Potts model

In statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a crystalline lattice. By studying the Potts model, one may gain insight into the behaviour of ferromagnets and certain other phenomena of solid-state physics. The strength of the Potts model is not so much that it models these physical systems well; it is rather that the one-dimensional case is exactly solvable, and that it has a rich mathematical formulation that has been studied extensively. In statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a crystalline lattice. By studying the Potts model, one may gain insight into the behaviour of ferromagnets and certain other phenomena of solid-state physics. The strength of the Potts model is not so much that it models these physical systems well; it is rather that the one-dimensional case is exactly solvable, and that it has a rich mathematical formulation that has been studied extensively. The model is named after Renfrey Potts, who described the model near the end of his 1951 Ph.D. thesis. The model was related to the 'planar Potts' or 'clock model', which was suggested to him by his advisor, Cyril Domb. The four-state planar Potts model is sometimes known as the Ashkin–Teller model, after Julius Ashkin and Edward Teller, who considered an equivalent model in 1943. The Potts model is related to, and generalized by, several other models, including the XY model, the Heisenberg model and the N-vector model. The infinite-range Potts model is known as the Kac model. When the spins are taken to interact in a non-Abelian manner, the model is related to the flux tube model, which is used to discuss confinement in quantum chromodynamics. Generalizations of the Potts model have also been used to model grain growth in metals and coarsening in foams. A further generalization of these methods by James Glazier and Francois Graner, known as the cellular Potts model, has been used to simulate static and kinetic phenomena in foam and biological morphogenesis. 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 n = 0, 1, ..., q-1 and that the interaction Hamiltonian be given by with the sum running over the nearest neighbor pairs (i, j) over all lattice sites. The site colors si take on values in {1, ..., q}. Here, Jc is a coupling constant, determining the interaction strength. This model is now known as the vector Potts model or the clock model. Potts provided the location in two dimensions of the phase transition, for q = 3 and 4. In the limit as q → ∞, this becomes the XY model. What is now known as the standard Potts model was suggested by Potts in the course of his study above, and uses a simpler Hamiltonian, given by: where δ(si, sj) is the Kronecker delta, which equals one whenever si = sj and zero otherwise. The q=2 standard Potts model is equivalent to the Ising model and the 2-state vector Potts model, with Jp = −2Jc. The q = 3 standard Potts model is equivalent to the three-state vector Potts model, with Jp = −(3/2)Jc.