XY model

The classical XY model (sometimes also called classical rotor (rotator) model or O(2) model) is a lattice model of statistical mechanics. It is the special case of the n-vector model for n = 2.

Given a D-dimensional lattice Λ, per each lattice site j ∈ Λ there is a two-dimensional, unit-length vector s_j = (cos θ_j, sin θ_j)

The spin configuration, s = (s_j)_{j ∈ Λ} is an assignment of the angle −π < θ_j ≤ π for each j ∈ Λ.

Given a translation-invariant interaction J_ij = J(i − j) and a point dependent external field ${\displaystyle \mathbf {h} _{j}=(h_{j},0)}$, the configuration energy is

The case in which J_ij = 0 except for ij nearest neighbor is called nearest neighbor case.

The configuration probability is given by the Boltzmann distribution with inverse temperature β ≥ 0:

where Z is the normalization, or partition function. The notation ${\displaystyle \langle A(\mathbf {s} )\rangle }$ indicates the expectation of the random variable A(s) in the infinite volume limit, after periodic boundary conditions have been imposed.

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