Classical XY model

Like the Ising model, the XY model is one of the many highly simplified models in the branch of physics known as statistical mechanics. In the XY model, classical spins are confined to some 2D lattice. The spins are vectors that obey O(2) (or U(1)) symmetry in the plane of the lattice. Mathematically, the Hamiltonian of the XY model with the above prescriptions is given by the following:

$H=-J\Sigma ^{'}\mathbf {s_{i}} \mathbf {s_{i'}}$

Where $\Sigma ^{'}$ indicates a summation over neighboring spins, and $\mathbf {s_{i}}$ is the spin vector. The continuous version of the XY model is often used to model systems that possess order parameters with the same kinds of symmetry, e.g. superfluid helium, hexatic liquid crystals. Topological defects in the XY model leads to a vortex-unbinding transition from the low-temperatured phase to the high-temperatured disordered phase.

See also Goldstone boson, Ising model, Potts model, Kosterlitz-Thouless transition, Topological defects