XY model

Example

Let $G = (V, E)$ be a finite graph, let $J > 0$ be a coupling constant and let $\beta \ge 0$ be the inverse temperature. A configuration assigns to each vertex $x$ an angle $\theta_x \in [0, 2\pi)$, that is, a unit vector $s_x = (\cos \theta_x, \sin \theta_x)$ in $\R^2$. Its energy is $$H(\theta) = -J \sum_{\set{x, y} \in E} \cos(\theta_x - \theta_y) = -J \sum_{\set{x, y} \in E} s_x \cdot s_y.$$ The XY model on $G$ is the probability measure on $[0, 2\pi)^V$ with density $e^{-\beta H(\theta)} / Z$ with respect to the product of the uniform measures. Here $Z$ is the normalizing constant. This is a Boltzmann distribution on a continuous spin space, so the XY model is a lattice model when $G$ is a box of $\Z^d$.