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))).
Q23456789
Tc(MF)2.01.441.211.080.990.930.880.84
Tc(exact)1.13460.994970.910240.851530.807610.773060.744900.72135
MF theory obviously leads to overestimated temperatures wrt actual transition temperatures.

 Bibliography :