$d$-wave Superconductivity in the Hubbard model on the isotropic triangular lattice and a possibility of the chiral $d+id$ pairing as a quasi-stable state

We study $d$-wave superconductivity(SC) in the Hubbard model on the isotropic triangular lattice described by the hopping parameter $t$ and on-site Coulomb repulsion $U$ at zero temperature and half-filling using the variational cluster approximation. We found that the $d_{xy}$ SC is the ground state below the Mott insulator phase $U\mathrm{/}t\lesssim 6$, and the energy of chiral $d+id$ SC is slightly higher than the $d_{xy}$ SC. The energy difference between the normal and $d_{xy}$ states is about $0.02t\sim 0.06t$ for $U\mathrm{/}t\simeq 5$. This result is semi-quantitatively consistent with the SC transition temperature $T_K=3.9$ K of $\kappa$-(BEDT-TTF)${}_2$Cu${}_2$(CN)${}_3$, where $t$ is estimated to be about $0.06$ eV, and the predicted pairing symmetry $d_{xy}$ agrees with the STM observations. The energy difference between the $d+id$ and $d_{xy}$ is about $0.01t\sim 0.03t$ for $U\mathrm{/}t\simeq 5$ so the transition from $d+id$ to $d_{xy}$, or some effects of $d+id$ in $d_{xy}$ phase may be observed in experiments for $\kappa$-(BEDT-TTF)${}_2$Cu${}_2$(CN)${}_3$.