Potts model

The Potts model, proposed by Renfrey B. Potts in 1952 ^[1]^[2], is a generalisation of the Ising model to more than two components. For a general discussion on Potts models see Refs ^[3]^[4]. In practice one has a lattice system. The sites of the lattice can be occupied by particles of different species, $S=1,2,\cdots ,q$ .

The energy of the system, $E$ , is defined as:

E=-K\sum _{\langle ij\rangle }\delta (S_{i},S_{j})

where $K$ is the coupling constant, $\langle ij\rangle$ indicates that the sum is performed exclusively over pairs of nearest neighbour sites, and $\delta (S_{i},S_{j})$ is the Kronecker delta. Note that the particular case $q=2$ is equivalent to the Ising model.

Phase transitions

Considering a symmetric situation (i.e. equal chemical potential for all the species):

\mu _{1}=\mu _{2}=\cdots =\mu _{q}

;

the Potts model exhibits order-disorder phase transitions. For space dimensionality $d=2$ , and low values of $q$ the transitions are continuous ( $E(T)$ is a continuous function), but the heat capacity, $C(T)=(\partial E/\partial T)$ , diverges at the transition temperature. The critical behaviour of different values of $q$ belong to (or define) different universality classes of criticality For space dimensionality $d=3$ , the transitions for $q\geq 3$ are first order ( $E$ shows a discontinuity at the transition temperature).