(Top) Adapted from F. Wang and D.-H. Lee [9]; (Bottom) Adapted from J. Hu and N. Hao [2]

Figure 1: (Top) Lattice structure of the iron-based superconductors. Note that the atomic positions are symmetric under a rotation by 90 degrees in the $ab$ plane, followed by a reflection $c→-c$. This is the basic operation of the $S4$ symmetry. (Bottom) The gauge transformation consists of multiplying by a minus sign the electronic creation and annihilation operators on the orange iron sites. The transformation changes the pairing symmetry from $s$ wave to $d$ wave, as shown.