Tetragonal magnetic order in iron pnictides: phase diagram, spin-wave dispersion, and competition with superconductivity

Recent experiments in hole-doped iron pnictides have found a magnetic state near optimal doping that displays magnetic Bragg peaks at $(\pi,0)$ and $(0,\pi)$ but no splitting of the lattice Bragg peaks, implying that the system remains tetragonal. Here we use a multi-band itinerant microscopic model to discuss the origin, the phase diagram, and the manifestations of this tetragonal magnetic state. Starting from perfect nesting, we find that increasing the doping concentration leads to a change in the magnetic state from a single-Q orthorhombic ($C_2$) phase to a double-Q tetragonal ($C_4$) phase and then back to a single-Q $C_2$ phase. As temperature is lowered, we find that the $C_4$ phase is in general unstable towards the $C_2$ phase. Furthermore, we show that the superconducting state tends to be more strongly suppressed by the $C_4$ phase due to the interplay between tetragonal symmetry-breaking and the near-degeneracy of the superconducting ground state. We compare our results with the phase diagram of Na- and K-doped Ba122 materials, and discuss the theoretical spin-wave spectrum of the tetragonal magnetic phase, highlighting the features that allow for an unambiguous experimental distinction between the $C_4$ and $C_2$ phases.