Magnetic degeneracy and intertwined orders in iron-based superconductors

Recent experiments in hole-doped iron-based materials have unveiled a surprising competition between different magnetic ground states near optimal doping, where the superconducting transition temperature acquires its largest value. In particular, besides the usual stripe-type magnetic phase, so-called C₄ phases have been observed in a variety of different materials, challenging our understanding of the magnetic phase diagram of these unconventional superconductors. Here, we present a model that captures all these distinct magnetic phases and attributes the experimental observations to an emergent magnetic degeneracy arising from the interplay between spin-orbit coupling and quantum magnetic fluctuations. This results in a rich magnetic landscape with stripe spin density waves, charge-spin density waves, and spin-vortex crystals intertwined with composite vestigial orders that break different symmetries of the system and transform as B_2g, B_2u, and A_2u irreducible representations of the point group of the lattice. We argue that, while a charge-spin density wave is realized in hole-doped 122 materials, a spin-vortex crystal emerges in the newly discovered 1144 system due to its peculiar crystalline structure, which provides a conjugate field to the composite order that is intertwined with the spin-vortex crystal phase. Finally, we also discuss the potential implications of this magnetic degeneracy to the superconducting properties of these systems.