$-odd/$\mathcal{T}$-even classification of physical responses and argue that the $\mathcal{T}$ and $\mathcal{PT}$ symmetries play complementary roles.

The response formula is generally given by

$$X_{i}=\chi_{ij}^{XY}F_{j}^{Y},$$

(20)

where the physical quantity $\bm{X}$ responds to the external field $\bm{F}^{Y}$ conjugate to a physical quantity $\bm{Y}$ by the perturbed Hamiltonian $H_{\text{ex}}=\bm{Y}\cdot\bm{F}^{Y}$. Based on the Kubo formula, we derive the ac response function $\chi_{ij}^{XY}(\omega)$ in the Lehmann representation

$$\chi_{ij}^{XY}(\omega)=\sum_{a,b}\frac{\rho_{a}-\rho_{b}}{\omega+i\eta+\epsilon_{a}-\epsilon_{b}}X_{ab}^{i}Y_{ba}^{j},$$

(21)

with the summation over the many-body eigenstates $(a,b)$ for the unperturbed Hamiltonian. We introduced the eigenenergy $\epsilon_{a}$, Boltzmann factor $\rho_{a}=\exp{(-\epsilon_{a}/T)}/Z$ ($Z$: partition function), and infinitesimal positive parameter $\eta=+0$. Physical quantities are given by the matrix element for the eigenstates $X_{ab}^{i}=\Braket{a}{X_{i}}{b}$. The response function is classified by the parity $\tau_{g}$ under a symmetry operation $g$ such as $\mathcal{P}$, $\mathcal{T}$, and $\mathcal{PT}$ operations. In the case of magnetization response to the electric field [$(\bm{X},\bm{Y})=(\bm{M},\bm{E})$] known as the magnetoelectric effect, the response function $\hat{\chi}^{ME}$ has odd- ($\tau_{I}=-1$), odd- ($\tau_{\theta}=-1$), and even-parity ($\tau_{I\theta}=+1$) under the $\mathcal{P}$, $\mathcal{T}$, and $\mathcal{PT}$ operations, respectively. Note that the parity satisfies the relation $\tau_{I\theta}=\tau_{I}\cdot\tau_{\theta}$.

The response function is further divided into the symmetric and antisymmetric parts under the permutation of $X_{i}$ and $Y_{j}$ as

$$\chi_{ij}^{XY}=\chi_{ij}^{XY;\text{s}}+\chi_{ij}^{XY;\text{a}}.$$

(22)

The components are respectively given by

$$\chi_{ij}^{XY;\text{a}}(\omega)=\sum_{a,b}\left(\rho_{a}-\rho_{b}\right)\frac{\omega+i\eta}{\left(\omega+i\eta\right)^{2}-\left(\epsilon_{a}-\epsilon_{b}\right)^{2}}X_{ab}^{i}Y_{ba}^{j},$$

(23)

and

$$\chi_{ij}^{XY;\text{s}}(\omega)=\sum_{a,b}\left(\rho_{a}-\rho_{b}\right)\frac{\epsilon_{b}-\epsilon_{a}}{\left(\omega+i\eta\right)^{2}-\left(\epsilon_{a}-\epsilon_{b}\right)^{2}}X_{ab}^{i}Y_{ba}^{j}.$$

(24)

We may gain an intuitive understanding of the symmetric and antisymmetric terms by considering the dc limit ($\omega\to 0$). By replacing $\eta$ with the phenomenological scattering effect $\gamma>0$, each contribution is given as

$$\hat{\chi}^{XY;\text{a}}\sim\frac{\gamma}{\gamma^{2}+\left(\epsilon_{a}-\epsilon_{b}\right)^{2}},~{}~{}~{}\hat{\chi}^{XY;\text{s}}\sim\frac{\epsilon_{a}-\epsilon_{b}}{\gamma^{2}+\left(\epsilon_{a}-\epsilon_{b}\right)^{2}}.$$

(25)

The antisymmetric contribution explicitly depends on the sign of $\gamma$ and is proportional to $\gamma^{-1}$ for the equi-energy transitions ($\epsilon_{a}=\epsilon_{b}$). As a result, the response is in an intimate relation with the dissipative phenomenon accompanied by energy absorption. On the other hand, the symmetric term does depend on $|\gamma|$ and remains finite even in the limit of $\gamma\to+0$. This property indicates that the symmetric term is generically irrelevant to the scattering process and may occur without dissipation. In the light of these contrasting aspects with respect to dissipation, the antisymmetric and symmetric terms are also called dissipative and dissipation-less responses, respectively Freimuth et al. (2014); Železný et al. (2017a); Watanabe and Yanase (2017).

Following the symmetry arguments in Ref. Watanabe et al. (2024), we finally obtain the $\mathcal{T}$-parity decomposition of the antisymmetric and symmetric responses in Eqs. (23) and (24). We here introduce the $\mathcal{T}$-even and $\mathcal{T}$-odd contributions which appear with and without $\mathcal{T}$-symmetry, respectively. As displayed in Table 1, the symmetric and antisymmetric parts of a given response function are classified by the $\mathcal{T}$-parity $\tau_{\theta}$; for a response with $\tau_{\theta}=+1$, only the symmetric part is allowed in the $\mathcal{T}$-symmetric system, while the antisymmetric contribution is admixed by the $\mathcal{T}$-symmetry breaking. On the other hand, for the case of $\tau_{\theta}=-1$, the response in the $\mathcal{T}$-symmetric system is solely attributed to the antisymmetric part, but it contains both antisymmetric and symmetric contributions when the $\mathcal{T}$ symmetry is lost. For instance, the dc electric conductivity tensor ($J_{a}=\sigma_{ab}E_{b}$) formally shows the odd-parity under the $\mathcal{T}$ operation, and thus $\tau_{\theta}=-1$. Therefore, only the antisymmetric part is finite in the $\mathcal{T}$-symmetric system, consistent with the fact that the allowed longitudinal response is dissipative as formulated in the Boltzmann’s semiclassical theory of transport phenomena. The symmetric part is relevant to the $\mathcal{T}$-symmetry breaking. This argument agrees with the fact that the electric conductivity tensor hosts the symmetric part such as the Hall conductivity which can generically be free from dissipation. ²²2The symmetric and antisymmetric terms of the electric conductivity do not indicate the tensor symmetry of $\sigma_{ab}$; for instance, the antisymmetric term we introduced is not the Hall response ($\sigma_{ab}=-\sigma_{ba}$) but the dissipative and longitudinal conductivity ($\sigma_{ab}=\sigma_{ba}$). This is because we relate the electric conductivity with the inverse response written by $P_{a}=\chi_{ab}A_{b}$ ($\bm{P}$ is the electric polarization, $\bm{A}$ is the vector potential). The antisymmetric term therefore satisfies $\sigma_{ab}=-\chi_{ba}$. The well-known Onsager relation is reproduced by using the relations $\bm{J}=-i\omega\bm{P}$ and $\bm{E}=i\omega\bm{A}$. Similarly, one can derive the symmetric-antisymmetric decomposition by using the linear response formula in the canonical-correlation representation, by which the Onsager reciprocity is obtained more explicitly Watanabe et al. (2024).

Similarly, the response function with $\tau_{\theta}=+1$ is decomposed into the $\mathcal{T}$-even and $\mathcal{T}$-odd contributions. An example is the nonmagnetic and magnetic spin Hall effects which are of high interest in recent studies on spintronic effects. The spin-polarized current response to the electric fields is given by the formula

$$J_{a}^{s_{c}}=\sigma_{ab}^{c}E_{b},$$

(26)

where $\bm{J}^{s_{c}}$ denotes the current with the spin polarization along the $c$ direction. Specifically, the spin Hall conductivity is defined by the off-diagonal elements $\sigma_{ab}^{c}+\sigma_{ba}^{c}\neq 0$. The $\mathcal{T}$-even component can make contributions through the symmetric part of the tensor [Eq. (24)], consistent with the dissipationless nature of the spin-Hall effect, as demonstrated in intensive studies of the spin Hall effect of nonmagnetic semiconductors Sinova et al. (2015). On the other hand, the $\mathcal{T}$-symmetry violation gives rise to the antisymmetric counterpart [Eq. (23)] of spin Hall response called magnetic spin Hall effect Kimata et al. (2019). Since its dissipative aspect is closely linked with the metallic conductivity, the magnetic spin Hall effect is characteristic of magnetic ($\mathcal{T}$-odd) metals Železný et al. (2017b); Mook et al. (2020).

It is noteworthy that either $\mathcal{T}$-even or $\mathcal{T}$-odd contribution may appear without being concomitant with the other due to the additional symmetry constraint. For example, the magnetoelectric effect is $\mathcal{P}$-odd ($\tau_{I}=-1$) and can occur in parity-violating materials. The electric parity violation allows the system to have the antisymmetric part of $\hat{\chi}^{ME;\text{A}}$ because of the preserved $\mathcal{T}$ symmetry. This response is accompanied by dissipation and is called the inverse magnetogalvanic effect (Edelstein effect). On the other hand, those with the pure magnetic parity violation manifest only the symmetric counterpart not being admixed with the antisymmetric contribution, because the magnetoelectric effect is incompatible with the space-time symmetry of magnetic parity violation due to the $\mathcal{PT}$-odd nature. Consequently, the $\mathcal{T}$- and $\mathcal{PT}$-symmetries play complementary roles in $\mathcal{P}$-odd physical responses due to the $\mathcal{P}$-symmetry constraint. The symmetry argument is applicable to the nonlinear responses as well as the linear response, as we see later in Sec. III.3.

The piezoelectric effect is the interconversion between the stress and electric polarization allowed in noncentrosymmetric materials. The effect has been widely used for applications such as the transducer. The response formula reads as

$$P_{a}=\bar{e}_{abc}s_{bc},~{}~{}~{}\varepsilon_{ab}=e_{abc}E_{c},$$

(27)

with electric polarization $\bm{P}$, stress $\hat{s}$, and strain $\hat{\varepsilon}$. Typically, the piezoelectric-active materials are limited to insulators such as ferroelectric ceramics to hold electric polarization. By taking $\bm{X}=\bm{P}$ and $\bm{F}^{\bm{Y}}=\hat{s}$ in Eq. (20), the time-reversal parity $\tau_{\theta}=+1$ indicates that the $\mathcal{T}$-even part is the symmetric part (see Table 1) and may be attributed to the equilibrium property in the DC limit, being consistent with the piezoelectric effect. On the other hand, the $\mathcal{T}$-parity classification in Table 1 also makes us aware of the possibility of the $\mathcal{T}$-odd and antisymmetric contribution to the electric-elastic coupling. Notably, owing to the $\mathcal{P}$-odd property of the response ($\tau_{I}=-1$), the $\mathcal{T}$-odd contribution is allowed without being admixed with the conventional piezoelectric effect in the $\mathcal{PT}$-symmetric systems.

Let us take the response function given by $\bm{X}=\hat{\varepsilon}$ and $\bm{F}^{Y}=\bm{E}$ and corroborate the DC limit. Owing to the $\mathcal{PT}$-symmetry constraint, the surviving term is purely the antisymmetric part as

$$e^{\text{m}}_{abc}\equiv\left(\hat{\chi}^{\varepsilon P;\text{a}}\right)_{abc}=\sum_{p,q}\left(\rho_{p}-\rho_{q}\right)\frac{-i\gamma}{\gamma^{2}+\left(\epsilon_{p}-\epsilon_{q}\right)^{2}}\varepsilon_{pq}^{ab}P_{qp}^{c}.$$

(28)

We introduced the phenomenological scattering effect parametrized by $\gamma$ similarly to Eq. (25). In the clean limit ($\gamma\to+0$), the response is dominated by the equi-energy transition process ($\epsilon_{a}=\epsilon_{b}$) giving rise to the response as much as $O(\gamma^{-1})$.

In the independent particle approximation, the formula in the clean limit is recast as Watanabe and Yanase (2017)

$$e^{\text{m}}_{abc}=-\frac{e}{\gamma}\int\frac{d{\bm{k}}}{(2\pi)^{d}}\sum_{p}\varepsilon_{pp}^{ab}v_{pp}^{c}\frac{\partial f(\varepsilon)}{\partial\varepsilon}\bigg{|}_{\varepsilon=\varepsilon_{{\bm{k}}p}},$$

(29)

where $e>0$ is the elementary charge of electrons and the summation is over the momentum (${\bm{k}}$) and the band indices ($p$). We introduced the velocity operator $\bm{v}$ for electrons and the Fermi-Dirac distribution function $f(\varepsilon)$. The strain and velocity operators are evaluated by the Bloch states $\ket{\psi_{{\bm{k}}p}}$ having the eigenvalue $\varepsilon_{{\bm{k}}p}$ for the Hamiltonian. The Fermi-surface factor $\partial_{\varepsilon}f(\varepsilon)$ indicates that metallic conductivity is required for this piezoelectric-like response in sharp contrast to the conventional piezoelectric response allowed even in insulators.

This unconventional type of “piezoelectricity” is termed as magnetopiezoelectric effect and was predicted by theories with a semiclassical theory including the quantum-geometrical effect Varjas et al. (2016) and with full-quantum treatment based on the linear response theory Watanabe and Yanase (2017). When the stimulus $E_{z}$ is replaced with the electric current $J_{z}$, the response formula is rewritten by

$$\varepsilon_{ab}=\kappa_{abc}J_{c},$$

(30)

where the response function is obtained as $\kappa_{abc}=\varepsilon_{abc}^{\text{m}}/\sigma_{cc}$ by using the formula for longitudinal conductivity $J_{a}=\sigma_{aa}E_{a}$. The obtained magnetopiezoelectric response function $\kappa_{abc}$ is not sensitive to the phenomenological scattering parameter. This is because the Drude-type conductivity $\sigma_{aa}=\text{O}(\gamma^{-1})$ leads to the scattering-rate dependence of $\kappa_{abc}=e_{abc}^{\text{m}}/\sigma_{cc}=\text{O}(\gamma^{0})$ with Eq. (28). As a result, the response function $\hat{\kappa}$ does depend on the generic material properties as in the case of inverse magneto-galvanic effect Levitov et al. (1985); Edelstein (1990). It indicates the fact that the electric current plays an essential role in this magnetic counterpart of the piezoelectric response rather than the electric field. It follows that the magnetopiezoelectric response occurs under the electric current flow and is inevitably accompanied by the Joule heating. The energy loss may be unfavorable for future applications based on the magnetic metals. The undesirable heating effect may be alleviated by utilizing the superconducting property (see Sec. V). The metallic property could also be an advantage of the magnetopiezoelectric effect for applications as we discuss at the end of this subsection. We summarize the contrasting properties of the known piezoelectric effect and the magnetopiezoelectric effect in Table 2. For the switchability of the $\mathcal{PT}$-symmetric magnetic order and the magnetopiezoelectric effect, one can refer to the discussions in Sec. IV.

The magnetopiezoelectric effect can also be observed in nonmagnetic materials with the electric parity violation by applying the external magnetic field. In this case, not only the $\mathcal{T}$ symmetry but also the $\mathcal{PT}$ symmetry is broken, and the elastic-electric coupling is obtained as the combination of the conventional piezoelectric response and the magnetopiezoelectric response. The admixture may prevent us from specifying the magnetopiezoelectric response Varjas et al. (2016). On the other hand, there is no such concern in the case of the $\mathcal{PT}$-symmetric magnets because the $\mathcal{PT}$ symmetry exactly forbids the conventional piezoelectric response.

In the following, we explain the microscopic grounds for the magnetopiezoelectric effect by taking a $\mathcal{PT}$-symmetric magnetic metal, hole-doped BaMn₂Pn₂ (Pn=As, Sb, Bi) Watanabe and Yanase (2017). BaMn₂As₂, for instance, undergoes the G-type antiferromagnetic order at high Néel temperature $T_{\text{N}}\sim 600\,\mathrm{K}$ which breaks the $\mathcal{P}$ symmetry. The $\mathcal{P}$ violation originates from the coupling between the antiferromagnetic order and the locally noncentrosymmetric environment of Mn sites [Fig. 8(a)]. Considering the site symmetry at Mn atoms, one can derive the antisymmetric spin-orbit coupling as

$$\bm{g}_{\bm{k}}=\left(\alpha_{1}\sin{k_{y}},\alpha_{1}\sin{k_{x}},\alpha_{3}\sin{\frac{k_{x}}{2}}\sin{\frac{k_{y}}{2}}\sin{\frac{k_{z}}{2}}\right),$$

(31)

where $\alpha_{1},\alpha_{3}$ denote the coupling constants of the sublattice-dependent anti-symmetric spin-orbit coupling. The locally noncentrosymmetric Mn atoms show hidden spin polarization described by Eq. (12) with Eq. (31), and it couples to the antiferromagnetic order.

The ordered state manifests the magnetic parity violation. Specifically, the magnetic point group symmetry is

$$\mathbf{G}=4^{\prime}/m^{\prime}m^{\prime}m=\bar{4}2m\cup I\theta\cdot\bar{4}2m,$$

(32)

consisting of the unitary symmetry with the point group $\bar{4}2m$ and of the anti-unitary symmetry including the $\mathcal{PT}$ symmetry ($I\theta$). Although the parent compound BaMn₂As₂ shows insulating behavior with a small gap $\sim 10$ (meV), the metallic conductivity is acquired by doping hole carriers or by applying high pressure. Thus, the material is a promising candidate for $\mathcal{PT}$-symmetric magnetic metals by which we can demonstrate the interplay between the metallic conductivity and magnetic parity violation.

By imposing the unitary symmetry $\bar{4}2m$ in Eq. (32), the allowed components of the response tensor Eq. (28) are

$$e^{\text{m}}_{xyz},~{}~{}e^{\text{m}}_{zxy}=e^{\text{m}}_{yzx}.$$

(33)

We again stress that the preserved $\mathcal{PT}$ symmetry forbids the conventional (inverse) piezoelectric effect of Eq. (27). Taking the independent-particle approximation, the microscopic origin can be inferred from the formula Eq. (29). To grasp the intuitive picture of the magnetopiezoelectric effect in BaMn₂As₂, let us focus on the electronic structure unique to it. The metallic conductivity is ascribed to the hole pocket placed at $\Gamma$ point, which is expected to realize the interplay between the magnetic parity violation and itinerant property. According to the magnetic point group symmetry of Eq. (32), the active odd-parity magnetic multipoles are such as the magnetic quadrupole [Eq. (17)] and magnetic hexadecapole moment Watanabe and Yanase (2018a). The group-theoretical argument introduced in Sec. II.3 allows us to identify the anti-symmetric modulation in the electronic structure given by the tetrahedral modulation of the energy spectrum $\delta\varepsilon_{{\bm{k}}p}\sim k_{x}k_{y}k_{z}$ [Eq. (19)].

The obtained asymmetric electronic band structure plays an essential role in the magnetopiezoelectric effect. When applying the electric field to metals, the Fermi surface is shifted along the applied direction as obtained in the semiclassical theory of transport as

$$\delta f(\varepsilon_{{\bm{k}}p})\sim\gamma^{-1}\bm{E}\cdot\bm{v}_{pp}\partial_{\varepsilon}f(\varepsilon_{{\bm{k}}p})=\gamma^{-1}E_{k}\cdot\partial_{\bm{k}}\varepsilon_{{\bm{k}}p}\partial_{\varepsilon}f(\varepsilon_{{\bm{k}}p}).$$

(34)

The field-induced shift ($\delta f$) is coupled to the asymmetry in the band dispersion ($\delta\varepsilon_{{\bm{k}}p}$) and may induce additional anisotropy in the Fermi surface. In the case of BaMn₂As₂, the coupling between the tetrahedral modulation and the field-induced shift along the $z$-axis results in the nematic anisotropy in the $k_{x}$-$k_{y}$ plane. The resultant nematic anisotropy is absent in equilibrium according to the magnetic symmetry of Eq. (32).

By Letting $\varepsilon_{xy}^{({\bm{k}})}$ be $k_{x}k_{y}$-type nematic anisotropy in the Fermi surface, the response formula reads as

$$\varepsilon_{xy}^{({\bm{k}})}=e_{xyz}^{\text{m}}({\bm{k}})E_{z},$$

(35)

where the electric-elastic coupling is defined with the electronic strain $\varepsilon_{xy}^{({\bm{k}})}\sim k_{x}k_{y}$. The strain is induced by the dissipative electrical stimulus leading to the shift in Eq. (34). Then, the response formula may be better to be rewritten by using the electric current $\bm{J}$ as

$$\varepsilon_{xy}^{({\bm{k}})}=\overline{e}_{xyz}^{\text{m}}({\bm{k}})J_{z},$$

(36)

agreeing with Eq. (30). The current-induced electronic strain $\varepsilon_{xy}^{({\bm{k}})}$ may subsequently induce the lattice strain $\varepsilon_{xy}$ through the electron-phonon coupling.

The theoretical prediction of magnetopiezoelectric responses has been confirmed by a series of experiments Shiomi et al. (2019b, a, 2020). The experiments with the $\mathcal{PT}$-symmetric magnets such as EuMnBi₂ and CaMn₂Bi₂ are consistent with theories. It has been verified that the elastic response is proportional to the applied current density and follows the aforementioned symmetry analysis Shiomi et al. (2020) (Fig. 8). While extensive research has been devoted to maximizing the magnitude of the conventional piezoelectric effect, the magnetopiezoelectric effect has not been fully explored for its material dependence and potential applications. For instance, the magnetopiezoelectric effect varies by the magnetic order and may apply to future elastic devices due to its compatibility with the functionality unique to metals (e.g., giant magnetoresistance Aoyama et al. (2022); Huynh et al. (2019); Ogasawara et al. (2021); Sun et al. (2021)) and nontrivial temperature dependence coming from that of magnetic moments. Notably, the switchable property of $\mathcal{PT}$-symmetric magnetic order (see Sec. IV) may be favorable for that purpose. More explorations by experiments and quantitative estimates by the first-principles calculations are highly desirable.

The $\mathcal{T}$-parity classification introduced in Sec. III.1 is generalized to the nonlinear responses, while the parameter dependence of each contribution gets more complicated than the linear response functions. For an example of the nonlinear response, let us consider the electric current response to double electric fields

$$J_{a}(\omega)=\int\frac{d\omega_{1}d\omega_{2}}{2\pi}\delta(\omega-\omega_{1}-\omega_{2})~{}\sigma_{a;bc}(\omega_{1},\omega_{2})E_{b}(\omega_{1})E_{c}(\omega_{2}).$$

(37)

The generated current is not flipped under the reversal of the electric field. Thus, the response can be termed as the nonreciprocal current generation and is characteristic of parity-violating materials Tokura and Nagaosa (2018). The formula covers various phenomena attracting a lot of interest for a long time such as the nonreciprocal conductivity ($\omega_{1}=\omega_{2}=0$), second-harmonic generation ($\omega_{1}=\omega_{2}$), photocurrent generation ($\omega_{1}=-\omega_{2}$), and parametric generation ($\left|\omega_{1}\right|\neq\left|\omega_{2}\right|$). In the following, we mainly discuss the nonreciprocal conductivity and photocurrent generation from the viewpoint of the $\mathcal{PT}$ symmetry. ³³3 The photocurrent may respond to the irradiating light in materials without the parity violation such as due to the drag effect Plank et al. (2016). In this review, we consider only the nonreciprocal photocurrent response arising from the parity violation.