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Journal of Physics A Mathematical and General

Critical dynamics and the potential moving approximation

Y Achiam

Published Apr 1, 1980• 15 citations
Publisher

Abstract

The real-space time-dependent renormalisation group approach is applied to the study of a square lattice. A decimation and a spin-block transformation are performed according to the potential moving trick suggested by Kadanoff (see Ann. Phys. Lpz., vol.100, p.359, 1976). A value z=2.2 is found for the dynamic index characterising the two-dimensional lattice.

DOI
10.1088/0305-4470/13/4/026

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Journal of Physics A Mathematical and General•Feb 1, 1981•7 citations
The real-space time-dependent renormalisation group approach is applied to the study of a square lattice. A decimation and a spin-block transformation are performed according to the potential moving trick suggested by Kadanoff (see Ann. Phys. Lpz., vol.100, p.359, 1976). A value z=2.2 is found for the dynamic index characterising the two-dimensional lattice.
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.
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.