Potts model
The Potts model is a fundamental model in statistical mechanics that generalizes the Ising model to describe systems of interacting particles or spins on a lattice, where each site can take one of q discrete states (with q ≥ 2) and neighboring sites contribute a lower energy if they share the same state.[1] Formally, for a graph G = (V, E) representing the lattice, the Hamiltonian is given by H(σ) = -∑{<x,y> ∈ E} δ{σ_x, σ_y}, where σ_x ∈ {1, 2, ..., q} is the spin at vertex x, and δ is the Kronecker delta; the probability distribution is proportional to e^{-β H(σ)}, with β the inverse temperature and the partition function Z = ∑_σ e^{-β H(σ)} encoding thermodynamic properties like phase transitions.[1] Introduced by Renfrey B. Potts in 1952 as part of his work on order-disorder transformations, the model extends the binary-spin Ising model of 1925 (recoverable as the q=2 case, up to a rescaling of β by 1/2) to study ferromagnetism and symmetry breaking in multi-state systems.[2][3][1] Key features of the Potts model include its ability to exhibit continuous or first-order phase transitions depending on q and dimensionality, with exact solvability in two dimensions for all q via duality relations and connections to the eight-vertex model, as established in the 1970s.[4] For q=1, it reduces to bond percolation, linking it to connectivity properties in random graphs, while the Fortuin-Kasteleyn random-cluster representation (developed in 1972) reformulates the partition function in terms of weighted spanning subgraphs, facilitating proofs of critical behavior and correlations.[1] In higher dimensions, such as three-dimensional cubic lattices, numerical methods like tensor renormalization group approaches reveal phase diagrams with ferromagnetic ordering at low temperatures.[5] Beyond physics, the Potts model has broad applications: in materials science for modeling alloy ordering and grain growth, in biology for simulating cell adhesion and tumor dynamics, and in computer science, where the antiferromagnetic variant relates to graph coloring problems, as the number of zero-temperature ground states equals the value of the chromatic polynomial at q, and more generally through the Fortuin–Kasteleyn random-cluster representation.[6] Experimental realizations include magnetic systems like dilute antiferromagnets and structural transitions in crystals, underscoring its role in bridging theory and observation.[4]Definition
q-State Potts Model
The q-state Potts model is a discrete spin model defined on a lattice in statistical mechanics, serving as a generalization of the two-state Ising model to an arbitrary number q ≥ 2 of possible states per site. Each lattice site hosts a spin variable that can take one of q discrete values, conventionally labeled as 1 through q and interpretable as distinct colors, phases, or atomic species in physical systems. The model's interaction rule is fully symmetric under permutations of the state labels: neighboring sites incur no energy contribution when occupying the same state, but a fixed positive penalty when their states differ, thereby encouraging local alignment without privileging any particular state. R. B. Potts introduced this formulation in 1952 to investigate order-disorder transformations in mixtures of alloys comprising q types of atoms, where the states represent different atomic components and the interactions capture preferences for segregation or clustering into ordered phases.[2] This setup has since become a cornerstone for modeling multi-phase coexistence and symmetry-breaking phenomena in diverse condensed matter systems. When q = 2, the model precisely recovers the Ising model, with states corresponding to up/down spins or two alloy components, highlighting its role as a unifying framework. For q = 3, it illustrates a ternary system akin to a three-coloring analogy on the lattice, where ferromagnetic interactions promote domains of uniform color rather than strict avoidance.Hamiltonian Formulation
The q-state Potts model is defined through its Hamiltonian, which describes the energy of a system of spins on a lattice graph G = (V, E) with N = |V| sites, where each site i \in V hosts a spin variable \sigma_i taking values in the set \{1, 2, \dots, q\}. The standard ferromagnetic Hamiltonian is given by H(\boldsymbol{\sigma}) = -J \sum_{\{i,j\} \in E} \delta_{\sigma_i, \sigma_j}, where J > 0 is the coupling constant, the sum runs over all edges connecting nearest-neighbor sites, and \delta_{\sigma_i, \sigma_j} is the Kronecker delta function, equal to 1 if \sigma_i = \sigma_j and 0 otherwise.[4] This form favors aligned spins on adjacent sites, generalizing the Ising model (which corresponds to q=2) by allowing q discrete states per spin rather than just \pm 1.[4] The partition function of the model, which encodes the statistical mechanics of the system at inverse temperature \beta = 1/(k_B T) (with k_B Boltzmann's constant and T temperature), is Z = \sum_{\boldsymbol{\sigma} \in \{1,\dots,q\}^N} \exp\left[-\beta H(\boldsymbol{\sigma})\right], where the sum (or trace) is over all q^N possible spin configurations \boldsymbol{\sigma} = (\sigma_1, \dots, \sigma_N).[4] This expression provides the basis for computing thermodynamic quantities such as the free energy via F = -k_B T \ln Z. In the antiferromagnetic case, J < 0, the interaction instead penalizes aligned neighboring spins, promoting diversity in spin values, as originally motivated for certain lattice geometries like the triangular lattice with q=3.[2][4] Although the standard Potts model omits external fields, a magnetic field term can be incorporated as H_h(\boldsymbol{\sigma}) = -h \sum_{i \in V} \delta_{\sigma_i, s} for a preferred state s and field strength h, which breaks the q-state symmetry; however, such extensions are not part of the baseline formulation.[4] In the thermodynamic limit N \to \infty, the finite-volume Gibbs measures \mu_{\Lambda, \beta, J} on finite subsets \Lambda \subset V converge (under suitable boundary conditions) to infinite-volume Gibbs measures on the full lattice, describing equilibrium properties of the infinite system and enabling the study of phase transitions via limiting behaviors of correlation functions and spontaneous symmetry breaking.[4]Historical Context
Origins and Development
The Potts model was introduced by R. B. Potts near the end of his 1951 D.Phil. thesis at the University of Oxford, published in 1952, to study generalized order-disorder transformations by extending the binary spin variables of the Ising model to q discrete states (q ≥ 2).[2] This development drew direct inspiration from Lars Onsager's 1944 exact solution of the two-dimensional Ising model, which demonstrated a continuous phase transition in a lattice system without an external field and established key techniques like transfer matrices for solvable models. By positioning the Potts model as a q-state extension of the Ising case (recovering Ising for q=2), Potts aimed to extend Onsager's insights to more complex ordering processes in lattice systems. Early applications centered on metallurgy, where the model described the thermodynamics of phase separation in binary alloy mixtures under thermal equilibrium, predicting critical temperatures for ordering transitions based on interaction energies between unlike atoms. By the 1960s, interest expanded beyond metallurgy into general statistical mechanics, as researchers recognized the model's utility for simulating cooperative phenomena in diverse lattice systems, including extensions to higher dimensions and varying interaction ranges.[7] A pivotal early advancement came in the 1970s with R. J. Baxter's exact solutions for the two-dimensional q-state Potts model on square and triangular lattices, achieved through duality relations and star-triangle mappings that yielded closed-form expressions for the partition function and critical points. These results connected the model to broader classes of integrable systems, highlighting its role in understanding universality classes of phase transitions.[8]Key Milestones
In the early 1970s, a significant advancement came from the introduction of the random-cluster representation by C. M. Fortuin and P. W. Kasteleyn, which reformulated the Potts model in terms of bond percolation on the lattice, enabling deeper connections to percolation theory and facilitating analysis of phase transitions across different values of q. Building on this, R. J. Baxter provided the exact solution for the two-dimensional q-state Potts model on the square lattice in 1973, deriving the partition function and spontaneous magnetization through transfer matrix methods and identifying the self-dual point where the model exhibits criticality for q ≤ 4, given by e^{\beta J} - 1 = \sqrt{q}. In the 1980s, F. Y. Wu's comprehensive review synthesized these developments, emphasizing duality transformations that map high-temperature expansions to low-temperature ones and establishing the link between the zero-temperature Potts partition function and the chromatic polynomial of the lattice graph, which counts proper q-colorings. More recently, numerical renormalization group techniques, particularly tensor network methods, have addressed the longstanding challenge of three-dimensional criticality; for instance, studies using higher-order tensor renormalization group transformations on the cubic lattice have computed critical temperatures and exponents for q=2,3,4 with high precision, confirming first-order transitions for q>4 and continuous ones otherwise up to 2025.[9]Phase Transitions
Order-Disorder Transitions
The Potts model exhibits order-disorder phase transitions characterized by spontaneous symmetry breaking in the low-temperature phase and restoration of the full symmetry in the high-temperature phase.[4] In the ferromagnetic case on a two-dimensional lattice, the low-temperature ordered phase features a preference for one of the q states, breaking the underlying S_q symmetry, while the high-temperature disordered phase maintains full rotational symmetry among the states.[4] This transition occurs at a critical temperature T_c, where thermodynamic quantities such as the specific heat and magnetic susceptibility exhibit divergences, signaling the onset of long-range order.[4] The order parameter quantifying this symmetry breaking is defined as a magnetization-like quantity, m = \frac{q \delta_{\max} - 1}{q - 1}, where \delta_{\max} is the fraction of sites occupying the dominant state in the thermodynamic limit.[4] In the ordered phase below T_c, m approaches a finite value greater than zero, reflecting the emergence of a preferred state, whereas it vanishes in the disordered phase above T_c due to equal occupation probabilities across states.[4] For the two-dimensional square lattice ferromagnetic Potts model, the critical inverse temperature satisfies \beta_c J = \ln(1 + \sqrt{q}), marking the precise location of the transition. The nature of the transition—continuous (second-order) or discontinuous (first-order)—depends critically on the number of states q. In two dimensions, the transition is continuous for q ≤ 4, allowing the order parameter to vary smoothly through zero at T_c, while it becomes discontinuous for q > 4, with a latent heat and jump in the order parameter indicative of a first-order transition.[4] These qualitative differences arise from the model's underlying duality properties, which sharpen the transition for larger q.[4]Critical Exponents and Universality
The critical exponents of the Potts model characterize the singular behavior of thermodynamic quantities near the critical point, such as the specific heat ~ |t|^α, order parameter ~ |t|^β, susceptibility ~ |t|^γ, and correlation length ~ |t|^ν, where t is the reduced temperature. These exponents satisfy scaling relations like α + 2β + γ = 2 - α, with their values depending on q and the spatial dimension d. In two dimensions, exact expressions for the exponents are known for second-order transitions (1 ≤ q ≤ 4), derived from the solution of the model's duality relations or conformal field theory descriptions, as established in seminal works.[4] For the 2D q-state Potts model, the exponents vary continuously with q, indicating that each value of q belongs to a distinct universality class, except for q=2 which coincides with the Ising universality class. The correlation length exponent ν, for instance, takes the value ν = 1 for q=2, ν = 5/6 for q=3, and ν = 2/3 for q=4. Other exponents follow similarly, as summarized in the table below for representative cases (derived from exact solutions via transfer matrix methods and duality).[4]| q | α | β | γ | ν | η |
|---|---|---|---|---|---|
| 2 | 0 | 1/8 | 7/4 | 1 | 1/4 |
| 3 | 1/3 | 1/9 | 13/9 | 5/6 | 4/15 |
| 4 | 2/3 | 1/12 | 7/6 | 2/3 | 1/2 |