Classical XY model

Like the famous Ising and Heisenberg models, the XY model is one of the many highly simplified models in statistical mechanics. It is a special case of the n-vector model.

Hamiltonian and Symmetry

In the XY model, 2D classical spins $\mathbf {s} _{i}$ are confined to some lattice. The spins are 2D unit vectors that obey O(2) (or U(1)) symmetry, (as they are classical spins). The spin at lattice position ${\vec {x}}$ is represented as a vector on a 2-D plane, which when parameterized in polar coordinates, assuming spin takes unit values, takes the form ${\vec {S}}=\left(\cos \theta ({\vec {x}}),\sin \theta ({\vec {x}})\right)$, where $\theta ({\vec {x}})$ is the orientation of the spin at lattice position ${\vec {x}}$. The spin can point along any direction, corresponding to some value of $\theta$ between 0 and $2\pi$. Notice that $\theta ({\vec {x}})$ for all ${\vec {x}}$ would define the entire model and we expect the HamiltonianI of this model to be a function of $\theta$ only . The Hamiltonian of the XY model with the above prescriptions is:

$H=-J{\sum }_{\langle i,j\rangle }\mathbf {s} _{i}\cdot \mathbf {s} _{j}=-J{\sum }_{\langle i,j\rangle }\cos(\theta _{i}-\theta _{j})$

where the $i$-th spin phase $\theta _{i}$ is measured e.g. from the horizontal axis in the counter-clockwise direction and the sum runs over all pairs of neighboring spins.

Topological Defects in XY model

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. This is what makes them peculiar from other phase transitions which are always accompanied with a symmetry breaking. Topological defects in the XY model leads to a vortex-unbinding transition from the low-temperature phase to the high-temperature disordered phase. In two spatial dimensions the XY model exhibits a Kosterlitz-Thouless transition from the disordered high-temperature phase into the quasi-long range ordered low-temperature phase.

See also

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

References

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