Intermediate phase for a classical continuum model.

A number of years ago, Widom and Rowlinson proposed a continuum model of interacting spherical particles and provided compelling evidence that this model has a phase transition. A rigorous proof of the existence of a two-phase regime in this model emerged shortly thereafter and recently, this phase transition has been understood in geometric terms—as a percolation phenomenon. Nevertheless, progress in the rigorous study of phase transitions in continuum systems has been slow and primarily confined to models of the Widom-Rowlinson ~WR! type. The q-component generalizations of the WR model and their perturbations have been studied in Ref. 5 and the existence of a region with multiple phases have been demonstrated at large activities. The present work does not really provide any further significant insight for the general problem of liquid-gas transition. However, we have discovered a modified type of penetrable sphere model that exhibits two phase transitions; the intermediate phase in this model is quite distinct from the highor low-density phase of the usual WR systems. The q-component WR model may be loosely described as a continuum version of the q-state Potts model ~or, somewhat more accurately, the annealed dilute Potts models!. In this sense, the model we study is a continuum Ashkin-Teller model. On the lattice, the Ashkin-Teller model is known to have an intermediate phase and, as will be demonstrated in this work, such a phase survives in the continuum version. The reader will recall that the Ashkin-Teller model consists of four species of particles, here denoted by Y ~yellow!, R ~red!, G , and B . In the most interesting region of the phase diagram, each particle type is strongly attracted to particles of their own species, weakly attracted to one other species and repelled by the other two. In our case, we will take Y and R as allies against the G-B team. Here we will assume for simplicity that the Y -G and Y -B ~repulsive! interaction are the same. In all other respects it will be assumed that the four species are identical under relabeling, e.g., Y→B→R→G→Y . With this description it is not difficult to imagine that as the ~common! fugacity is increased, or, in the lattice versions, as the temperature is lowered, there will come a point where one team dominates at the expense of the other. Explicitly, there will be two phases, a YR-rich phase and a GB-rich phase, within which YR and GB symmetry is still respected. Finally, at lower temperatures and/or higher fugacity, cooperation between the allied pairs is forsaken in favor of single color dominance. Here, there are four distinct ~but equivalent! phases. To implement this scheme in the continuum, we consider spherical particles with two interaction radii, a and A.a .The outer radius A serves as a hard-core interaction distance for phobic pairs, e.g., any G and Y particles are forbidden to come within a radius 2A of one another. The inner radius a then serves as a hard core for the phillic pairs. Finally, all the species are unaffected by the members of their own color group. As is the case in the usual WR models, one can conceive of integrating out all colors save a single species, Y , and regard all of this as a machinery for generating interactions between the Y particles. ~In the two-component model, the interaction can be written in a closed form.! In the present context, the resulting effective interaction does not immediately yield any intuitive features and hence will not be discussed further. Nevertheless, the single species description of the model is of interest and will be used alongside the fourcolor picture in describing the various phases. As in the case of the two-component model, we may describe the various phases in terms of different percolation properties. Indeed, in Refs. 3 and 4, it was shown that percolation is necessary and sufficient for the existence of distinct high-density phases. In the present context, we may envision two types of percolation: ~i! inner-core percolations and ~ii! outer-core percolation. The former is analogous to the high-density phase in the usual q-component WR models; there is an infinite cluster of particles that are connected in the sense that the union of spheres of radius a surrounding each particle form an infinite component. Observe, by the ~inner! hard-core rule such a cluster must be of a single color; indeed, in the usual q-component models, these particles constitute the density excess in the two-phase regime and are hence identified with the condensate. Percolation of type ~ii! constitutes the feature of the present model. Namely, consider a situation where the innercore percolation does not occur and yet there is an infinite cluster of outer cores connected in the sense of overlapping spheres of radius A . Here, the outer-core rule requires only that the infinite cluster belong to one of the two teams, it thus represents the excess density of ~say! YR over GB . However, the fact that there is no percolation of the inner cores indicates that within the YR infinite cluster, the densities of yellow and red particles are equal. It turns out that by using the methods of Ref. 3, the abovedescribed picture can ~for certain values of parameters! be transformed into precise statements concerning coexisting PHYSICAL REVIEW B 1 OCTOBER 1996-I VOLUME 54, NUMBER 13