Series Expansions for the Classical Vector Model

Abstract

In the. last lecture, we dealt with the ${\rm spin}-{1\over2}$ Ising model, with the Hamiltonian, ${\rm {\cal H}_I = -J \sum\limits_{<ij>}\sigma_i \sigma_j - mH\sum\sigma_i},$ where the coupling between the spins is confined to the z-components, with σ₁ = ± 1. We shall now consider a more general Hamiltonian of the form, ${\rm {\cal H}=-{1\over {S^2}} \sum\limits_{<ij>} J_{ij} {\underline S}_i\cdot {\underline S}_j -{m\over S}\sum\limits_{i=1}^N {\underline S}_i\cdot{\underline H}_i}.$ The spins ${\rm {\underline S}_i}$ are now D-dimensional vectors given by, ${\rm {\underline S}_i= \{S_i^1\ S_i^2\ \cdot\cdot\cdot\ S_i^D\}}$ The interaction between the spins takes the form of a scalar product ${\rm {\underline S}_i\cdot}{\rm {\underline S}_j}$ and (2) defines the D-vector model.