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Path-integral Monte Carlo calculations have been performed to study the$^4$He adsorption on both sides of a single $\alpha$-graphyne sheet. Forinvestigation of the interlayer correlation between the upper and the lowermonolayer of $^4$He adatoms, the $^4$He-substrate interaction is described bythe sum of the $^4$He-C interatomic pair potentials, for which we use bothLennard-Jones and Yukawa-6 anisotropic potentials. When the lower $^4$He layeris a C$_{4/3}$ commensurate solid, the upper-layer $^4$He atoms are found toform a Kagom\'e lattice structure at a Mott insulating density of 0.0706\AA$^{-2}$, and a commensurate solid at an areal density of 0.0941 \AA$^{-2}$for both substrate potentials. The correlation between upper- and lower-layerpseudospins, which were introduced in Ref. [1] for two degenerateconfigurations of three $^4$He atoms in a hexagonal cell, depends on thesubstrate potential used; With the substrate potential based on the anisotropicYukawa-6 pair potentials, the Ising pseudo-spins of both $^4$He layers arefound to be anti-parallel to each other while the parallel and anti-parallelpseudo-spin alignments between the two $^4$He layers are nearly degenerate withthe Lennard-Jones potentials. This is attributed to the difference in theinterlayer distance, which is $\sim 4$ \AA~ with the Yukawa-6 substratepotential but as large as $\sim 4.8$ \AA~with the Lennard-Jones potential.  [1] Y. Kwon, H. Shin, and H. Lee, Phys. Rev. B 88, 201403(R) (2013)

Interlayer correlation between two $^4$He monolayers adsorbed on both sides of $\alpha$-graphyne