Physics Department - Theory of Magnetic Octupoles in Crystalline Solids

Abstract
Magnetic multipoles serve as order parameters for characterizing magnetic structures in crystalline solids and provide a systematic framework for classifying symmetry-allowed physical responses. Recently, magnetic octupoles have been proposed as order parameters for time-reversal-symmetry-breaking centrosymmetric antiferromagnets, including altermagnets. However, a microscopic formulation of magnetic octupoles in crystalline solids remains elusive because their classical definition is given by the second-order moment of the magnetic moment density, leading to gauge dependence in a direct quantum-mechanical evaluation.
In this study, we derive gauge-invariant expressions for spin and orbital magnetic octupoles in periodic crystals based on quantum mechanics and thermodynamics. Applying these expressions to minimal models of d-wave altermagnets, we show that magnetic octupoles capture the essential properties of the altermagnetic order. We further demonstrate their direct relations to intrinsic responses induced by electromagnetic-field gradients and identify a heuristic relation between the spin magnetic octupole and dissipative spin conductivity. These results not only enable a quantitative characterization of unconventional magnetic order but also provide a comprehensive and microscopic understanding of its associated physical responses.