Classical ferromagnetic chain in a field: Spin waves versus nonperturbative interactions

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. 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. 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. At very low temperatures and for H? 0, non-interacting spin-wave theory predicts results in good agreement with the numerical data. Larger and larger discrepancies, however, appear as the field is decreased, until for H=0 spin-wave calculations predict a magnetisation->-?, in sharp contrast with the exact result < Sz > = 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.