Unravelling incommensurate magnetism and its emergence in iron-based superconductors

We focus on a broad class of tetragonal itinerant systems sharing a tendency towards the spontaneous formation of incommensurate magnetism with ordering wavevectors $\mathbf{Q}_{1,2}=(\pi-\delta,0)/(0,\pi-\delta)$ or $\mathbf{Q}_{1,2}=(\pi,\delta)/(-\delta,\pi)$. Employing a Landau approach, we obtain the generic magnetic phase diagram and identify the leading instabilities near the paramagnetic-magnetic transition. Nine distinct magnetic phases exist that either preserve or violate the assumed $C_4$-symmetry of the paramagnetic phase. These are single- and double-$\mathbf{Q}$ phases consisting of magnetic stripes, helices and whirls, either in an individual or coexisting manner. These nine phases can be experimentally distinguished by polarized neutron scattering, or, for example, by combining measurements of the induced charge order and magnetoelectric coupling. Within two representative five-orbital models, suitable for BaFe$_2$As$_2$ and LaFeAsO, we find that the incommensurate magnetic phases discussed here are accessible in iron-based superconductors. Our investigation unveils a set of potential candidates for the unidentified $C_2$-symmetric magnetic phase that was recently observed in Ba$_{1-x}$Na$_x$Fe$_{2}$As$_{2}$. Among the phases stabilized we find a spin-whirl crystal, which is a textured magnetic $C_4$-symmetric phase. The possible experimental observation of textured magnetic orders in iron-based superconductor, opens new directions for realizing intrinsic topological superconductors.