Kekul\'e superconductivity and antiferromagnetism on the graphene lattice

We investigate superconducting order in the extended Hubbard model on the two-dimensional graphene lattice using the variational cluster approximation (VCA) with an exact diagonalization solver at zero temperature. Building on the results of Ref.$[1]$, which identified triplet $p$- and $p+ip$-wave superconductivity as the most favored pairing symmetries in that model, we place uniform SC solutions in competition with the proposed nonuniform Kekul\'e ($p+ip$-K) superconducting pattern proposed in Ref.$[2]$. We find that the $p+ip$-K solution is in fact the most favored pairing in most of the phase diagram. In addition, we show that antiferromagnetism can co-exist with the $p+ip$-K state and that both orders are enhanced by their coexistence.