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Communications in Theoretical Physics

Improvement on the Migdal-Kadanoff Renormalization Transformation

Kun-fa Tang, Qing Ye, Jia-Zhen Hu

Published Sep 1, 1988• 0 citations
Publisher

Abstract

A Migdal-Kadanoff transformation is improved by using the mean-field approximation to find the relation between the coupling constants of the restructured lattice and the original one. Calculations of the critical coupling and exponents for the Ising model showed much improvement, even for small clusters, compared with the results of MK calculations.

DOI
10.1088/0253-6102/10/2/135

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A Migdal-Kadanoff transformation is improved by using the mean-field approximation to find the relation between the coupling constants of the restructured lattice and the original one. Calculations of the critical coupling and exponents for the Ising model showed much improvement, even for small clusters, compared with the results of MK calculations.
A modification of the Migdal-Kadanoff (MK) real-space renormalization technique is studied and applied to the q-state Potts model on the square and the simple cubic lattices. A parameter x which describes the boundary condition is introduced to the cluster-decimation (CD) approximation. When x=2, the present method is the same as the CD approximation, and in the limit x\ensuremath{\rightarrow}\ensuremath{\infty} this method reduces to the MK technique. Critical temperatures and exponents of the Potts model are calculated for 0<q\ensuremath{\le}4 by using various boundary conditions. Very good estimates of critical properties are obtained. The choice of the boundary condition is discussed.
A two-step renormalisation group approach-a Migdal-Kadanoff (or decimation) transformation (MK) followed by a mean-field renormalisation group (MFRG)-has been applied to random bond Ising models on d-dimensional hypercubic lattices. Calculations of the critical couplings and percolation concentrations for two and three dimensions show much improvement, even for small size clusters, compared with the results of separate MK or MFRG calculations. Critical exponents nu T and nu p are also improved in some cases. Estimates of critical curves in the concentration-temperature plane are also obtained for the two- and three-dimensional models.
The simple, but ad hoc, prescription for bond shifting in the Migdal-Kadanoff approximation is replaced by a systematic calculational procedure. In this method, the free energy is required to be preserved exactly by means of matching series expansions. The resulting renormalization group is characterized by a small parameter, and can be extended by working to higher order. A significant quantitative improvement over the standard Migdal-Kadanoff method is obtained with little increase in calculational effort. This approach is ultimately limited, however, by being restricted to a one-dimensional interaction parameter space.
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.