Nematic Bogoliubov Fermi surfaces from magnetic toroidal order in FeSe 1 − x S x

Recently it has been shown experimentally that the superconducting state of ${\mathrm{FeSe}}_{1\ensuremath{-}x}{\mathrm{S}}_x$ exhibits Bogoliubov Fermi surfaces for $x>0.17$. These Bogoliubov Fermi surfaces appear together with broken time-reversal symmetry and surprisingly demonstrate nematic behavior in a structurally tetragonal phase. Here, through a comprehensive analysis that deduces the structure of Bogoliubov Fermi surfaces from all symmetry-allowed types of translation invariant broken time-reversal symmetry, we find that the origin of nematic Bogoliubov Fermi surfaces is due to magnetic toroidal order that belongs to an ${E}_u$ irreducible representation of the ${D}_4h$ point group. We further show that this magnetic toroidal order appears as a consequence of either static N\'eel antiferromagnetic order or due to the formation of a spontaneous pair density wave superconducting order. Finally, we reveal that independent of the presence of Bogoliubov Fermi surfaces, supercurrents will induce N\'eel magnetic order in many Fe-based superconductors.