Editing Potts model (section)

== Definition ==

=== Vector Potts model ===
The Potts model consists of ''spins'' that are placed on a [[lattice (group)|lattice]]; the lattice is usually taken to be a two-dimensional rectangular [[Euclidean space|Euclidean]] lattice, but is often generalized to other dimensions and lattice structures.

Originally, Domb suggested that the spin takes one of <math>q</math> possible values {{Citation needed|date=May 2022}}, distributed uniformly about the [[circle]], at angles
: <math>\theta_s = \frac{2\pi s}{q},</math>
where <math>s = 0, 1, ..., q-1</math> and that the interaction [[Hamiltonian mechanics|Hamiltonian]] is given by
: <math>H_c = J_c\sum_{\langle i, j \rangle} \cos \left( \theta_{s_i} - \theta_{s_j} \right)</math>
with the sum running over the nearest neighbor pairs <math>\langle i,j \rangle</math> over all lattice sites, and <math>J_c</math> 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 <math>q = 3,4</math>. In the limit <math>q \to \infty</math>, this becomes the [[XY model]].

=== Standard Potts model ===
What is now known as the standard '''Potts model''' was suggested by Potts in the course of his study of the model above and is defined by a simpler Hamiltonian:
: <math>H_p = -J_p \sum_{(i,j)}\delta(s_i,s_j) \,</math>
where <math>\delta(s_i, s_j)</math> is the [[Kronecker delta]], which equals one whenever <math>s_i = s_j</math> and zero otherwise.

The <math>q=2</math> standard Potts model is equivalent to the [[Ising model]] and the 2-state vector Potts model, with <math>J_p = -2J_c</math>. The <math>q=3</math> standard Potts model is equivalent to the three-state vector Potts model, with <math>J_p = -\frac{3}{2}J_c</math>.

=== Generalized Potts model ===
A generalization of the Potts model is often used in statistical inference and biophysics, particularly for modelling proteins through [[direct coupling analysis]].<ref name=":1">{{Cite journal |last1=Shimagaki |first1=Kai |last2=Weigt |first2=Martin |date=2019-09-19 |title=Selection of sequence motifs and generative Hopfield-Potts models for protein families |url=https://link.aps.org/doi/10.1103/PhysRevE.100.032128 |journal=Physical Review E |volume=100 |issue=3 |pages=032128 |arxiv=1905.11848 |bibcode=2019PhRvE.100c2128S |doi=10.1103/PhysRevE.100.032128 |pmid=31639992 |s2cid=167217593}}</ref><ref>{{Cite journal |last1=Mehta |first1=Pankaj |last2=Bukov |first2=Marin |last3=Wang |first3=Ching-Hao |last4=Day |first4=Alexandre G. R. |last5=Richardson |first5=Clint |last6=Fisher |first6=Charles K. |last7=Schwab |first7=David J. |date=2019-05-30 |title=A high-bias, low-variance introduction to Machine Learning for physicists |journal=Physics Reports |volume=810 |pages=1–124 |arxiv=1803.08823 |bibcode=2019PhR...810....1M |doi=10.1016/j.physrep.2019.03.001 |issn=0370-1573 |pmc=6688775 |pmid=31404441}}</ref> This generalized Potts model consists of 'spins' that each may take on <math>q</math> states: <math>s_i \in \{1,\dots,q\}</math> (with no particular ordering). The Hamiltonian is,
: <math>
H = \sum_{i < j} J_{ij}(s_i,s_j) + \sum_i h_i(s_i),
</math>
where <math>J_{ij}(k,k')</math> is the energetic cost of spin <math>i</math> being in state <math>k</math> while spin <math>j</math> is in state <math>k'</math>, and <math>h_i(k)</math> is the energetic cost of spin <math>i</math> being in state <math>k</math>. Note: <math>J_{ij}(k,k') = J_{ji}(k',k)</math>. This model resembles the [[Sherrington-Kirkpatrick model]] in that couplings can be heterogeneous and non-local. There is no explicit lattice structure in this model.