A pr 1 99 4 Exact thermostatic results for the n - vector model on the harmonic chain

Here H0 is the Hamiltonian for the nearest-neighbor coupled harmonic chain which consists of a one-dimensional lattice lZ with lattice constant l > 0 and a set of N+1 point particles of mass m > 0 distributed along the Euclidean line IR at positions jl + qj , j = 1, 2, . . . , N + 1. The momentum of the j-th particle is denoted by pj and the spring constant of this chain is mω 0. We assume now that each particle carries a set of internal rotational degrees of freedom which we collectively represent by a classical spin, that is, by an n-component Euclidean unit vector ~ Sj ∈ IR. The Hamiltonian H1 then models the simplest rotational invariant interaction between the spins of two nearest neighboured particles. The interaction strength between two spins is described by the real-valued even function W : x 7→ W (x) and depends on the actual interparticle distance as indicated in (3). In accordance with the harmonic approximation it is sufficient to consider only the first two terms in a Taylor expansion of W , W (l+ qj+1 − qj) ≈ J + (qj+1 − qj)η, (4) where J := W (l) and η := W ′(l). Within this approximation it is possible to decouple the vibrational and rotational degrees of freedom by introducing shifted particle positions (cf. [1, 2] for n = 1)