Theory of magnetic octupoles in crystalline solids

Magnetic octupoles have been proposed as order parameters characterizing the magnetic structure of time-reversal-symmetry breaking centrosymmetric antiferromagnets. In classical electromagnetism, the magnetic octupole is defined as the second-order moment of the magnetic moment density. Hence, its straightforward quantum mechanical expression contains position operators, and the expectation value depends on the choice of gauge. In this talk, we present a gauge-invariant expression of spin and orbital magnetic octupoles in periodic crystals based on quantum mechanics and thermodynamics. The obtained magnetic octupole is a reducible rank-3 time-reversal-odd axial tensor, which can be decomposed into contributions from a totally symmetric magnetic octupole, a magnetic toroidal quadrupole, and two magnetic dipoles. We show a direct relationship between certain physical response tensors and the magnetic octupole in insulators at T=0, going beyond symmetry arguments. Through model calculations, we demonstrate that the spin magnetic octupole captures the nonrelativistic properties of d-wave altermagnets, while the orbital one captures the relativistic properties that require finite spin-orbit coupling. We also discuss the role of the magnetic dipole contribution of the magnetic octupole tensor in ferromagnetism-related physical responses in antiferromagnets, such as the anomalous Hall effect.