Theory of the intrinsic optical anomalous Hall responses in time-reversal symmetry breaking superconductors

Determining the symmetry of the Cooper pairing in superconductors is a central challenge in condensed matter physics. Time-reversal symmetry breaking (TRSB) is among the most distinctive symmetry-breaking pattern of a superconductor. An important subset of TRSB pairing states, including the topological p x+ip y pairing, is expected to exhibit the optical anomalous Hall effect. However, the microscopic ingredients that enable the Hall response and their underlying organizing principle in clean superconductors remain unelucidated. Here, we develop a unified theory for TRSB multiband superconductors possessing inversion symmetry, and establish their intrinsic optical Hall response as fundamentally a quantum-geometric phenomenon. We distinguish unitary and nonunitary TRSB states. Diagrammatically, we identify two classes of processes: single-electron tunneling and Cooper-pair tunneling. Both involve at least one interband tunneling event driven by the interband velocity-which is closely tied to the interband Berry connection. We rigorously prove that interband pairing is a prerequisite for unitary states to generate intrinsic Hall response, whereas it is not absolutely needed for nonunitary states. In nonunitary states with intraband pairing alone, the Hall effect can arise through single-electron tunneling, provided spin-orbit coupling is present. An effective theory obtained by integrating out the hole degrees of freedom reproduces these results and shows how pairing may endow electrons with effective Berry curvature. Finally, through symmetry analysis, we show that intrinsic resonances near the superconducting gap edge are allowed in nonunitary states but forbidden in unitary states. Our work thus establishes quantum geometry as a unifying principle for the intrinsic Hall response in superconductors and provides a guide for distinguishing different candidate TRSB pairings.