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Concept# Classical XY model

Summary

The classical XY model (sometimes also called classical rotor (rotator) model or O(2) model) is a lattice model of statistical mechanics. In general, the XY model can be seen as a specialization of Stanley's n-vector model for n = 2.
Definition
Given a D-dimensional lattice Λ, per each lattice site j ∈ Λ there is a two-dimensional, unit-length vector sj = (cos θj, sin θj)
The spin configuration, s = (sj)j ∈ Λ is an assignment of the angle −π < θj ≤ π for each j ∈ Λ.
Given a translation-invariant interaction Jij = J(i − j) and a point dependent external field \mathbf{h}_{j}=(h_j,0), the configuration energy is
: 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
The case in which Jij = 0 except for ij nearest neighbor is called nearest neighbor case.
The configuration probability is giv

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