Phase diagram in the generalized chiral clock models

Using the results of the integrable chiral Potts model, we speculate about the possible phase diagram of the generalized chiral clock model. In the integrable model, it is found that the chiral fields are almost proportional to one another. Here we consider a chiral model with all the chiral fields exactly proportional to one another with fixed proportionality constants given by the integrable case. This reduces the chiral field variables to just one denoted by Δ. The model thus defined includes the generalized clock model or ZN model (at Δ = 0 or 12) and the chiral clock models as special cases. Using the results of many authors on the ZN model in the so-called symmetric case (with vertical and horizontal interactions equal), and the behavior of the system at T = 0, we can get a pretty good picture of the phase diagram of this chiral model. The exact results show that the commonly held belief that the massless phase in the ZN model for N ≥ 5 is in the same universality class as the classical XY model and the belief that the commensurate-incommensurate phase transition is of the Pokrovskii—Talapov type and the incommensurate-fluid transition is of Kosterlitz—Thouless type may not be correct, as universality may not even hold on these phase boundaries.