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Communications in Theoretical Physics

New Generalizations of Migdal—Kadanoff Bond-Moving Recursion Procedures and Their Applications

Chun-Yang Wang, Wen-Xian Yang, Zhi-Wei Yan, Hong Du, Xiang-Mu Kong, Yu-Qi Zhang, Lingyu Zhang

Published Apr 1, 2012• 1 citation
Publisher

Abstract

Considering in symmetrical half-length bond operations, we present in this paper two types of newly-developed generalizations of the remarkable Migdal-Kadanoff bond-moving renormalization group transformation recursion procedures. The predominance in application of these generalized procedures are illustrated by making use of them to study the critical behavior of the spin-continuous Gaussian model constructed on the typical translational invariant lattices and fractals respectively. Results such as the correlation length critical exponents obtained by these means are found to be in good conformity with the classical results from other previous studies.

DOI
10.1088/0253-6102/57/4/27

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Critical Properties of S4S^{4}S4 System Restudied via Generalized Migdal-Kadanoff Bond-moving Renormalization
We study the critical properties of the spin-continuous S4S^{4}S4 system on the typical translational invariant triangular lattices by combining the recently-developed generalized Migdal-Kadanoff bond-moving recursion procedures with the cumulative expansion technique. In three different cases of nearest-neighbor, next nearest neighbor and external field we obtain the critical points and further calculate the critical exponents according to the scaling theory. In all case it is found that there exists three fixed points. The correlation length critical exponents obtained near the Wilson-Fisher fixed points are found getting smaller and smaller with the increasing of the system complexity. Others are found similar to the results of the classical Gaussian model and different from those of the Ising system.
The superiority of the generalized Migdal–Kadanoff bond-moving renormalization (GMKBR) method in studying phase transition of complex spin systems is revealed by a sampled application on the classical S4model. In three different cases of nearest-neighbor, next-nearest neighbor and external field, we obtain the critical points and further calculate the critical exponents according to the scaling theory. In all the cases, it is shown that the method is of great convenience in obtaining results which are in good conformity with the previous ones.
The critical properties of Potts models with random bonds are considered in two dimensions. A position-space renormalization-group procedure, based on the Migdal-Kadanoff method, is developed. While all previous position-space calculations satisfied the Harris criterion and the resulting scaling relation only approximately, we found conditions under which these relations are exactly satisfied, and constructed our renormalization-group procedure accordingly. Numerical results for phase diagrams and thermodynamic functions for various random-bond Potts models are presented. In addition, some exact results obtained using a duality transformation, as well as an heuristic derivation of scaling properties that correspond to the percolation problem are given.
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Renormalization group recursions based on mean-field approximations [J. O. Indekeu, A. Maritan, and A. L. Stella, J. Phys.A15, L291 (1982)], commonly referred to as mean-field renormalization group methods (MFRG), have proven to be efficient and easily applicable for computing non-classical critical properties of lattice models. We give a fairly complete bibliography of applications to date, and extend previous test calculations of bulk, surface, and corner critical exponents in the two-dimensional Ising model to larger cluster sizes on triangular, square (including crossing bonds), and honeycomb lattices. Without much effort the exact value of the critical exponent ratioyH/yT is reproduced systematically with a precision of 2%. This ratio turns out to be the most accurate probe of non-classical critical behaviour that is available in the MFRG method.