The classical XY model (sometimes also called classical rotor (rotator) model or O(2) model) is a lattice model of statistical mechanics. In general, the XY model can be seen as a specialization of Stanley's n-vector model for . Definition Given a -dimensional lattice , per each lattice site there is a two-dimensional, unit-length vector The spin configuration, is an assignment of the angle for each . Given a translation-invariant interaction and a point dependent external field , the configuration energy is The case in which except for nearest neighbor is called nearest neighbor case. The configuration probability is given by the Boltzmann distribution with inverse temperature : where is the normalization, or partition function. The notation indicates the expectation of the random variable in the infinite volume limit, after periodic boundary conditions have been imposed. Rigorous results The existence of the thermodynamic limit for the free energy and spin correlations were proved by Ginibre, extending to this case the Griffiths inequality. Using the Griffiths inequality in the formulation of Ginibre, Aizenman and Simon proved that the two point spin correlation of the ferromagnetics XY model in dimension , coupling and inverse temperature is dominated by (i.e. has an upper bound given by) the two point correlation of the ferromagnetic Ising model in dimension , coupling and inverse temperature Hence the critical of the XY model cannot be smaller than the double of the critical temperature of the Ising model One dimension As in any 'nearest-neighbor' n-vector model with free (non-periodic) boundary conditions, if the external field is zero, there exists a simple exact solution. In the free boundary conditions case, the Hamiltonian is therefore the partition function factorizes under the change of coordinates This gives where is the modified Bessel function of the first kind. The partition function can be used to find several important thermodynamic quantities. For example, in the thermodynamic limit (), the free energy per spin is Using the properties of the modified Bessel functions, the specific heat (per spin) can be expressed as where , and is the short-range correlation function, Even in the thermodynamic limit, there is no divergence in the specific heat. Indeed, like the one-dimensional Ising model, the one-dimensional XY model has no phase transitions at finite temperature. The same computation for periodic boundary condition (and still ) requires the transfer matrix formalism, though the result is the same. This transfer matrix approach