Correlated Random-Chemical-Potential Model for the Phase Transitions of Helium Mixtures in Porous Media

A lattice model is constructed for ${}_{}{}^3{}_{}{}^{}\mathrm{He}$-${}_{}{}^4{}_{}{}^{}\mathrm{He}$ mixtures in porous media, characterized by correlated random chemical-potential differences between isotopes. Sites with high chemical-potential differences form a random connected porous structure, similar to silica strands in aerogel. We find by Monte Carlo simulation that the bulk tricritical phase diagram is replaced: The $\lambda$ line extends to zero temperature, as had been theoretically predicted some time ago, and phase separation occurs within the superfluid phase. These findings and the behavior of the order parameter agree with recent experiments.