Critical Exponents of the Four-State Potts Model

The critical exponents of the four-state Potts model are directly derived from the exact expressions for the latent heat, the spontaneous magnetization, and the correlation length at the transition temperature of the model. PACS numbers: 05.50.+q, 05.70.–a, 64.60.Cn, 75.10.Hk Typeset using REVTEX 1 The q-state Potts model [1,2] is a generalization of the Ising model that is the twostate Potts model. Although the Potts model has not been solved exactly, there have been several exact results at the critical point for this model in two dimensions. In 1952 Potts [1] conjectured the exact critical temperatures of his model on the square lattice for all q by a Kramers-Wannier [3] type duality argument. In 1971 Temperley and Lieb [4] showed that the Potts model can be expressed as a staggered six-vertex model. Following the equivalence [5] between the Potts model and a staggered six-vertex model, in 1973 Baxter [6] calculated the free energy of the Potts model at the critical temperature, and showed that the model has a continuous phase transition for q ≤ 4, and has a first-order phase transition (i.e. has latent heat) for q > 4. In 1979 den Nijs [7] conjectured the thermal scaling exponent for q ≤ 4 by considering relation between the eight-vertex model and the Potts model. In 1980 Nienhuis et al. [8] and Pearson [9] conjectured independently the magnetic scaling exponent for q ≤ 4 from numerical results. In 1981 Black and Emery [10] showed the den Nijs conjecture to be asymptotically exact by using the Coulomb-gas representation [11] of the Potts model and renormalization-group methods. In 1982 Baxter [12] calculated the spontaneous magnetization of the model at the transition point for q > 4. In 1983 den Nijs [13] verified a conjecture for the magnetic scaling exponent for q ≤ 4 from the scaling behavior of the correlation function in the Coulomb-gas representation. In 1984 Dotsenko [14] again verified the conjectures for the thermal and magnetic scaling exponents for q ≤ 4 using conformal field theory. Recently Buffernoir and Wallon [15] obtained an exact expression for the correlation length of the Potts model at the critical temperature for q > 4 by using Temperley-Lieb algebra [5] and a Bethe ansatz [16]. In this paper we derive the critical exponents of the four-state Potts model directly from the three main exact results of the Potts model which are Baxter’s calculation of the latent heat and the spontaneous magnetization and Buffernoir and Wallon’s calculation of the correlation length. The Hamiltonian for the q-state Potts model on the isotropic square lattice is