Spherical Model

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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 which 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 <math> mathbb </math> which contains *N* sites. For each site *j* of <math> mathbb </math>, a spin <math> sigma_j </math> which interacts only with its nearest neighbours and an external field *H*. It differs from the Ising model in that the <math> sigma_j </math> are no longer restricted to <math> sigma_j in </math>, 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 <math> delta </math> is the Dirac delta function, <math> langle jl rangle</math> are the edges of the lattice, and...

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- <math> sum_^N sigma_j^2 = N </math>

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

- <math> Z_N = int_^ ldots int_^ dsigma_1 ldots dsigma_N exp left delta left </math>

where <math> delta </math> is the Dirac delta function, <math> langle jl rangle</math> are the edges of the lattice, and...

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