Classical XY model

Lattice model of statistical mechanics 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, at each lattice site j ∈ Λ there is a two-dimensional, unit-length vector sj = (cos θj, sin θj). A spin configuration, s = (sj)j ∈ Λ is an assignment of the angle −π < θj ≤ π for each j ∈ Λ. Given an interaction Jij (often assumed to be translation-invariant, so Jij = J(i − j)) and a point-dependent external field h j = ( h j , 0 ) {\displaystyle \mathbf {h} _{j}=(h_{j},0)} , the configuration energy of a given spin configuration is H ( s ) = − ∑ i ≠ j J i j s i ⋅ s j − ∑ j h j ⋅ s j = − ∑ i ≠ j J i j cos ⁡ ( θ i − θ j ) − ∑ j h j cos ⁡ θ j . {\displaystyle {\begin{aligned}H(\mathbf {s} )&=-\sum _{i eq j}J_{ij}\;\mathbf {s} _{i}\cdot \mathbf {s} _{j}-\sum _{j}\mathbf {h} _{j}\cdot \mathbf {s} _{j}\\&=-\sum _{i eq j}J_{ij}\;\cos(\theta _{i}-\theta _{j})-\sum _{j}h_{j}\cos \theta _{j}.\end{aligned}}} Often one considers