Superfluid density in inhomogeneous 4He with applications to thin films.

Abstract

A general perturbative method for finding the zero-temperature superfluid density ${\mathrm{\ensuremath{\rho}}}_{\mathit{s}}$ in $^{4}\mathrm{He}$ in the presence of weak inhomogeneous external potentials is developed. The lack of translational invariance reduces ${\mathrm{\ensuremath{\rho}}}_{\mathit{s}}$ below the average mass density \ensuremath{\rho}\ifmmode\bar\else\textasciimacron\fi{}, leading to an inert, or normal, part, even at temperature T=0. A variational formula for 1-${\mathrm{\ensuremath{\rho}}}_{\mathit{s}}$/\ensuremath{\rho}\ifmmode\bar\else\textasciimacron\fi{} is given in terms of the density response function for uniform $^{4}\mathrm{He}$. Approximations lead to a result in terms of the uniform fluid structure factor and the single-atom effective mass. Explicit numerical predictions are given for thin $^{4}\mathrm{He}$ films adsorbed on ${\mathrm{H}}_{2}$ and other substrates.