Onset of superfluidity inHe4films adsorbed on disordered substrates

We have studied ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ films adsorbed in two porous glasses, aerogel and Vycor, using high-precision torsional oscillator and dc calorimetry techniques. Our investigation focused on the onset of superfluidity at low temperatures as the ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ coverage is increased. Torsional oscillator measurements of the ${}_{}{}^4{}_{}{}^{}\mathrm{aerogel}$ system were used to determine the superfluid density of films with transition temperatures as low as 20 mK. Heat capacity measurements of the ${}_{}{}^4{}_{}{}^{}\mathrm{Vycor}$ system probed the excitation spectrum of both nonsuperfluid and superfluid films for temperatures down to 10 mK. Both sets of measurements suggest that the critical coverage for the onset of superfluidity corresponds to a mobility edge in the chemical potential, so that the onset transition is the bosonic analog of a superconductor-insulator transition. The superfluid density measurements, however, are not in agreement with the scaling theory of an onset transition from a gapless, Bose glass phase to a superfluid. The heat capacity measurements show that the nonsuperfluid phase is better characterized as an insulator with a gap.