Editing Classical XY model (section)

==Definition==
Given a {{mvar|D}}-dimensional [[Lattice model (physics)|lattice]] {{math|Λ}}, per each lattice site {{math|''j'' ∈ Λ}} there is a two-dimensional, [[Unit vector|unit-length vector]] {{math|'''s'''<sub>''j''</sub> {{=}} (cos ''θ<sub>j</sub>'', sin ''θ<sub>j</sub>'')}}

The ''spin configuration'', {{math|'''s''' {{=}} ('''s'''<sub>''j''</sub>)<sub>''j'' ∈ Λ</sub>}} is an assignment of the angle {{math|−''π'' < ''θ<sub>j</sub>'' ≤ ''π''}} for each {{math|''j'' ∈ Λ}}.

Given a ''translation-invariant'' interaction {{math|''J<sub>ij</sub>'' {{=}} ''J''(''i'' − ''j'')}} and a point dependent external field <math>\mathbf{h}_{j}=(h_j,0)</math>, the ''configuration energy'' is

:<math> H(\mathbf{s}) = - \sum_{i\neq j} J_{ij}\; \mathbf{s}_i\cdot\mathbf{s}_j -\sum_j \mathbf{h}_j\cdot \mathbf{s}_j =- \sum_{i\neq j} J_{ij}\; \cos(\theta_i-\theta_j) -\sum_j h_j\cos\theta_j </math>

The case in which {{math|''J<sub>ij</sub>'' {{=}} 0}} except for {{mvar|ij}} nearest neighbor is called ''nearest neighbor'' case.

The ''configuration probability'' is given by the [[Boltzmann distribution]] with inverse temperature {{math|''β'' ≥ 0}}:

:<math>P(\mathbf{s})=\frac{e^{-\beta H(\mathbf{s})}}{Z} \qquad Z=\int_{[-\pi,\pi]^\Lambda} \prod_{j\in \Lambda} d\theta_j\;e^{-\beta H(\mathbf{s})}.</math>

where {{mvar|Z}} is the [[Normalizing constant|normalization]], or [[partition function (statistical mechanics)|partition function]].<ref name=chaikin>{{cite book|last1=Chaikin|first1=P.M.|last2=Lubensky|first2=T.C.|title=Principles of Condensed Matter Physics|year=2000|publisher=Cambridge University Press|isbn=978-0521794503|url=https://books.google.com/books?id=P9YjNjzr9OIC}}</ref> The notation <math>\langle A(\mathbf{s})\rangle</math> indicates the expectation of the random variable {{math|''A''('''s''')}} in the infinite volume limit, after ''[[periodic boundary conditions]]'' have been imposed.