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 $(\ensuremath{\pi},0)$ and $(0,\ensuremath{\pi})$ are possible in these systems: one ${C}₂$ (i.e., orthorhombic) symmetric stripe magnetic phase and two ${C}₄$ (i.e., tetragonal) symmetric magnetic phases. While the spin anisotropic interactions allow the magnetic moments to point in any direction in the ${C}₂$ phase, they restrict the possible moment orientations in the ${C}₄$ phases. As a result, an interesting scenario arises in which the spin anisotropic interactions favor a ${C}₂$ phase, but the other spin isotropic interactions favor a ${C}₄$ 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 ${C}₂,{C}₄$, and mixed ${C}₂$--${C}₄$ 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 ${C}₂$ and ${C}₄$ phases. We argue that this frustration is the reason why most ${C}₄$ phases in the iron pnictides only appear inside the ${C}₂$ phase, and discuss additional manifestations of this frustration in the phase diagrams of these materials.