Bose-Einstein condensation and superfluidity of a dilute Bose gas in a random potential

There is a growing interest in the relation between Bose-Einstein condensation (BEC) and the superfluidity. A Bose system confined in random media such as porous glass is suitable for studying this relation because BEC and superfluidity can be suppressed and controlled in such a disordered environment. However, it is not clear how this relation is affected by disorder and there are few theoretical studies that can be quantitatively tested by experiment. In this work we develop a dilute Bose gas model with a random potential that takes into account the pore-size dependence of porous glass. Then we compare our model with the measured low-temperature specific heat, condensate density, and superfluid density of ${}^4\mathrm{He}$ in Vycor glass. This comparison uses no free parameters. We predict phenomena at low temperatures: First, the random potential causes a T-linear specific heat instead of the $T^3$ dependence that is usually caused by phonons. Second, the BEC can remain even when the superfluidity disappears at low densities. Third, the system makes a reentrant transition at low densities; that is, the superfluid phase changes to the normal phase again as the temperature is reduced. This reentrant transition is more likely to be observed when the strength of the random potential is increased.