p-wave superconductivity on monolayer and bilayer honeycomb lattice

We derive ground state wave functions of superconducting instabilities on the honeycomb lattice induced by nearest-neighbor attractive interactions. They reflect the Dirac nature of electrons in the low-energy limit. For the order parameter that is the same irrespective of the direction to any of the nearest neighbors we find weak pairing (slowly decaying) behavior in the orbital part of the Cooper pair with no angular dependence. At the neutrality point, in the spin-singlet case, we recover a strong pairing behavior. We also derive ground state wave functions for the superconductivity on the bilayer honeycomb lattice, with strong interlayer coupling, induced by attractive interactions between sites that participate in a low-energy description. Without these interactions, free electrons are described by a Dirac equation with a quadratic dispersion. This unusual feature, similarly to $^3$He - B phase, leads to the description with two kinds of Cooper pairs, with $p_x + i p_y$ and $p_x - i p_y$ pairing, in the presence of the attractive interactions. We discuss the edge modes of such a spin-singlet superconductor and find that it represents a trivial topological superconductor.