Quasiclassical theory of $C_4$-symmetric magnetic order in disordered multiband metals

Recent experimental studies performed in the normal state of iron-based superconductors have discovered the existence of the $C_4$-symmetric (tetragonal) itinerant magnetic state. This state can be described as a spin density wave with two distinct magnetic vectors ${\vec Q}_1$ and ${\vec Q}_2$. Given an itinerant nature of magnetism in iron-pnictides, we develop a quasiclassical theory of tetragonal magnetic order in disordered three-band metal with anisotropic band structure. Within our model we find that the $C_4$-symmetric magnetism competes with the $C_2$-symmetric state with a single ${\vec Q}$ magnetic structure vector. Our main results is that disorder promotes tetragonal magnetic state which is in agreement with earlier theoretical studies.