Spin Systems

The following sections are included:

• Preface

• Acknowledgements

• Contents

In recent years a considerable amount of research has been done in the field of low-dimensional quantum-spin systems. The aim of the present work is to derive some generic results from this research and to grasp the qualitative differences between systems of different dimensions. Dimension is mainly understood in the sense of dimension of the lattice (dimension d), but also “dimension” (D) of the spins will be taken into consideration. The last property of individual spins is connected with the type of interaction with its surroundings. Apart from dimension D the spin value S is an important property, which determines strongly the collective behaviour of large systems…

There exists an extensive literature on the physical properties of magnetic materials. The group of magnetic insulators with an antiferromagnetic interaction between neighbour pairs is discussed in this monograph from a theoretical viewpoint. In this respect the discussion of the relevance of the models used is of central importance…

The topic of this monograph is the behaviour, in particular the ordering, of spin systems in their role as models of magnetic materials. As a work of theoretical physics its main accents are in the realm of many-particle physics and not in that of fundamental quantum mechanics. A proper introduction of spin variables, or more in general the concept of angular momentum, however, seems to be in order…

In the previous chapter we have laid particular stress on the form of the hamiltonian (2.1), which describes the quantum mechanics of a system of magnetic ions with their mutual interaction. These ions are also subjected to an external magnetic field B and crystalline fields, the effects of the latter being given by the second term. All the magnetic ions in this model are supposed to be identical and are located on equivalent lattice sites, circumstances resulting in the translational invariance of (2.1). This translational invariance is not a fundamental restriction but it simplifies the theoretical models used and the (partial) solution of the corresponding secular problem. However, extension to more complicated systems that still have an ideal periodicity can be easily given in many instances but is usually not interesting from a theoretical point of view. So the relation between real physical types of interaction and a model hamiltonian will be discussed in terms of the hamiltonian (2.1) throughout this chapter…

This chapter is completely devoted to exact results for onedimensional arrays (d=1) of quantum spins (D=2,3). For these results one cannot give a simple, intuitive interpretation: This situation occurs for typical quantum-mechanical calculations, especially for many-particle phenomena. One may ask the principal question whether or not this is a consequence of a possible lack of understanding of quantum mechanics, a deficiency that may be overcome by future generations. But it could also be possible that an intuitive interpretation always results in a classical model that shows a dynamical behaviour one could also imagine to be shown by a macroscopic system. Maybe this is a philosophical question or a problem in the realm of didactics, but this monograph is also intended for use by experimentalists with some interest in the structure of models in solid state physics and the author is seriously interested in the usefulness of his work…

In this chapter we want to expose the general ideas that form the base of our (partial) understanding of magnetic ordering, with a restriction to d=1 and 2, because for these lower dimensions the effects are most interesting. Our language will be that of quantum mechanics, as is appropriate for atomic systems, and antiferromagnetic lattices will be in the centre of our attention, because the typical effects of quantization are most outspoken in these arrays…

The physical picture of 3d ordering in ferromagnetic or antiferromagnetic systems is quite clear. One can safely state that the classical, “intuitive” idea of such an ordering is adequate, and that quantum effects only show up as corrections of the main features of the model. This is the state of the art for the ground state and the elementary excitations: If one looks at the theoretical picture of critical phenomena one discovers a richness of quantum effects that have been intensively studied in the last 25 years but are out of the scope of this work. These effects, however, clearly demonstrate that the fluctuations in a quantummechanical model of magnetic ordering behave quite differently from their classical counterparts. These differences also show up in the ground state and in the elementary excitations in 3d models, but they are not able to change the general picture of the ordered state drastically, as in the case of d=1 or 2…

In this final chapter we discuss some interesting recent developments that do not properly fit into the picture we have given in this monograph. The main accent in the preceding chapters was on our endeavour to understand the true character of quantummechanical ordering in AF systems of spins S=1/2. We think that two tendencies could be observed, one for which the classical Néel structure dominates and one for which the saturation of the coupling between neighbour spins is the main feature of the ordering. It should be clear that a higher dimension favours the classical ordering…

The following sections are included:

• Appendix (Section V.3). The 9-Spin State ||e>>_j

• List of symbols

• References

• Index