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arXiv:1110.5957

Generalized Migdal-Kadanoff Bond-moving Renormalization Recursion Procedure I: Symmetrical Half-length Bond Operation on Translational Invariant Lattices

Chunyang Wang, Wen-Xian Yang, Zhi-Wei Yan, Hong Du, Xiang-Mu Kong, Yuqi Zhang, Lingyu Zhang

Published Oct 27, 2011• 0 citations
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Abstract

We report in a series of papers two types of generalized Migdal-Kadanoff bond-moving renormalization group transformation recursion procedures. In this first part the symmetrical operation of half length bonds on translational invariant lattices are considered. As an illustration of their predominance in application, the procedures are used to study the critical behavior of the spin-continuous Gaussian model constructed on the triangular lattices. Results such as the correlation length critical exponents obtained by this means are found to be in good conformity with the classical results from other studies.

DOI
10.48550/arxiv.1110.5957

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We report in a series of papers two types of generalized Migdal-Kadanoff bond-moving renormalization group transformation recursion procedures. In this first part the symmetrical operation of half length bonds on translational invariant lattices are considered. As an illustration of their predominance in application, the procedures are used to study the critical behavior of the spin-continuous Gaussian model constructed on the triangular lattices. Results such as the correlation length critical exponents obtained by this means are found to be in good conformity with the classical results from other studies.
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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 critical temperature, thermal and magnetic exponents, and the graphs of dTeffdl\frac{d{T}_{\mathrm{eff}}}{\mathrm{dl}}dldTeff​​ versus Teff{T}_{\mathrm{eff}}Teff​ (where Teff{T}_{\mathrm{eff}}Teff​ denotes "effective" temperature of the system and lll the renormalization-group parameter) for the d=3d=3d=3, n=2,3n=2, 3n=2,3 models are obtained using the Migdal-Kadanoff scheme. Results of finite-lattice approximations for the d=2d=2d=2, n=2n=2n=2 model are also presented.
The spin-(1/2 anisotropic Heisenberg model is studied by a modified Migdal-Kadanoff method, called the cluster-decimation approximation. The free energy for a small cluster of spins with an appropriate boundary condition is assumed to be preserved in the bond-moving step and the approximate decimation of Suzuki and Takano is adopted in the site-decimation step. Critical temperatures obtained by the present method are in general much better than those determined by preserving the high-temperature series expansion of the free energy. In particular the XY fixed point and the isotropic Heisenberg fixed point for the square lattice are not found in the present calculation.