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Journal of Physics A Mathematical and General

Optimised renormalisation group transformations of lattice spin models

Andrew F. Bennett

Published Oct 1, 1987• 1 citation
Publisher

Abstract

The optimisation of real space renormalisation group transformations is explored in the framework of series expansions. A block-spin transformation (in addition to decimation) is constructed which exactly maps the one-dimensional nearest-neighbour Ising model onto itself. The same construction can also be applied to the q-state Potts model. In higher dimensions, similar optimisation can be achieved only if the weight function has interactions of range comparable to those generated in the series expansion.

DOI
10.1088/0305-4470/20/14/027

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The optimisation of real space renormalisation group transformations is explored in the framework of series expansions. A block-spin transformation (in addition to decimation) is constructed which exactly maps the one-dimensional nearest-neighbour Ising model onto itself. The same construction can also be applied to the q-state Potts model. In higher dimensions, similar optimisation can be achieved only if the weight function has interactions of range comparable to those generated in the series expansion.
Using a continuous class of real-space renormalisation transformations the authors study the critical behaviour of the Ising model in a variety of two-dimensional lattices. They use the simplest possible cluster, with two cells, and a single renormalised coupling. Certain of these transformations-well separated from the majority spin rule-produce sharply more accurate critical properties than the rest. From this study they note an optimal set of conditions which characterise the successful transformations. The optimal transformations appear to have a certain 'locality' property: renormalised couplings beyond nearest neighbour are especially small.
The renormalisation group method of Kadanoff is applied to the spin- 1 / 2 Ising model in two dimensions using a transformation applicable to both the square and triangular lattices. Two types of cluster are studied for the triangular lattice.The first cluster yields a new fixed point but the results for the critical temperature and critical exponents are in poor agreement with the known exact values. The second cluster maps directly onto the fixed point found by Kadanoff for the square lattice yielding excellent values for both the critical exponents and the critical temperature.
The real space renormalisation group of Niemeijer and van Leeuwen (1974) is applied to a mixed-spin Ising model on a simple quadratic lattice. The motivation is the investigation of critical phenomena in Ising models with less than the usual translational symmetry. The models in question are relevant to the study of ferrimagnetism. Two calculations, characterised by different block constructions, are performed and compared. Exponent values are found to be in good agreement with those suggested by the universality hypothesis. The utility of the renormalisation group for dealing with ferrimagnetism is demonstrated, but the high degree of labour involved in such an exercise is indicated.
A real-space renormalization-group transformation is constructed between blocks of linear dimensions bL\mathrm{bL}bL and LLL to study the ground-state properties of quantum lattice models. In this scheme it is possible (i) to estimate the errors due to the irrelevant variables, including boundary effects, and (ii) to test scaling explicitly, so that an accurate calculation can be performed over the entire range of the parameter space. Application to the one-dimensional Ising model in a tranverse field yields excellent results.