Localization versus inhomogeneous superfluidity: Submonolayer He4 on fluorographene, hexagonal boron nitride, and graphene

We study a submonolayer $^{4}\mathrm{He}$ adsorbed on fluorographene (GF) and on hexagonal boron nitride (hBN) at low coverage. The adsorption potentials have been computed ab initio with a suitable density functional theory including dispersion forces. The properties of the adsorbed $^{4}\mathrm{He}$ atoms have been computed at finite temperature with path integral Monte Carlo and at $T=0\phantom{\rule{0.28em}{0ex}}\mathrm{K}$ with variational path integral. From both methods we find that the lowest energy state of $^{4}\mathrm{He}$ on GF is a superfluid. Due to the very large corrugation of the adsorption potential this superfluid has a very strong spatial anisotropy, the ratio between the largest and smallest areal density being about 6, the superfluid fraction at the lowest $T$ is about 55%, and the temperature of the transition to the normal state is in the range 0.5--1 K. Thus, GF offers a platform for studying the properties of a strongly interacting highly anisotropic bosonic superfluid. At a larger coverage $^{4}\mathrm{He}$ has a transition to an ordered commensurate state with occupation of 1/6 of the adsorption sites. This phase is stable up to a transition temperature located between 0.5 and 1 K. The system has a triangular order similar to that of $^{4}\mathrm{He}$ on graphite but each $^{4}\mathrm{He}$ atom is not confined to a single adsorption site and the atom visits also the nearest neighboring sites giving rise to a novel three-lobed density distribution. The lowest energy state of $^{4}\mathrm{He}$ on hBN is an ordered commensurate state with occupation of 1/3 of the adsorption sites and triangular symmetry. A disordered state is present at lower coverage as a metastable state. In the presence of an electric field the corrugation of the adsorption potential is slightly increased but up to a magnitude of 1 V/\AA{} the effect is small and does not change the stability of the phases of $^{4}\mathrm{He}$ on GF and hBN. We have verified that also in the case of graphene such electric field does not modify the stability of the commensurate $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}R{30}^{\ensuremath{\circ}}$ phase.