The Potts model
The Q-state Potts model is a lattice model. Each spin on the lattice can take on an integer value between 1 and Q. The particular value Q=2 reduces to the Ising model. The model Hamiltonian is given by: where the sum is over every pair of nearest-neighbors, and the minus sign ensures that interactions are indeed ferromagnetic.
Experiments and applications : (in the following, D denotes the spatial dimension) (D=2, Q=3) transitions in monolayers adsorbed on a surface (lattice gas); (D=2) the Q=0 and Q=1 limits describe infinite resistor networks ("spanning trees") and percolation processes, respectively. (D=3) magnetism in some alloys (e.g., CoCs2Br5, CoCs3Cl5, DyPO4); (D=3) structural transitions (e.g., SrTiO3)
Thermodynamic properties : the two-dimensional Potts model exhibits a first-order phase transition whenever q > 4, and a second-order transition if q < 4 In two dimensions, i.e., with a coordinance number z=4, mean-field theory (Kihara et al., 1954) yields the following transition temperature: Q=2 (Ising) : Tc = 2.0 Q>2 : Tc = 2(q-2)/[(q-1)log(q-1)] (to be compared with the exact result 1/Tc = ln(1+sqrt(q))).
Q | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
Tc(MF) | 2.0 | 1.44 | 1.21 | 1.08 | 0.99 | 0.93 | 0.88 | 0.84
Tc(exact) | 1.1346 | 0.99497 | 0.91024 | 0.85153 | 0.80761 | 0.77306 | 0.74490 | 0.72135
MF theory obviously leads to overestimated temperatures wrt actual transition temperatures.
Bibliography : F.Y. Wu Rev. Mod. Phys.54, 235 (1982) P.M. Chaikin, T.C. Lubensky, Principles of condensed matter physics, (Cambridge: Cambridge University Press, 1995).