Nonreciprocal responses from non-centrosymmetric quantum materials — TEI-tagged view

Yoshinori TokurapersName, Naoto NagaosapersName (2018date). “Nonreciprocal responses from non-centrosymmetric quantum materials.” Nature Communicationstitle·j.

Object of study — nonreciprocal responsesterm·dom of noncentrosymmetricterm·key quantum materials, including magnetochiral effects and magnon transport

Research question — how nonreciprocity generalises to the nonlinear regime, which is far less explored than linear response

Contribution — exploring nonreciprocal responsesterm·dom in quantum and topological materials as a direction toward emergent functions

Full text — tagged

Nonreciprocal responsesterm·dom from non-centrosymmetric quantum materials Abstract Directional transport and propagation of quantum particle and current, such as electron, photon, spin, and phonon, are known to occur in the materials system with broken inversion symmetry, as exemplified by the diode in semiconductor p–n junction and the natural optical activity in chiralterm·dom materials. Such a nonreciprocal responseterm·dom in the quantum materials of noncentrosymmetry occurs ubiquitously when the time-reversal symmetry is further broken by applying a magnetic field or with spontaneous magnetization, such as the magnetochiral effect and the nonreciprocal magnon transport or spin current in chiralterm·dom magnets. In the nonlinear regime responding to the square of current and electric field, even a more variety of nonreciprocal phenomena can show up, including the photocurrent of topological origin and the unidirectional magnetoresistance in polar/chiralterm·dom semiconductors. Microscopically, these nonreciprocal responsesterm·dom in the quantum materials are frequently encoded by the quantum Berry phaseterm·key, the toroidal momentterm·dom, and the magnetoelectric monopole, thus cultivating the fertile ground of the functional topological materials. Here, we reviewterm·mth the basic mechanisms and emergent phenomena and functions of the nonreciprocal responsesterm·dom in the noncentrosymmetricterm·key quantum materials.

Introduction Chiralityterm·key, which characterizes right and lift, is an important and fundamental notion in whole sciences, including physics, chemistry, and biology 1 . For example, a molecule is called chiralterm·dom when its mirror image does not overlap with the original molecule. In the language of symmetry, it has no inversion symmetry Î nor mirror symmetry M ^ . The inversion Î can be expressed in terms of the product of the mirror symmetry M ^ and the 180° rotation C 2 around the axis perpendicular to the mirror plane. Therefore, Î and M ^ are identical once C 2 is assumed in free space. (Hereafter, we use Î symmetry representing either Î or M ^ depending on the crystal symmetry.) It often determines the biological functions and is crucial for the pharmaceutical biotechnology. In addition to these issues on static structures, the dynamics of the systems is the main interest in physics. Namely, the directional responses may occur in physical systems with broken Î . In solids, the crystal structure often breaks the Î symmetry, but they do not necessarily guarantee the directional response. To see this, one can look at the one-dimensional scattering problem with the asymmetric potential barrier V ( x )(≠ V (− x )). The transmission and reflection probabilities are identical, irrespective of the left-going or right-going incident wave. Namely, the directional dependence is missing here. It comes from the unitary nature of the scattering ( S )-matrix. Furthermore, the time-reversal symmetry T ^ of the Schroedinger equation imposes the condition S = S T ( S T means the transpose of S ). It is natural that the right mover and left mover are exchanged by both Î and T ^ , and hence these two symmetries play essential roles in the nonreciprocal responsesterm·dom.

Accordingly, as shown in Box 1 , there are four categories of nonreciprocal responsesterm·dom depending on (i) the linear and nonlinear response, and (ii) T ^ -unbroken and T ^ -broken. While the physical mechanisms of the phenomena and effects listed in this table will be discussed below in detail, we stress here that nonreciprocal responsesterm·dom touch the most fundamental issues of condensed matter physics such as symmetries, quantum geometrical nature of electrons, and electron correlation.

Box 1 The classification of nonreciprocal responsesterm·dom in noncentrosymmetricterm·key materials according to the time-reversal symmetry and linear/nonlinear nature of the responses The nonreciprocal responsesterm·dom are classified into four categories, as shown in Table depending on (i) the linear or nonlinear responses, and (ii) with time-reversal symmetry or broken time-reversal symmetry. Note that the broken inversion symmetry Î is always assumed. The nonreciprocal linear response is described by the difference of the response function between q and –q ( q : the wavevector of the electromagnetic field), and is subject to Onsager’s reciprocal theorem Eqs. 1 and 2 in Section “Onsager reciprocal theorem” of the main text. Therefore, the diagonal response, i.e., α = β in Eq. 2 , the time-reversal symmetry T ^ must be broken for the different response between q and –q . An example is the directional magnetochiral effect, where the dielectric function ε ( q , ω , B ) differs from ε (− q , ω , B ) (Section “Directional dichroismterm·dom and birefringence in magnetoelectrics”). The constraint ε( q , ω , B ) = ε (− q , ω ,− B ) from Onsager’s theorem requires that B should be finite for the magnetochiral effect. As for the off-diagonal linear response, the rotation of the polarization of light in the opposite directions between q and – q is possible for noncentrosymmetricterm·key systems even without the broken T ^ , i.e., the natural circular dichroism. Once the T ^ -symmetry is broken, we expect the directional dependence of the propagation of excitations in general. Magnon propagation in noncentrosymmetricterm·key magnets is an example, where the q -linear term in the energy dispersion appears (Section “Nonreciprocal dynamics of magnons”).

On the other hand, nonlinear nonreciprocal responsesterm·dom are characterized by the directional I–V characteristics, typically realized in p–n junction. In this case, the role of the wavevector q is replaced by the current I as first suggested by Rikken 40 , which leads to the resistance R expressed by where the resistance depends on the direction of the current I (Section “Unidirectional magnetoresistance”). Here it is evident that the broken T ^ -symmetry due to B is needed for the directional resistance. Also, various nonreciprocal nonlinear optical responses are realized when both Î - and T ^ -symmetries are broken, e.g., in multiferroicsterm·key (Section “Nonlinear optical effects in multiferroicsterm·key”). However, this argument is a heuristic one, and it should be noted that the breaking of T ^ -symmetry is not needed in the nonreciprocal responseterm·dom in sharp contrast to the linear response. A classic example of the nonreciprocal nonlinear response is the rectification effect of p–n junction, and another example is the shift currentterm·dom in noncentrosymmetricterm·key materials, where the optical excitation creating interband transitions induces the spontaneous dc photocurrent depending on the polarity of the crystal (Section “Shift currentterm·dom in polar insulators”). This effect does not require the T ^ -symmetry breaking, and is related to the Berry phaseterm·key connection of the Bloch wavefunction.

Linear nonreciprocal responsesterm·dom Onsager reciprocal theorem In many-body systems in thermodynamic limit, the dissipative process occurs commonly, which determines the direction of time due to its irreversible nature. However, this is distinct from the time-reversal symmetry T ^ breaking due to the magnetic field B or the spontaneous magnetization M . Onsager recognized the role of T ^ -symmetry in the microscopic dynamics of the system appearing in the macroscopic response functions linear to the external stimuli 2 , 3 . It is formulated by Kubo formula for linear response function K AB ( q , ω , B ) corresponding to the input B and output A with the wavevector q and frequency ω , which satisfies the relation 4 where ε A = ±1, ε B = ±1 specifies the even–odd nature of the quantity A ( B ) for the time-reversal operation. This Onsager’s reciprocal theorem, when applied to the conductivity tensor, gives The magnetochiral effect described in Box 1 corresponds to the formula that σ αα ( q , ω , B ) = σ αα ( ω ) + κBq , which is consistent with Eq. (2) . Another application of Eq. ( 2 ) is the natural circular dichroism which is expressed by σ αβ ( q , ω , B = 0) = ζε αβγ q γ with ε αβγ being the totally antisymmetric tensor, which describes the rotation of the light polarization in a chiralterm·dom crystal. This natural circular dichroism should be distinguished from the Kerr and Faraday rotation which follows from σ αβ ( q = 0, ω , B ) = ξε αβγ B γ , and does not depend on the direction of light, i.e., q . Below, we discuss the linear nonreciprocal responsesterm·dom in more detail describing their microscopic mechanism.

Directional dichroismterm·dom and birefringence in magnetoelectrics The light wave-vector- ( q ) dependent optical response, typically termed directional dichroismterm·dom or directional birefringence, can be observed ubiquitously in materials with broken symmetries of both space-inversion Î and time-reversal T ^ and in a wide photon-energy region ranging from a microwave to hard X-ray. One of the early observations is the directional light emission (luminescence) from a chiralterm·dom molecule with luminescent Eu 3+ ion under the magnetic field reported by Rikken et al. 5 ; depending on the relative direction of parallel and antiparallel light-emission q and the applied magnetic field B , i.e., q·B > 0 or <0, the light-emission intensity is modulated, typically up to a few % for B of the order of a few tesla. This case is called the magnetochiral dichroism. Magnetoelectric (ME) directional anisopropy has been first observed in Er 1.5 Y 1.5 Al 5 O 12 with the modulation of the refractive index δn = γ q· ( E × B ) 6 . When luminescent rare-earth ions are incorporated into a polar medium, e.g., in ferroelectric BaTiO 3 doped with Er 7 , modulation of luminescence intensity occurs depending on the case, q· ( P × B )>0 or <0; here P denotes the electric polarization. When the breaking of T ^ is brought about by spontaneous magnetization M instead of applied magnetic field B , even a larger directional effect can show up. The directional dichroismterm·dom is phenomenologically described by the Maxwell equations plugged in with the ME susceptibility tensor α; P = αB or M = t αE . In case of multiferroicsterm·key where the spontaneous P and M coexist, the diagonal and off-diagonal components of α are outcomes of the free energy terms proportional to E·B 8 and E × B 6 , respectively.

Figure 1a–d exemplifies the directional dichroismterm·dom observed for the ME spin excitation, termed electromagnonterm·dom, which shows both magnetic and electric transition-dipole activities 9 . The helical or conical spin order as shown in Fig. 1a can produce the electronic chiralityterm·key depending on the magnetic helicity. In this particular compound, Ga-doped CuFeO 2 with a triangular lattice of Fe 3+ ions 10 , the hybridization of the metal (Fe 3+ ) d state, and ligand (O 2− ) p state under the influence of spin–orbit interaction can produce the electric polarization P along the screw wavevector ( Q ) direction, as described by the d–p orbital hybridization model 11 . This multiferroicterm·key character can allow one to obtain the single spin-helicity domain, or equivalently the single chiralityterm·key domain, via the ME cooling procedure under B // E , since the sign of P depends on the spin helicity in this mechanism. Thus, when the magnetic field is applied along the Q direction (or the Q is directed along the applied magnetic field direction), the magnetochiral dichroism for the q ω // Q (here q ω being the light q vector with the frequency of ω , set parallel to the z axis) is anticipated to stem from the off-diagonal ME coefficient (α xy ) since the orthogonal light electric and mangetic fields are along x or y . Figure 1b, c exemplifies the large directional dichroismterm·dom spectra, i.e., the spectra of real part ( n ) and imaginary part ( κ ) of the refractive index depending on the q ω // P or – P 12 . The change of complex refractive index is given by the off-diagonal component of the ME susceptibility, ± α ij ( ω ), in conjunction with the Maxwell wave equation. The α xy ( ω ) = Δ n ( ω )+ i Δ κ ( ω ) shows the characteristic anomalous dispersion spectrum due to the resonance of the electromagnonterm·dom.

Magnetochiral dichroism and gyrotropic birefringenceterm·dom (GBterm·dom) in multiferroicsterm·key. a Schematic illustration of the conical screw spin structure under an external magnetic field ( H ex ) in multiferroicterm·key CuFe 1– x Ga x O 2 ( x = 0.035). The H ex parallel to the in-plane magnetic modulation q m (e.g., // [110]) modifies the proper screw spin structure to the conical one. The ferroelectric polarization ( P ) driven by the screw spin structure also points to the in-plane q m direction ([110]). The electromagnonterm·dom spectra of ( b ) real part n ( ω ) and ( c ) imaginary part κ ( ω ) of the complex refractive index at 4.6 K and 7 T ( H ex || [110]) with the light configuration in ( a ). Spectra with both signs of spin helicity ( γ m ( + ) or γ m (–)) and magnetization ( + M or – M ) are shown. In the respective lower panels, magnetoelectric spectra are shown defined as ( b ) Δ n ( ω ) = n ( + k ω )– n ( –k ω ) and ( c ) Δ k ( ω ) = κ (+ k ω ) – κ (– k ω ) 12 . d Schematic of GBterm·dom for light with propagation vector + k ω (upper) and − k ω (bottom) perpendicular to the c axis in the case of (Fe; Zn) 2 Mo 3 O 8 ; P s and M s stand for the spontaneous electric polarization and magnetization. P ω and P ME show the electric polarizations induced by the electric field ( E ω ) and magnetic field ( B ω ) of light, respectively. The green plate and red and blue bars denote the sample and rotating principal optical axes, respectively. Reprinted figure with permission from ref 18 . Copyright (2017) by the American Physical Society. e Axion insulator potentially showing the topological magnetoelectric effect and associated GBterm·dom, composed of the magnetic topological insulatorterm·dom with the opposite magnetization directions on the top and bottom layers. Figure from ref. 21 with permission from Nature Publishing Group The diagonal ME susceptibility arises from the E·B -type coupling term in the free energy and is found in the system endowed with the ME monopoles. Here the ME monopole is defined by the magnetic structure with the inward or outward configuration of spin moment ( m ), i.e., div m ≠ 0 13 . A classic example of this is the antiferromagnetically ordered state in Cr 2 O 3 , in which Cr dimers formed along the c ( z ) direction show the up and down spin moment configuration, i.e., the ME monopole, and the linear ME effect was first predicted 14 and detected 15 . While Cr 2 O 3 shows both the diagonal and off-diagonal ME effect, the diagonal component leads to the specific nonreciprocal optical effect, termed gyrotropic birefringenceterm·dom (GBterm·dom). The GBterm·dom represents the rotation of the optical principle axes depending on the light incident direction, as schematized in Fig. 1d . When the light propagates along the y direction ( q ω // y ) with the light E ω // z and H ω // x , the additional P ω x and M ω z components are induced inside the material by the diagonal ME susceptibility of α xx and α zz , respectively. Therefore, the rotation of the optical principle axes is described in terms of the dynamic ME susceptibility α GBterm·dom ( ω ) = α xx ( ω )–α zz ( ω ). This was first confirmed for the Cr d–d excitation region in the visible range 16 .

The large GBterm·dom has been observed for the electromagnonterm·dom in multiferroicterm·key (Fe, Zn) 2 Mo 3 O 8 with the ferromagnetic moment M and the inherent electric polarization P both along the c axis (Fig. 1d ) 17 , 18 . The ferrimagnetic spin alignment along the c axis also produces the ME monopoles. The electromagnonterm·dom locating at 1.4 THz (5.6 meV) shows the resonant dispersion of α GBterm·dom ( ω ), giving rise to the 0.02-rad rotation and 0.04-rad ellipticity of the light polarization for the 430-μm-thick crystal at 5 K. The electromagnonterm·dom resonance is turned out to contribute to about 8% of the DC ( ω = 0) linear ME susceptibility, pointing to the important role of the electromagon excitation as the ME fluctuation in multiferroicsterm·key 18 .

Another important example of multiferroicsterm·key to potentially show the GBterm·dom is the axion insulator state constituted from the topological insulatorterm·dom (TIterm·dom). In the TIterm·dom, the Lagrangian of the electromagnetic field can contain the axion coupling term, α 4 π 2 θ E ⋅ B , in addition to the conventional Maxwell term 19 , 20 . Here, α = e 2 / ħc is the fine structure constant and θ is equal to π (0 or 2π) in the topological (non-topological) insulator. When the top and bottom surface-state dispersions both show mass gaps due to the exchange coupling with the opposite magnetizations (i.e., ±1 Chern number, Fig. 1e ) and the Fermi level is inside these mass gaps, the compound can be an axion insulator with the topological ME effect, as characterized by the diagonal ME susceptibility α zz ~ α . This state can also be viewed as possessing the ME monopole like Cr 2 O 3 , and be anticipated to show the topological GBterm·dom in a quantized manner 21 .

As exemplified above, the nonreciprocal dichroism/birefringence can be ubiquitously observed in a broad family of materials system with both broken Î and T ^ in a wide photon-energy region ranging from μeV (microwave) 22 to keV (hard X-ray) region 23 . Apart from the case of the ME excitation resonance like the electromagnonterm·dom 24 , 25 , their magnitudes usually remain at most a few % to the original absorption intensity or to the π/2 rotation for GBterm·dom. A notable exception is the case of sharp resonant crystal-field d–d excitations in the canted antiferromagnetic phase of CuB 2 O 4 with noncentrosymmetricterm·key space group I 4 2d 26 . There, zero-phonon line of the d–d transition is observed to show the huge q -directional change of the absorption or the nearly perfect cloaking at high magnetic field, which is ascribed to the strong interference between E1 (electric-dipole) and M1 (magnetic-dipole) transitions caused by the spin–orbit coupling 26 , 27 .

Nonreciprocal dynamics of magnons When the quasiparticle with the wave vector k propagates along the magnetic field B (or magnetization M ) in the chiral-structure compound, the relevant response function adds the additional term A MCh , which is in proportion with σ [sgn( k·H )], to the conventional term A 0 ; here σ = ±1 depends on the lattice chiralityterm·key, i.e., right or left handedness of the crystal. This relation generally holds for photon, magnon, and other quasiparticles which can be sensitive to the B or M . The case of photon is the magnetochiral dichroism as described in the previous section. Here we reviewterm·mth the studies done on the magnon (spin wave) propagation.

The most archetypal example is the spin wave propagation in the chiral-lattice ferromagnet; this was recently observed in chiralterm·dom magnets such as LiFe 5 O 8 28 , Cu 2 OSeO 3 29 , and other intermetallic compounds 30 , 31 . The effective Hamiltonian of the chiralterm·dom magnet can be described as the sum of symmetric ( J ) and antisymmetric ( D ) exchange interactions, cubic anisotropy ( K ) term, and Zeeman term proportional to H . There the asymmetric exchange interactionterm·dom, called Dzyaloshinskii–Moriya (DM) interaction, originates from the relativistic spin–orbit coupling in the noncentrosymmetricterm·key (here, chiralterm·dom) lattice and works as a source of nonreciprocity of magnon transportterm·dom. The chiralterm·dom magnet generally shows the helical (screw) spin state whose magnetic periodicity is given as aD/J with a being lattice constant. The application of H turns the helical spin ground state to the spin-collinear (ferromagnetic) state, where the spin wave dispersion is given by Here, C sym = JSV 0 κ 2 + 2 KV 0 S 3 + γħμ 0 H is an even function of k ; γ, μ 0 , V 0 , and S are gyromagnetic ratio, vacuum magnetic permeability, unit cell volume, and vector spin density, respectively. The first k -linear term causes the shift of the parabolic spin wave dispersion ( C sym ) and hence the different group velocity between + k and – k , as shown in Fig. 2a , leading to the nonreciprocal propagation of the spin wave. (In the case of the plate-shaped sample, the dispersion is modified by the magnetic dipolar interaction in the k ~0 region, as discerned by the sharp spike around k ~0 in Fig. 2a .) Spin wave spectroscopy on the nonreciprocal magnon transport in a chiralterm·dom magnet. a Asymmetric spin wave dispersions for the collinear ferromagnetic state in a chiralterm·dom magnet MnSi according to Eq. ( 3 ); a sharp spike around k~0 is due to the dipolar interaction. b The experimental setup to detect the nonreciprocity for the magnon transport between the coplanar microwave guides (ports 1 and 2). c Imaginary part spectra of the mutual inductance Δ L 12 (Δ L 21 ), representing the transport of the magnons from 1 to 2 (from 2 to 1) for magnetic fields applied parallel (+ H ) or antiparallel (– H ) to the propagation direction; the result for the enantiomer D-body crystal. Another enantiomer L-body crystal shows the opposite behaviors for the exchange between Δ L 12 and Δ L 21 . d Magnetic field dependence of spin wave nonreciprocity between Δ L 12 and Δ L 21 (i.e., + k and − k ), measured for the D-body crystal of Cu 2 OSeO 3 with the H // k // [001] configuration at 30 K. Here, ν p indicates the magnetic resonance frequency giving the peak value of |Δ Lnm |, and Δ ν p the difference between ± k , respectively. |Δ L p nm | and v p g indicate the corresponding peak value and the group velocity at the frequency ν p . The right panel shows the schematic illustration of collinear ferromagnetic state and helical spin state, respectively. The nonreciprocal spin wave propagation between ± k is discerned only in the former spin state. Reprinted figure with permission from ref 29 . Copyright (2016) by the American Physical Society The spin wave spectroscopy can directly demonstrate the nonreciprocal magnon transport. As the typical experimental setup (Fig. 2b ), a pair of coplanar wave guides (ports 1 and 2) placed beneath the chiralterm·dom magnet play roles of generator and detector of magnon. The wave number ( k ) distribution of the generated spin wave is represented by the Fourier transform of the wave guide pattern with the spacing of λ , i.e., k ~2 π / λ . The spectra of the mutual inductance L 12 ( L 21 ) represent the propagation of the spin wave or magnon from terminal 1 to 2 (2 to 1). Figure 2c exemplifies the imaginary part of the inductance spectra L 12 and L 21 with λ ∼12 μm, taken for the specific-chirality (D-body or σ = +1) crystal of chiralterm·dom (space group: P 2 1 3) magnet Cu 2 OSeO 3 with the spin-collinear state induced by the magnetic field of ±740 Oe applied parallel/antiparallel to the magnon k direction 29 . The magnetochiral nonreciprocal propagation can be clearly discerned as the frequency shift between L 12 and L 21 and also as the reversal of the shift upon the reversal of the magnetic field direction. The results for the L-body ( σ = −1) crystal show the reversed relation, confirming the magnetochiral nature. Figure 2d summarizes the result of the spin wave spectroscopy; in the relatively high magnetic field (hatched) region where the spin-collinear ferromagnetic state is stable, the difference Δ v p of the peak frequency of L 12 and L 21 (representing the magnon energy difference propagating from terminal 1-to-2 and 2-to-1) is opposite in sign in the positive and negative magnetic fields. The magnon group velocity v g deduced by the L 12 and L 21 data differs between the 1-to-2 and 2-to-1 propagation and the difference is also reversed upon the reversal of magnetization. All these features point to the magnetochiral nonreciprocity of magnon transportterm·dom, which is well accounted for with Eq. ( 3 ). By contrast, the helical spin state in the low magnetic field shows the minimal nonreciprocity although the symmetry argument would allow the nonreciprocal effect. The magnon energy difference between + k and – k is directly related to the DM interaction D , and in turn its observation can give the quantitative estimate of D , as demonstrated in a series of the chiral-lattice magnets hosting the magnetic skyrmion near room temperature 31 .

The k -linear term in the magnon dispersion caused by the DM interaction can be probed by other means of spin wave or magnon spectroscopy such as inelastic neutron spectroscopy 30 and light Brillouin scattering spectroscopy 32 , 33 . For example, the inelastic neutron-scattering spectroscopy on chiral-lattice magnet MnSi ( P 2 1 3 space group) has directly demonstrated the asymmetric magnon dispersion as described by Eq. ( 3 ) 30 . The DM interaction produced at the interface between the heavy-element (i.e., large spin–orbit interaction) metal and ferromagnet can play an important role in spintronics function and also generate the nonreciprocal magnon transport along the lateral direction under the magnetic field applied laterally and normal to the k direction. This was also detected as the difference of the spin wave resonance frequency between + k and – k in terms of the light Brillouin scattering spectroscopy 32 , 33 .

Nonlinear/nonreciprocal transport Theorem of fluctuation Compared with the linear response theory, its nonlinear generalization of nonreciprocity is far more difficult and not well explored. However, the generalization of Onsager’s reciprocal theorem has been intensively discussed recently, and is called “fluctuation theoremterm·dom”, which also originates from the time-reversal symmetry of the microscopic dynamics of the system 34 – 37 . Let p ( R ) be the probability that the entropy change is R . Then the fluctuation theoremterm·dom claims that where R > 0 means the entropy production, while R < 0 the entropy reduction. B represents the magnetic field, and the denominator and numerator of the left-hand side of Eq. ( 4 ) are related by the time-reversal symmetry. In the thermodynamics limit, R becomes very large, and hence the right-hand side of Eq. ( 4 ) is infinite, and the probability of the entropy reduction is zero, i.e., the second law of thermodynamics. This theorem imposes the constraint on the fluctuation around the average, and the linear response theory as well as the Onsager’s reciprocal theorem can be derived from this theorem. Especially, the fluctuation phenomena in the mesoscopic systems are an ideal laboratory to study this theorem. As an application of the fluctuation theoremterm·dom, the analysis of the counting statistics gives the relation between the nonlinear transport coefficient and noise in quantum transport of a mesoscopic conductor 38 . Let I be the current and V the voltage between the electrodes. The nonlinear I–V characteristics can be written as and the current-noise S = <(δ I ) 2 >/Δ f ( δI : the deviation of the current from the average, Δ f : the frequency range) is expressed as The fluctuation-dissipation theorem in the linear response is S 0 = 4 k B TG 1 , which determines the Johnson–Nyquist noise S 0 in thermal equilibrium. The generalization to nonlinear response reads S 1 = 2 k B TG 2 . The analysis including the magnetic field B as well as its experimental test has been done, and the readers would refer to the literature 38 , 39 . Fluctuation theoremsterm·dom can provide the basis of the nonlinear responses beyond the Onsager’s reciprocal theorem, but their applications to the bulk transport phenomena are not well explored as yet.

Unidirectional magnetoresistance The directional nonlinear dc resistance has been discussed by Rikken 40 , who gave a heuristic argument generalizing the Onsager’s reciprocal theorem into the nonlinear regime, and suggested Eq. ( B1 ) in Box 1 for the current ( I )-dependent resistivity R , where the second term with coefficient β represents the usual magnetoresistance, while the third term with coefficient γ the directional resistance, i.e., magnetochiral anisotropyterm·key. Although Eq. ( B1 ) does not describe the vector nature of the current and magnetic field, there are two types of magnetochiral anisotropyterm·key according to the crystal structures with broken inversion symmetry. One is the polar structure where the directions of the polarization P , magnetic field B , and the current I (electric field E ) are orthogonal to each other; R = R 0 [1+ γ ′( P × B ) ·I ]. The upper part of Fig. 3 presents the list of examples, i.e., Si FET 41 , magnetic bilayer 42 , BiTeBr 43 , and surface state of TIterm·dom 42 , in this class. One typical example is the rectification effect in BiTe X ( X = I, Br) in which Bi, Te, and X layers are stacking alternately so that the mirror symmetry along the c axis is broken, i.e., P // c . This material shows the giant spin splitting of the band structure due to the Rashba spin–orbit interaction 44 . However, the time-reversal symmetry leads to the relation with k being the crystal momentum and σ is the spin component. Therefore, when the spin components are summed up as in the case of charge transport, the directional dependence disappears. Once the external magnetic field is applied in plane, e.g., along the y -axis, the energy dispersion becomes asymmetric between k x and –k x as shown in Fig. 4a , and I–V characteristic along a -direction becomes nonreciprocal. Analysis in terms of the Boltzmann transport theory concludes that the coefficient γ in Eq. ( B1 ) is independent of the lifetime of the electrons in the relaxation time approximation, similar to the case of Hall coefficient, and is an intrinsic quantity to the band structure. Therefore, one can predict quantitatively the magnitude of γ as a function of the electron density and temperature. Figure 4b shows the dependence of the nonlinear resistivity on the direction of the magnetic field in BiTeBr, which is consistent with the relation that R = R 0 [1+ γ ′( P × B ) ·I ] 43 . Figure 4c presents the electron density dependence of γ measured by experiments and calculated theoretically without any fitting parameters since the band structure of this material has been already determined 43 . Excellent agreement between theory and experiment indicates that the microscopic origin of the magnetochiral anisotropyterm·key in this material is the asymmetric deformation of the energy dispersion.

Systems showing the unidirectional magnetoresistance. There are two classes according to the crystal symmetry, i.e., polar and chiralterm·dom structures. Polar systems include Si FET 41 , magnetic bilayer 42 , BiTeBr 43 , and surface state of topological insulatorterm·dom (TIterm·dom) 44 . In this class of systems, the resistivity R showing the magnetochiral anisotropyterm·key has the form R = R 0 [1+ γ ( P × B ) ·I ] where the directions of the polarization P , magnetic field B , and the current I (electric field E ) are orthogonal to each other. There are several microscopic mechanisms of the magnetochiral anisotropyterm·key. For Si FET 41 , the relativistic Lorentz transformation and associated correction due to the factor v/c ( v : velocity of electrons, c : velocity of light) was proposed 41 . For magnetic bilayer Ta|Co and Pt|Co systems, it has been proposed that the current-induced spin accumulation modifies the resistivity through the spin–orbit interaction and spin-dependent scattering 42 . The asymmetric band dispersion under the in-plane magnetic field in the presence of the Rashba spin–orbit interaction is identified as the origin of the nonreciprocal resistivity in BiTeBr 43 , while the asymmetric scattering of electrons by magnons is the origin in the surface state of topological insulatorterm·dom (TIterm·dom) 44 . The examples of the other class, i.e., the chiralterm·dom structure, are shown in the lower part, which shows the behavior R = R 0 [1+ γ ( B·I )] called the electrical magnetochiral effectterm·dom. Examples of this class include Bi helix 40 , 46 , molecular solid 47 , and MiSi 48 . The helix structure is a representative example of this class. It has been discussed that the magnetic field b produced by the current is combined to the external magnetic field B , and the magnetoresistance for b + B results in the electrical magnetochiral effectterm·dom in Bi helix 40 . Similar effect was also observed in a molecular semiconductor [DM-EDT-TTF] 2 ClO 4 (middle of the lower panel) 47 . In a ferromagnet, the time-reversal symmetry is spontaneously broken, and the chiralterm·dom ferromagnet can show the electrical magnetochiral effectterm·dom. MnSi is a representative example, and that effect is enhanced above the helical transition temperature T c 48 . This suggests that the spin fluctuation of chiralterm·dom nature is the origin of the electrical magnetochiral effectterm·dom. Reprinted figures with permission from ref 40 , 41 , 45 . Copyright (2001, 2005, 2016) by the American Physical Society. Figures from ref. 42 – 44 , 47 , 48 with permission from Nature Publishing Group Magnetochiral anisotropyterm·key in polar systems. a The distorted band structure of Rashba system under the external magnetic field. When the polar axis is along the c -direction, the spin splitting occurs with the spin polarization for each band lying in the a–b plane. When the magnetic field is applied along b axis, the asymmetry between two directions along a -axis appears 43 . b The dependence of the nonlinear resistivity corresponding to the γ term in Eq. (B 1 ) on the direction of the external magnetic field in BiTeBr, which is consistent with the formula R = R 0 [1+ γ ′( P × B ) ·I ] 43 . c The coefficient γ as a function of electron density in BiTeBr. The black curve is the theoretical calculation without any fitting parameters. Figure from ref 43 . Reprinted with permission from Nature Publishing Group. d Schematics of the surface state of a three-dimensional topological insulatorterm·dom, which is described by the two-dimensional Weyl Hamiltonian with spin-momentum locking. With the in-plane magnetization along the y -axis, the inelastic scattering due to magnons to the right and left directions is different, leading to the nonreciprocal transport 45 . e The schematic experimental setup (left), and the obtained magnetic field dependence of the nonlinear resistivity 45 . f Magnetic field dependence of nonlinear resistivity at several temperatures. Reprinted figure with permission from ref 45 . Copyright (2016) by the American Physical Society Another example of the (P × B) · I -type nonreciprocity is the case of the TIterm·dom, as shown in Fig. 4d–f 45 . In the TIterm·dom thin film (Bi, Sb) 2 Te 3 (BST), the upper layer part (denoted CBST) is doped with magnetic Cr ions to enhance the magnetic response as well as to introduce the asymmetry between the top and bottom surface layers where the cone-like dispersion of the conduction electron is formed with strong spin-momentum locking (Fig. 4d ). When the magnetic field B is applied along in-plane and perpendicular to the current I direction, the large nonlinear component Δ R xx shows up (Fig. 4e ); note that the spontaneous magnetization is originally (at zero field) perpendicular to the plane but inclined to the in-plane by the external in-plane field. The important feature for Δ R xx is its sign reversal upon the reversal of B and also upon the reversal of the stacking sequence structure of BST (nonmagnetic) and CBST (magnetic) layer; the latter procedure corresponds to the reversal of P , thus satisfying the condition of the magnetochiral anisotropyterm·key. Under the in-plane B field, the Weyl cone should show a parallel shift in the k -space, yet this would give no effect on the nonlinear conduction if the quadratic term of the band dispersion could be neglected, being contrary to the case of the above example of the Rashba system. The unidirectional nonlinear magnetoresistance Δ R xx observed in this semi-magnetic TIterm·dom system shows the characteristic magnetic field and temperature dependence; as increasing B to several-tesla region and as elevating temperature across the ferromagnetic ordering temperature, Δ R xx shows a steep reduction. These observed features are all explained in terms of the relevance of the magnon excitations, i.e., their emission and absorption, in the originally forbidden backward-scattering process of the spin-momentum-coupled Weyl electrons (Fig. 4a ). The application of B opens the spin wave gap, resulting in the reduction of the magnon scattering events at low temperatures. Furthermore, the tuning of the Fermi level around the Dirac (band crossing) point is found to enhance Δ R xx because the magnon’s q vector and energy to mediate the backward scattering of the conduction electron can be small and more effectively thermally populated.

The other class of the materials showing unidirectional magnetoreistance is the chiralterm·dom structure shown in the lower part of Fig. 3 ; in this case, the nonlinear nonreciprocal conduction is sometimes called the electrical magnetochiral effectterm·dom in analogy to the above-described optical magnetochiral effect. Examples of this class include Bi helix 40 , 46 , molecular solid 47 , and MiSi 48 . The helix is a representative example of this class, and an early experiment on Bi-helix wire found a finite γ value 40 . The suggested mechanism of this effect is the magnetoresistance due to the magnetic induction b by the current I circulating along the helix, i.e., the effective magnetic field felt by the electrons is B + b , the magnitude of which depends on the direction of I . In this case, the nonlinear conduction is maximized when magnetic field B and the current I (electric field E ) are parallel to each other, namely taking the form of B·I . A similar effect was also observed in a molecular semiconductor 47 . Ferromagnets with chiralterm·dom crystal structures are also expected to show the magnetochiral anisotropyterm·key.

Beyond the phenomenological or symmetry argument, there is known the case where a specific scattering mechanism can cause the electrical magnetochiral effectterm·dom. In MnSi with the cubic chiral-lattice structure ( P 2 1 3), in which spin-helical or skyrmion phase forms below T c = 35 K because of Dzyaloshinskii–Moriya interaction inherent to the lattice chiralityterm·key, the B·I -type nonlinear conduction is observed with the opposite sign for the two enantiomers 48 . Symmetry allows the electrical magnetochiral effectterm·dom in every T–B region. In reality, however, the effect becomes appreciable not in the helical magnetic order region but immediately above T c , where the spin fluctuation is observed to retain the chiralterm·dom character. This means that the scattering process of the conduction electron by the chiralterm·dom spin fluctuation is proven to be the major source of the electrical magnetochiral effectterm·dom in this case.

Nonreciprocal responseterm·dom in Weyl semimetalsterm·dom Weyl semimetalterm·dom is a class of materials where the Weyl fermions are at the Fermi energy typically described by the Hamiltonian 49 where σ = (σ x ,σ y ,σ z ) are Pauli matrices, v is the velocity, k 0 is the band crossing point, and η = ±1 specifies the chiralityterm·key. It has been known that the Weyl fermion acts as the magnetic monopole or antimonopole of the emergent magnetic field in the momentum space for η = +1 and η = −1, respectively. The Nielsen–Ninomiya theoremterm·dom 50 dictates the equal number of the Weyl fermions with opposite chiralities. One of the peculiar phenomena associated with Weyl fermions is the chiral anomalyterm·dom, which is shared also by the Dirac fermion with 4 × 4 Hamiltonian. This is the phenomenon where the fermions are transferred between opposite chiralities in proportion to E·B under the external electromagnetic field. This results in the large negative magnetoresistanceterm·dom when the electric and magnetic fields are parallel or antiparallel to each other 49 , 51 .

Note that, as for the Weyl fermions with 2 × 2 Hamiltonian, either the time-reversal T ^ or the space-inversion Î must be broken to lift the Kramer’s degeneracy except at k 0 . Accordingly, there are two types of Weyl semimetalsterm·dom. One is the T ^ -broken, but Î -symmetric Weyl semimetalterm·dom. Î -symmetry relates the Weyl fermions at k 0 and − k 0 , and the chiralities of these are opposite to each other. On the other hand, in the Î -symmetry broken Weyl semimetalsterm·dom, the Weyl fermions at k 0 and − k 0 have the same chiralityterm·key. Therefore, according to the Nielsen–Ninomiya theoremterm·dom 50 , there must be at least another pair of Weyl fermions at k 1 and − k 1 . Namely, there are at least four Weyl fermions at ± k 0 and ± k 1 . The energy dispersions and also the Fermi energy shift at ± k 0 and ± k 1 are different and contribute differently to the conductivity. Therefore, once the external magnetic and electric fields shift the electrons in proportion to E·B between positive and negative chiralityterm·key Weyl fermions due to the chiral anomalyterm·dom, there occurs the current proportional to ( E·B ) E , which indicates the magnetochiral anisotropyterm·key 52 . Because the Weyl fermions in Î -broken system are solely due to the spin–orbit interaction, and also the Fermi energy shift from the Weyl point is usually small, the smallness of the perturbation discussed in the previous section is avoided, leading to the large nonreciprocal responseterm·dom. Quantitatively, γ value in Eq. ( B1 ) can be four orders of magnitude larger than those discussed in the literature 40 .

Nonreciprocal transport in a noncentrosymmetricterm·key superconductor The magnetochiral anisotropyterm·key is usually a small effect because it requires both the spin–orbit interaction λ , which reflects the inversion symmetry breaking and the magnetic field B breaking the time-reversal symmetry. For each perturbation, the energy denominator is typically the Fermi energy ε F , and γ is expected to be proportional to ( μ B B /ε F )( λ /ε F ) and is usually very small. In the case of BiTeBr discussed above, the giant Rashba splitting leads to λ / ε F ∼ 1 and the reasonably large γ . However, even larger γ is realized in the noncentrosymmetricterm·key superconductors 53 as shown below. The physical picture for this enhancement is that the Cooper pairs with the large coherence length ξ feel the noncentrosymmetricterm·key nature of the potential sensitively in the low- energy region below the superconducting gap Δ. In the I –V characteristics, the voltage is zero in the low current region. However, in 2D superconductors with an external magnetic field, the unpinned vortices produce the finite resistance and γ can be defined 54 . In this case, the energy denominator in the expression of γ becomes Δ replacing ε F , leading to the enhancement of γ by some power of the factor ( ε F /Δ). An example is the noncentrosymmetricterm·key superconductor MoS 2 composed of the stacked layers with weak van der Waals interaction 55 . The monolayer system is a 2D network with D 3h symmetry, leading to the trigonal-warping of the Fermi surfaces and out-of-plane spin polarization with effective Zeeman fields at K -points. Measurement of γ as a function of temperature shows its rapid growth below the temperature slightly higher than the mean field transition temperature T 0 , the mean field transition temperature, reaching the value γ ~8000 T −1 A −1 well below T 0 56 .

Theoretical analysis has been restricted to the paraconductivity above T 0 57 , concluding the enhancement of γ as γ S γ N ~ ε F Δ 3 , and predicted γ ~400 T −1 A −1 . Although the analysis well below T 0 has not yet been done, the rapid increase of γ observed experimentally is consistent with the theoretical prediction. The dynamics of vortices should be relevant to the nonreciprocal resistivity well below T 0 , which is left for future studies.

Nonlinear/nonreciprocal photonic responses Nonlinear optical effects in multiferroicsterm·key The nonlinear and nonreciprocal optical effects are also expected generically for the materials system with broken symmetries of space-inversion and time-reversal. One important source of such nonlinear optics (NLO) in the magnetic system is the toroidal momentterm·dom T 58 as defined by T = 1 2 ∑ i r i × S i ; here S i is the spin moment on the site r i . Since the effective spin–orbit coupling term in Hamiltonian is described as λ L·S = λ( r × p ) ·S = − λ ( r × S ) ·p = e A eff ·p where e and p are electron’s charge and momentum, the toroidal momentterm·dom T can be viewed as the built-in vector potential A eff under the presence of spin–orbit interaction. Then T can mix with the ac vector potential A ω from the light (frequency ω , i.e., T + A ω ). The compound may not show the second-order nonlinear optical (NLO) effect responding to A ω 2 , but generically does the third-order term responding to A ω 3 . When the T is mixed in, this third-order term can effectively generate the second-order response, i.e., responding to T A ω 2 . Namely, when the light electric field E ω (or A ω ) is parallel to T of the material, the effective second-order NLO, typically second-harmonic generation (SHG with frequency of 2 ω ) is observed. Conversely, such magnetization-induced SHG can be used as a probe for the toroidal momentterm·dom.

Figure 5 exemplifies the toroidal moment-induced SHG in multiferroicsterm·key 59 . FeGaO 3 (Fig. 5a ) shows the polar structure along the b axis ( P // b ), while the magnetic moments on two Fe sites (Fe1 and Fe2) is antiferromagnetically coupled with their moments parallel to the c axis. Then Fe1 and Fe2 jointly form the toroidal momentterm·dom T // a 60 . The respective sublattice magnetization on Fe1 and Fe2 shows the unbalance, allowing the ferromagnetic moment M , which enables to control the sign of T //( P × M ) by application of an external field. The SHG is originally active to produce E 2 ω // b light along the polar axis, while it is also activated by the toroidal momentterm·dom for E ω // a . In the case of oblique incidence of the light with E ω // a (s-polarized) on the ac plane sample, both the SHG components ( E 2 ω // b from the original crystal polarity and E 2 ω // a from the toroidal momentterm·dom) can mix to generate the rotation of the polarization of the SHG light, as shown in Fig. 5b . This is called nonlinear Kerr rotation 61 and the effect is usually much larger than the conventional Kerr rotation angle in ferromagnets. The reversal of either P or M can cause the sign reversal of the nonlinear Kerr rotation. The parallel component (say, x -component) of the oblique incident light k -vector is either parallel or antiparallel to the applied magnetic field or magnetization direction (along x ), and hence the reversal of k x shows also the sign reversal of the nonlinear Kerr rotation. The phase (π) change of the E 2ω of the SH light upon the sign change of the toroidal momentterm·dom enables one to image and map the toroidal momentterm·dom domains in a multi-domain state, as exemplified in Fig. 5c . In some multiferroicsterm·key, the ferrotoroidic/antiferrotoroidic domains can be distinguished from ferromagnetic/antiferromagnetic domains, as exemplified for the case of multiferroicterm·key LiCoPO 4 62 . This method has been also applied to probe the magnetization at the heterointerface with the effective polarity along the normal 63 , 64 . It is also noted that the spin toroidization in periodic crystals has been formulated recently from the viewpoint of the Berry phaseterm·key of Bloch wavefunctions 65 .

Second-harmonic (SH) generation from the toroidal momentsterm·dom in multiferroicsterm·key. a Crystal and magnetic structures of GaFeO 3 . In the right panel, outward (inward) arrows represent the spin direction of Fe1 (Fe2) atoms antiparallel (parallel) to the c axis. Horizontal solid lines represent the shifted positions of Fe ions along the b axis from the symmetric positions (dashed lines) of FeO 6 octahedra. Note that the shifted directions of Fe1 and Fe2 atoms are opposite to each other. Reprinted figure with permission from ref 60 . Copyright (2005) by the Physical Society of Japan. b Experimental configuration for the measurement of the Kerr rotation of SH light in a GaFeO 3 crystal (left panel). Analyzer-angle dependence of SH intensity at 180 K ( <T C ). Analyzer angles θ of 0 and 90° stand for p - and s -polarized SH light, respectively. Solid and open circles indicate the SH intensity for + z and –z directions of the magnetic field, respectively. c (Left) topographic image of the ac surface of the GaFeO 3 crystal as taken by the SH light in the nonmagnetic configuration, i.e., P in- P out. Magnetic domains aligned along the c axis were observed by reflected SH intensity in (center) S in ( S + P ) out and (right) S in ( S –P ) out configurations. Reprinted figure with permission from ref 59 . Copyright (2004) by the American Physical Society Shift currentterm·dom in polar insulators Up to now, we assume the broken time-reversal symmetry to obtain finite nonreciprocal responseterm·dom, e.g., the current proportional to BE 2 . However, one can ask if there is a possible mechanism of the current proportional E 2 even without the broken T ^ -symmetry. From this viewpoint, it is known that the photoexcitation produces the current even without the potential gradient, i.e., the external electric field, in noncentrosymmetricterm·key materials. This photocurrent is called shift currentterm·dom. Figure a of Box 2 shows the experiment on a perovskite ferroelectrics [KNbO 3 ] 1–x [BaNi 1/2 Nb 1/2 O 3–δ ] x (KBNNO), where the direction of the shift currentterm·dom is switched by that of the electric polarization 66 .

Then, an important question is how the nonreciprocal nonlinear responses are possible without the broken T ^ -symmetry? To answer to this question, one notes that the previous discussion was based on the energy dispersion satisfying the relation Eq. (7). Therefore, when the current does depend not only on the energy dispersion but also on the wavefunction, it is possible to realize the nonreciprocal responsesterm·dom without T ^ -breaking. This means that one must go beyond the Boltzmann transport theory where only the energy dispersion ε n ( k ) of the n th band and consequent group velocity appear in the equation. Namely, the group velocity v n ( k ) = ∂ ε n ( k )/∂ k corresponds to the intraband matrix element of the current operator. On the other hand, the interband matrix elements of the current operator are related to the quantum geometry of the Bloch wavefunction and Berry phaseterm·key 67 , playing essential roles in the various quantum transport phenomena including quantum Hall effect, quantum charge pumping, anomalous Hall effect, and spin Hall effect 68 , 69 . The Berry connectionterm·dom a n ( k ) is defined as a n ( k ) = i < u n ( k )|∂/∂ k | u n ( k )> with | u n ( k )> being the periodic part of the Bloch wavefunction. a n ( k ) is related to the overlap of the two Bloch wavefunctions neighboring in the momentum space as < u n k ∣ u n k + Δ k > = e iΔ k ⋅ a n k , and has the geometrical meaning of “connection” of manifold in Hilbert space. Therefore a n ( k ) is called Berry connectionterm·dom acting as the vector potential in the momentum space. From the vector potential, one can define the “magnetic field”, i.e., Berry curvature b n ( k ) as b n ( k ) = ∇ k × a n ( k ). It is known that the Berry curvature causes the anomalous velocity of the electrons v n an k = F × b n k under the force F acting on the electrons. This anomalous velocity is regarded as the origin of the intrinsic anomalous Hall effect in ferromagnets 69 . When the symmetries Î and T ^ are valid, there occurs the Kramer’s doublet at each k -point, and therefore the Berry phaseterm·key should be defined as the 2 × 2 matrix, i.e., SU(2), instead of the U(1) phase as discussed before. In other words, the U(1) Berry curvature is zero in this case. On the other hand, when Î - symmetry is broken, the Kramer’s degeneracy is lifted at each k -point in the presence of the spin–orbit interaction, and U(1) Berry phaseterm·key can be nonzero. Therefore, the noncentrosymmetry is encoded in the U(1) Berry phaseterm·key. (Note that T ^ -breaking also produces the Berry phaseterm·key in the presence of the spin–orbit interaction.) Since the real-space position r of the wavepacket made from the Bloch wavefunctions is given by the gauge covariant derivative as r n ( k ) = i ∇ k + a n ( k ), a n ( k ) has the meaning of intracell coordinate 70 . The interband transition from n -band to m -band induced by the incident light results in the change of this intracell coordinate from a n ( k ) to a m ( k ) and the corresponding shift of the electron by the amount of r nm ( k ) = ∇ k φ nm ( k ) + a n ( k )– a m ( k ) (shift vector) where the first term makes the r nm gauge invariant with φ nm ( k ) being the phase of the interband matrix element of the current operator. Therefore, in the steady state under the light irradiation, the continuous shift of the electrons by r nm gives the dc current called shift currentterm·dom 66 , 71 – 75 . This shift currentterm·dom has been estimated as a function of incident light energy for BaTiO 3 by the first-principles calculation, and a good agreement was obtained between theory and experiment 73 .

Shift currentterm·dom is essentially different from the conventional transport currentterm·dom in the sense that the former comes from the interband matrix elements of the current J , while the latter from the intraband ones 71 – 75 . Therefore, it is analogous to the polarization current in ferroelectrics, but the remarkable feature is that it can be the dc current. The experimental results on tetrathiafulvalene- p -chloranil (TTF-CA) which is a quasi-one-dimensional compound consisting of mixed stacks of alternating donor (TTF) and acceptor (CA) molecules are shown in Box 2 76 . Shift currentterm·dom is a rather ubiquitous phenomenon in noncentrosymmetricterm·key systems. Actually, shift currentterm·dom has been studied or discussed for GaAs 77 , SbSI 78 , and warped surface state of the three-dimensional TIterm·dom 79 . Furthermore, the shift spin current can be also considered, which can have an application to spintronics 79 .

Box 2 Shift currentterm·dom has been proposed theoretically 71 , 72 , and recently attracts intensive interest as a possible mechanism of the highly efficient solar cell action in noncentrosymmetricterm·key crystals such as perovskite oxides. Figure a shows a representative example, i.e., ferroelectric oxide [KNbO 3 ] 1– x [BaNi 1/2 Nb 1/2 O 3 –δ ] x (KBNNO) 66 . This materials shows the ferroelectric behavior, and the spontaneous electric polarization can be switched by an external electric field. Correspondingly, the direction of the photocurrent changes as shown in Fig. b . Figures c – e show another example, i.e., charge transfer organic semiconductor tetrathiafulvalene-p-chloranil (TTF-CA), where the shift currentterm·dom is considered to be the main mechanism of photovoltaic effect 76 . Figure c shows its crystal structure made of alternating donor and acceptor molecules, and Fig. d the temperature dependence of the photocurrent without the applied electric field (short-circuit current). It is seen that the photocurrent grows rapidly below T c. Figure e shows the light irradiation position dependence of the photocurrent at above and below T c. Above T c, it shows the maximum near the electrodes and the sign is the opposite between the right and left. This can be interpreted naturally that the photoinduced electrons and holes diffuse and are easily captured by the electrodes near the ends of the sample. Below T c, this behavior changes dramatically as well as the magnitude of the photocurrent. It is almost independent of the local position of light irradiation, which indicates the nonlocal nature of the shift currentterm·dom. A recent theoretical study reproduces this behavior on the basis of the numerical simulation of Rice–Mele model under the local light irradiation 89 . Shift currentterm·dom in polar perovskite system and charge transfer organic compound. a Crystal structure of the ferroelectric oxide [KNbO 3 ] 1 –x [BaNi 1/2 Nb 1/2 O 3– δ ] x (KBNNO) used in DFT band structure calculation. K and Ba are shown by blue and green spheres, respectively. b The time-dependence of the photocurrent induced by the pulse excitation without the applied electric field. Red and blue curves correspond to the opposite directions of the poling electric field and electric polarization. The inset shows the applied voltage dependence of the photocurrent. Figure from ref. 66 with permission from Nature Publishing Group. c Molecular structures of TTF (donor) and CA (acceptor) (left), and the crystal structure of TTF-CA, which consists of alternating donor and acceptor molecules along a -axis. This material shows a ferroelectric transition at T C = 81 K due to the dimerization. d The temperature dependence of the shift currentterm·dom in TTF-CA, which becomes finite below T C . The inset shows the temperature dependence of the electric polarization. e The dependence of the photocurrent on the position of the local optical excitation. Above T C (blue), it shows maximum near the electrodes with the opposite signs, and nearly zero at the center of the sample. This is consistent with the diffusion model of photocarriers created by the optical excitation. Below T C , the magnitude of the photocurrent is enhanced and also shows the almost constant behavior in the middle region the sample in sharp contrast to that above T C . This is interpreted as the onset of the shift currentterm·dom below T C 76 Giant second-harmonic generation in Weyl semimetalsterm·dom The shift currentterm·dom is the dc current induced by the optical excitation, and is the second-order process where the current is proportional to E ( ω ) E (− ω ). Similarly, the ac current with frequency 2 ω proportional to E ( ω ) E ( ω ) can be produced, i.e., second-harmonic generation (SHG). It has been discussed that SHG has the similar expression in terms of the shift vector r nm ( k ) = ∇ k φ nm ( k )+ a n ( k )– a m ( k ) defined above, and again has the geometrical meaning. Especially, the Weyl point corresponds to the magnetic (anti)monopole of the Berry connectionterm·dom 49 , and hence can be a source of large SHG. Actually as shown in Fig. 6 , in noncentrosymmetricterm·key Weyl semimetalterm·dom TaAs, the large SHG has been observed with the magnitude of | d |≈3600 pmV −1 ( d represents a component of the third-rank tensor representing SHG). This large value is to be compared with corresponding values in other materials such as | d |≈350 pmV −1 for GaAs, | d |≈250 pmV −1 for ZnTe, and | d |≈15 pmV −1 for BaTiO 3 80 . SHG of a model with two Weyl fermions has been analyzed, and reasonably consistent values with the experiment were obtained.

Second-harmonic generation (SHG) in TaAs. a Schematic of the SHG experimental setup. b SHG intensity as a function of angle of incident polarization at 20 K. c Color plot of the momentum-resolved contribution to the SHG for a model for a simplified mode for TaAs. d The photon-energy dependence of SHG for a simplified model. Figures from ref 80 . Reprinted with persmission from Nature Publishing Group Nonlinear photocurrent induced by Berry curvature The photocurrent discussed above is driven by the Berry connectionterm·dom a n ( k ) rather than the Berry curvature b n ( k ). Then it is natural to ask if there is a photocurrent induced by b n ( k ). This question has been studied by Moore and Orenstein, who predicted the helicity-dependent photocurrent due to Berry curvature 81 . The idea is that the deviation δf n ( k , t ) of the electron distribution function occurs in the presence of the electric field E and is proportional to τ(∂ f (ε)/∂ε) v n ( k ) E with being the relaxation time and v n ( k ) the group velocity. The integral of the anomalous velocity v n an k = - e E × b n k over the modified distribution function results in the photocurrent proportional to E 2 . The photocurrent is induced by this mechanism for the quantum wells structure with the quantum confinement of electrons 81 . A similar idea has also been applied to the nonlinear Hall effect in time-reversal symmetric systems induced by Berry curvature dipole 82 .

Perspective In this article, we have seen ample examples of nonreciprocal responsesterm·dom from quantum materials with broken space-inversion symmetry Î . The directional transport/propagation of the quanta, such as electron, photon, spin, and phonon, enables the one-way transmission of the information carrier. As shown in Box 1 , the diagonal linear nonreciprocal responseterm·dom is only allowed for the system with broken time-reversal symmetry, namely in a magnetic field or in a magnetically ordered state such as the multiferroicsterm·key. As the linear response, the nonreciprocal transport of magnon can substantiate the diode effect of spin current, which may append an important spintronic function. The directional dichroismterm·dom response is also an important nonreciprocal function in the multiferroicsterm·key that can support the ME excitations, typically electromagnonsterm·dom. The directional dichroismterm·dom can realize the cloaking function; contrary to the conventional Faraday rotation, it does not need the additional analyzer of the light polarization to achieve the one-way transmission of light, if the effect can be tuned so as to nearly equalize the electric-dipole and magnetic-dipole transition moments.

Let us turn our eyes to the nonreciprocal transport of an electron. The p–n junction (diode) is the most successful and spectacular example for this. Although we did not touch on this classic example in this article, it originates from the change in the width of the depletion layers at the junction, and can be attributed to the Coulombic interaction of electrons, i.e., the electron correlation effect. Recently, this issue has been revisited from the viewpoint of the multiband effect 83 . As for the dc electric field, only the inclusion of the electron–electron interaction can produce the nonreciprocal current proportional to E 2 , while the non-interacting electron system cannot.

The shift currentterm·dom is another example that does not need breaking of time-reversal symmetry, but even the electron correlation is not required in this case. Its topological nature may ensure the ballistic photocurrent upon the photoexcitation. In addition to its importance as the initial process of the photovoltaic action, the shift currentterm·dom may also work as potentially ultrafast information transfer. As exemplified by the p–n junction and the shift currentterm·dom, the nonlinear nonreciprocity is the key to realize the ac-to-dc conversion. On this basis, the dc spin current generation is also possible by the ac (photo) excitation on the surface state of TIterm·dom 79 . In the magnetic system with broken time-reversal symmetry T ^ , an even more variety of the nonlinear and nonreciprocal current controls are possible, as described in this article for the nonlinear conduction for the Rashba system under a magnetic field, the semi-magnetic TIterm·dom, and the chiral-lattice magnet. There, one mechanism of unidirectional magnetoresistance is the field (magnetization)-induced electronic structural change even with the constant scattering time as in the Rashba system, and the other is due to the carrier scattering itself, for example, the backward scattering via magnon emission/absorption in a TIterm·dom and the directional scattering due to chiralterm·dom spin fluctuation in the chiral-lattice magnet (MnSi). Thus, the study on the nonlinear and nonreciprocal charge transport can bring about the important information about the underlying charge dynamics in the simultaneously broken symmetries of time-reversal and space-inversion.

One interesting question is whether the nonreciprocal responsesterm·dom are possible when the product of the symmetries Î T ^ is intact while both Î and T ^ are broken. One can easily see that this product symmetry does not exclude the linear nonreciprocal responsesterm·dom. For example, the toroidal momentterm·dom, which produces the directional dichroismterm·dom, breaks both Î and T ^ symmetries, while it does not break Î T ^ . . Of particular interest is the Î T ^ symmetry in non-Hermitian systems. (Usually it is called PT -symmetry 84 .) It has been found that PT -symmetry plays a crucial role in enhancing the optical isolator function 85 , 86 . In addition to optics, the nonreciprocal propagation of heat is also a hot topic 87 . Also the relation to the quantum Ratchet is an interesting problem to be explored more in depth 88 . In this case, there is no T ^ symmetry breaking and the soliton-like nonlinear excitations are regarded as the source of the thermal rectification effect.

From the viewpoint of the basic principles, the noncentrosymmetry is encoded by the Berry phaseterm·key, and the Bloch wavefunctions acquire the quantum geometric properties as discussed in section “Nonlinear/nonreciprocal photonic responses.” Berry phaseterm·key is related to the interband matrix elements of the current operator, and produces new types of current in sharp contrast to the conventional transport currentterm·dom due to the intraband matrix elements which allows the particle picture through the wavepacket formalism. This “interband current” manifests itself as the shift currentterm·dom appearing as the photovoltaic effect. Furthermore, it should be noted that the electron correlation combined with the interband current matrix elements plays an essential role in the nonreciprocal responsesterm·dom. Therefore, the physics of nonreciprocal responsesterm·dom touch the most important elements of modern condensed matter physics, i.e., the symmetry, quantum geometry or topology, electron correlation, and also irreversibility. Reducing the symmetries in space and/or time in quantum materials have explored a fertile ground for condensed matter physics and electronics. The recent research of multiferroicsterm·key exemplifies this. Exploring the nonreciprocal responsesterm·dom in quantum and topological materials as described here is a promising direction of the research in search for emergent functions.

Acknowledgements This work was partly supported by the Japan Society for the Promotion of Science through the Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST Program) on “Quantum Science on Strong Correlation” initiated by the Council for Science and Technology Policy and CREST, JST (Grant No. JPMJCR16F1).

Author contributions Y.T. and N.N. jointly discussed the content of this article and wrote the manuscript.

Competing interests The authors declare no competing interests.

Footnotes Contributor Information Yoshinori TokurapersName, Email: tokurapersName@riken.jp.

Naoto NagaosapersName, Email: nagaosapersName@riken.jp.

References