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

Path-integral Monte Carlo calculations have been performed to study the $^4$He adsorption on both sides of a single $\alpha$-graphyne sheet. For investigation of the interlayer correlation between the upper and the lower monolayer of $^4$He adatoms, the $^4$He-substrate interaction is described by the sum of the $^4$He-C interatomic pair potentials, for which we use both Lennard-Jones and Yukawa-6 anisotropic potentials. When the lower $^4$He layer is a C$_{4/3}$ commensurate solid, the upper-layer $^4$He atoms are found to form 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-layer pseudospins, which were introduced in Ref. [1] for two degenerate configurations of three $^4$He atoms in a hexagonal cell, depends on the substrate potential used; With the substrate potential based on the anisotropic Yukawa-6 pair potentials, the Ising pseudo-spins of both $^4$He layers are found to be anti-parallel to each other while the parallel and anti-parallel pseudo-spin alignments between the two $^4$He layers are nearly degenerate with the Lennard-Jones potentials. This is attributed to the difference in the interlayer distance, which is $\sim 4$ \AA~ with the Yukawa-6 substrate potential 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)