Superfluidity and phase separation in helium films

A vectorial generalization of the Blume-Emery-Griffiths model is proposed to describe superfluidity in films of ${}_{}{}^3{}_{}{}^{}\mathrm{He}$-${}_{}{}^4{}_{}{}^{}\mathrm{He}$ mixtures, and is solved by an approximate renormalization scheme due to Migdal. In contrast to bulk mixtures, the line of superfluid transitions is connected to the phase-separation curve by a critical end point. The universal jump of the superfluid density, present in the pure ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ system, is preserved with increasing ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ concentrations $x$ until the critical end point occurs at $x\lesssim 0.12$. At smaller $x$, phase separation causes a kink in the superfluid density versus temperature curve. No tricritical point occurs for any value of the model parameters, although an effectively tricritical phase diagram is obtained in a certain limit. Lines of constant superfluid density bunch up near the effective tricritical point, as predicted by tricritical scaling theory. This treatment also describes superfluidity in pure ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ films in the presence of two-dimensional liquid-gas phase separation. In addition we calculate the specific heat of the pure ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ system, using the recursion relations of Kosterlitz. This specific heat has a broad maximum above the superfluid transition temperature, corresponding to a gradual dissociation of vortex pairs with increasing temperature.