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Journal of Physics A Mathematical and General

Optimal real-space renormalisation

Thomas A. Witten, Jeffrey J. Prentis

Published Feb 1, 1981• 7 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/14/2/020

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Physical Review E•Apr 18, 2005•4 citations
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 authors introduce a new real space renormalisation method which is valid not only on the real axis but also everywhere in the complex plane of temperatures. The method requires the knowledge of the partition function on small lattices and can in principle be improved systematically. They test this method on the two-dimensional Ising model. They find that it gives a satisfactory picture for the set of zeros of the partition function. When restricted to the real axis, the estimates for the renormalisation transformation, the critical point and the thermal exponent are also very good.
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.
A simple renormalisation-group approach based on self-dual clusters is proposed for the two-dimensional nearest-neighbour spin- 1 / 2 Ising model on the square lattice; it reproduces the exact critical point. The authors calculate the internal energy and the specific heat for vanishing external magnetic field, spontaneous magnetisation and the thermal (y T ) and magnetic (y H ) critical exponents. The results obtained from the first four smallest cluster sizes strongly suggest the convergence towards the exact values when the cluster size increases. Even for the smallest cluster, where the calculation is very simple, the results are quite accurate, particularly in the neighbourhood of the critical point.
We test the performance of the Monte Carlo renormalization method in the context of the Ising model on a triangular lattice. We apply a block-spin transformation which allows for an adjustable parameter so that the transformation can be optimized. This optimization purportedly brings the fixed point of the transformation to a location where the corrections to scaling vanish. To this purpose we determine corrections to scaling of the triangular Ising model with nearest- and next-nearest-neighbor interactions by means of transfer-matrix calculations and finite-size scaling. We find that the leading correction to scaling just vanishes for the nearest-neighbor model. However, the fixed point of the commonly used majority-rule block-spin transformation appears to lie well away from the nearest-neighbor critical point. This raises the question whether the majority rule is suitable as a renormalization transformation, because the standard assumptions of real-space renormalization imply that corrections to scaling vanish at the fixed point. We avoid this inconsistency by means of the optimized transformation which shifts the fixed point back to the vicinity of the nearest-neighbor critical Hamiltonian. The results of the optimized transformation in terms of the Ising critical exponents are more accurate than those obtained with the majority rule.