Gap symmetries from the neighbor coupling in square-lattice superconductors

The gap symmetries of superconductivity are studied in this work. It is found that the gap symmetries are simply determined by the 4-fold rotational symmetries of the coupling potential on neighbor sites. A local on-site coupling potential results in the on-site pairing with the conventional s-wave symmetry, but a coupling potential between the nearest neighbors or the next-nearest neighbors results in the pairing on neighbor sites with the $s^-$, $d_{x^2-y^2}$, $d_{xy}$, or $s_{x^2y^2}$ gap symmetries. It is proved that both isotropic and anisotropic gap functions are allowed by the 4-fold rotational symmetries of the coupling potential. Finally a numerical computation is performed to demonstrate the gap symmetries. This neighbor coupling provides a unified picture for the gap functions of the conventional and the high Tc superconductivity.