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Most iron-based superconductors undergo a transition to a magneticallyordered state characterized by staggered stripes of parallel spins. Withordering vectors $(\pi,0)$ or $(0,\pi)$, this magnetic state breaks thehigh-temperature tetragonal symmetry of the system, which is manifested by asplitting of the lattice Bragg peaks. Remarkably, recent experiments inhole-doped iron arsenides reported an ordered state that displays magneticBragg peaks at $(\pi,0)$ and $(0,\pi)$ but remains tetragonal. Despite beinginconsistent with a magnetic stripe configuration, this unusual magnetic phasecan be described in terms of a double-$\mathbf{Q}$ magnetic structureconsisting of an equal-weight superposition of the ordering vectors $(\pi,0)$and $(0,\pi)$. Here we show that a non-collinear double-$\mathbf{Q}$ magneticconfiguration, dubbed \emph{orthomagnetic}, arises naturally within anitinerant three-band microscopic model for the iron pnictides. In particular,we find that strong deviations from perfect nesting and residual interactionsbetween the electron pockets favor the orthomagnetic over the stripe magneticstate. Using an effective low-energy model, we also calculate the spin-wavespectrum of the orthomagnetic state. In contrast to the stripe state, there arethree Goldstone modes, manifested in all diagonal and one off-diagonalcomponent of the spin-spin correlation function. The total magnetic structurefactor displays two anisotropic spin-wave branches emerging from both $(\pi,0)$and $(0,\pi)$ momenta, in contrast to the case of domains of stripe order,where only one spin-wave branch emerges from each momentum. We propose thatthese unique features of the orthomagnetic state can be used to unambiguouslydistinguish it from the stripe state via neutron scattering experiments, anddiscuss the implications of its existence to the nature of the magnetism of theiron arsenides.

Magnetic order without tetragonal-symmetry-breaking in iron arsenides: microscopic mechanism and spin-wave spectrum