Lack of correlation-length scaling for an array of boxes

Finite-size scaling theory predicts that uniformly small critical systems that have the same dimensionality and belong to the same universality class will scale as a function of the ratio of the spatial length L to the correlation length xi. This should occur for all temperatures within the critical region. Measurements of the heat capacity of liquid He-4 confined to a two-dimensional (2D) planar geometry agree well with this prediction when the He-4 is normal but disagree near the specific heat maximum where the confined He-4 becomes superfluid. Data for He-4 confined to 1D structures show a similar behavior (however the lack of data collapse is not as dramatic). Recent measurements of the heat capacity from two 0D confinements, which differ by a factor of two in size, fail to scale at any temperature within the critical region. This lack of scaling may be due to the interaction of neighboring boxes through the shallow channels used to fill them. This is quite surprising since the liquid in the channels is not superfluid at the temperatures of interest for the helium in the boxes. Further more, measurements of the superfluid density of the helium within the channels reveal a critical temperature that is higher than expected suggesting that the normal fluid is affected by the already superfluid regions at each end of these channels. Both of these anomalies might be explained by a proximity effect analogous to what is seen when normal metals are sandwiched between two superconductors.