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# XY model

Like the famous Ising and Heisenberg models, 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 $\mathbf{s}_i$ 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 sum runs over all pairs of neighboring spins and $\cdot$ denotes the standard Euclidean inner product.