Editing Potts model (section)

== Physical properties ==
=== Phase transitions ===
Despite its simplicity as a model of a physical system, the Potts model is useful as a model system for the study of [[phase transition]]s. For example, for the standard ferromagnetic Potts model in <math>2d</math>, a phase transition exists for all real values <math>q \geq 1</math>,<ref name=":0">{{Cite journal |last1=Beffara |first1=Vincent |last2=Duminil-Copin |first2=Hugo |date=2012-08-01 |title=The self-dual point of the two-dimensional random-cluster model is critical for q ≥ 1 |journal=Probability Theory and Related Fields |language=en |volume=153 |issue=3 |pages=511–542 |doi=10.1007/s00440-011-0353-8 |s2cid=55391558 |issn=1432-2064|doi-access=free }}</ref> with the critical point at <math>\beta J = \log(1 + \sqrt{q})</math>. The phase transition is continuous (second order) for <math>1 \leq q \leq 4</math> <ref>{{Cite journal |last1=Duminil-Copin |first1=Hugo |last2=Sidoravicius |first2=Vladas |last3=Tassion |first3=Vincent |date=2017-01-01 |title=Continuity of the Phase Transition for Planar Random-Cluster and Potts Models with <math>{1 \le q \le 4}</math> |url=https://doi.org/10.1007/s00220-016-2759-8 |journal=Communications in Mathematical Physics |language=en |volume=349 |issue=1 |pages=47–107 |doi=10.1007/s00220-016-2759-8 |arxiv=1505.04159 |s2cid=119153736 |issn=1432-0916}}</ref> and discontinuous (first order) for <math>q > 4</math>.<ref>{{cite journal
| last1=Duminil-Copin | first1=Hugo | authorlink1=Hugo Duminil-Copin
| last2=Gagnebin | first2=Maxime
| last3=Harel | first3=Matan
| last4=Manolescu | first4=Ioan
| last5=Tassion | first5=Vincent
| date=2021
| title=Discontinuity of the phase transition for the planar random-cluster and Potts models with <math>q>4</math>
| arxiv=1611.09877
| journal=Annales Scientifiques de l'École Normale Supérieure
| volume=54
| issue=6
| pages=1363-1413
| doi=10.24033/asens.2485}}</ref>

For the clock model, there is evidence that the corresponding phase transitions are infinite order [[BKT transition]]s,<ref name="lyxt19" /> and a continuous phase transition is observed when <math>q \leq 4</math>.<ref name="lyxt19" /> Further use is found through the model's relation to [[percolation theory|percolation]] problems and the [[Tutte polynomial|Tutte]] and [[chromatic polynomial]]s found in combinatorics. For integer values of <math>q \geq 3</math>, the model displays the phenomenon of 'interfacial adsorption' <ref>{{Cite journal |last1=Selke |first1=Walter |last2=Huse |first2=David A. |date=1983-06-01 |title=Interfacial adsorption in planar potts models |url=https://doi.org/10.1007/BF01304093 |journal=Zeitschrift für Physik B: Condensed Matter |language=en |volume=50 |issue=2 |pages=113–116 |doi=10.1007/BF01304093 |bibcode=1983ZPhyB..50..113S |s2cid=121502987 |issn=1431-584X}}</ref> with intriguing critical [[wetting]] properties when fixing opposite boundaries in two different states {{Clarify|reason=is this for the standard potts or clock?|date=May 2022}}.

=== Relation with the random cluster model ===

The Potts model has a close relation to the Fortuin-[[Pieter Kasteleyn|Kasteleyn]] [[random cluster model]], another model in [[statistical mechanics]]. Understanding this relationship has helped develop efficient [[Markov chain Monte Carlo]] methods for numerical exploration of the model at small <math>q</math>, and led to the rigorous proof of the critical temperature of the model.<ref name=":0" />

At the level of the partition function <math>Z_p = \sum_{\{s_i\}} e^{-H_p}</math>, the relation amounts to transforming the sum over spin configurations <math>\{s_i\}</math> into a sum over edge configurations <math>\omega=\Big\{(i,j)\Big|s_i=s_j\Big\}</math> i.e. sets of nearest neighbor pairs of the same color. The transformation is done using the identity<ref>{{cite book |last=Sokal |first=Alan D. |title=Surveys in Combinatorics 2005 |chapter=The multivariate Tutte polynomial (alias Potts model) for graphs and matroids |year=2005 |arxiv=math/0503607 |pages=173–226 |doi=10.1017/CBO9780511734885.009|isbn=9780521615235 |s2cid=17904893 }}</ref> 
: <math>
e^{J_p\delta(s_i,s_j)} = 1 + v \delta(s_i,s_j) \qquad \text{ with } \qquad v = e^{J_p}-1 \ . 
</math> 
This leads to rewriting the partition function as 
: <math>
Z_p = \sum_\omega v^{\#\text{edges}(\omega)} q^{\#\text{clusters}(\omega)}
</math>
where the ''' FK clusters''' are the connected components of the union of closed segments <math>\cup_{(i,j)\in\omega}[i,j]</math>. This is proportional to the partition function of the random cluster model with the open edge probability <math>p=\frac{v}{1+v}=1-e^{-J_p}</math>. An advantage of the random cluster formulation is that <math>q</math> can be an arbitrary complex number, rather than a natural integer.

Alternatively, instead of FK clusters, the model can be formulated in terms of '''spin clusters''', using the identity 
: <math>
e^{J_p\delta(s_i,s_j)} = (1 - \delta(s_i,s_j)) + e^{J_p} \delta(s_i,s_j)\ . 
</math>
A spin cluster is the union of neighbouring FK clusters with the same color: two neighbouring spin clusters have different colors, while two neighbouring FK clusters are colored independently.