Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical review. B, Condensed matter

Exact preservation of the free energy in a modified Migdal-Kadanoff approximation

James S. Walker

Published Oct 1, 1982• 16 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.26.3792

Related papers

Preserving the free energy in a Migdal-Kadanoff approximation for the q -state Potts model

David Andelman, James S. Walker

We study a new modification of the Migdal-Kadanoff (MK) approximation, in which the bond-shifting step is generalized so that the resulting transformation preserves the free energy. This method is then utilized to study the $q$-state Potts model in two and three dimensions. While this method is just as easy to implement as the standard MK approximation, significant quantitative improvement is achieved by preserving the free energy even to very low order in a series expansion. We also discuss the limitations inherent in employing a single-parameter renormalization-group transformation.

Physical review. B, Condensed matter•Jan 1, 1983•14 citations

Real-space renormalization-group study of the exchange-interaction model

H. H. Chen, Felix Lee

Through use of an approximation introduced by Suzuki and Takano, the Migdal-Kadanoff (MK) renormalization-group transformation with the rescaling length b=2 is derived for the exchange-interaction (EI) model. Both the standard MK method and its modification, which preserves the free energy in the renormalization transformation, are used to determine critical temperatures and thermal exponents of the EI model on cubic lattices for various spins.

Physical review. B, Condensed matter•Dec 1, 1990•6 citations

Modified Migdal-Kadanoff renormalization for the spin-(1/2 anisotropic Heisenberg model

Felix Lee, H. H. Chen, H. C. Tseng

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.

Physical review. B, Condensed matter•Jul 1, 1988•7 citations

Preservation of the free energy in a Migdal-Kadanoff approximation for the spin-1/2 anisotropic Heisenberg model

Felix Lee, H. H. Chen, H. C. Tseng

A modification of the Migdal-Kadanoff approximation is applied to study the spin-(1/2 anisotropic Heisenberg model. The approximate decimation introduced by Suzuki and Takano, and the method of preserving the free energy developed by Walker are adopted. Fixed points and critical lines are determined for cubic lattices. The dependence of fixed points on the number of terms used in the series expansion of the free energy is analyzed and discussed.

Physical review. B, Condensed matter•Apr 1, 1988•5 citations

Direct calculation of the derivatives of the free energy for Ising models by a modified Kadanoff's variational method

Chin‐Kun Hu, Peter H. Kleban

We present a systematic procedure for the direct calculation of the free energy and its first and second derivatives for the Ising model in one to three dimensions with a wide class of symmetry properties. This renormalization group method is based on the modified Kadanoff's variational method (MKUM). This method is not only general but also very accurate numerically both near and far from the critical point. Further it includes correction to scaling effects not present in the standard linearized renormalization group treatment. This work describes the techniques and presents some illustrative results for the square latttice and body-centered cubic lattice with nearest neighbor interactions. A full treatment of our results will be given elsewhere. (Author)

Unknow•Feb 1, 1982•0 citations

Improvement on the Migdal-Kadanoff Renormalization Transformation

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

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.

Communications in Theoretical Physics•Sep 1, 1988•0 citations
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.
We study a new modification of the Migdal-Kadanoff (MK) approximation, in which the bond-shifting step is generalized so that the resulting transformation preserves the free energy. This method is then utilized to study the qqq-state Potts model in two and three dimensions. While this method is just as easy to implement as the standard MK approximation, significant quantitative improvement is achieved by preserving the free energy even to very low order in a series expansion. We also discuss the limitations inherent in employing a single-parameter renormalization-group transformation.
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.
We present a systematic procedure for the direct calculation of the free energy and its first and second derivatives for the Ising model in one to three dimensions with a wide class of symmetry properties. This renormalization group method is based on the modified Kadanoff's variational method (MKUM). This method is not only general but also very accurate numerically both near and far from the critical point. Further it includes correction to scaling effects not present in the standard linearized renormalization group treatment. This work describes the techniques and presents some illustrative results for the square latttice and body-centered cubic lattice with nearest neighbor interactions. A full treatment of our results will be given elsewhere. (Author)