# 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, ${\displaystyle S=1,2,\cdots ,q}$.

The energy of the system, ${\displaystyle E}$, is defined as:

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

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

## Phase transitions

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

${\displaystyle \mu _{1}=\mu _{2}=\cdots =\mu _{q}}$;

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