Title: Magnetic phase diagram of the iron pnictides in the presence of spin-orbit coupling: Frustration between $C_2$ and $C_4$ magnetic phases,
Abstract: 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 $(\pi,0)$
and $(0,\pi)$ 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 novel $C_2$, $C_4$, 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.