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Journal of Hubei University

Revision on solution of the renormalization group for Ising model with weight factor

Xia Liu

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

Two dimensional tetragonal Ising model is studied with PSRG method in the Renormalization Group(RG) by introducing a energy dependent weight factor, when the Kadanoff clusters are constructed with five-point cluster. The new solution to the RG method and its corresponding critical indices are deduced, which approach exact solution more than those of before revision, as well as a larger Kadanoff cluster. 

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Two dimensional tetragonal Ising model is studied with PSRG method in the Renormalization Group(RG) by introducing a energy dependent weight factor, when the Kadanoff clusters are constructed with five-point cluster. The new solution to the RG method and its corresponding critical indices are deduced, which approach exact solution more than those of before revision, as well as a larger Kadanoff cluster. 
Spin glass model is discussed with real space renormalization group(RSRG) method,three kinds of fixed points and critical indices are deduced.Since a gap exists between the solution and the exact solution,revision is required.Some literatures indicate that it approaches exact solution if Kadanoff clusters are constructed with lager points,but the numeration increases greatly at the same time.Then a more efficient method is needed.The spin glass model is discussed with RSRG method by introducing an energy and temperature dependent weight factor.The new solution and the corresponding critical indices approach the exact solution more than those of before revision.
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.
We present a self consistent method based on cluster algorithms and Renormalization Group on the lattice to study critical systems numerically. We illustrate it by means of the 2D Ising model. We compute the critical exponents ννν and ηηη and the renormalization group flow of the probability density function of the magnetization. The results, compared to the standard Monte Carlo Renormalization Group proposed by Swendsen [1], are very accurate and the method works faster by a factor which grows monotonically with the lattice size. This allows to simulate larger lattices in reachable computational times.