Introduction

An inductor is a component that exhibits an inductive counter-electromotive force, V, under a time-varying electric current, I, following

$$V=L\frac{{{{\rm{d}}}}I}{{{{\rm{d}}}}t},$$
(1)

where L denotes the inductance. The electric work done by the external power supply, IV, is hence

$$\int\,{{{\rm{d}}}}t\,IV=\int\,{{{\rm{d}}}}t\,IL\frac{{{{\rm{d}}}}I}{{{{\rm{d}}}}t}=\int\,{{{\rm{d}}}}\left(\frac{1}{2}LI{(t)}^{2}\right),$$
(2)

which shows that the inductor stores an energy of \(\frac{1}{2}L{I}^{2}\). Thus, it can also be said that an inductor is a component that can store an energy of \(\Delta E=\frac{1}{2}L{I}^{2}\), under the application of an electric current. A textbook example is a solenoidal inductor, which stores energy as a magnetic-field energy1. Other inductors possess similar energy-storing properties. An established example is the so-called kinetic inductor, in which the energy is stored as the kinetic energy of mobile charge carriers. When considering the Drude model of conduction electrons, one can immediately find that the inductance defined using the imaginary part of the angular-frequency (ω)-dependent resistivity, ρ(ω), agrees with the inductance defined using the total kinetic energy of electrons2.

Recently, a new class of inductors, now referred to as emergent inductors, has been proposed theoretically3 and confirmed experimentally4,5,6. In these inductors, the flowing conduction electrons exert a spin-transfer torque (STT)7,8,9,10 on the underlying magnetic texture; as a result, the magnetic texture exhibits time-dependent elastic deformations under an AC electric current in the linear-response regime. Such current-induced magnetic-texture dynamics exert a back action on the flowing conduction electrons, yielding the so-called spin-motive force or emergent electric field (EEF)11,12,13,14,15. This phenomenon can be derived microscopically in terms of the so-called spin-Berry phase or the effective U(1) gauge field, and the resulting EEF can be described by

$${e}_{i}({{{\boldsymbol{r}}}},t)=\frac{\hslash }{2| e| }{{{\boldsymbol{m}}}}({{{\boldsymbol{r}}}},t)\cdot [{\partial }_{i}{{{\boldsymbol{m}}}}({{{\boldsymbol{r}}}},t)\times {\partial }_{t}{{{\boldsymbol{m}}}}({{{\boldsymbol{r}}}},t)],$$
(3)

where e(>0) is the elementary charge, m(r, t) is the unit vector of the local magnetic moment at position r and time t, and ∂i (i = x, y, z) and ∂t denote spatial and time derivatives, respectively (when the conduction-electron spins are not fully polarized, the so-called spin-polarization factor P is further multiplied on the right-hand side of Eq. (3)12,15). It has been numerically demonstrated that in the so-called adiabatic limit (i.e., β = 0; see the “Methods” section), the inductance value defined using the EEF under an AC electric current quantitatively agrees with that defined using the current-induced magnetic-texture-deformation energy16. Thus, in the adiabatic limit, the emergent inductance is well defined, and both the electric and energetic responses are correctly captured by Eq. (1). On the other hand, when nonadiabaticity is concerned (i.e., β ≠ 0), the inductance values derived independently from the two definitions do not match, implying that the system responses are beyond the framework of Eq. (1) and hence the inductance interpretation does not apply16.

An interesting aspect of emergent inductors is that the inductive electric response is potentially not limited to the applied current direction but may also appear along the perpendicular directions, as inferred from Eq. (3). Thus, in general, the emergent inductance, when it is well defined, should be represented by a tensor: \({V}_{i}={L}_{ij}\frac{{{{\rm{d}}}}{I}_{j}}{{{{\rm{d}}}}t}(i,j=x,y)\) or

$$\left(\begin{array}{c}{V}_{x}\\ {V}_{y}\end{array}\right)=\left(\begin{array}{ll}{L}_{xx}&{L}_{xy}\\ {L}_{yx}&{L}_{yy}\end{array}\right)\frac{{{{\rm{d}}}}}{{{{\rm{d}}}}t}\left(\begin{array}{c}{I}_{x}\\ {I}_{y}\end{array}\right).$$
(4)

In classical electrodynamics, such an inductance tensor with i, j = 1, 2 may be introduced to describe two mutually coupled coils, 1 and 2. It, therefore, appears that an emergent inductor possesses a function similar to that of a coupled classical inductor system. However, such an intuitive analogy requires careful consideration because the microscopic mechanism is quite different between classical and emergent inductors. For instance, in a coupled classical inductor system, one can analytically express the mutual inductance and find L12 = L21 ≡ M1; moreover, the fact that the coupled system stores a positive energy in the quadratic form of \(\frac{1}{2}{L}_{11}{I}_{1}^{2}+\frac{1}{2}{L}_{22}{I}_{2}^{2}+M{I}_{1}{I}_{2}\) for arbitrary values of I1 and I2 leads to a constraint, L11L22 ≥ M217, in addition to the obvious one, L11, L22 ≥ 0. Such classical electrodynamics considerations, however, are not helpful for emergent inductors consisting of an arbitrary spin texture including disorder, and thus, the relation between Lxy and Lyx appears to be nontrivial.

When considering the nature of Lij, it is instructive to review the resistivity tensor, ρij, as a textbook example. Note that any second-order tensor, Kij, can always be decomposed into a symmetric part, \({K}_{ij}^{{{{\rm{S}}}}}\), and an antisymmetric part, \({K}_{ij}^{{{{\rm{A}}}}}\); namely, \({K}_{ij}={K}_{ij}^{{{{\rm{S}}}}}+{K}_{ij}^{{{{\rm{A}}}}}\) with \({K}_{ij}^{{{{\rm{S}}}}}=({K}_{ij}+{K}_{ji})/2\) and \({K}_{ij}^{{{{\rm{A}}}}}=({K}_{ij}-{K}_{ji})/2\). In the case of ρij, the symmetric part represents dissipative transport, whereas the antisymmetric part represents nondissipative transport, that is, the Hall resistivity. Thus, the symmetric and antisymmetric parts of ρij have their own physical meanings with quite different characteristics. Therefore, the symmetry of the emergent inductance tensor is also an important issue in understanding the underlying physics.

In this paper, focusing on the adiabatic limit, in which the emergent inductance is well defined by Eq. (1)16, we aim to reveal the symmetry of the emergent inductance tensor and discuss the physical implications of the revealed symmetry. Our approach is two-fold. First, we consider the tensor-expressed circuit equation [Eq. (4)] in detail and draw a conclusion regarding the symmetry of Lij: this also enables us to discuss how the inductor tensor should behave under the time-reversal operation. Second, we numerically investigate Lij for various magnetic textures using micromagnetic simulations. These two approaches consistently show that Lij is a symmetric tensor (that is, Lxy = Lyx) and Lij is even under the time-reversal operation. By combining the numerical results and symmetry arguments, we also find what kinds of magnetic textures can or cannot exhibit a transverse emergent inductance, Lyx. We note that the present conclusion is for the case where the emergent inductance is well-defined (i.e., the adiabatic limit, β = 0). The effect of nonadiabaticity (i.e., β ≠ 0), which makes the emergent inductance ill-defined16, is discussed in the Supplementary Note 1.

Results

Considerations for the circuit equation

We discuss the consequences that are prescribed in the tensor-expressed circuit equation, Eq. (4). Following the general arguments on a second-order tensor, we decompose Lij into symmetric and antisymmetric parts: \({L}_{ij}={L}_{ij}^{{{{\rm{S}}}}}+{L}_{ij}^{{{{\rm{A}}}}}\), or explicitly,

$$\left(\begin{array}{ll}{L}_{xx}&{L}_{xy}\\ {L}_{yx}&{L}_{yy}\end{array}\right)=\left(\begin{array}{ll}{L}_{xx}^{{{{\rm{S}}}}}&{L}_{xy}^{{{{\rm{S}}}}}\\ {L}_{xy}^{{{{\rm{S}}}}}&{L}_{yy}^{{{{\rm{S}}}}}\end{array}\right)+\left(\begin{array}{cc}0&{L}_{xy}^{{{{\rm{A}}}}}\\ -{L}_{xy}^{{{{\rm{A}}}}}&0\end{array}\right).$$
(5)

To gain insight into the physical meaning of the symmetric and antisymmetric tensors, we consider the work done by the power source along a closed loop in the Ix–Iy plane, which is expressed as:

$$\begin{array}{lll}{\displaystyle\oint} \,{{{\rm{d}}}}t\,({I}_{x}{V}_{x}+{I}_{y}{V}_{y}) &=& {\displaystyle\oint} \,{{{\rm{d}}}}\left(\dfrac{1}{2}{L}_{xx}^{{{{\rm{S}}}}}{I}_{x}^{2}+\dfrac{1}{2}{L}_{yy}^{{{{\rm{S}}}}}{I}_{y}^{2}+{L}_{xy}^{{{{\rm{S}}}}}{I}_{x}{I}_{y}\right)\\ &&+\,{\displaystyle\oint} \,{{{\rm{d}}}}t\,{L}_{xy}^{{{{\rm{A}}}}}\left({I}_{x}\dfrac{{{{\rm{d}}}}}{{{{\rm{d}}}}t}{I}_{y}-{I}_{y}\dfrac{{{{\rm{d}}}}}{{{{\rm{d}}}}t}{I}_{x}\right)\end{array}.$$
(6)

Note that the first term in the right-hand side consists of only the symmetric tensor components and the integrand takes the form of a total derivative; hence, the contour integral results in zero. The expression, \(\frac{1}{2}{L}_{xx}^{{{{\rm{S}}}}}{I}_{x}^{2}+\frac{1}{2}{L}_{yy}^{{{{\rm{S}}}}}{I}_{y}^{2}+{L}_{xy}^{{{{\rm{S}}}}}{I}_{x}{I}_{y}\), is essentially the same as that derived for mutually coupled classical inductors, representing energy stored in the emergent inductor under a current. In contrast, the second term consists of only the antisymmetric-tensor components, and the integrand is not the form of a total derivative, indicating that the second term is non-zero and dependent on the path. These features imply that \({L}_{xy}^{{{{\rm{A}}}}}\) is associated with a non-conserved quantity.

To see the consequences of the antisymmetric component \({L}_{xy}^{{{{\rm{A}}}}}\) more clearly, it is helpful to consider a specific closed path for the integral(s) of Eq. (6). Suppose \({L}_{xy}^{{{{\rm{A}}}}} \,>\, 0\); we consider a specific cycle C that consists of three paths, C1, C2, and C3, as shown in Fig. 1: \(({I}_{x},{I}_{y})=(0,0)\mathop{\longrightarrow }\limits^{{C}_{1}}({I}_{0},{I}_{0})\mathop{\longrightarrow }\limits^{{C}_{2}}({I}_{0},0)\mathop{\longrightarrow }\limits^{{C}_{3}}(0,0)\) with constraints of Ix = Iy on C1, Ix = I0 on C2, and Iy = 0 on C3. Thus, taking the contour integral along the cycle in the clockwise direction results in:

$${\oint_{C}}{{{\rm{d}}}}t\,({I}_{x}{V}_{x}+{I}_{y}{V}_{y})=-{L}_{xy}^{{{{\rm{A}}}}}{I}_{0}^{2} \,<\, 0.$$
(7)

The result indicates that if a positive \({L}_{xy}^{{{{\rm{A}}}}}\) were present, the power source could acquire energy by cycling the closed loop. Such behavior is obviously not allowed for a passive element, such as a stable material. Similarly, one can consider the case of \({L}_{xy}^{{{{\rm{A}}}}}\, <\, 0\), and the same conclusion can be drawn by considering the same closed loop C but in the counterclockwise direction. Thus, Eq. (4) concludes that even for the case of an emergent inductor, the inductance tensor cannot have an antisymmetric component; that is, \({L}_{xy}^{{{{\rm{A}}}}}=0\), and an emergent inductor tensor should be a symmetric tensor (below, we therefore omit the superscript, S),

$${L}_{xy}={L}_{yx}\equiv {L}_{tr}.$$
(8)

Hence, the energy stored in an emergent inductor under current is found to be expressed by \(\frac{1}{2}{L}_{xx}{I}_{x}^{2}+\frac{1}{2}{L}_{yy}{I}_{y}^{2}+{L}_{tr}{I}_{x}{I}_{y}\), and for this quadratic form to be nonnegative, Lij should satisfy

$${L}_{xx}{L}_{yy}\ge {({L}_{tr})}^{2},$$
(9)

in addition to Lxx, Lyy ≥ 016.

Fig. 1
figure 1

A specific closed loop used to prove the absence of the antisymmetric components of an inductance tensor.

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Thus, although the microscopic mechanism is quite different between classical and emergent inductors, it turns out that there is no difference in the constraints that inductance tensors should satisfy. These characteristics are implicitly prescribed by the relation between voltage and current, Eq. (4), not depending on the microscopic mechanism for inductors.

Having established the symmetry of Lij, we can discuss the behavior of Lij under the time-reversal operation. Since an emergent inductance arises from a magnetic texture {m(r)}, the behavior of an emergent inductance under the time-reversal operation is an interesting issue. In fact, in experiments, the magnetic-field (B)-dependence of an emergent inductance has been frequently investigated4,5,6. To incorporate a case where {m(r)} shows hysteretic behavior with respect to changes in B, Lij may be expressed as a function with {m(r)} and B as variables. Note that regardless of the details of the variables, Lij should be a symmetric tensor as discussed above, and hence, Lij(B, {m(r)}) = Lji(B, {m(r)}) should always be satisfied. Moreover, with respect to the complex resistivity, Onsager’s reciprocal theorem concludes \({{\mathrm{Im}}}\,{\rho }_{ij}(\omega ,{{{\boldsymbol{B}}}},\{{{{\boldsymbol{m}}}}({{{\boldsymbol{r}}}})\})={{\mathrm{Im}}}\,{\rho }_{ji}(\omega ,-{{{\boldsymbol{B}}}},\{-{{{\boldsymbol{m}}}}({{{\boldsymbol{r}}}})\})\) (the real part also satisfies the same relation)18; hence, the inductance tensor should also satisfy Lji(B, {m(r)}) = Lij( −B, { − m(r)}). By combining the two relations regarding Lij, one can thus conclude

$${L}_{ij}({{{\boldsymbol{B}}}},\{{{{\boldsymbol{m}}}}({{{\boldsymbol{r}}}})\})={L}_{ij}(-{{{\boldsymbol{B}}}},\{-{{{\boldsymbol{m}}}}({{{\boldsymbol{r}}}})\}).$$
(10)

This relation indicates that Lij is even under time reversal, or equivalently, Lij is a polar symmetric tensor. In particular, we note Lyx(B, {m(r)}) = Lyx( −B, { − m(r)}), distinct from the Hall resistivity, which satisfies ρyx(B, {m(r)}) = −ρyx(−B, { − m(r)}). For this reason, we call Lyx the transverse inductance, not the Hall inductance.

Micromagnetic simulations

To observe the symmetry of the inductance tensor of emergent inductors, we consider magnetic textures that slowly vary in space; for such magnetic textures, the EEF can be calculated according to Eq. (3). We further consider the pinned regime, in which a magnetic texture does not exhibit a steady flow under a DC electric current19,20,21,22,23,24,25,26,27,28. The procedure for calculating the emergent inductance arising from a slowly varying magnetic texture in the pinned regime is detailed in the literature16 and also in the “Methods” section. We consider a spin Hamiltonian based on the continuum approximation that can exhibit helical and skyrmion lattice (SkL)29,30,31,32 magnetic textures and calculate the current-induced dynamics of a magnetic texture by numerically solving the Landau–Lifshitz–Gilbert (LLG) equation33 (see the “Methods” section). To be more specific, the magnetic-texture dynamics under the application of an AC current along the x-direction are calculated by micromagnetic simulation; then, by referring to Eq. (3), the time-dependent EEFs along both the x- and y-directions are further derived; and finally, by referring to Eq. (4), Lxx and Lyx are obtained. Similarly, we obtain Lyy and Lxy by simulating the case of an AC current along the y-direction. The emergent inductance Lij depends on the system size in the form of \({L}_{ij}={\tilde{L}}_{ij}\frac{\ell }{S}\), where \({\tilde{L}}_{ij},\ell\), and S represent the normalized inductance (we call it “inductivity”), system length, and system cross-section area. Below, we, therefore, present \({\tilde{L}}_{ij}\), rather than the system-size-dependent Lij. The inductivity tensor may be defined by

$${e}_{i}={\tilde{L}}_{ik}\frac{{{{\rm{d}}}}{j}_{k}}{{{{\rm{d}}}}t},$$
(11)

where j represents the current density. The following simulation results are obtained for the case of β = 0 (i.e., the adiabatic limit).

Figure 2 summarizes the magnetic textures investigated in this study and the corresponding inductance tensors. We studied four examples of helical magnetic textures, for which the helical q-vector forms approximately an angle θ = 0°, ± 20°, and ± 45° with respect to the x-direction (Fig. 2a–e, respectively); a maze-helix texture (Fig. 2f); and an SkL (Fig. 2g). The intensity and concentration of disorder were minimized as much as possible while confirming the linear response of the pinned dynamics. As a result, more disorders had to be included when examining the maze-helix and SkL, as summarized in Table 1: selecting a much lower current density while keeping the disorder density as low as 0.3% was not appropriate in terms of the required numerical accuracy. As shown in Fig. 2, we find that \({\tilde{L}}_{xy}={\tilde{L}}_{yx}\) invariably holds within the numerical error, consistent with the conclusion derived from the circuit equations.

Fig. 2: Various metastable magnetic textures and corresponding inductance tensors.
figure 2

a–e Helical magnetic textures with the helical q-vector that forms approximately an angle θ = 0° (a), 20° (b) −20° (c), 45° (d), and −45° (e) with respect to the x-direction. f Maze-helix. g Skyrmion lattice. The corresponding fast-Fourier-transform (FFT) images are also shown in each panel. Color wheels specify the x–y plane magnetization direction. The brightness of the color represents the z component of the magnetization, and white represents the local magnetizations pointing toward the z-direction. The current-induced magnetic-texture dynamics are calculated under the application of a weak AC current. The parameters used for the simulation are tabulated in Table 1; they are chosen so that the resulting emergent voltage is in the linear response and low-frequency regimes (see the “Methods” section). The simulations were done for β = 0.

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Table 1 Parameters used for the micromagnetic simulations displayed in Fig. 2.
Full size table

The diagonal components of the inductance tensor are invariably positive, whereas the off-diagonal components can be either positive or negative. Nevertheless, we emphasize that the inductance tensor retains energetic interpretations; that is, the energy increase in the magnetic system, ΔE(Ix, Iy), caused by the application of an electric current agrees with \(\frac{1}{2}{L}_{xx}{I}_{x}^{2}\)\(+\frac{1}{2}{L}_{yy}{I}_{y}^{2}+{L}_{tr}{I}_{x}{I}_{y}\). As an example, we discuss the results for the helical texture with θ = − 20°, in which the off-diagonal components of \({\tilde{L}}_{ij}\) are negative. The ΔE(Ix, Iy) is calculated for the following three cases independently: (i) (Ix ≠ 0, Iy = 0), (ii) (Ix = 0, Iy ≠ 0), and (iii) (Ix ≠ 0, Iy ≠ 0). Then, by solving the three simultaneous equations regarding \(\Delta E({I}_{x},{I}_{y})=\frac{1}{2}{L}_{xx}{I}_{x}^{2}+\frac{1}{2}{L}_{yy}{I}_{y}^{2}+{L}_{tr}{I}_{x}{I}_{y}\), we can obtain: \(({\tilde{L}}_{xx},{\tilde{L}}_{yy},{\tilde{L}}_{tr})=(2.67,0.63,-1.00)\times 1{0}^{-21}\) H m. These values are in quantitative agreement with \({\tilde{L}}_{ij}\) calculated from the EEF (Fig. 2c), indicating that the emergent inductivity is well defined by Eq. (11). We also confirmed \({\tilde{L}}_{xy}({{{\boldsymbol{B}}}},{{{\boldsymbol{m}}}}({{{{\boldsymbol{r}}}}}_{k}))={\tilde{L}}_{xy}(-{{{\boldsymbol{B}}}},-{{{\boldsymbol{m}}}}({{{{\boldsymbol{r}}}}}_{k}))\) numerically (Fig. 3), in which the disorder density and strength Kimp (see “Methods”) are fixed to 3% and 1.0 × 107 J m−3, respectively, and the single-q-helix with θ = 45° was considered. Thus, our numerical study confirms that \({\tilde{L}}_{ij}\) for an emergent inductor is a polar symmetric tensor.

Fig. 3: Magnetic-field dependence of the transverse inductivity.
figure 3

H, SkL, and F denote the single-q-helix with θ = 45°, skyrmion lattice, and ferromagnetic state, respectively. The disorder density and strength Kimp (see “Methods”) is fixed to 3% and 1.0 × 107 J m−3, respectively. The simulations were done for β = 0. When β is non-zero, the effective inductivity tensor \({\tilde{L}}_{ij}^{{{{\rm{eff}}}}}\) should be discussed (\({\tilde{L}}_{ij}^{{{{\rm{eff}}}}}={{\mathrm{Im}}}\,[{\rho }_{ij}(\omega )-{\rho }_{ij}(0)]/\omega\)): The magnetic-field dependence of the transverse components of \({\tilde{L}}_{ij}^{{{{\rm{eff}}}}}\) at finite β is shown in Fig. S1 and discussed in Supplementary Note 2. Note that \({\tilde{L}}_{ij}^{{{{\rm{eff}}}}}\) is a different quantity from the inductivity tensor defined by Eq. (11).

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When comparing the three helical textures quantitatively, one can find that as the θ increases from 0° to 45°, \({\tilde{L}}_{xx}\) decreases, whereas \({\tilde{L}}_{xy}\) increases. We also note that in the maze-helix and SkL textures, the transverse component, \({\tilde{L}}_{xy}\), is more than one order of magnitude smaller than the longitudinal components, \({\tilde{L}}_{xx}\) and \({\tilde{L}}_{yy}\). As discussed below, these observations can be explained by considering an orthogonal transformation of \({\tilde{L}}_{ij}\) and the rotational symmetry that each magnetic texture has.

Discussion

In the following, we aim to categorize inductance tensors of a magnetic-texture origin and consider how our numerical results obtained in the adiabatic limit can be explained in terms of the symmetry operations that each magnetic system has. Note that because an inductance tensor is real and symmetric, it can be diagonalized by performing an appropriate orthogonal transformation, R, or equivalently by choosing appropriate Cartesian coordinates:

$$\left(\begin{array}{ll}{\tilde{L}}_{xx}&{\tilde{L}}_{tr}\\ {\tilde{L}}_{tr}&{\tilde{L}}_{yy}\end{array}\right)\,\mathop{\to }\limits^{R}\left(\begin{array}{cc}{\lambda }_{1}&0\\ 0&{\lambda }_{2}\end{array}\right),$$
(12)

where λ1 and λ2 (λ1, λ2 ≥ 0) represent the eigenvalues of the inductance tensor. Hence, to classify an emergent inductance tensor, it is sufficient to consider the diagonalized form. This approach does not lose generality because a representation in different Cartesian coordinates can be immediately obtained by performing the corresponding orthogonal transformation. Following the group theory arguments for a polar symmetric tensor, one can conclude that: (i) when the system has three-fold or higher rotational symmetry with respect to the z-axis (i.e., C3z, C4z, C6z or C∞z), λ1 and λ2 should be equal, whereas (ii) when the system has only two-fold with respect to the z-axis (C2z) or no rotational symmetry, λ1 and λ2 should be inequivalent: For the details of the derivation, see Supplementary Note 3. Thus, the diagonalized two-by-two tensor can be classified as one of the two categories, which are characterized by λ1 = λ2 and λ1 ≠ λ2, respectively.

The first category, λ1 = λ2, is represented by an isotropic tensor \(\left(\begin{array}{rc}1&0\\ 0&1\end{array}\right)\), and thus, the off-diagonal components are always zero for arbitrarily chosen Cartesian coordinates; that is, the transverse inductance response does not appear. From group theory, a magnetic texture that has C3z, C4z, C6z, or C∞z symmetry should belong to this category. Note that our numerical calculations deal with finite-size systems including randomly distributed disorder, and therefore, the simulated magnetic textures do not have any rotational symmetry in a strict sense. Nevertheless, we numerically find that the inductance tensors of the maze-helix and SkL textures satisfy \({\tilde{L}}_{xx}\approx {\tilde{L}}_{yy}\) and \({\tilde{L}}_{xy},{\tilde{L}}_{yx}\ll {\tilde{L}}_{xx},{\tilde{L}}_{yy}\) (Fig. 2f, g, respectively), indicating that the obtained tensors are close to isotropic. These results appear reasonable, considering that in a macroscopic system, the maze-helix and SkL textures have global approximate C∞z and C6z symmetries, respectively. The symmetry of the macroscopic systems can be imagined by looking at the corresponding fast-Fourier-transform (FFT) images. To be precise, the rotation symmetry of the SkL confined in the finite-size system is C2z, rather than C6z, as indicated by the FFT image (Fig. 2g): This perturbative symmetry lowering from C6z to C2z explains the small but finite symmetric off-diagonal component, which is originally prohibited under C6z symmetry. When nonadiabaticity is not negligible, the effective inductivity tensor defined by \({\tilde{L}}_{ij}^{{{{\rm{eff}}}}}={{\mathrm{Im}}}\,[{\rho }_{ij}(\omega )-{\rho }_{ij}(0)]/\omega\) is discussed, but it should be noted that \({\tilde{L}}_{ij}^{{{{\rm{eff}}}}}\) is a different quantity from the inductivity tensor in Eq. (11). For instance, the \({\tilde{L}}_{ij}^{{{{\rm{eff}}}}}\) of the SkL has antisymmetric off-diagonal components when β ≠ 0, although the antisymmetric component in \({\tilde{L}}_{ij}\) is energetically prohibited; for more details, see Supplementary Notes 1 and 3.

The second category consists of tensors that have two inequivalent components, \(\left(\begin{array}{rc}{\lambda }_{1}&0\\ 0&{\lambda }_{2}\end{array}\right)\), and thus, off-diagonal components can appear if arbitrary Cartesian coordinates are chosen. For instance, the matrix R(θ) that rotates Cartesian coordinates clockwise by θ transforms the diagonalized tensor into a nondiagonal form:

$$\left(\begin{array}{cc}{\lambda }_{1}&0\\ 0&{\lambda }_{2}\end{array}\right)\mathop{\longrightarrow }\limits^{R(\theta )}\left(\begin{array}{ll}{\lambda }_{1}{\cos }^{2}\theta +{\lambda }_{2}{\sin }^{2}\theta &({\lambda }_{1}-{\lambda }_{2})\sin \theta \cos \theta \\ ({\lambda }_{1}-{\lambda }_{2})\sin \theta \cos \theta &{\lambda }_{1}{\sin }^{2}\theta +{\lambda }_{2}{\cos }^{2}\theta \end{array}\right).$$
(13)

Thus, \({\tilde{L}}_{xy}={\tilde{L}}_{yx}\) can be either positive or negative depending on the selection of Cartesian coordinates. An example of this category is a single-q-helix, in which the inductance tensor is diagonalized, for instance, when the x-axis is chosen parallel to the helical q-vector. An important feature of an ideal single-q-helix is that the local magnetic moments show no modulation along the direction perpendicular to q. Hence, no STT effect is expected for the current along the y-axis, resulting in λ2 = 0. Thus, for the case of an ideal single-q-helix, the diagonalized form and its orthogonal transformation are given as:

$$\left(\begin{array}{cc}1&0\\ 0&0\end{array}\right)\mathop{\longrightarrow }\limits^{R(\theta )}\left(\begin{array}{cc}{\cos }^{2}\theta &\sin \theta \cos \theta \\ \sin \theta \cos \theta &{\sin }^{2}\theta \end{array}\right).$$
(14)

Figure 4 displays the comparison between the numerically obtained inductivity tensors of various q-direction helices and the orthogonal transformation of λ1 = 3.11 × 10−21 H m and λ2 = 0, which is an approximate inductivity tensor of the single-q-helix with θ = 0° (Fig. 2a). Although the simulated single-q-helices are more or less affected by random disorder and the open boundaries of the system, the overall tendency is well reproduced by orthogonal transformation. This observation demonstrates that the above arguments based on the orthogonal transformation of the symmetric inductivity tensor are helpful when considering a single-q-helix with arbitrary q-direction.

Fig. 4: Numerically obtained inductivity tensors of helical magnetic textures for various q-directions.
figure 4

The θ = 0 represents the q-direction parallel to the x-axis. The solid symbols are the data obtained by the micromagnetic simulations, and the solid curves represent the corresponding trigonometric functions multiplied by λ1 = 3.11 × 10−21 H m. The simulations were done for β = 0.

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For a more complicated magnetic texture, positive λ1 and λ2 with λ1 ≠ λ2 may be expected. For instance, a multidomain of single-q-helices obviously belongs to this category. In contrast, a long-range ordered state belonging to this category may not be so clear. A candidate of this category is likely a magnetic texture that has multiple q vectors with different wavenumbers, such as those observed in EuAg4As234 and EuAl435; however, such an anisotropic magnetic texture is beyond the scope of our model Hamiltonian based on the continuum approximation [see Eq. (15) in the “Methods” section].

To conclude, we have revealed analytically and numerically the symmetry of the emergent inductance tensor exhibited by pinned magnetic textures. We focused on the adiabatic limit, where the inductance tensor is well defined by \({e}_{i}={\tilde{L}}_{ik}\frac{{{{\rm{d}}}}{j}_{k}}{{{{\rm{d}}}}t}\). We thus found that the inductance tensor is a real symmetric tensor, and hence, the presence and magnitude of the transverse component are determined by the degree to which the measurement axis is tilted from the principal axis that generates the diagonalized tensor. As a natural consequence of the real symmetric tensor, the transverse component does not change sign with respect to the magnetic-field reversal. These fundamental aspects of the emergent inductance tensor will be useful when exploring the transverse inductive response in a magnetic texture. However, it must also be noted that when nonadiabaticity is not negligible, the electric response produced by magnetic textures may not be described by an inductance tensor in the strict sense defined by Eq. (4).

Methods

Numerical model

In this study, we consider long-period helical magnetic textures that are stabilized by the Dzyaloshinskii–Moriya (DM) interaction36,37. We consider dirty systems including randomly distributed disorder. Our model Hamiltonian is:

$$\begin{array}{ll}{{{\mathscr{H}}}}\,=\,{\displaystyle\int}\,\dfrac{{{{{\rm{d}}}}}^{3}r}{{a}^{3}}\left[\dfrac{J}{2}{(\nabla {{{\boldsymbol{m}}}})}^{2}+D{{{\boldsymbol{m}}}}\cdot (\nabla \times {{{\boldsymbol{m}}}})\right]\\ \qquad\quad- \displaystyle\mathop{\sum}\limits_{k\in \Lambda }{\displaystyle\int}_{\!{V}_{k}}{{{{\rm{d}}}}}^{3}r\,{K}_{{{{\rm{imp}}}}}{({{{{\boldsymbol{m}}}}}_{k}\cdot {{{{\boldsymbol{n}}}}}_{{{{\rm{imp}}}},k})}^{2}\end{array}$$
(15)

where J is the Heisenberg exchange energy, D is the DM interaction and a is the lattice constant. The extrinsic pinning effect is controlled by the last term of Eq. (15), which is introduced to randomly selected cells to break the translational symmetry: Kimp(>0) represents the magnetic-easy-axis anisotropy along a randomly chosen direction, nimp,k, at the k-th cell (the cell volume Vk is 33 nm3), and Λ is a set of random numbers. The disorder density displayed in Table 1 represents the ratio of the number of cells with finite Kimp to the total number of cells (243 × 243).

When simulating the current-induced dynamics of a given helical magnetic structure, we insert the spin Hamiltonian into the following Landau–Lifshitz–Gilbert (LLG) equation33:

$$\frac{{{{\rm{d}}}}{{{{\boldsymbol{m}}}}}_{{{{\boldsymbol{r}}}}}(t)}{{{{\rm{d}}}}t}\,=\,-\frac{| \gamma | }{1+{\alpha }^{2}}\frac{{{{\rm{d}}}}{{{\mathscr{H}}}}}{{{{\rm{d}}}}{{{{\boldsymbol{m}}}}}_{{{{\boldsymbol{r}}}}}}\times {{{{\boldsymbol{m}}}}}_{{{{\boldsymbol{r}}}}}-\frac{\alpha | \gamma | }{1+{\alpha }^{2}}\left[{{{{\boldsymbol{m}}}}}_{{{{\boldsymbol{r}}}}}\times \left(\frac{{{{\rm{d}}}}{{{\mathscr{H}}}}}{{{{\rm{d}}}}{{{{\boldsymbol{m}}}}}_{{{{\boldsymbol{r}}}}}}\times {{{{\boldsymbol{m}}}}}_{{{{\boldsymbol{r}}}}}\right)\right]\\ \qquad\qquad\quad+\frac{1}{1+{\alpha }^{2}}\left\{(1+\beta \alpha )\right.{{{{\boldsymbol{m}}}}}_{{{{\boldsymbol{r}}}}}\times [{{{{\boldsymbol{m}}}}}_{{{{\boldsymbol{r}}}}}\times ({{{\boldsymbol{u}}}}\cdot {{{\boldsymbol{\nabla }}}}){{{{\boldsymbol{m}}}}}_{{{{\boldsymbol{r}}}}}]\\ \qquad\qquad\quad +(\beta -\alpha )\left.[{{{{\boldsymbol{m}}}}}_{{{{\boldsymbol{r}}}}}\times ({{{\boldsymbol{u}}}}\cdot {{{\boldsymbol{\nabla }}}}){{{{\boldsymbol{m}}}}}_{{{{\boldsymbol{r}}}}}]\right\},$$
(16)

where u represents the spin drift velocity, α is the Gilbert damping constant, β is a dimensionless constant that characterizes the nonadiabatic electron spin dynamics, and γ ( > 0) is the gyromagnetic ratio; u is related to the electric current density j by \({{{\boldsymbol{u}}}}=\frac{P{\mu }_{{{{\rm{B}}}}}}{2| e| {M}_{{{{\rm{s}}}}}(1+{\beta }^{2})}{{{\boldsymbol{j}}}}\), where μB is the Bohr magneton and Ms is the saturation magnetization. When implementing the micromagnetic simulation, we use the open software MuMax3 (https://mumax.github.io/download.html)38. We choose the following parameter set: J/(2a3) = 1.8 × 10−11 J m−1, D/a3 = 2.8 × 10−3 J m−2, Ms = 2.45 × 105 A m−1, P = 1, and α = 0.04.

In the simulation, we apply a current density of a sufficiently small magnitude so that the magnetic system is certainly in the linear-response regime; that is, with respect to the input alternating electric current along the x- or y-direction, \({j}_{i}(t)={j}_{0,i}\sin \omega t\) (i = x, y), the magnetic system is in the pinned regime, and the output AC emergent voltage, Ve,i(t), is \(\propto {j}_{0,k}\omega \cos \omega t\,(i,k=x,y)\). Based on these observations, Lij is derived from the following equations:

$${V}_{e,i}(t)=\langle {e}_{i}(t)\rangle \ell ={L}_{ij}\frac{{{{\rm{d}}}}({I}_{j}(t))}{{{{\rm{d}}}}t},$$
(17)

where 〈⋯〉 denotes a spatially averaged value, the system length ℓ is 243 × 3 nm, and I = jS with the cross-section area S = 243 × 1 × 32 nm2. In the present frequency range (≤100 MHz), it is confirmed that the inductivity is independent of ω (i.e., 〈ei〉 ∝ ω) (Fig. S2) and the α dependence of the numerical results is negligibly small (Fig. S3) (see also Supplementary Note 4). The numerical accuracy of MuMax3 is Δm/∣m∣~10−7, and the typical increment of Δm/∣m∣ in one time step (4 ps) is ~10−5 under the current application of ~1010 A m−2. This finite accuracy eventually gives rise to an uncertainty of ~10−23 H m in the calculated inductivity.

In the numerical simulation, a uniform current density is considered to understand fundamental aspects of the inductivity tensor. On the other hand, the local ρxx and ρyx may be non-uniform in real material, reflecting spatial variations in magnetic textures. Nevertheless, the uniform current is a good approximation as long as 〈ρxx〉 ≫ δρxx, δρyx, where δρxx and δρyx represent the magnitude of the spatial variations. For instance, in the chiral magnet MnSi at 10 K, the presence or absence of the metastable skyrmion lattice changes ρxx and ρyx by ≈ 50 nΩ cm and ≈ 30 nΩ cm, respectively, whereas ρxx ≈ 5 μΩ cm39. Such magnetic-texture-dependent ρxx and ρyx imply that 〈ρxx〉 ≫ δρxx, δρyx holds, although the precise estimation of the spatial variations is experimentally difficult; thus, the current uniformity is well expected. If δρxx and δρyx are significant, the current distribution should be determined self-consistently; for instance, see ref. 40.

Initial-state preparation

To obtain various metastable magnetic textures, a pristine helical texture with a different oblique angle of the helical q-vector, a random spin configuration, or a SkL is prepared as an initial state and then relaxed under zero current. Note that imposing the open-boundary condition and introducing disorder are key in obtaining the intended magnetic textures.