Potts Model - Physical Description

Physical Description

The Potts model consists of spins that are placed on a lattice; the lattice is usually taken to be a two-dimensional rectangular Euclidean lattice, but is often generalized to other dimensions or other lattices. Domb originally suggested that the spin take one of q possible values, distributed uniformly about the circle, at angles

, where

and that the interaction Hamiltonian be given by

with the sum running over the nearest neighbor pairs over all lattice sites. The site colors take on values, ranging from . Here, is a coupling constant, determining the interaction strength. This model is now known as the vector Potts model or the clock model. Potts provided a solution for two dimensions, for q=2,3 and 4. In the limit as q approaches infinity, this becomes the XY model.

What is now known as the standard Potts model was suggested by Potts in the course of the solution above, and uses a simpler Hamiltonian, given by:

where is the Kronecker delta, which equals one whenever and zero otherwise.

The q=2 standard Potts model is equivalent to the Ising model and the 2-state vector Potts model, with . The q=3 standard Potts model is equivalent to the three-state vector Potts model, with .

A common generalization is to introduce an external "magnetic field" term, and moving the parameters inside the sums and allowing them to vary across the model:

where the inverse temperature, k the Boltzmann constant and T the temperature. The summation may run over more distant neighbors on the lattice, or may in fact be an infinite-range force.

Different papers may adopt slightly different conventions, which can alter and the associated partition function by additive or multiplicative constants.