Commensurate-Incommensurate Transition in $^{4}$He Monolayer Adsorbed on a C$_{60}$ Molecule

Path-integral Monte Carlo calculations have been performed to study adsorption of $^{4}$He on a single C$_{60}$ fullerene molecule. In order to account for helium corrugations on the molecular surface, the sum of all interatomic pair potentials between a carbon atom and a $^{4}$He atom is used for the $^{4}$He-C$_{60}$ interaction. The radial density distributions reveal a layer-by-layer growth of $^{4}$He with the first adlayer being located at a distance of $\sim$ 6.2 $\AA$ from the center of a C$_{60}$ molecule. This first layer is found to exhibit various quantum states as the number of adsorbed $^{4}$He atoms $N$ varies. For $N$=32 the helium layer shows a commensurate solid structure with twenty helium atoms being localized on the tops of the hexagon centers of the C$_ {60}$ surface and the other twelve atoms above the pentagon centers. As more $^{4}$He atoms are added, a commensurate-incommensurate transition is observed. After going through various domain wall states the first layer is crystallized into an incommensurate solid for $N \sim 52$. We find that solid states observed for $N$=32,44, and 48 do not show any superfluid response even below 0.2 K while domain-wall fluids formed with 45 to 47 $^4$He atoms show significant superfluid fractions below 0.6 K. Finally different quantum states oberseved in the first $^4$He layer around a C$_{60}$ are compared with phase diagrams determined for the helium monolayer on a graphite surface.