Nonlinear Transverse Magnetic Susceptibility under Electric Toroidal Dipole Ordering

An electric toroidal dipole (ETD) moment is one of the fundamental dipole moments as well as electric and magnetic ones. Although it directly couples to neither an electric nor magnetic field due to its spatial inversion and time-reversal parities, its ordered state leads to unconventional transverse responses of the conjugate physical quantities. We here theoretically investigate nonlinear transverse magnetic susceptibility under the ETD ordering. By performing a self-consistent mean-field calculation for a five $d$-orbital model under a tetragonal crystalline electric field and using the nonlinear Kubo formula, we show that a third-order transverse magnetic susceptibility corresponding to a uniform magnetization perpendicular to the external magnetic field becomes nonzero once the ETD moment is ordered under tetragonal crystalline electric field. Moreover, we find that spin-orbital entanglement and a low-lying first excited crystal-field level are important for realizing large transverse responses.