Correlated Dirac Particles and Superconductivity on the Honeycomb Lattice

We investigate the properties of the nearest-neighbor singlet pairing and the emergence of d-wave superconductivity in the doped honeycomb lattice considering the limit of large interactions and the $t-J_1-J_2$ model. First, by applying a renormalized mean-field procedure as well as slave-boson theories which account for the proximity to the Mott insulating state, we confirm the emergence of d-wave superconductivity in agreement with earlier works. We show that a small but finite $J_2$ spin coupling between next-nearest neighbors stabilizes d-wave symmetry compared to the extended s-wave scenario. At small hole doping, to minimize energy and to gap the whole Fermi surface or all the Dirac points, the superconducting ground state is characterized by a $d+id$ singlet pairing assigned to one valley and a $d-id$ singlet pairing to the other, which then preserves time-reversal symmetry. The slightly doped situation is distinct from the heavily doped case (around 3/8 and 5/8 filling) supporting a pure chiral $d+id$ symmetry and breaking time-reversal symmetry. Then, we apply the functional Renormalization Group and we study in more detail the competition between antiferromagnetism and superconductivity in the vicinity of half-filling. We discuss possible applications to strongly-correlated compounds with Copper hexagonal planes such as In$_3$Cu$_{2}$VO$_9$. Our findings are also relevant to the understanding of exotic superfluidity with cold atoms.