Formulation
The model describes a set of particles on a lattice which contains N sites. For each site j of, a spin which interacts only with its nearest neighbours and an external field H. It differs from the Ising model in that the are no longer restricted to, but can take all real values, subject to the constraint that
which in a homogenous 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 is the Dirac delta function, are the edges of the lattice, and and, 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 -summation in the Ising model can be viewed as a sum over all corners of an N-dimensional hypercube in -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.
Read more about this topic: Spherical Model
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