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

Journal of Physics C Solid State Physics

Migdal renormalisation-group transformation for the kinetic Ising model in arbitrary dimensions

Michel Droz, Anna‐Sapfo Malaspinas

Published Aug 20, 1980• 8 citations
Publisher

Abstract

The critical behaviour of the kinetic Ising model is analysed in the framework of the Migdal dynamic renormalisation group, for arbitrary dimensions. The dynamic exponent z, correlation length exponent nu and critical fixed point K* are computed for d=1 to 6, in the Suzuki approximation. By analytic continuation, the case d=1+ epsilon is considered within this approximation. It is shown that already to leading order in epsilon , the predictions for K*, nu and z given by dynamics differ from those given by statics. Accordingly, the two-dimensional case is revisited by taking into account higher-order correlation functions. This modification of the renormalisation-group transformation improves significantly the preceding results.

DOI
10.1088/0022-3719/13/23/021

Related papers

Migdal-type renormalization-group calculation for the kinetic Ising model

S. T. Chui, Gabor Forgács, H. L. Frisch

We make use of the calculation of Achiam and Kosterlitz and investigate the generalization of the Migdal-type renormalization-group calculation to critical dynamics, by looking at the one- and two-dimensional kinetic Ising model with no conserved magnetization. In the two-dimensional case the dynamical critical index ${Z}_{M}(\mathrm{magnetic}\mathrm{perturbation})=2.064$ and ${Z}_{E}(\mathrm{energylike}\mathrm{perturbation})=1.819$ for scale factor $\ensuremath{\lambda}=1$. ${Z}_{M}$ involves the static $\frac{\ensuremath{\beta}}{\ensuremath{\nu}}$ exponent, while ${Z}_{E}$ involves $\frac{1}{\ensuremath{\nu}}$, and therefore, it is not surprising that ${Z}_{M}$ is closer to the high-temperature expansion results, since $\frac{\ensuremath{\beta}}{\ensuremath{\nu}}$ in the Migdal approximation for statics is much closer to the exact value than $\frac{1}{\ensuremath{\nu}}$. In one dimension we obtain ${Z}_{M}={Z}_{E}=2$, which is the exact result of Glauber.

Physical review. B, Condensed matter•Jul 1, 1979•17 citations

Real-Space Renormalization Group Approach to Critical Dynamics: -- Migdal Approximation and Other New Methods --

Masuo Suzuki, Kiyoshi Sogo, Ikuo Matsuba +3 more

This paper represents some general aspects of real-space renormalization group approach to critical dynamics. The main idea is to combine Kadanoff's cell analysis with coarse graining of time corresponding to Markoffian approximation, which has been proposed by Suzuki to derive the dynamic scaling law. In order to realize this idea explicitly, the Migdal approximation is extended to the two-dimensional kinetic Ising model, to obtain z=1.96 for the dynamical critical exponent z. This result agrees very well with the value z=2.0±0.05 obtained by Yahata and Suzuki in a high temperature expansion method. Other new methods for real-space RGA are also proposed.

Progress of Theoretical Physics•Mar 1, 1979•28 citations

Two-step renormalisation group approach for randomly diluted Ising models

L. R. Evangelista, V. K. Saxena

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.

Journal of Physics A Mathematical and General•May 11, 1985•6 citations

Dynamical real-space renormalization group calculations with a highly connected clustering scheme on disordered networks

Duygu Balcan, Ayşe Erzan

We have defined a type of clustering scheme preserving the connectivity of the nodes in a network, ignored by the conventional Migdal-Kadanoff bond moving process. In high dimensions, our clustering scheme performs better for correlation length and dynamical critical exponents than the conventional Migdal-Kadanoff bond moving scheme. In two and three dimensions we find the dynamical critical exponents for the kinetic Ising model to be z=2.13 and z=2.09 , respectively, at the pure Ising fixed point. These values are in very good agreement with recent Monte Carlo results. We investigate the phase diagram and the critical behavior of randomly bond diluted lattices in d=2 and 3 in the light of this transformation. We also provide exact correlation exponent and dynamical critical exponent values on hierarchical lattices with power-law and Poissonian degree distributions.

arXiv:cond-mat/0408619•Physical Review E•Feb 28, 2005•0 citations

Monte Carlo estimate of the dynamical critical exponent of the 2D kinetic Ising model

John Kino Yamaguchi Williams

This paper describes a Monte Carlo study of the 2D kinetic Ising model. The long-time behaviour of various time-delayed correlation functions is investigated by renormalisation group methods. This long-time behaviour can be described in terms of a simple exponential relaxation of the magnetisation which enables a reliable estimate of the dynamical critical exponent, z, to be made.

Journal of Physics A Mathematical and General•Jan 11, 1985•61 citations

Ising model at an edge: a position space renormalisation group approach

Thomas A. Larsson

The critical behaviour of a three-dimensional Ising model, bounded by two plane surfaces meeting at an angle alpha , is studied using the Migdal-Kadanoff and two-terminal cluster renormalisation methods. The authors obtain expressions for the edge fixed point and for the independent edge exponent y he at the various transitions. They find that y he is not completely universal but depends on the angle alpha , and that it shows some unexpected features.

Journal of Physics A Mathematical and General•Jun 21, 1986•6 citations
The critical behaviour of the kinetic Ising model is analysed in the framework of the Migdal dynamic renormalisation group, for arbitrary dimensions. The dynamic exponent z, correlation length exponent nu and critical fixed point K* are computed for d=1 to 6, in the Suzuki approximation. By analytic continuation, the case d=1+ epsilon is considered within this approximation. It is shown that already to leading order in epsilon , the predictions for K*, nu and z given by dynamics differ from those given by statics. Accordingly, the two-dimensional case is revisited by taking into account higher-order correlation functions. This modification of the renormalisation-group transformation improves significantly the preceding results.
We make use of the calculation of Achiam and Kosterlitz and investigate the generalization of the Migdal-type renormalization-group calculation to critical dynamics, by looking at the one- and two-dimensional kinetic Ising model with no conserved magnetization. In the two-dimensional case the dynamical critical index ZM(magneticperturbation)=2.064{Z}_{M}(\mathrm{magnetic}\mathrm{perturbation})=2.064ZM​(magneticperturbation)=2.064 and ZE(energylikeperturbation)=1.819{Z}_{E}(\mathrm{energylike}\mathrm{perturbation})=1.819ZE​(energylikeperturbation)=1.819 for scale factor $\ensuremath{\lambda}=1$. ZM{Z}_{M}ZM​ involves the static $\frac{\ensuremath{\beta}}{\ensuremath{\nu}}$ exponent, while ZE{Z}_{E}ZE​ involves $\frac{1}{\ensuremath{\nu}}$, and therefore, it is not surprising that ZM{Z}_{M}ZM​ is closer to the high-temperature expansion results, since $\frac{\ensuremath{\beta}}{\ensuremath{\nu}}$ in the Migdal approximation for statics is much closer to the exact value than $\frac{1}{\ensuremath{\nu}}$. In one dimension we obtain ZM=ZE=2{Z}_{M}={Z}_{E}=2ZM​=ZE​=2, which is the exact result of Glauber.
This paper represents some general aspects of real-space renormalization group approach to critical dynamics. The main idea is to combine Kadanoff's cell analysis with coarse graining of time corresponding to Markoffian approximation, which has been proposed by Suzuki to derive the dynamic scaling law. In order to realize this idea explicitly, the Migdal approximation is extended to the two-dimensional kinetic Ising model, to obtain z=1.96 for the dynamical critical exponent z. This result agrees very well with the value z=2.0±0.05 obtained by Yahata and Suzuki in a high temperature expansion method. Other new methods for real-space RGA are also proposed.
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.
We have defined a type of clustering scheme preserving the connectivity of the nodes in a network, ignored by the conventional Migdal-Kadanoff bond moving process. In high dimensions, our clustering scheme performs better for correlation length and dynamical critical exponents than the conventional Migdal-Kadanoff bond moving scheme. In two and three dimensions we find the dynamical critical exponents for the kinetic Ising model to be z=2.13 and z=2.09 , respectively, at the pure Ising fixed point. These values are in very good agreement with recent Monte Carlo results. We investigate the phase diagram and the critical behavior of randomly bond diluted lattices in d=2 and 3 in the light of this transformation. We also provide exact correlation exponent and dynamical critical exponent values on hierarchical lattices with power-law and Poissonian degree distributions.