On the non-existence of magnetic order in one and two dimensions

Ion Stancu

We have investigated the dynamics of the isotropic N-vector model with long-range exchange couplings in random magnetic fields using a 1/N expansion. The leading order is exactly solved, showing the existence of a ferromagnetic phase separated from the disordered paramagnetic phase by a line. The critical behaviour of the system has been examined in the next-to-leading order of the 1/N expansion, showing that the critical exponents are by no means related to the ones of the pure system in d-2 dimensions. The ordered phase has been also investigated in the next-to-leading order, revealing a typical Goldstone behaviour of the non time-persistent part of the transverse fluctuations. For the longitudinal fluctuations, two different types of coexistence singularities emerge, one from the non time-persistent (as in the pure systems), vanishing with the temperature, and a more divergent one from the time-persistent part of the correlation.

Xue-Yang Song

Edmund Bennett

This thesis explores the theoretical representation of localised electrons in magnetic systems, using Majorana fermions. A motivation is provided for the Majorana fermion representation, which is then developed and applied as a mean-field theory and in the path-integral formalism to the Ising model in transversal-field (TFIM) in one, two and three dimensions, on an orthonormal lattice. In one dimension the development of domain walls precludes long-range order in discrete systems; this is as free energy savings due to entropy outweigh the energetic cost of a domain wall. An argument due to Peierls exists in 2D which allows the formation of domains of ordered spins amidst a disordered background, however, which may be extended to 3D. The forms of the couplings to the bosons used in the Random Phase Analysis (RPA) are considered and an explanation for the non-existence of the phases calculated in this thesis is discussed, in terms of spare degrees of freedom in the Majorana representation. This thesis contains the first known application of Majorana fermions at the RPA level.

Satoshi Takada

The X-Y model in one- and two-dimensions is investigated by means of the path integral. The low temperature properties are shown to be described by the phase mode. In particu-lar, the quantum model in the limit S--'fco is found to be well expressed by the planer model. The spin-correlation functions are calculated explicitly. As a result it is found that at zero temperature the quasi long-range order in one-dimension and the long-range order in two-dimen-sions appear. The spontaneous magnetization is explicitly given at zero temperature in two-dimensions. At finite temperatures the dynamical behaviour is the same as that studied by Nelson and Fisher. The divergence of the susceptibility at T=O in one-dimension is shown to be different from that of the classical model. Considerable progress has been made recently 111 understanding the nature of the phase transition in the two-dimensional planer model without long-range or-cler.l)~81 It has been revealed that below the transition temperature there exists a topological order in which bound pairs of vortex and antivortex are formed, the

Daniel Reyes, M. A. Contínentino