Chern number landscape of spin-orbit coupled chiral superconductors

Chiral superconductors are one of the predominant quantum electronic states of matter where topology, symmetry, and Fermiology intertwine. This is pushed to a new limit by further invoking the coupling between spin and charge degrees of freedom, which fundamentally affects the principal nature of the Cooper pair wave function. We investigate the onset of superconductivity in the Rashba-Hubbard model on the triangular lattice, which is symmetry classified by the associated irreducible representations (irrep) of the hexagonal point group. From an instability analysis by means of the truncated-unity functional renormalization group (TUFRG) we find the ${E}_{2}$ irrep to dominate a large fraction of phase space and to lead up to an energetically preferred gapped, chiral superconducting state. The topological phase space classification associated with the superconducting order parameter obtained from TUFRG reveals a fragmentation of the ${E}_{2}$ domain into different topological sectors with vastly differing Chern numbers. It hints at a potentially applicable high sensitivity and tunability of chiral superconductors with respect to topological edge modes and phase transitions.