Theory Of Antisymmetric Spin-Pair-Dependent Electric Polarization In Multiferroics

We investigate magnetoelectric couplings between an electric polarization and an antisymmetric spin pair, S-i x S-j, in a multiorbital Hubbard model on a distorted lattice. We microscopically derive a generic form of the electric polarization, PAS = (d) over cap (S-i x S-j), with a tensor, (d) over cap, which includes the electric polarization induced by the Katsura-Nagaosa-Balatsky formula as a subset. The origin and nature of these magnetoelectric couplings are clarified in a unified way; the results indicate that various noncollinear magnetic structures, such as canted antiferromagnetic, proper screw, and 120 degrees. spin structures, show multiferroic behaviors owing to non-KatsuraNagaosa-Balatsky coupling.