Superfluid and Supersolid Phases of ${}^4\mathrm{He}$ on the Second Layer of Graphite

We revisited the phase diagram of the second layer of ${}^4\mathrm{He}$ on top of graphite using quantum Monte Carlo methods. Our aim was to explore the existence of the novel phases suggested recently in experimental works, and determine their properties and stability limits. We found evidence of a superfluid quantum phase with hexatic correlations, induced by the corrugation of the first Helium layer, and a quasi-two-dimensional supersolid corresponding to a $7/12$ registered phase. The $4/7$ commensurate solid was found to be unstable, while the triangular incommensurate crystals, stable at large densities, were normal.

Figure 1

Top panel: Full squares: Energy per ${}^4\mathrm{He}$ atom in a single layer of a triangular solid as a function of surface per helium atom. Open squares: Same for an arrangement comprising two helium layers, the density of the first one being $0.115{\AA }^{-2}$. A least-squares fitting polynomial to the first layer data and the Maxwell construction line between the two arrangements are also shown. Bottom panel: Data for the energy of the second layer system after subtracting the values given by the Maxwell construction line.

Figure 2

Same as in the previous figure for the two layer system at higher densities. Data for different phases are displayed.

Figure 3

Superfluid density for different second layer arrangements. Full squares, a translationally invariant phase with $\sigma =0.170{\AA }^{-2}$. Open circles, same but for $\sigma =0.165{\AA }^{-2}$. Open squares, results for a $7/12$ phase with $\sigma =0.186{\AA }^{-2}$. Full circles, same for a triangular solid with $\rho =0.196{\AA }^{-2}$. The dotted line corresponds to a perfect superfluid with no normal component.

Figure 4

Hexatic order parameter for translationally invariant phases of $\sigma =0.170{\AA }^{-2}$ (full squares) and $\sigma =0.163{\AA }^{-2}$ (open squares). Full circles correspond to a phase with $\sigma =0.165{\AA }^{-2}$, but in which the effect of the first solid layer have been smoothed out. When not displayed, the error bars are similar to the ones shown.