Spherical Model as the Limit of Infinite Spin Dimensionality

The Berlin-Kac spherical model (or"spherical approximation to the Ising model") HSM=-JΣ〈ij〉μᵢμⱼ, with ΣNj=1uⱼ²=N, is found to be equivalent to the ν→∞ limit of the Hamiltonian ${H}^{({ν})}={-}J{Σ}{}{〈ij〉}{{S}ᵢ}^{({ν})}·{}{{S}ⱼ}^{({ν})},$ where ${{S}ᵢ}^{({ν})}$ are isotropically interacting ${ν}$-dimensional classical spins.