Superfluidity of Dense ${}^4\mathrm{He}$ in Vycor

We calculate properties of a model of ${}^4\mathrm{He}$ in Vycor using the path integral Monte Carlo method. We find that ${}^4\mathrm{He}$ forms a distinct layered structure with a highly localized first layer, a disordered second layer with some atoms delocalized and able to give rise to the observed superfluid response, and higher layers of nearly perfect crystals. The addition of a single ${}^3\mathrm{He}$ atom was enough to bring down the total superfluidity by blocking the exchange in the second layer. Our results are consistent with the persistent liquid-layer model to explain the observations. Such a model may be relevant to the experiments on bulk solid ${}^4\mathrm{He}$, if there is a fine network of grain boundaries in those systems.

Figure 1

The external potential $V(z)$ experienced by the helium atoms in “cell 2.” The Vycor is on the left side ($z=-12.5\AA$) and solid helium on the right side ($z=12.5\AA$). The inset is a 3D representation of the rough Vycor surface: The black dots are the positions of the Vycor impurities placed randomly at 1 Å away from the Vycor wall. The rugged surface shows the positions of the helium atoms (located at the vertices) in the first layer.

Figure 2

The density and superfluid density as a function of $z$. The left scale (solid line) corresponds to pure ${}^4\mathrm{He}$ and the right to the mixture: one ${}^4\mathrm{He}$ atom replaced by a ${}^3\mathrm{He}$ atom. Top: The density in the $z$ direction for cell 1 at 0.2 K. The split peak in the first (leftmost) layer is due to the rough Vycor surface. The dotted lines show the ${}^3\mathrm{He}$ density at 0.2 and 0.7 K. Bottom: Local superfluid density of ${}^4\mathrm{He}$ determined by recording which layers the winding paths visit; Eq. (2).

Figure 3

Top: Contour plots of the helium density in the $x\mathrm{\text{−}}y$ plane in the first three layers at 0.2 K for cell 1. In the first layer above the Vycor, the atoms are pinned by the strong Vycor interaction, in the second layer they are relatively delocalized, whereas the higher levels show a density distribution characteristic of bulk solid ${}^4\mathrm{He}$. Bottom: The layer-specific structure factor. The first layer has an amorphous structure, the second is still distorted but with a much smaller peak, while the third and higher layers have a single large peak at ${\mathrm{k}}_o=2.04{\AA }^{-1}$ characteristic of a 2D quantum solid.

Figure 4

The superfluid fraction ${\rho }_s/\rho$ vs temperature. The upper curve is for pure ${}^4\mathrm{He}$. The lower curve is for the system with a ${}^4\mathrm{He}$ atom replaced by a ${}^3\mathrm{He}$ atom.