Heisenberg Model

Abstract

In this chapter, the predictions of Heisenberg model, which were physically justified in Chap. 5, will be investigated from the point of view of magnetic phenomena. Magnetic systems, where the permanent magnetic moments arise due to localized electrons, are described very realistically by this model.

We begin with an in-depth discussion of the model Hamiltonian, in particular, discuss a few often applied transformations of the spin operators (Pauli, Holstein– Primakoff, Dyson-Maléev) and finally list out the most important extensions of the actual model.

The Heisenberg model is in general not exactly solvable; only for some special cases, it is possible to make rigorous statements. Here we discuss the Mermin– Wagner theorem which excludes spontaneous magnetization in one- or two-dimensional isotropic Heisenberg model. Further, we show that one-magnon states are exact eigenstates of the model Hamiltonian.

With the help of the simple molecular field approximation, we handle approximately the ferromagnet, antiferromagnet and ferrimagnet, without imposing any restrictions on the external parameters such as the temperature T and field B ₀. As results, we obtain quantities such as magnetization, susceptibility and specific heat as functions of T and B ₀. The next considered spin wave theories are lowtemperature approximations. In particular the linear spin wave theory, which is developed for the ferromagnets, antiferromagnets and ferromagnets with dipole interaction, is reliable only at low temperatures, since non-interacting spin waves (magnons) are assumed. The renormalized spin wave theory, which is discussed for ferromagnets, takes into account in a first approximation the interaction among the magnons and therefore can encompass somewhat higher temperatures.

While the spin wave theories are restricted to low-temperature region, the molecular field approximation, which is valid in principle for all temperatures, is rather coarse. At the end of this chapter, a large amount of space is devoted to the Green’s functions of the Heisenberg spin system with whose help detailed information for high (T ≫ T _C), for low (T ≪ T _C) and for critical (T ≈ T _C) temperatures is extracted.