  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.
