Classical Ferromagnetic Chain in a Field; Spin-Waves Versus Non-Perturbative Interactions

Abstract

One of the more interesting features of magnetic chains is the lack of spontaneous magnetisation at any finite temperature.In one dimension (1-D) this result is quite general and holds irrespective of the quantum or classical nature of the system, as well as of the presence of exchange or single-ion anisotropies [1]. In the classical Heisenberg case, which we shall explicitly consider in the following, all the thermodynamic quantities of the system can be evaluated analytically at any temperature in zero external field [2]. Unfortunately, in the presence of a magnetic field H, the statistical properties of the model can be determined only in a numerical way, e.g. by transfer matrix techniques [3]. At very low temperatures and for H≠ 0, non-interacting spin-wave theory predicts results in good agreement with the numerical data [4, 5]. Larger and larger discrepancies, however, appear as the field is decreased, until for H=0 spin-wave calculations predict a magnetisation<S^z>→−∞, in sharp contrast with the exact result < S^z > = 0. This dramatic failure of an approach which is so successful in the usual 3-D magnets clearly indicates a breakdown in the conventional many-body techniques based on perturbation theory, and requires a new treatment.