Comment on ‘ Equilibrium crystal shape of the Potts model at the first - order transition point ’

We comment on the article by Fujimoto (1997 J. Phys. A: Math. Gen. 30 3779) where the exact equilibrium crystal shape (ECS) in the critical Q-state Potts model on the square lattice was calculated, and its equivalence with ECS in the Ising model was established. We confirm these results, giving their alternative derivation applying the transformation properties of the one-particle dispersion relation in the six-vertex model. It is shown, that this dispersion relation is identical with that in the Ising model on the square lattice. In paper [1] Fujimoto determined the equilibrium crystal shape (ECS) in the Q-state Potts model on the square lattice at the first order transition temperature. Fujimoto claimed [1, 2], that ECS is universal for a wide class of models, including the eightvertex model [2, 3], and the Ising model on the square lattice [4]. The origin of this universality is still not well understood. The subject of the this comment is to show, that the one-particle spectrum in the six-vertex model (which is known to be equivalent to the critical Q-state Potts model), and in the Ising model on the square lattice are also the same. So, one can say, that the universality of ECS in different square-lattice models reflects the universality in the one-particle dispersion relation. Consider the Q-state Potts model (Q > 4) on the square lattice, shown in figure 1. The sites of the Potts lattice are depicted by open circles. The model Hamiltonian is given by β E = −K1 ∑ j, l δ(σj,l+1 − σj, l)−K2 ∑ j, l δ(σj+1, l − σj, l). (1) Here we use the same notations as in [1], constants K1, K2 obey the critical temperature condition [3]: (expK1−1)(expK2−1) = Q. Let ψ be a vector associated with a horizontal row of the Potts lattice, ψ = ψ(..., σ1, σ2, ..., σl, ...). Denote by T1 and T2 the two shift operators [1, 2], shown schematically in figure 1. Operator T2 is the row-to-row transfer matrix of the Potts model, and T1 acts on the vector ψ as T1 ψ(..., σ1, σ2, ..., σl, ...) = ψ(..., σ2, σ3, ..., σl+1, ...). (2) It is well known, that the critical Potts model is equivalent to the six-vertex model [3, 5], and to the interaction round a face (IRF) model [1, 6]. Consider for definiteness the associated six-vertex model. Its sites are shown by full circles in figure 1. One can define two shift operators Tx and Ty corresponding to translations along the directions x, and y. Note, that operator Ty is the row-to-row transfer matrix of the six-vertex Comment on ’ECS of the Potts model’ 2