Topological superconductivity in the extended Kitaev-Heisenberg model

We study superconducting pairing in the doped Kitaev-Heisenberg model by taking into account the recently proposed symmetric off-diagonal exchange $\Gamma$. By performing a mean-field analysis, we classify all possible superconducting phases in terms of symmetry, explicitly taking into account effects of spin-orbit coupling. Solving the resulting gap equations self-consistently, we map out a phase diagram that involves several topologically nontrivial states. For $\Gamma<0$, we find a competition between a time-reversal symmetry breaking chiral phase with Chern number $\pm1$ and a time-reversal symmetric nematic phase that breaks the rotational symmetry of the lattice. On the other hand, for $\Gamma \geq 0$ we find a time-reversal symmetric phase that preserves all the lattice symmetries, thus yielding clearly distinguishable experimental signatures for all superconducting phases. Both of the time-reversal symmetric phases display a transition to a $\mathbb{Z}_2$ non-trivial phase at high doping levels. Finally, we also include a symmetry-allowed spin-orbit coupling kinetic energy and show that it destroys a tentative symmetry protected topological order at lower doping levels. However, it can be used to tune the time-reversal symmetric phases into a $\mathbb{Z}_2$ non-trivial phase even at lower doping.