Like the Ising model, the XY model is one of the many highly simplified models in the branch of physics known as statistical mechanics. It is a special case of the n-vector model.In the XY model, 2D classical spins 
are confined to some lattice. The spins are 2D unit vectors that obey O(2) (or U(1)) symmetry. (remember that they are classical spins) Mathematically, the Hamiltonian of the XY model
with the above prescriptions is given by the following:
where
- the sum runs over all pairs of neighboring spins
denotes the standard Euclidean inner product.
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-temperature phase to the high-temperature disordered phase.
See also
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