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Journal of Physics A Mathematical and General

A real-space renormalisation group study of a semi-infinite Ising model

H. C. Tseng, H H Chen

Published Aug 21, 1989• 0 citations
Publisher

Abstract

A modified Migdal-Kadanoff renormalisation group scheme, called the cluster-decimation approximation, is generalised and applied to study surface critical properties of the Ising model with a free surface. The authors determine the fixed points and calculate surface critical exponents of this model. Comparisons of their results with other real-space renormalisation group estimates are given.

DOI
10.1088/0305-4470/22/16/024

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A modified Migdal-Kadanoff renormalisation group scheme, called the cluster-decimation approximation, is generalised and applied to study surface critical properties of the Ising model with a free surface. The authors determine the fixed points and calculate surface critical exponents of this model. Comparisons of their results with other real-space renormalisation group estimates are given.
Using the Migdal-Kadanoff renormalisation method, the authors study the semi-infinite two-dimensional Ising model with inhomogeneous nearest-neighbour coupling constants, that deviate from the bulk coupling by Am -y for large m, m being the distance from the edge. The approximation correctly predicts irrelevance of the inhomogeneity for y>v -1 , a surface magnetic phase at the bulk critical temperature for y<v -1 , A>0, and a non-universal A-dependent surface magnetic index in the marginal case y=v -1 .
A semi-infinite Ising model is studied by renormalization-group techniques in the position-space formulation due to Niemeijer and van Leeuwen. We show how to set up the calculation of thermodynamic properties (both critical and noncritical) of a sample with a free surface by iteration of a set of recursion relations for spatially inhomogeneous couplings and magnetic fields. Surface and bulk properties are clearly distinguished. The method is illustrated by a calculation (using a two-cell cluster approximation) of the magnetic-surface critical exponent in two dimensions, where exact results are available for comparison.
A simple hybrid of real space renormalisation group theory and finite-lattice calculations is described. The method is based on a lattice-restructuring transformation, with the criterion that the free energies of the original and restructured lattices be equal. The authors approximate the free energy per spin of the infinite lattice by an exact evaluation of the partition function for a small cluster. As a result of this transformation, the coupling constant of the restructured lattice is found to be temperature dependent. The method is illustrated by studying the Ising model in two and three dimensions, and two-dimensional Potts models. The calculations involve little more than is required for a Migdal-Kadanoff transformation, yet, the critical properties are found to be significantly improved in comparison. Detailed comparisons are also made with the global thermodynamic properties calculated within this approximation (free energy, internal energy, specific heat, spontaneous magnetisation) and exact or best known results for the models. The simplicity of the method allows for the study of more complex lattice models, and several possible applications are discussed.
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.