Anomalous Elasticity of $^4$He Films at the Quantum Phase Transition

$^4$He films on solid substrates exhibit a quantum phase transition between localized (nonsuperfluid) and superfluid states by changing coverage $n$. We have made torsional oscillator (TO) studies for $^4$He films adsorbed on nanoporous glasses. A TO with localized films showed an apparent "supersolid" behavior, an increase in TO frequency $f$ with broad peak in $Q^{-1}$. Combining with FEM analyses for TO's with different designs, we conclude that the behavior results from the softening of adsorbed $^4$He films at high temperatures. The features in $f$ and $Q^{-1}$ are fitted well to a Debye-like activation with a distributed energy gap $\Delta$, so the elasticity is accounted by thermal excitation of localized atoms to an "extended" state. As the critical coverage $n_{c}$ approaches the gap decreases to zero with a powerlaw $\Delta \propto (n-n_{c})^{1.2}$. Assuming that the $^4$He chemical potential $\mu(n)$ is located in the middle of the gap, we can estimate the elastic constant $\kappa^{-1} = n^2 \partial \mu / \partial n$. The elasticity agrees with shear moduli of $^4$He films obtained from the FEM analysis within factor of three. The energetics proposed from the elastic behavior naturally explains other properties of He films adsorbed on disordered substrates.