One-dimensional model systems: Theoretical survey

In the early 1960’s one-dimensional model systems were regarded as amusing toys with the advantage of being far more easily solvable than their ’’real’’ three-dimensional counterparts. Now essentially 1-D (quasi-1-D) magnets can be ’’tailor-made’’ in the laboratory. Even more popular is the field of organic conductors like TTF⋅TCNQ, which are naturally quasi-1-D. Currently solitons and related solutions of non-linear, dispersive 1-D differential equations are ubiquitous in physics, including the area of 1-D magnetism. These developments are discussed in the Introduction. The rest of this paper is concerned with model Hamiltonians, model comparisons, critical singularities in 1-D (quasi-1-D) systems, accuracy of numerical techniques in comparison with exact solutions, brief accounts of dilute and disordered 1-D systems, and 1-D spin dynamics. Finally, a comment is made on a variety of interesting isomorphisms between 1-D magnets and phenomena in several other areas of physics, for example 2-D ferroelectrics, field-theoretic models, and realistic fluids. Comparison of theory and experiment has been the subject of several excellent reviews and is therefore not discussed here.