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 transtions

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

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

the Potts models exhibit 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 behavior 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)