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## Space Group Symmetry, Spin-orbit Coupling and the Low-energy Effective Hamiltonian for Iron-based Superconductors

Iron-based superconductors are, in general, multi-band semi-metals with competing instabilities (itinerant spin-density wave, nematic structural transition, superconductivity). This should be a motivation to use the method of invariants, originally developed for semi-conductors, when constructing an effective theory for these materials. Surprisingly, such a theory had not been developed until now. In the construction of the low-energy effective theory we use the space group, which, being non-symmorphic, leads to peculiar consequences at the Brillouin zone corner, the M-point, precisely where the low-energy states reside. Our model displays good quantitative agreement with the multi-band tight-binding models, while obeying all the symmetries.

The spin-orbit coupling in iron is known to be significant and our model can easily incorporate it. We predict several consequences of the spin-orbit coupling on the spin-density wave orders. We analyze the spectrum in the presence of collinear or coplanar (C4) SDW. On the symmetry grounds, we show that, even with the spin-orbit coupling, each state is Kramers degenerate in the symmetry unbroken and the collinear SDW phase. On the other hand, the coplanar SDW breaks the Kramers degeneracy with important physical consequences. The nodal collinear SDW is unstable toward any finite spin-orbit coupling.

We study the quasiparticle dispersion of the low-energy effective model in the presence of an A1g (s-wave) spin singlet superconducting order. In the absence of the spin-orbit coupling, this phase can be characterized with three k-independent pairing amplitudes. This minimal model yields isotropic gaps on both hole Fermi surfaces. The gap structure (anisotropy, the relative sign, and the presence of nodes) on the electron Fermi surfaces is determined by the ratio of the pairing parameters. The results obtained for the s-wave can directly be used to analyze the B2g (d-wave) spin singlet pairing. In the presence of the spin-orbit interaction, two additional k-independent spin triplet pairing terms are allowed. These result in the gap anisotropy on the hole Fermi surfaces and a qualitative change of the gap structure on the electron Fermi surfaces.