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Wuhan University Journal

Kadanoff Blocks and Renormalization Group Approach to Two-Dimensional Lattice

Yao Chen

Published Jan 1, 2003• 0 citations
OpenAlex

Abstract

The renormalization group transformation and Kadanoff construction are applied to two\|dimensional lattices. The general rule for Kadanoff construction and permitted site’s number are given. The fixed points and critical exponents are then calculated for the blocks with 9 and 13 sites. The comparison is made between the results and those obtained by the exact solution. The result shows that the calculation precision tends to be raised with increasing of the site’s number in the construction.

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The renormalization group transformation and Kadanoff construction are applied to two|dimensional lattices. The general rule for Kadanoff construction and permitted site’s number are given. The fixed points and critical exponents are then calculated for the blocks with 9 and 13 sites. The comparison is made between the results and those obtained by the exact solution. The result shows that the calculation precision tends to be raised with increasing of the site’s number in the construction.
A renormalization-group approach using a scaling transformation in real space is applied to the critical behavior of two-dimensional percolation systems. In various approximations for the triangular-site lattice and the square-bond lattice, the location of the fixed point and the correlation-length exponent are calculated by determining the behavior of the probabilities under a scale transformation. The fixed points for all approximations for the two lattices are found to be in complete agreement with known exact results for the critical percolation probability.
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 fixed-point topology of Kadanoff's lower-bound renormalization transformation is discussed for the d=2d=2d=2 square-lattice Ising model in the three-dimensional space of coupling constants for the nearest-neighbor, next-nearest neighbor, and four-spin interactions. An obstacle one encounters in trying to calculate a critical surface is pointed out. In the three-dimensional space of coupling constants Kadanoff's fixed point has two relevant vectors and can only be reached from a critical line, not from a critical surface. Another fixed point is reported which does have only one relevant vector and can be reached from a critical surface. Another fixed point is reported which does have only one relevant vector and can be reached from a critical surface. The critical exponents of this fixed point are close to the exact values, though not nearly so close as for Kadanoff's fixed point. The properties of both fixed points as a function of the variational parameter ppp are described. Critical values of the nearest-neighbor couplings leading to the fixed points are compared.
Accurate calculations of the thermodynamic properties of a system near its critical point are now possible using finite-lattice approximations to renormalization groups. In order to investigate some features of such approximations, we introduce a linear-renormalization-group transformation appropriate to a system of continuous spins like the Gaussian model of Berlin and Kac. We solve the renormalization-group equations for the Gaussian model exactly, and then study in detail a finite-lattice approximation to the renormalization group for the three-dimensional case. The renormalization-group transformation contains a parameter bbb. The exact transformation has fixed points only for $b={2}^{\ensuremath{-}\frac{5}{2}}$. The approximate transformation has fixed points for a range of values of bbb around ${2}^{\ensuremath{-}\frac{5}{2}}$. Eigenvalues of the transformation, which determine critical exponents, depend on the value of bbb used in the calculation. We study the problem of identifying the correct value of bbb.
The Kadanoff lower-bound renormalization transformation is used to obtain approximate values of the critical exponents for the spin-1/2 Ising model on a simple hypercubic ddd-dimensional lattice for all integral values of z=2dz={2}^{d}z=2d between 3 and 16. The results for $\ensuremath{\gamma}, \ensuremath{\delta}$, and the thermal and magnetic correction to scaling exponents are compared with corresponding results obtained from the $\ensuremath{\epsilon}$ expansion and series calculations.
Abstract Until the early 1970s all calculations of critical exponents were made either by exactly solving a model for its thermodynamic properties and then examining its behaviour in the critical regime (as with the Ising model in one and two dimensions) or by direct numerical simulations (as with the three-dimensional Ising model) or by extrapolation from approximate solutions which become invalid in the vicinity of a phase transition (high-temperature expansions, for example). However, in a now-famous paper published in 1966, Kadanoff presented arguments which, he maintained, would allow one to simplify calculations in the critical regime to the point at which critical exponents could be extracted, without ever working out the partition function for the problem. However, though Kadanoff’s ideas showed great physical intuition about the processes giving rise to critical phenomena, they lacked the mathematical precision which might have given one faith in his results.