Origin of the anomalous Hall effect in two-band chiral superconductors

We consider the origin of the anomalous Hall effect in a general model of a clean two-band chiral superconductor. Within the Kubo formalism we derive an analytic expression for the high-frequency ac Hall conductivity valid close to the critical temperature. This expression involves two distinct gauge-invariant time-reversal-odd bilinear (TROB) functions involving the pairing potential and its Hermitian conjugate. We argue that the existence of at least one of these TROBs generically implies a nonzero ac Hall conductivity. The TROBs allow us to clarify the roles of intra- and interband pairing, and provide a straightforward criterion for a superconducting state to exhibit the anomalous Hall effect. We briefly exemplify our results with model calculations for a chiral $p$-wave pairing state in strontium ruthenate and a chiral $d$-wave pairing state on the honeycomb lattice.