# Potts model

The Potts model is a lattice model in statistical mechanics in which each site of a lattice carries a spin variable that can take one of q discrete states, and neighboring sites gain an energy −J when they are in the same state and zero interaction energy otherwise. Introduced by [Renfrey Potts](https://www.edgepedia.org/renfrey-potts) in 1952 as a multi-color generalization of the [Ising model](https://www.edgepedia.org/ising-model), it has become a standard testing ground for the analytical and numerical study of phase transitions.<sup>[1](https://ftp.ussg.iu.edu/CRAN/web/packages/potts/index.html)</sup><sup> • </sup><sup>[2](https://pure.coventry.ac.uk/ws/portalfiles/portal/3939241/0012026v1.pdf)</sup>

| Key fact | Detail |
|---|---|
| Definition | Spins \( s_i \in \{1, \ldots, q\} \) on lattice sites interact through \( \mathcal{H} = -J \sum_{\langle i,j \rangle} \delta_{s_i, s_j} \); the symmetry is the permutational group \( S_q \)<sup>[3](https://springerlink.fh-diploma.de/article/10.1140/epjb/s10051-021-00076-0)</sup> |
| Origin | Potts (1952), as the multi-color generalization of the Ising model<sup>[1](https://ftp.ussg.iu.edu/CRAN/web/packages/potts/index.html)</sup> |
| Ising case | At \( q = 2 \) the model is equivalent to the spin-1/2 Ising model with a rescaled coupling and a shifted ground-state energy<sup>[4](https://archive.nptel.ac.in/content/storage2/courses/115103028/module4/lec3/2.html)</sup> |
| 2D transition order | Continuous for \( q \le 4 \), first order for \( q > 4 \)<sup>[5](https://harvest.aps.org/v2/journals/articles/10.1103/RevModPhys.54.235/fulltext)</sup> |
| Higher dimensions | First order for \( q > 2 \) in dimensions larger than two<sup>[6](https://ethz.ch/content/dam/ethz/special-interest/phys/theoretical-physics/itp-dam/documents/compstatphys/CompStatPhysPartI_slides.pdf)</sup> |
| Exact 2D critical point | \( \beta_c = \ln(1 + \sqrt{q}) \) on the square lattice<sup>[2](https://pure.coventry.ac.uk/ws/portalfiles/portal/3939241/0012026v1.pdf)</sup> |
| Cluster algorithms | The Swendsen-Wang and Wolff algorithms, developed using the Potts model as the example, update whole clusters in one step and are efficient near critical points<sup>[7](https://www.sciencedirect.com/science/article/pii/037843719090275W)</sup> |
| Applications | Adsorption physics (krypton on graphite as a \( q = 3 \) system) and image segmentation<sup>[4](https://archive.nptel.ac.in/content/storage2/courses/115103028/module4/lec3/2.html)</sup><sup> • </sup><sup>[8](https://arxiv.org/abs/0906.4745)</sup>  |

## Definition

On a graph or lattice, each site \( i \) carries a variable \( s_i \) taking one of \( q \) values, often described as colors. The zero-field Hamiltonian is

\[ \mathcal{H}_{\text{Potts}} = -J \sum_{\langle i,j \rangle} \delta_{s_i, s_j}, \]

where the sum runs over nearest-neighbor pairs and \( \delta_{s_i, s_j} \) equals 1 when the two spins match and 0 otherwise. A pair of aligned neighbors therefore lowers the energy by \( J \), while unlike neighbors pay no interaction energy. The Hamiltonian is invariant under global permutations of the \( q \) values, so the symmetry group is \( S_q \); the \( \mathbb{Z}_2 \) symmetry of the Ising model is recovered at \( q = 2 \).<sup>[3](https://springerlink.fh-diploma.de/article/10.1140/epjb/s10051-021-00076-0)</sup> The ground state is \( q \)-fold degenerate, one configuration for each color filling the whole lattice.<sup>[4](https://archive.nptel.ac.in/content/storage2/courses/115103028/module4/lec3/2.html)</sup>

At \( q = 2 \) the [Kronecker delta](https://www.edgepedia.org/kronecker-delta) can be rewritten in terms of spin-1/2 Ising variables, showing that the two-state Potts model is an Ising model with a rescaled interaction strength and a constant shift in ground-state energy. The \( q = 3 \) model is not equivalent to the spin-1 Ising model, which has a different ground-state degeneracy structure.<sup>[4](https://archive.nptel.ac.in/content/storage2/courses/115103028/module4/lec3/2.html)</sup>

## Phase transitions and the role of q

The order of the transition depends on \( q \) and on dimensionality, which is a central reason the model is used as a test case. In two dimensions the transition is continuous for \( q \le 4 \) and first order for \( q > 4 \), a result established by Baxter and collaborators.<sup>[5](https://harvest.aps.org/v2/journals/articles/10.1103/RevModPhys.54.235/fulltext)</sup> In dimensions larger than two the transition is first order already for \( q > 2 \).<sup>[6](https://ethz.ch/content/dam/ethz/special-interest/phys/theoretical-physics/itp-dam/documents/compstatphys/CompStatPhysPartI_slides.pdf)</sup> A first-order transition shows a bimodal distribution of the order parameter, stochastic switching between the two states in small systems, and hysteresis when the field is changed.<sup>[6](https://ethz.ch/content/dam/ethz/special-interest/phys/theoretical-physics/itp-dam/documents/compstatphys/CompStatPhysPartI_slides.pdf)</sup>

**Interaction range matters.** The order of the transition is not fixed by \( q \) alone. In the two-dimensional \( q = 3 \) model with equivalent neighbors, increasing the number of interacting neighbors \( z \) changes the transition from second order to first order; for \( q = 4 \) the transition becomes discontinuous for \( z > 16 \).<sup>[9](https://journals.aps.org/pre/abstract/10.1103/t156-9vyr)</sup><sup> • </sup><sup>[10](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.94.052103)</sup>  Disorder has a similar effect in three dimensions: the pure \( q = 4 \) model and a weakly diluted one at spin concentration \( p = 0.90 \) show first-order transitions, while strong dilution at \( p = 0.65 \) produces a second-order transition.<sup>[11](https://www.scientific.net/SSP.168-169.357.pdf)</sup>

## Exact results

The two-dimensional model is one of the few interacting lattice systems with substantial exact knowledge. On the square lattice the ferromagnetic critical temperature satisfies \( \beta_c = \ln(1 + \sqrt{q}) \), with exactly known latent heat in the first-order regime.<sup>[2](https://pure.coventry.ac.uk/ws/portalfiles/portal/3939241/0012026v1.pdf)</sup> Baxter solved the square-lattice ferromagnetic model at the criticality condition \( V = \sqrt{q} \) and the antiferromagnetic model at \( V = \sqrt{4 - q} \), where \( V = e^{K} - 1 \) with \( K = J/kT \).<sup>[12](https://www.sciencedirect.com/topics/physics-and-astronomy/pott-model)</sup> For the antiferromagnetic square-lattice model the criticality condition is \( \exp(J/kT) = -1 + \sqrt{4 - q} \); as with the ferromagnet the transition is continuous for \( q \le 4 \) and first order for \( q > 4 \), but the critical temperature is real only for \( q \le 3 \).<sup>[13](https://www.researchgate.net/publication/252149774_Critical_antiferromagnetic_square-lattice_Potts_model)</sup>

The partition function also has a graph-theoretic meaning. Expressed as a polynomial \( Z(G, q, v) \) in \( q \) and a temperature-dependent Boltzmann variable \( v \), the zero-temperature antiferromagnetic case \( v = -1 \) gives \( Z(G, q, -1) = P(G, q) \), the chromatic polynomial counting the ways to color the vertices of \( G \) with \( q \) colors so that no two adjacent vertices share a color.<sup>[14](https://dspace.sunyconnect.suny.edu/items/8a3bdf19-6a5f-4423-88aa-6e9242a72785)</sup>

A quantum version exists as well. The quantum \( q \)-state Potts Hamiltonian has \( Z_q \) symmetry, is ferromagnetic for \( \lambda < 1 \) and paramagnetic for \( \lambda > 1 \), and undergoes a continuous quantum phase transition for \( q \le 4 \) but a discontinuous one for \( q > 4 \) at \( \lambda = 1 \), where the model is exactly solved. At \( q = 2 \) it reduces to the transverse-field Ising model.<sup>[15](https://pmc.ncbi.nlm.nih.gov/articles/PMC4286789/)</sup>

## Monte Carlo simulation and cluster algorithms

The Potts model served as the example system for the cluster algorithms that transformed [Monte Carlo](https://www.edgepedia.org/monte-carlo) simulation of spin systems. The Swendsen-Wang algorithm occupies each bond between like spins with probability \( p = 1 - e^{-\beta J} \), identifies the resulting clusters, and reassigns each cluster a new spin value in one global move; for \( q > 2 \) each cluster simply chooses a new state.<sup>[6](https://ethz.ch/content/dam/ethz/special-interest/phys/theoretical-physics/itp-dam/documents/compstatphys/CompStatPhysPartI_slides.pdf)</sup> Unlike local single-spin algorithms, cluster algorithms update dynamical variables globally, making large changes in a single step, and they are especially efficient near the critical point of a second-order transition, where they avoid critical slowing down.<sup>[16](https://arxiv.org/abs/cond-mat/9510082)</sup>

Robert G. Edwards and Alan D. Sokal, physicists then at New York University, gave a rigorous explanation of the algorithm in 1988 in terms of a joint model of Potts spin variables interacting with bond occupation variables, building on the Fortuin-Kasteleyn representation, and showed how to generalize both the representation and the algorithm to arbitrary models.<sup>[17](https://doi.org/10.1103/PhysRevD.38.2009)</sup> The Wolff algorithm, a single-cluster variant, was described together with Swendsen-Wang in a 1990 review using the Potts model as the worked example.<sup>[7](https://www.sciencedirect.com/science/article/pii/037843719090275W)</sup> Antiferromagnetic Potts models on the square lattice have been studied with the Wang-Swendsen-Kotecky algorithm, which is ergodic at zero temperature on bipartite lattices.<sup>[18](https://arxiv.org/abs/cond-mat/9811345)</sup>

## Variants

**Antiferromagnetic couplings.** With \( J < 0 \), unlike neighbors are favored. On the square lattice the critical behavior differs from the ferromagnet: high-precision Monte Carlo work on the \( q = 3 \) and \( q = 4 \) cases found correlation lengths controllable up to about \( \xi \sim 5000 \) for \( q = 3 \).<sup>[18](https://arxiv.org/abs/cond-mat/9811345)</sup>

**Potts glasses.** Models with competing or random interactions, such as the fully connected \( p \)-state Potts glass, show strong finite-size effects in static quantities and a dynamical transition: the relaxation time diverges as \( \tau \propto (T - T_D)^{-\Delta} \) with \( \Delta \approx 2 \), and at \( T_D \) it scales as \( N^{z^{\star}} \) with \( z^{\star} \approx 1.5 \).<sup>[19](https://arxiv.org/abs/cond-mat/0009475)</sup>

**Nonequilibrium drive.** A three-state Potts model coupled to two heat reservoirs at different temperatures, simulated with [Glauber dynamics](https://www.edgepedia.org/glauber-dynamics), keeps its second-order transition in the nonequilibrium steady state, and the critical properties fall in the same universality class as the equilibrium model.<sup>[20](https://doi.org/10.1103/PhysRevE.49.5871)</sup>

## Applications

Because the \( q \) states can represent distinct labels, the model appears outside bulk statistical mechanics. In adsorption physics, krypton on the basal plane of graphite is described by a \( q = 3 \) Potts model, and two-dimensional adsorbed systems can realize various Potts universality classes by choosing the substrate array and coverage.<sup>[4](https://archive.nptel.ac.in/content/storage2/courses/115103028/module4/lec3/2.html)</sup><sup> • </sup><sup>[5](https://harvest.aps.org/v2/journals/articles/10.1103/RevModPhys.54.235/fulltext)</sup>  In image analysis, segmentation is formulated as a Potts model in which spin interactions are defined by gray-scale differences within the image, and cluster-update algorithms provide fast segmentation via superparamagnetic clustering.<sup>[21](https://doi.org/10.1103/physreve.62.r1461)</sup> A drawback of conventional Potts-based segmentation is that it processes in exponential time, motivating linear-time Q-Ising formulations.<sup>[8](https://arxiv.org/abs/0906.4745)</sup>

## References

1. [Potts model R package documentation (citing Potts, 1952, doi:10.1017/S0305004100027419), CRAN](https://ftp.ussg.iu.edu/CRAN/web/packages/potts/index.html)
2. [The Strength of First and Second Order Phase Transitions, University of Coventry repository](https://pure.coventry.ac.uk/ws/portalfiles/portal/3939241/0012026v1.pdf)
3. [Particles, conformal invariance and criticality in pure and disordered systems, The European Physical Journal B](https://springerlink.fh-diploma.de/article/10.1140/epjb/s10051-021-00076-0)
4. [Module 4, Lecture 3: q-state Potts model, NPTEL course notes](https://archive.nptel.ac.in/content/storage2/courses/115103028/module4/lec3/2.html)
5. [The Potts model, F. Y. Wu, Reviews of Modern Physics 54, 235 (1982), full text via APS](https://harvest.aps.org/v2/journals/articles/10.1103/RevModPhys.54.235/fulltext)
6. [Computational Statistical Physics, Part I, ETH Zurich lecture notes](https://ethz.ch/content/dam/ethz/special-interest/phys/theoretical-physics/itp-dam/documents/compstatphys/CompStatPhysPartI_slides.pdf)
7. [Cluster Monte Carlo algorithms, J.-S. Wang and R. H. Swendsen, Physica A 167, 565 (1990), ScienceDirect](https://www.sciencedirect.com/science/article/pii/037843719090275W)
8. [A Q-Ising model application for linear-time image segmentation, arXiv:0906.4745](https://arxiv.org/abs/0906.4745)
9. [Change in the order of a phase transition in the two-dimensional Potts model with equivalent neighbors, Physical Review E](https://journals.aps.org/pre/abstract/10.1103/t156-9vyr)
10. [Equivalent-neighbor Potts models in two dimensions, Physical Review E 94, 052103 (2016)](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.94.052103)
11. [Phase Transitions in 3D Site-Diluted Potts Model with Spin States q=4, Solid State Phenomena 168-169, 357 (2010)](https://www.scientific.net/SSP.168-169.357.pdf)
12. [Potts Model, ScienceDirect Topics overview](https://www.sciencedirect.com/topics/physics-and-astronomy/pott-model)
13. [Critical antiferromagnetic square-lattice Potts model, R. J. Baxter, Proceedings of the Royal Society A 383, 43 (1982)](https://www.researchgate.net/publication/252149774_Critical_antiferromagnetic_square-lattice_Potts_model)
14. [Potts Model and Generalizations: Exact Results and Statistical Physics, SUNY repository](https://dspace.sunyconnect.suny.edu/items/8a3bdf19-6a5f-4423-88aa-6e9242a72785)
15. [Universal Order Parameters and Quantum Phase Transitions: A Finite-Size Approach, PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC4286789/)
16. [Cluster Monte Carlo algorithms and their applications, J.-S. Wang, arXiv:cond-mat/9510082](https://arxiv.org/abs/cond-mat/9510082)
17. [Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm, R. G. Edwards and A. D. Sokal, Physical Review D 38, 2009 (1988)](https://doi.org/10.1103/PhysRevD.38.2009)
18. [Antiferromagnetic Potts Models on the Square Lattice: A High-Precision Monte Carlo Study, arXiv:cond-mat/9811345](https://arxiv.org/abs/cond-mat/9811345)
19. [Finite-size scaling at the dynamical transition of the mean-field 10-state Potts glass, arXiv:cond-mat/0009475](https://arxiv.org/abs/cond-mat/0009475)
20. [Phase transitions in a nonequilibrium Potts model, K. E. Bassler and R. K. P. Zia, Physical Review E 49, 5871 (1994)](https://doi.org/10.1103/PhysRevE.49.5871)
21. [Cluster update algorithm and recognition, Physical Review E 62, R1461 (2000)](https://doi.org/10.1103/physreve.62.r1461)

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