Dissipationless Spin-Orbit Torque and Higher-Order Angular Response in Magnetic Insulators

Spin–orbit torque (SOT) has emerged as a leading mechanism for all-electrical control of magnetization, converting charge currents into spin currents through spin–orbit coupling. In conventional metallic heterostructures, a charge current flowing through a heavy metal generates a transverse spin current via the spin Hall effect; this spin current diffuses into an adjacent ferromagnet, exerting a torque that drives magnetization dynamics. Such observations have reinforced the notion that SOT necessarily relies on mobile conduction electrons.

In this work, we challenge that assumption by demonstrating a dissipationless SOT in a purely insulating ferromagnet. Using a minimal inversion-breaking magnetic-insulator model, we show that the time-reversal-even component of the torque remains finite deep within the band gap, despite the absence of itinerant carriers. The in-gap torque persists even in topologically trivial insulators (Chern number = 0), indicating that it is not directly tied to electronic topology. Nevertheless, its magnitude is strongly enhanced in topological phases with comparable gaps.

Furthermore, we reveal that the angular dependence of this dissipationless torque exhibits distinct higher-order harmonics beyond the conventional form. By introducing altermagnetic d-wave exchange pairing, we demonstrate a robust enhancement of these higher-order spin–orbit torque components. These results uncover a new pathway for symmetry-engineered, energy-efficient SOT control in insulating magnets, opening opportunities for next-generation spin–orbitronic devices.