Magnetic phase diagram of the iron pnictides in the presence of spin-orbit coupling: Frustration between C2 and C4 magnetic phases

We investigate the impact of spin anisotropic interactions, promoted by spin-orbit coupling, on the magnetic phase diagram of the iron-based superconductors. Three distinct magnetic phases with Bragg peaks at (π,0) and (0,π) are possible in these systems: one C2 (i.e., orthorhombic) symmetric stripe magnetic phase and two C4 (i.e., tetragonal) symmetric magnetic phases. While the spin anisotropic interactions allow the magnetic moments to point in any direction in the C2 phase, they restrict the possible moment orientations in the C4 phases. As a result, an interesting scenario arises in which the spin anisotropic interactions favor a C2 phase, but the other spin isotropic interactions favor a C4 phase. We study this frustration via both mean-field and renormalization-group approaches. We find that, to lift this frustration, a rich magnetic landscape emerges well below the magnetic transition temperature, with interesting C2,C4, and mixed C2-C4 phases. Near the putative magnetic quantum critical point, spin anisotropies promote a stable Gaussian fixed point in the renormalization-group flow, which is absent in the spin isotropic case, and is associated with a near-degeneracy between C2 and C4 phases. We argue that this frustration is the reason why most C4 phases in the iron pnictides only appear inside the C2 phase, and discuss additional manifestations of this frustration in the phase diagrams of these materials.