The Potts model is a simple physical model in statistical mechanics. It is a generalization of the Ising model.
The model is named after R. B. Potts who described the model near the end of his 1952 PhD thesis. The model was related to the "planar Potts" or "clock model", which was suggested to him by his PhD advisor C. Domb. The Potts model is sometimes known as the Ashkin-Teller model, as they considered a four component version in 1943.
The Potts model consists of spins that are placed on a lattice. The spin variables si take on values, or colors, ranging from 1...q where q defines the number of states. The Hamiltonian is given by:
-
where Σ(i,j) indicates a summation over all pairs of neighboring spins.
Often the model is considered on a 2D lattice, but any neighborhood relationship is possible. Other generalisations include adding an external "magnetic field" term h, and moving the parameters inside the sums and allowing them to vary across the model:
| H = - | ∑ | Jijδ(si,sj) - | ∑ | hisi |
| (i,j) | | i | |
Different papers may adopt slightly different conventions, which can alter H and the associated partition function by additive or multiplicative constants.
Despite its simplicity as a model of a physical system, the Potts model is useful as a model system for the study of phase transitions. When q > 3 the model has a first order transition, otherwise a continuous transition is observed, as in the Ising model where q = 2. Further use is found through the models relation to percolation problems and the Tutte and chromatic polynomials found in combinatorics.
The model has a close relation to the Fortuin-Kasteleyn random cluster model, another model in statistical mechanics. Understanding this relationship has helped develop efficient Markov chain Monte Carlo techniques for numerical exploration of the model at small q.