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Riemannian Geometric Classification and Emergent Phenomena of Magnetic Textures

Koki Shinada1,* and Naoto Nagaosa1,2,†

  • *Contact author: koki.shinada@riken.jp
  • †Contact author: nagaosa@riken.jp
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Phys. Rev. X 16, 031026 – Published 4 August, 2026

DOI: https://doi.org/10.1103/2szj-s2nx

Abstract

We propose a new classification of magnetic textures from a differential-geometric viewpoint. Magnetic textures are conventionally classified into collinear, coplanar, and noncoplanar magnets according to the relative arrangement of spins. These classes are typically characterized by the vector spin chirality (VSC) and the scalar spin chirality (SSC), which serve as indicators of 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 the differential geometry of curves on surfaces: the former is associated with the geodesic curvature, which quantifies the deviation from a geodesic, while the latter is related to the torsion. Based on these chiralities, we identify three distinct classes of noncoplanar magnetic textures. Furthermore, in close analogy with 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 that incorporates nonadiabatic effects and higher-order gradients of the spatial modulation of spins, we demonstrate that the geodesic SSC induces an emergent band asymmetry, leading to a nonreciprocal response 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.

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Popular Summary

Classifying complex magnetic textures has become increasingly challenging because conventional frameworks based solely on vector and scalar spin chiralities cannot capture their full structural variety or the phenomena they generate. We addressed this limitation by introducing a classification framework for magnetic structures rooted in differential geometry, identifying two new geometric quantities called the geodesic and torsional scalar spin chiralities. Our framework refines the taxonomy of magnetic textures and reveals previously overlooked geometric contributions to electron transport. We demonstrated that the geodesic scalar spin chirality enters directly into the semiclassical equations of motion of electrons as a distinct quantum geometric effect, playing a role analogous to how conventional scalar spin chirality acts as an effective magnetic field to drive the topological Hall effect. These results provide an analytical language capable of describing emergent electronic phenomena that fall outside the scope of conventional emergent electrodynamics. Our work establishes differential geometry as a predictive tool for analyzing complex ordered states and engineering nonreciprocal electronic responses in magnetic materials.

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Phys. Rev. X 16, 031026– Published 4 August, 2026
  • Received 16 March 2026
  • Revised 12 June 2026
  • Accepted 25 June 2026
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