 XY model

The XY model assigns a planar unit spin ${\mathbf{s}}_i=(\cos {\theta }_i,\sin {\theta }_i)$ to each lattice site, with a continuous angle ${\theta }_i$. Its usual Hamiltonian is $H=-J{\sum }_{⟨ij⟩}\cos ({\theta }_i-{\theta }_j)$. For $J>0$, alignment lowers energy, but a common rotation of all spins leaves every angle difference and hence the energy unchanged.

This continuous orthogonal group $O(2)$ symmetry distinguishes it from a finite-state clock model. Small angular variations produce a spin wave description, while large windings can introduce vortices. The XY model is useful for studying collective phase coherence as well as planar magnetism because the angle can represent an order parameter's phase rather than a literal magnetic arrow. Dimension, interaction range and temperature remain essential when deciding which kind of order the model can sustain.