Optimization of Renormalization Group Transformations.

Available from UMI in association with The British Library. Requires signed TDF. This thesis examines the optimization of renormalization group transformations. To do this I have examined simple spin models where some understanding of the problems which arise and particularly the systematic errors can be acquired. In the first chapter I have given a brief description of critical phenomena which provides the motivation for the study of the renormalization group in the rest of this thesis. The second section of this chapter introduces the ideas of the renormalization group, stressing the parts most relevant to optimization. The last part of this chapter contains a discussion of redundant operators and their role in optimization. Chapter two consists of an examination of a renormalization group transformation applied to the Gaussian model. This is solved both on an infinite lattice and in four dimensions on a finite lattice using the finite lattice approximation. The results from this calculation can be compared with a Monte Carlo renormalization group analysis of a varphi ^4 model on a similar size lattice to determine if varphi^4 theory is trivial. Chapter three explores various aspects important to optimization of the renormalization group transformation used in chapter two. Chapter four examines an optimization scheme in which the renormalization group transformation is chosen so that the the fixed point has a nearest-neighbour interaction alone. In particular I describe some renormalization group transformations (different from decimation) which map the one dimensional nearest-neighbour Ising and Potts models and also a hierarchical model onto themselves. I have also examined optimizing a renormalization group transformation for the two dimensional Ising model where I have found strong indications that it is impossible to exactly block a nearest-neighbour model onto itself using a purely local transformation. Chapter five examines, a possible optimization scheme for the approximate recursion formula, which removes the six-spin interaction at second order within an epsilon-expansion. The final chapter summarizes some of the ideas developed throughout the rest of the thesis.