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The geometry of magnetic textures provides a fundamental framework for understanding magnetism and emergent electrodynamics. Conventionally, magnetic textures have been classified into collinear, coplanar, and noncoplanar magnets, characterized primarily by the vector spin chirality (VSC) and scalar spin chirality (SSC). These geometric quantities also underlie various emergent phenomena, including transport, optical responses, and cross correlations, as exemplified by the SSC-induced topological Hall effect in magnetic skyrmions. Recently, however, nonreciprocal transport phenomena have been observed in noncoplanar magnets, such as conical magnets, revealing limitations of the conventional framework. In particular, the SSC vanishes identically in one-dimensional textures, even when the spin configuration is noncoplanar.

In this talk, we revisit magnetic textures from a differential-geometric viewpoint and introduce two new geometric quantities: the geodesic SSC and torsional SSC [1]. Based on these quantities, we establish a refined classification of noncoplanar magnets into three distinct classes. Furthermore, by extending conventional semiclassical theory, we show that the geodesic SSC induces an asymmetry in the electronic band structure, giving rise to nonreciprocal transport even in the absence of spin-orbit coupling. These results extend the conventional SSC-based framework of emergent electrodynamics, establishing a parallel role for the geodesic SSC in governing electronic responses.

 

[1] K. Shinada and N. Nagaosa, Phys. Rev. X 16, 031026 (2026).

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