Potts model

The Potts model is a lattice spin system in which each site carries one of $q$ discrete states (“colors”). It generalizes the Ising model from two spin values to $q\ge 2$ values.

Let $\Lambda$ be a finite subset of $\mathbb{Z}^d$ (or a finite graph) and fix an integer $q\ge 2$. A spin configuration is a map $\sigma:\Lambda\to\{1,2,\dots,q\}$, i.e. the spin space is $\{1,\dots,q\}$. With boundary condition $\eta$ on $\Lambda^c$, a standard nearest-neighbor Hamiltonian is

where $J\in\mathbb{R}$ controls the interaction and $(h_1,\dots,h_q)$ is an external field favoring certain states.

At inverse temperature $\beta$, the associated finite-volume Gibbs measure is proportional to $\exp(-\beta H_\Lambda(\sigma\mid\eta))$, normalized by the partition function .

Key properties

• Ferromagnetic vs antiferromagnetic.

  • If $J>0$, equal neighboring spins are favored (ferromagnetic Potts).
  • If $J<0$, unequal neighbors are favored (antiferromagnetic Potts), tying the model to graph coloring constraints at low temperature.

• Symmetry. With $h_a\equiv 0$, the model is invariant under permutations of the $q$ states (the symmetry group is $S_q$). Ordered phases correspond to selecting a preferred state, captured by an order parameter such as the deviation of state densities from $1/q$.

• Reduction to Ising at $q=2$. For $q=2$, the Potts model is equivalent (after a simple reparameterization and additive energy shift) to the Ising model .

• Random-cluster (Fortuin–Kasteleyn) representation. In the ferromagnetic case, the Potts model is tightly linked to the random-cluster model : the Potts partition function can be rewritten as a sum over bond configurations with a cluster weight $q$. This connection is a key tool for studying phase transitions and correlations.

• Correlations and criticality. Two-point correlations (see two-point correlation functions ) can exhibit long-range order or critical decay depending on dimension, $q$, and the sign of $J$. In two dimensions, the ferromagnetic model has especially rich and well-understood critical behavior.

Physical interpretation

The Potts model describes systems with multiple equivalent local states, such as:

• multi-orientational “spins” in magnetic or structural phase transitions,
• domain formation with $q$ possible labels,
• simplified models of ordering where the order parameter selects one of several symmetry-related states.

The ferromagnetic model captures domain alignment (neighboring sites prefer the same state), while the antiferromagnetic model captures competition (neighbors prefer different states), often leading to frustration and highly constrained low-temperature structure.