  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.
