On-site attractive multiorbital Hamiltonian ford-wave superconductors

We introduce a two-orbital Hamiltonian on a square lattice that contains on-site attractive interactions involving the two ${e}_g$ orbitals. Via a canonical mean-field procedure similar to the one applied to the well-known negative-$U$ Hubbard model, it is shown that the model develops $d$-wave (${B}_1g$) superconductivity with nodes along the diagonal directions of the square Brillouin zone. This result is also supported by exact diagonalization of the model in a small cluster. The expectation is that this relatively simple attractive model could be used to address the properties of multiorbital $d$-wave superconductors in the same manner that the negative-$U$ Hubbard model is widely applied to the study of the properties of $s$-wave single-orbital superconductors. In particular, we show that by splitting the ${e}_g$ orbitals and working at three-quarters filling, such that the ${x}²\ensuremath{-}{y}²$ orbital dominates at the Fermi level but the $3{z}²\ensuremath{-}{r}²$ orbital contribution is nonzero, the $d$-wave pairing state found here phenomenologically reproduces several properties of the superconducting state of the high ${T}_c$ cuprates.