Supersolidity in quantum films adsorbed on graphene and graphite

Using quantum Monte Carlo we have studied the superfluid density of the first layer of ${}^4$He and H${}_2$ adsorbed on graphene and graphite. Our main focus has been on the equilibrium ground state of the system, which corresponds to a registered $\sqrt{3}\times \sqrt{3}$ phase. The perfect solid phase of H${}_2$ shows no superfluid signal whereas ${}^4$He has a finite but small superfluid fraction (0.67%). The introduction of vacancies in the crystal makes the superfluidity increase, showing values as large as 14% in ${}^4$He without destroying the spatial solid order.