The Potts Model

The Potts model [#!POTTS!#,#!WU!#] encompasses a number of problems in statistical physics and lattice theory. It generalizes the Ising model so that each spin can have more than two components, and has been a subject of increasingly intense research interest in recent years. It includes the ice-rule vertex and bond percolation models as special cases.

The $Q$-state Potts model consists of a lattice of spins $\sigma_i$, which can take $Q$ different values, and whose Hamiltonian is

$$H \; = \; K \; \sum_{(i,j)} \; \delta_{\sigma_i\sigma_j},$$ (6.8)

where the spins take on the values $\sigma_i = 0,1,2,...,Q-1$, and the sum is over nearest neighbor pairs on sites on a lattice. For $Q = 2$, this is equivalent to the Ising model. The Potts model is thus a simple extension of the Ising model; however, it has a richer phase structure, which makes it an important testing ground for new theories and algorithms in the study of critical phenomena.

Figure 6.2: The Main Procedure of Potts Model Simulation using Metropolis Algorithm: One starts with an initial configuration of spins and repeats these procedures.