Series expansion method based on the droplet description of ferromagnetic and fully frustrated q-state Potts models.

Abstract

We develop an alternative formalism, based upon the droplet description of critical point phenomena, for calculating the high-temperature (low-density) series expansions for (i) ferromagnetic and (ii) fully frustrated q-state Potts models. For both (i) and (ii), we apply this formalism to explicitly calculate, for the square lattice, the first 20 terms in the series expansion of the general-q partition function. We then obtain from the partition function the series for the mean number of clusters and the specific heat, and analyze the series using Pad\'e approximants. For case (i), the ferromagnetic Potts model, we verify the existence of a geometric transition at the same temperature as the well-known ferromagnetic transition temperature for all q. For case (ii), the fully frustrated Potts model, our analysis reveals the presence of a geometric (percolation) transition at a finite temperature ${\mathit{T}}_{\mathrm{perc}}$(q), which is well above the critical temperature ${\mathit{T}}_{\mathit{c}}$=0 for these models, but somewhat below the corresponding ferromagnetic transition temperature.