The spherical model in statistical mechanics 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 when applied to systems of 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.
The model describes a set of particles on a lattice containing 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 homogeneous system ensures that the average of the square of any spin is one, as in the usual Ising model.
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.
Equation of state
for the function g defined as
The internal energy per site is given by
an exact relation relating internal energy and magnetization.
For 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 and 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.
- M. Kac and C. J. Thompson, Spherical model and the infinite spin dimensionality limit, Physica Norvegica, 5(3-4):163-168, 1971.
- R. J. Baxter, Exactly solved models in statistical mechanics, London, Academic Press, 1982