Riemannian geometric classification and emergent phenomena of magnetic textures

Abstract

We propose a new classification of magnetic textures from the viewpoint of differential geometry. Magnetic textures are conventionally classified into collinear, coplanar, and noncoplanar magnets. These classes are typically characterized by the vector spin chirality (VSC) and the scalar spin chirality (SSC), which indicate noncollinearity and noncoplanarity, respectively. However, this conventional classification is incomplete: in particular, noncoplanar textures cannot be fully characterized by the SSC alone, as exemplified by conical magnets. To refine this classification, we analyze the curves and surfaces traced by spins in real space using differential geometry and introduce two novel scalar spin chiralities that properly characterize noncoplanarity: the geodesic scalar spin chirality and the torsional scalar spin chirality. These quantities are directly connected to differential geometry: the former reflects the geodesic curvature while the latter is related to the torsion. Based on these chiralities, we identify three distinct classes of noncoplanar magnetic textures. Furthermore, analogous to the roles of the VSC and the conventional SSC in emergent electrodynamics, the geodesic SSC gives rise to novel emergent phenomena. By constructing a semiclassical theory including nonadiabatic effects and higher-order spatial gradients of magnetic textures, we demonstrate that the geodesic SSC induces an emergent band asymmetry, leading to nonreciprocal responses as a quantum geometric effect. This mechanism is a purely orbital effect, requiring no spin-orbit coupling, and the resulting discussion runs in parallel with the conventional picture of the topological Hall effect driven by the SSC. The geometric viewpoint developed here will provide broad new insights into classification, quantum geometry, emergent electrodynamics, and a wider variety of emergent phenomena.

Figures (6)

• Figure 1: Schematic illustrations of the vector spin chirality (VSC), the scalar spin chirality (SSC), the geodesic SSC, and the torsional SSC in (a), (b), (c), and (d), respectively. The geometric meanings of these quantities on $\mathbb{S}^2$ are as follows: (a) VSC corresponds to the Riemannian metric $g_{ij}$ on $\mathbb{S}^2$, representing the infinitesimal distance $ds$ between nearby distinct points on $\mathbb{S}^2$ to which spins point, shown by the red segment. (b) SSC corresponds to an oriented infinitesimal area spanned by two tangent vectors. The blue shaded region represents the area element. (c) Geodesic SSC corresponds to the geodesic curvature $\kappa_g$. A curve satisfying $\kappa_{g} = 0$ is called a geodesic and lies on a great circle on $\mathbb{S}^2$. The blue shaded plane is a tangent space on a point on a great circle and the projected curve to the tangent space is straight, implying the intrinsic curvature of the curve itself is zero ($\kappa_g = 0$). On the other hand, for a curve with finite $\kappa_g$, the projected curve is intrinsically curved, as shown in the green shaded tangent space. (d) Torsional SSC corresponds to the torsion $\tau$. A curve with $\tau = 0$ lies in a single plane and, when constrained on $\mathbb{S}^2$, is a circle. If a curve deviates from a circle, it no longer lies in a single plane and acquires a finite torsion.
• Figure 2: Mapping of helical and conical magnetic textures to the unit sphere $\mathbb{S}^2$. For a helical magnet, the spin trajectory traces a geodesic (a great circle) on the unit sphere, and hence the geodesic curvature $\kappa_g$ vanishes. In contrast, the trajectory for a conical magnet is not a geodesic; therefore its geodesic curvature $\kappa_g$ is finite.
• Figure 3: (a) $\theta$-dependence of the skyrmion number $N_{\mathrm{sk}}$. (b) $\theta$ dependence of the spatial average of $\gamma_{xxx}$ (blue), $\gamma_{yyy}$ (red), and $\gamma_{xxy}$ (green) in the unit cell. We set $\braket{\gamma_{ijk}}$ in units of $Q^3$.
• Figure 4: (a), (f) Spin textures described by the normalized spin $\bm{n}(\bm{x}) = \bm{N}(\bm{x})/|\bm{N}(\bm{x})|$ in Eq. (\ref{['eq_2d_skyrmion']}) within a unit cell at $\theta =0$ and $\theta = \pi/4$. Spatial distributions of $\chi_{xy}$ [(b), (g)], $\gamma_{xxy} - \gamma_{yxx}$ [(c), (h)], $\gamma_{xxx}$ [(d), (i)], and $\gamma_{yyy}$ [(e), (j)]. We set $\chi_{xy}$ and $\gamma_{ijk}$ in units of $Q^2$ and $Q^3$, respectively.
• Figure 5: Energy dispersion $E(k) = \braket{H}(\hbar k)$ for the conical magnet. We fix $k_F = 1 \mathrm{\AA}^{-1}$ and $E_F = \hbar^2 k_F^2 / 2m$, and the energy dispersion is shown near $\pm k_F$ and $E_F$. The solid and dashed lines correspond to the energy dispersions near $+k_F$ and $-k_F$, respectively.