Microscopic mechanism for intrinsic nonlinear Hall conductivity in noncollinear antiferromagnetic metals

Nonlinear responses in magnetic materials have recently been attracted, since they are induced by the interplay between magnetism and topology. In particular, we focus on the second-order electric conductivity in the antiferromagnetic (AFM) metals with space-time symmetry, where two contributions, the Drude and intrinsic terms, are important. The former is induced by an antisymmetric band modulation resulting from the effective coupling between magnetic ordering and antisymmetric spin-orbit interaction (ASOI), while the microscopic mechanism for the latter has not been fully elucidated.

We investigate the intrinsic term based on the symmetry and model analyses. First, we discuss the relationship between the conductivity tensor and the activated multipoles from the symmetry viewpoint. By separating the second-order conductivity into the Ohmic and Hall parts, we show that the intrinsic nonlinear Hall conductivity (INHE) corresponding to the intrinsic term in the Hall part is related to the emergence of the magnetic quadrupole and magnetic toroidal dipole. Next, we discuss the microscopic origin of the INHE by analyzing a minimal model on a layered four-sublattice tetragonal structure under the AFM ordering with magnetic quadrupole or magnetic toroidal dipole. We demonstrate the important model parameters for the INHE. Finally, we discuss the difference in the INHE between the magnetic orderings with magnetic quadrupole and magnetic toroidal dipole concerning the hopping parameter dependence.