Kac-type models of ferromagnetic thin films

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Abstract
The n-vector model of a thin film with free surfaces and Kac-type interactions is solved exactly. In zero field the system undergoes a phase transition of 'classical' type and, as conjectured by Costache (1975), the critical temperature is independent of n. Unexpectedly, an examination of the n to infinity limit yields a generalized spherical model and not the ordinary spherical model. Although investigated for purposes of comparison, the ordinary spherical model is shown to be distinguished in that the distribution of the magnetization across the film can be calculated explicitly in a vanishing field. Moreover, in the bulk limit, as the width of the film becomes unbounded, the total magnetization of this spherical model depends on the boundary conditions.

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