Spherical Model as the Limit of Infinite Spin Dimensionality

The Berlin-Kac spherical model (or"spherical approximation to the Ising model") ${\mathcal{H}}^{\mathrm{SM}}=-J\Sigma \stackrel{}{〈\mathrm{ij}〉}{\mu }_i{\mu }_j, \mathrm{with} \Sigma \stackrel{N}{j=1\mathrm{}}{u_j}^2=N,$ is found to be equivalent to the $\nu \to \infty$ limit of the Hamiltonian ${\mathcal{H}}^{\left(\nu \right)}=-J\Sigma \stackrel{}{〈\mathrm{ij}〉}{{\mathrm{S}}_i}^{\left(\nu \right)}·{{\mathrm{S}}_j}^{\left(\nu \right)},$ where ${S_i}^{\left(\nu \right)}$ are isotropically interacting $\nu$-dimensional classical spins.