Phase transition in a non-translationally invariant spherical model

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Abstract
A non-translationally invariant spherical model, in which only a finite number of spins interact, is solved exactly. The model exhibits a phase transition in a non-zero uniform field, without spontaneous magnetization. The anomalous transition is attributed to the finite number of interacting spins taking on abnormally large values of order N1/2 without contributing to the magnetization. The free energy of the model can be obtained from a spherical limit (n to infinity ) of a corresponding n-vector model. In zero field the free energy is of the Curie-Weiss (or mean-field) spherical form. The Curie-Weiss form can only be maintained in a field by admitting a non-uniform field of order N1/2. This modified spherical model is also accessible from an n to infinity limit of a corresponding n-vector model.

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