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$B_{1g}$-like pairing states in two-leg ladder iron superconductors

Motivated by the recent report of superconductivity in Fe-based ladder materials, we study the pairing state of a multi-orbital t-J model defined on two-leg ladders using the standard mean-field theory. We find that the superconducting order parameters change sign between the $d_{xz}$ and $d_{yz}$ orbitals in most of the phase diagram. By analogy with the two-dimensional Fe planes, we conclude that the leading pairing channel of this state belongs to the $B_{1g}$ symmetry class, which is distinct from the common $s_{\pm}$ gap with the $A_{1g}$ symmetry. By smoothly interpolating from planes into ladders, we show that a first-order transition occurs between these two competing phases when the dimension of the system is reduced.