Chiral superconductors from parent states with non-uniform Berry curvature: Momentum-space vortices, BdG topology, and thermal Hall conductivity

TL;DR

This paper explores how non-uniform Berry curvature in parent electronic states influences chiral superconductivity, leading to momentum-space vortices, with implications for thermal Hall measurements and topological properties.

Contribution

It introduces a continuum model to solve the full BCS gap equation considering Berry curvature effects, revealing vortex formation and topological constraints in chiral superconductors.

Findings

Non-uniform Berry curvature enriches the superconducting order parameter.

Momentum-space vortices form away from high-symmetry points.

Parent band Chern number constrains vortex nucleation.

Abstract

We investigate chiral superconductivity emerging from parent electronic states with non-uniform Berry curvature, motivated by recent experiments in rhombohedral graphene multilayers. Using the continuum $\lambda_N$-model-a tunable platform with independently controllable Berry curvature profiles-we solve the full BCS gap equation on a continuum Chern band beyond the weak-coupling limit. We find that a non-uniform Berry curvature of the parent band enriches the superconducting order parameter, leading to the formation of momentum-space vortices in the gap function away from high-symmetry points. By tuning the Berry curvature profile, we identify distinct regimes associated with vortex nucleation and vortex number saturation, and show that the nucleation of momentum-space vortices tends to lower the condensation energy. We then show analytically that the parent band Chern number constrains the number of momentum-space vortices that can nucleate in the gap-independent of details of the $\lambda_N$-model. We also provide a gauge-invariant formulation for computing the Bogoliubov-de Gennes (BdG) Berry curvature for continuum models, and find that it is determined by a momentum-space phase current. The winding of this current around vortices in the occupied region in turn determines the BdG Chern number. Finally, we discuss how thermal Hall measurements can be used to probe the formation of momentum-space vortices. Our results highlight the crucial role of Berry curvature in shaping chiral superconductivity, and offer guiding principles for its identification in systems such as rhombohedral graphene.