Unravelling incommensurate magnetism and the path to topological phases in iron-based superconductors

Motivated by recent experiments on iron-based superconductors hinting at incommensurate magnetic order, we investigate generic itinerant systems exhibiting a tendency towards incommensurate magnetism with ordering wavevectors $\mathbf{Q}_{1,2}=(Q,0)/(0,Q)$ and retrieve the complete phase diagram and leading instabilities near the paramagnetic-magnetic transition via a Landau approach. The aspect of incommensurability introduces a plethora of new exotic phases that can either preserve or violate C$_4$-symmetry. We additionally aim at unveiling prominent candidates for the recently observed, and yet unresolved, C$_2$ magnetic phase in Ba-doped iron-based superconductors (FeSCs). Within a representative five-orbital model we show that these nonstandard incommensurate magnetic phases become indeed favored in FeSCs. In fact, a C$_4$-preserving non-coplanar texture becomes stabilized and can be rendered skyrmionic by applying an infinitesimaly small external magnetic field. We illustrate how the microscopic coexistence of the latter with superconductivity, a feasible scenario for FeSCs, opens new perspectives for realizing intrinsic topological superconductivity.