Phase diagram of a model forHe3-He4mixtures in two dimensions

Abstract

A dilute two-dimensional classical $X-Y$ model is proposed to describe two-dimensional ${}_{}{}^3{}_{}{}^{}\mathrm{He}$-${}_{}{}^4{}_{}{}^{}\mathrm{He}$ mixtures. Using a Villain form for the potential, a Migdal-Kadanoff position-space renormalization is carried out. The resulting phase diagram plotted against $T$ and the ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ concentration $x$, is qualitatively similar to that for a three-dimensional mixture with some important differences: (i) The initial phase separation is into two normal fluids having different concentrations of ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ rather than into a normal fluid and a superfluid, so there is no real tricritical point. (ii) At lower temperatures, the phase separation is into a superfluid and a normal fluid, and as the phase-separation temperature is approached from the normal side, islands of superfluid appear, with the correlation length diverging with an exponent $\nu =1\mathrm{}$.