Tri-component-pairing chiral superconductivity on the honeycomb lattice with mixed $s$- and $d$-wave symmetries

In this work, we investigate chiral topological superconductors on a two-dimensional honeycomb lattice with coexisting $d_{x^2-y^2}$, $d_{xy}$, and $s$-wave pairing symmetries. Using a Ginzburg-Landau free energy analysis, the pairing gap function is shown to exhibit a tri-component form $s+d_{x^2-y^2}{e}^{i{\phi }_1}+d_{xy}{e}^{i{\phi }_2}$, where ${\phi }_1$ and ${\phi }_2$ are phase differences between the $d$- and $s$-wave pairing components, which spontaneously breaks both time reversal and $C_6$ rotational symmetries. Chern numbers of the energy bands are calculated to be nonzero, demonstrating the topologically nontrivial nature of the system. The anomalous AC Hall conductivity is computed, which is not invariant under $C_6$ rotations, reflecting the anisotropic nature of the pairing gap function. Fractional magnetic vortices are also discussed, arising from the multi-component nature of the pairing gap function.