Coexistence of superconductivity and a spin density wave in pnictides: Gap symmetry and nodal lines

We investigate the effect of a spin-density wave (SDW) on $s_{\pm}$ superconductivity in Fe-based superconductors. We show that, contrary to the common wisdom, no nodes open at the new, reconnected Fermi surfaces when the hole and electron pockets fold down in the SDW state, despite the fact that the $s_{\pm}$ gap changes sign between the two pockets. Instead, the order parameter preserves its sign along the newly formed Fermi surfaces. The familiar experimental signatures of an $s_{\pm}$ symmetry are still preserved, although they appear in a mathematically different way. For a regular $s$ case ($s_{++})$ the nodes do appear in the SDW state. This distinction suggests a novel simple way to experimentally separate an $s_{\pm}$ state from a regular $s$ in the pnictides. We argue that recently published thermal conductivity data in the coexisting state are consistent with the $s_{\pm},$ but not the $s_{++}$ state.