Nodal $d+id$ pairing and topological phases on the triangular lattice: unconventional superconducting state of Na$_x$CoO$_2\cdot y$H$_2$O

We show that the finite angular momentum pairing on the triangular lattice has point nodes in the complex gap function. A topological quantum phase transition takes place through a gapless critical state at a specific carrier density $x_c$ where the normal state Fermi surface crosses these isolated nodes. For spin singlet pairing, we show that the second nearest neighbor $d+id$ pairing is the dominate superconducting channel. The gapless critical state appears at $x_c\simeq0.25$ for the sodium cobaltates. It has six Dirac points and is topologically nontrivial with a $T^3$ spin relaxation rate below $T_c$. This theory provides a consistent explanation for the unconventional superconducting state of Na$_x$CoO$_2\cdot y$H$_2$O.