Crossover in the phase-coexistence between models with discrete and continuous variables

The relative weight of the distinct phases that coexist at a first-order phase transition in systems with discrete degrees of freedom is well understood. For example, in the q-state Potts model, it is characterized by a ratio R=1:q of the disordered vs. ordered regions. In models with continuous variables on the other hand, this ratio is generally unknown. Several recent instances however suggest that it equals R=1:I_O, where I_O denotes the integral measure of the space of extremal states of the ordered phase. In order to explore the emergence of this integral measure, we examine a system that realizes a crossover from discrete to continuous variables and study the behavior of R at its phase-coexistence points. In particular, we consider a generalized n-state clock-model on a three-dimensional simple cubic lattice with both bilinear and biquadratic exchange interactions. In the large-n (XY) limit, this model is known to harbor a first-order thermal phase transition, as does the 3-state Potts model, to which the model reduces in the limit of n=3. Here, we explore the phase-coexistence over the range of intermediate values of n using large-scale Monte Carlo simulations.