Emergent toroidal induction in a polar Weyl ferromagnet

Abstract

Spin-orbit coupling (SOC) underpins modern spintronics by enabling the electrical generation of spin torques. Its reciprocal counterpart, in which magnetization dynamics produce electromotive forces through a spin-dependent Berry phase, is known as emergent electromagnetic induction (EEMI). However, this effect has previously been observed only in magnetic textures with spatial gradients, such as domain walls, helices, and skyrmions. Here, we demonstrate that even a spatially uniform ferromagnet can host EEMI through a previously unrecognized Berry-phase mechanism inherent to noncentrosymmetric conductors. In the polar Weyl ferromagnet PrAlGe, an applied alternating current generates spin-orbit torques that drive collective magnetization dynamics. The resulting emergent toroidal moment (T = P \times M), where (P) is the crystal's polar axis and (M) is the net magnetization, acts as a gauge potential whose time derivative (dT/dt) induces a Hall voltage. This contribution appears specifically in the out-of-phase component of the AC Hall response and scales linearly with frequency, providing direct evidence for EEMI. First-principles calculations further reveal that this toroidal vector encodes the collective motion of Weyl nodes in momentum space. These findings establish "emergent toroidal induction" as a new manifestation of spin-orbit entanglement, unifying Berry phase, topology, and spin dynamics while opening a pathway toward intrinsic and energy-efficient spin-charge interconversion.