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$S_4$ Symmetric Microscopic Model for Iron-Based Superconductors

Although iron-based superconductors are multi-orbital systems with complicated band structures, we demonstrate that the low energy physics which is responsible for high-$T_c$ superconductivity is essentially governed by one effective Hamiltonianwith two almost decoupled orbitals near half filling. This underlining electronic structure is protected by the $S_4$ symmetry. With repulsive or strong next nearest neighbor antiferromagnetic exchange interactions, each single-orbital effective Hamiltonian results in a robust $A_{1g}$ s-wave pairing which can be exactly mapped to the d-wave pairing observed in cuprates. The classification of the superconducting(SC) states according to the $S_4$ symmetry leads to a natural prediction of the existence of two different phases named A and B phases. In the B phase, the superconducting order has an overall sign change along c-axis between the top and bottom As(Se) planes in a single Fe-(As)Se trilayer structure, which is an analogy of the sign change under the $90^\circ$ degree rotation in the d-wave SC state of cuprates. Our derivation provides a unified understanding of iron-pnictides and iron-chalcogenides, and suggests that cuprates and iron-based superconductors share identical high-$T_c$ superconducting mechanism.