Megann, Alexis Peter
(1987)
The many-body physics of some quasi-one dimension magnetic systems.
*University of Southampton, Doctoral Thesis*.

## Abstract

This thesis deals with original work on three crystalline magnetic systems, all of which posses one-dimensional equations of motion. The Heisenberg model, the most successful simple theory of insulating magnetic systems, and its phenomenology are introduced, with definitions of the quantities of interest and a discussion of the statics of the model. The important concepts in the treatment of the dynamics of magnetic systems are then described, and then the application of the coupled-mode formalism to this system is reviewed. The case of the Heisenberg model in one dimension, in which the coupled-mode theory fails conspicuously, is discussed, and a modified version of the theory is presented; this is shown to describe the dynamics of this system to a much better approximation at all temperatures, and in addition is exact in the low-temperature limit. Quasiperiodic order is the presence of long-range order in a system, despite the absence of any symmetry under translations. The distinction between this type of ordering and the cases of crystalline order and random disorder is discussed, and the important ideas in the study of such systems are introduced. The specific examples of two-dimensional Bloch electron states in a perpendicular magnetic field and of incommensurate spin phases in magnetic systems are reviewed, and then the most striking properties of quasiperiodically ordered systems are briefly summarised. The spectrum of Harper's equation, a quasiperiodic tight-binding equation which describes the wavefunctions of Bloch electrons in a magnetic field, is examined through the statistical distribution of the level spacings, normalised by the local density of states. It is shown that this distribution has a simple form in each regime of the system (that is, with localised, critical and extended eigenstates), in contrast to that of the unnormalised spacings, and these results are related to the concepts of level repulsion and spectral self-similarity. The theory of Ziman and Lindgaard for a single-Q modulated spin phase of a magnet is introduced, and the cases of commensurate and incommensurate modulation are contrasted. A continued-fraction approach to the evaluation of the dynamic susceptibility is presented, and it is shown that this gives exact results for modulation wave vectors close to the commensurate values Q = 2πmu/ν, thus providing values for the susceptibility for comparison with experimental results at any Q.

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