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Emergent Spin Hall phase at a Lifshitz transition from competing orders

The effects of competing orders, such as superconductivity and ferromagnetism, on a Fermi liquid are well established. A comprehensive understanding of such a competition in a metal whose Fermi surface has a non-trivial topology is yet to be achieved. Here, we address this question in a prototypical system: the 2D Rashba semimetal. We show that dominant superconductivity interplays with Rashba spin orbit interactions (SOI) in forming a novel intrinsic anomalous Hall effect (AHE) with gapless edge states of Bogoliubov-de Gennes (BdG) quasiparticles. As in the case of itinerant ferromagnets, the intrinsic AHE arises from Berry curvature effects in the band structure. This phenomenon is robust even as sub-dominant ferromagnetism dramatically changes the nature of pairing symmetry. An emergent spin Hall phase involving a change in Fermi-surface topology is found to accompany this Lifshitz quantum phase transition. We demonstrate the coexistence of the original and novel AHE in the presence of weak disorder. We offer a comparison of our results with experiments on the two dimensional electron gas at oxide hetero-interfaces as well as make some testable predictions.