XY model

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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. 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). Mathematically, the Hamiltonian of the XY model with the above prescriptions is given by the following:

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

where the $i$ -th spin phase $θ i$ is measured e.g. from the horizontal axis in the counter-clockwise direction and the sum runs over all pairs of neighboring spins. The dot $\cdot$ denotes the standard dot 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 two dimensions the XY model exhibits a Kosterlitz-Thouless transition from the disordered high-temperature phase into the quasi-long range ordered low-temperature phase.