Editing N-vector model (section)

{{DISPLAYTITLE:''n''-vector model}} 
In [[statistical mechanics]], the ''' ''n''-vector model''' or '''O(''n'') model''' is a simple system of interacting [[Spin (physics)|spins]] on a [[crystalline lattice]].  It was developed by [[H. Eugene Stanley]] as a generalization of the [[Ising model]], [[XY model]] and [[classical Heisenberg model|Heisenberg model]].<ref>{{cite journal|last=Stanley|first=H. E.|title=Dependence of Critical Properties upon Dimensionality of Spins|journal=Phys. Rev. Lett.|year=1968|volume=20|issue=12|pages=589–592|doi=10.1103/PhysRevLett.20.589|bibcode=1968PhRvL..20..589S}}</ref>  In the ''n''-vector model, ''n''-component unit-length classical [[Spin (physics)|spins]] <math>\mathbf{s}_i</math> are placed on the vertices of a ''d''-dimensional lattice.  The [[Hamiltonian mechanics|Hamiltonian]] of the ''n''-vector model is given by:

:<math>H = K{\sum}_{\langle i,j \rangle}\mathbf{s}_i \cdot \mathbf{s}_j</math>

where the sum runs over all pairs of neighboring spins <math>\langle i, j \rangle</math> and <math>\cdot</math> denotes the standard Euclidean inner product. Special cases of the ''n''-vector model are:

:<math>n=0</math>: The [[self-avoiding walk]]<ref>{{cite journal|last=de Gennes|first=P. G.|title=Exponents for the excluded volume problem as derived by the Wilson method|journal=Phys. Lett. A|year=1972|volume=38|issue=5|pages=339–340|doi=10.1016/0375-9601(72)90149-1|bibcode=1972PhLA...38..339D}}</ref><ref>{{cite journal|last1=Gaspari|first1=George|last2=Rudnick|first2=Joseph|title=n-vector model in the limit n→0 and the statistics of linear polymer systems: A Ginzburg–Landau theory|journal=Phys. Rev. B|year=1986|volume=33|issue=5|pages=3295–3305|doi=10.1103/PhysRevB.33.3295|pmid=9938709|bibcode=1986PhRvB..33.3295G}}</ref>
:<math>n=1</math>: The [[Ising model]]
:<math>n=2</math>: The [[XY model]]
:<math>n=3</math>: The [[classical Heisenberg model|Heisenberg model]]
:<math>n=4</math>: [[Toy model]] for the [[Higgs sector]] of the [[Standard Model]]

The general mathematical formalism used to describe and solve the ''n''-vector model and certain generalizations are developed in the article on the [[Potts model]].