$^4$He adsorption on $\alpha$-graphyne

Path-integral Monte Carlo calculations have been performed to study $^4$He adsorption on a single $\alpha$-graphyne sheet that is a hexagonal network of $sp$- and $sp^2$-bonded carbon atoms. Using the $^4$He-substrate interaction described by a pairwise sum of the helium-carbon inter-atomic potentials, we have found that each hexagon of a graphyne can accomodate one $^4$He atom at its in-plane center. The first layer of $^4$He atoms adsorbed on this $^4$He-attached graphyne sheet with a composite of C$_8$He$_1$, exhibits various quantum phases depending on the helium coverage. It is found to be in a Mott insulating state at a coverage of 0.0706~\AA$^{-2}$ with three $^4$He atoms occupying each unit cell while the helium atoms form a commensurate triangular solid at 0.0941~\AA$^{-2}$. With the introduction of Ising pseudospins for two degenerate configurations of three $^4$He atoms in a hexagonal cell, the transition from the Mott insulator to the triangular solid can be interpreted as a ferromagnetic transition. In addition we find stable formation of zero-point vacancies in the commensurate triangular solid and their roles in possible realization of supersolidity are under investigation