Potts model

The Potts model was proposed by Renfrey B. Potts in 1952 (Ref. 1). The Potts model is a generalisation of the Ising model to more than two components. For a general discussion on Potts models see Refs. 2 and 3. 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).

See also

• Ashkin-Teller model
• Kac model

References

1. Renfrey B. Potts "Some generalized order-disorder transformations", Proceedings of the Cambridge Philosophical Society 48 pp. 106−109 (1952)
2. F. Y. Wu "The Potts model", Reviews of Modern Physics 54 pp. 235-268 (1982)
3. F. Y. Wu "Erratum: The Potts model", Reviews of Modern Physics 55 p. 315 (1983)
4. Rodney J. Baxter "Exactly Solved Models in Statistical Mechanics", Academic Press (1982) ISBN 0120831821 Chapter 12 (freely available pdf)