Classical XY Model - Definition

Definition

Given a -dimensional lattice, per each lattice site there is a two-dimensional, unit-length vector .

The spin configuration, is an assignment of the angle per each site in the lattice.

Given a translation-invariant interaction and a point dependent external field, 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 except for nearest neighbor is called nearest neighbor case.

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

P(\mathbf{s}) ={e^{-\beta H(\mathbf{s})} \over Z} \qquad Z=\int_{^{\Lambda}}\!\prod_{j\in \Lambda}d\theta_j\;e^{-\beta H(\mathbf{s})} \,.

where is the normalization, or partition function. The notation indicates the expectation of the random variable in the infinite volume limit, after periodic boundary conditions have been imposed.

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