Spin Conductivity Based on Magnetic Toroidal Quadrupole Hidden in Antiferromagnets

We report our theoretical results on spin conductivity in antiferromagnets by focusing on the role of the magnetic toroidal quadrupole (MTQ) in electron systems. The MTQ is characterized as a time-reversal-odd rank-2 polar tensor degree of freedom in electrons, which is distinct from conventional rank-1 magnetic and magnetic toroidal dipoles. Based on a microscopic $sd$ model analysis for a tetragonal system under both collinear and noncollinear antiferromagnetic orderings, we clarify that the MTQ becomes a source of an extrinsic spin conductivity even with neither a uniform magnetization nor spin-orbit coupling. We also list all the magnetic point groups to accommodate the MTQs as a primary order parameter as well as the candidate antiferromagnetic materials.