Equivalence of Some Generalizations on the Spherical Model

Abstract

We define n‐spherical models as follows: (1) Divide a lattice of N points into n mutually exclusive subsets with N 1, ⋯, Nn points, respectively; ΣN α = N. With each lattice point j, associate a variable ε j , and impose the restrictions ∑ε j 2 =N α ,α=1,⋯,n ; the sums are over the sites of the αth subset. (2) Adopt the energy expression for the Ising model with the ε's playing the role of the ``scalar spins.'' We then prove that, in the thermodynamic limit (N → ∞, N α → ∞ for all α), the thermodynamic functions of the n‐spherical model are equal to those of the ordinary spherical model.