Pairing-induced Momentum-space Magnetism and Its Implication In Optical Anomalous Hall Effect In Chiral Superconductors

The intrinsic mechanisms of the magneto-optical Kerr signal in chiral superconductors often involve multi-orbital degree of freedom. Here by considering a generic single-orbital and spinful Hamiltonian, we generalize the Onsager's relation to obtain the necessary conditions for the optical anomalous Hall effect. Using the down-folding method, we identify two types of effective momentum-space magnetism responsible for the optical anomalous Hall conductivity from non-unitary and unitary pairing potentials respectively. The former is due to the angular momentum of Cooper pair, while the latter requires the participation of the spin-orbit coupling in the normal state and has been largely overlooked previously. Using concrete examples, we show that the unitary pairing can lead to both ferromagnetism and complicated antiferromagnetic spin texture in the momentum space, resulting in an in-plane optical anomalous Hall effect with the magnetism parallel to the Hall-deflection plane. Our work reveals the essential role of spin degree of freedom in the optical anomalous Hall effect.