On the spin wave problem in the Heisenberg model of Ferromagnetism

Abstract

The interacting spin system of the Heisenberg model is brought into contact with a system of free Bose particles which are independent of the spins. The partition function of the combined system differs from the original one only by a trivial factor. Without changing the thermodynamical properties, the total HamiltonianH is transformed by means of nonsingular operatorsT according toTHT−1. The interaction between the spins is thereby eliminated to lowest order whereas the energy of the free Bose particles changes to the well-known energy dispersion of ideal spin waves. The interaction part of the transformed Hamiltonian describes ordinary scattering processes between two particles at least. Furthermore it allows at once for a low-temperature expansion in analogy to diagrammatical expansions of normal many particle systems. The expansion is studied in more detail. It is shown that contact can be made between this expansion and the low-temperature treatment of the Heisenberg model byDyson. In particular, the lowest order contributions to the partition function may be calculated from a reduced Hamiltonian which coincides withDyson's Hamiltonian for ideal spin waves with dynamical interaction. Therefore it is proved once again thatDyson's kinematical interaction is negligible at lowest temperatures.