Spherical model

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 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 L {displaystyle mathbb {L} } containing N sites. For each site j of L {displaystyle mathbb {L} } , a spin σ j {displaystyle sigma _{j}} which interacts only with its nearest neighbours and an external field H. It differs from the Ising model in that the σ j {displaystyle sigma _{j}} are no longer restricted to σ j ∈ { 1 , − 1 } {displaystyle sigma _{j}in {1,-1}} , but can take all real values, subject to the constraint that