Spherical Model as the Limit of Infinite Spin Dimensionality

Abstract

The Berlin-Kac spherical model (or"spherical approximation to the Ising model") ${\mathcal{H}}^{\mathrm{SM}}=\ensuremath{-}J\ensuremath{\Sigma}\stackrel{}{〈\mathrm{ij}〉}{\ensuremath{\mu}}_{i}{\ensuremath{\mu}}_{j}, \mathrm{with} \ensuremath{\Sigma}\stackrel{N}{j=1}{{u}_{j}}^{2}=N,$ is found to be equivalent to the $\ensuremath{\nu}\ensuremath{\rightarrow}\ensuremath{\infty}$ limit of the Hamiltonian ${\mathcal{H}}^{(\ensuremath{\nu})}=\ensuremath{-}J\ensuremath{\Sigma}\stackrel{}{〈\mathrm{ij}〉}{{\mathrm{S}}_{i}}^{(\ensuremath{\nu})}\ifmmode\cdot\else\textperiodcentered\fi{}{{\mathrm{S}}_{j}}^{(\ensuremath{\nu})},$ where ${{S}_{i}}^{(\ensuremath{\nu})}$ are isotropically interacting $\ensuremath{\nu}$-dimensional classical spins.