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The polar Kerr effect and the closely related anomalous charge Hall effectare among the most distinguishing signatures of the superconducting state inSr$_2$RuO$_4$, as well as in several other compounds. These effects are oftenthought to be derived from chiral superconducting pairing, and differentmechanisms have been invoked for the explanation. However, the intrinsicmechanisms proposed previously often involve unrealistically strong interbandCooper pairing. In this study we show that, even without interband pairing,nonunitary superconducting states can support intrinsic anomalous charge Halleffect, thanks to the quantum geometric properties of the Bloch electrons. Thekey here is to have a normal-state spin Hall effect, for which a nonzerospin-orbit coupling is essential. A finite charge Hall effect then naturallyarises at the onset of a spin-polarized nonunitary superconducting pairing. Itdepends on both the spin polarization and the normal-state electron Berrycurvature, the latter of which is the imaginary part of the quantum geometrictensor of the Bloch states. Applying our results to the weakly-pairedSr$_2$RuO$_4$ we conclude that, if the reported Kerr effect is of intrinsicorigin, the superconducting state is most likely nonunitary and has odd-parity.Our theory may be generalized to other superconductors that exhibit polar Kerreffect.

Quantum-geometry-induced anomalous Hall effect in nonunitary superconductors and application to Sr$_2$RuO$_4$