XY model

Like the famous 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:

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