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, (as they are classical spins). Mathematically, the Hamiltonian of the XY model with the above prescriptions is given by the following: The Ising model, named after the physicist Ernst Ising, is a mathematical model in statistical mechanics. ... Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... The n-vector model or O(n) model is one of the many highly simplified models in the branch of physics known as statistical mechanics. ... In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is generated by the motion of its center of mass about an external point. ... In mathematics, a unit vector in a normed vector space is a vector (most commonly a spatial vector) whose length is 1. ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... In mathematics, the unitary group of degree n over the field F (which is either the field R of real numbers or the field C of complex numbers) is the group of n by n unitary matrices with entries from F, with the group operation that of matrix multiplication. ...

where the sum runs over all pairs of neighboring spins and 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. In quantum field theory and statistical mechanics, a system can be in two possible phases: an ordered phase and a disordered phase. ...

See also

• Goldstone boson
• Ising model
• Potts model
• Kosterlitz-Thouless transition
• Topological defect

In particle physics, Goldstone bosons are bosons that appear in models with spontaneously broken symmetry. ... The Ising model, named after the physicist Ernst Ising, is a mathematical model in statistical mechanics. ... In statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a crystalline lattice. ... The Kosterlitz-Thouless transition is a special transition seen in the the XY model for interacting spin systems. ... In cosmology, a topological defect is a (often) stable configuration of matter predicted by some theories to form at phase transitions in the very early universe. ...

External links

• Evgeny Demidov, Vorteces in the XY model (2004)