Spherical model
The spherical model is a model of ferromagnetism similar to the Ising model, which was solved in 1952 by T. H. Berlin and M. Kac. It has the remarkable property that for linear dimension "d" greater than four, the critical exponents that govern the behaviour of the system near the critical point are independent of "d" and the geometry of the system. It is one of the few models of ferromagnetism that can be solved exactly in the presence of an external field. Formulation. The model describes a set of particles on a lattice formula_1 containing "N" sites . For each site "j" of formula_1, a spin formula_3 which interacts only with its nearest neighbours and an external field "H". It differs from the Ising model in that the formula_3 are no longer restricted to formula_5, but can take all real values, subject to the constraint that which in a homogeneous system ensures that the average of the square of any spin is one, as in the usual Ising model. The partition function generalizes from that of the Ising model to where formula_8 is the Dirac delta function, formula_9 are the edges of the lattice, and formula_10 and formula_11 , where "T" is the temperature of the system, "k" is Boltzmann's constant and "J" the coupling constant of the nearest-neighbour interactions. Berlin and Kac saw this as an approximation to the usual Ising model, arguing that the formula_12-summation in the Ising model can be viewed as a sum over all corners of an "N"-dimensional hypercube in formula_12-space. The becomes an integration over the surface of a hypersphere passing through all such corners. It was rigorously proved by Kac and C. J. Thompson that the spherical model is a limiting case of the N-vector model. Equation  of state. Solving the partition function and using a calculation of the free energy yields an equation describing the magnetization "M" of the system for the function "g" defined as The internal energy per site is given by an exact relation relating internal energy and magnetization. Critical behaviour. For formula_17 the critical temperature occurs at absolute zero, resulting in no phase transition for the spherical model. For "d" greater than 2, the spherical model exhibits the typical ferromagnetic behaviour, with a finite Curie temperature where ferromagnetism ceases. The critical behaviour of the spherical model was derived in the completely general  circumstances that the dimension "d" may be a real non-integer dimension. The critical exponents formula_18 and formula_19 in the zero-field case which dictate the behaviour of the system close to were derived to be which are independent of the dimension of "d" when it is greater than four, the dimension being able to take any real value.