Superflow in Solid $^4$He

Kim and Chan have recently observed Non-Classical Rotational Inertia (NCRI) for solid $^4$He in Vycor glass, porous gold, and bulk. Using a microscopic theory where each atom has the same local superfluid velocity (which depends on the microscopic atomic mass density), we show that their low $T$ value of the superfluid fraction, $\rho_{s}/\rho_{0}\approx0.015$, is consistent with what is known of atomic delocalization in this system. In the macroscopic theory, we explicitly include a lattice mass density $\rho_{L}$ distinct from the normal fluid density $\rho_{n}$, thus making the superfluid hydrodynamics consistent with Galilean transformations, which implies that $\rho_{0}=\rho_{s}+\rho_{n}+\rho_{L}$. We also show that $\rho_{L}(T)=\rho_{0}(T)-\rho^{*}_{s}(T)$, where $\rho_{0}(T)$ is the average mass density and $\rho^{*}_{s}(T)$ is computed from the microscopic mass density. This added complexity makes determination of $\rho_{n}/\rho_{0}$ from the measured $\rho_{s}/\rho_{0}$ non-trivial, although an excitation energy of about 0.35~K is relevant as $\rho_{n}/\rho_{0}$ rises from its low temperature value of zero. The macroscopic phase inferred from the observation of NCRI suggests quantum vortices, whose cores must reside between the lattice sites.