Spin ordering in three-dimensional crystals with strong competing exchange interactions

In this paper we discuss magnetic ordering in three-dimensional crystals in which Heisenberg exchange interactions appear to dominate. Particular emphasis is placed on systems with strong competing exchange interactions, called frustrated spin systems. In the absence of such competition, one finds collinear spin ordering. However, strong competing interactions lead to non-collinear structures, often spirals and many variations thereof. The problem of understanding the origin and physical properties of such spin states within the classical Heisenberg model has been intensively addressed by a large community of researchers over the last 2–3 decades. The study of such problems actually began about five decades ago, and led to a large body of literature that has been overlooked in the publications of the last 2–3 decades (with two very recent exceptions). The early work established important fundamental concepts, including an unconventional theoretical approach, the generalized Luttinger–Tisza method and the idea of forced degeneracy. This resulted in the spin state called a ferrimagnetic spiral (FS), which enabled understanding of puzzling neutron diffraction data, plus NMR and ESR measurements, on the spinels XCr2O4, X = Mn, Co. Additional new aspects of this magnetic ordering have been uncovered in two very recent experimental works, indicating a partial spin-liquid-like behaviour and production of ferroelectricity by the pure spiral component of the FS. This exciting relevance of the early work to these modern results and the general lack of awareness of the early work has provided a source of motivation for this review, which covers that early body of work, both theoretical and experimental, and which also connects to the recent studies. A thorough discussion of the calculations, and comparison with experiments is presented. It will be seen that fundamental puzzles remain for these systems.