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

Acta Physica Sinica (Overseas Edition)

The critical dynamics of the kinetic Ising models on the systems with R min > 2

Zhou Jian, Chen Shi–gang

Published Dec 1, 1993• 0 citations
Publisher

Abstract

We perform the exact TDRG transformations on the kinetic Ising model of the braced ladder whose R min is 5. Within our knowledge, this is the only example that the exact TDRG transformations have been performed on a system with R min > 2. We show that under Achiam's linear response theory there must be important excitations of the multi-spin operators in the kinetic Ising models of the systems with R min > 2, the dynamic critical behaviours of the systems are greatly affected by these excitations. Moreover, we get the conclusion that for the kinetic Ising models of the systems with R min = ∞ exact TDRG calculations are impossible. The complete new and interesting results are given.

DOI
10.1088/1004-423x/2/12/003

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

The Universality of the Critical Dynamics of the Kinetic Ising Model on the Regular Fractals

Jian Feng Zhou

The form of the universal scaling law of the critical dynamic exponent, , is found on a family of regular fractals by the exact TDRG method. Here, we generate a regular fractal by an anisotropic growing process. Identifying the growing probabilities as the interactions between Ising spins on the fractals, we map the growing probability clouds as a group of the anisotropic Ising Hamiltonians. Applying the RG transformations, we find that the systems of this group of Ising Hamiltonians can be described by two universal static correlation exponents and . So, the growing processes proposed by us capture the essential features in the directed DLA simulations. The studies about their critical dynamic behaviours reveal that unlike the one-dimensional chain the critical dynamics of the kinetic Ising model on the regular fractals is universal. The further discussions show that there is a universal scaling law form of the critical dynamic exponent of the kinetic Ising model, , on the site models of the regular fractals with . Meanwhile, we discuss Daniel Kandal's correction to the formula of the, critical dynamic exponent in the TDRG method and show that our TDRG calculations are exact.

Communications in Theoretical Physics•Jul 30, 1994•0 citations

On kinetic Ising models in one dimension

J C Anglès d'Auriac, R. Rammal

It is shown that Glauber dynamics in 1D Ising spin systems is not universal. This is illustrated on a periodic model with a basic unit cell (J 1 ,. . ., J n ) containing an arbitrary set of n ferromagnetic coupling constants. The dynamic critical exponent z is calculated exactly as z=1+max(J i )/min(J i ), the known value z=2 is recovered only for n=1. The extension of this result to other types of dynamics is briefly discussed.

Journal of Physics A Mathematical and General•Feb 7, 1988•13 citations

Critical dynamics of the kinetic Glauber-Ising model on hierarchical lattices

Xiang-Mu Kong, Zhao‐Di Yang

The critical dynamics of the kinetic Glauber-Ising model is studied on a family of the diamond-type hierarchical lattices with various branches. By carrying out the time-dependent real-space renormalization-group transformation to the master equation of the systems considered, the dynamic exponent is calculated. We find that the dynamic exponent depends on fractal dimension d(f) or the branch number m in a generator, and that it increases with the increase of d(f) or m. We notice that for the case of m=1 (one-dimensional spin chain, d(f)=1) our result z=2 is the same as the exact result obtained by Glauber, and for the case of m=2 (the simplest one in the diamond-type hierarchical lattices, d(f)=2) the exponent z=2.626 is higher than those of the two-dimensional regular lattice and the triangular lattice.

Physical Review E•Jan 13, 2004•4 citations

Linear approximation in the kinetic Ising model on fractals

Jian Feng Zhou, X. T. He

We propose a scheme to study exactly the kinetic Ising model on fractals under the linear approximation. This enables us to obtain the critical slowing-down behavior of the longest relaxation time of the kinetic Ising model on a fractal exactly. As a result, we show that there are only two time scales in the relaxation process of the kinetic Ising model on a fractal cluster under the linear approximation; the critical dynamic exponent is always a constant that is unrelated to the temperature of the system and is equal to 2. The implication of our results to the linear approximation of the critical dynamics of the kinetic Ising model on fractals as well as to the time-dependent renormalization-group method is discussed.

Physical review. B, Condensed matter•Feb 1, 1995•0 citations

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

Michel Droz, Anna‐Sapfo Malaspinas

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.

Journal of Physics C Solid State Physics•Aug 20, 1980•8 citations
We perform the exact TDRG transformations on the kinetic Ising model of the braced ladder whose R min is 5. Within our knowledge, this is the only example that the exact TDRG transformations have been performed on a system with R min > 2. We show that under Achiam's linear response theory there must be important excitations of the multi-spin operators in the kinetic Ising models of the systems with R min > 2, the dynamic critical behaviours of the systems are greatly affected by these excitations. Moreover, we get the conclusion that for the kinetic Ising models of the systems with R min = ∞ exact TDRG calculations are impossible. The complete new and interesting results are given.
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.
The form of the universal scaling law of the critical dynamic exponent, , is found on a family of regular fractals by the exact TDRG method. Here, we generate a regular fractal by an anisotropic growing process. Identifying the growing probabilities as the interactions between Ising spins on the fractals, we map the growing probability clouds as a group of the anisotropic Ising Hamiltonians. Applying the RG transformations, we find that the systems of this group of Ising Hamiltonians can be described by two universal static correlation exponents and . So, the growing processes proposed by us capture the essential features in the directed DLA simulations. The studies about their critical dynamic behaviours reveal that unlike the one-dimensional chain the critical dynamics of the kinetic Ising model on the regular fractals is universal. The further discussions show that there is a universal scaling law form of the critical dynamic exponent of the kinetic Ising model, , on the site models of the regular fractals with . Meanwhile, we discuss Daniel Kandal's correction to the formula of the, critical dynamic exponent in the TDRG method and show that our TDRG calculations are exact.
The critical dynamics of the kinetic Glauber-Ising model is studied on a family of the diamond-type hierarchical lattices with various branches. By carrying out the time-dependent real-space renormalization-group transformation to the master equation of the systems considered, the dynamic exponent is calculated. We find that the dynamic exponent depends on fractal dimension d(f) or the branch number m in a generator, and that it increases with the increase of d(f) or m. We notice that for the case of m=1 (one-dimensional spin chain, d(f)=1) our result z=2 is the same as the exact result obtained by Glauber, and for the case of m=2 (the simplest one in the diamond-type hierarchical lattices, d(f)=2) the exponent z=2.626 is higher than those of the two-dimensional regular lattice and the triangular lattice.
We propose a scheme to study exactly the kinetic Ising model on fractals under the linear approximation. This enables us to obtain the critical slowing-down behavior of the longest relaxation time of the kinetic Ising model on a fractal exactly. As a result, we show that there are only two time scales in the relaxation process of the kinetic Ising model on a fractal cluster under the linear approximation; the critical dynamic exponent is always a constant that is unrelated to the temperature of the system and is equal to 2. The implication of our results to the linear approximation of the critical dynamics of the kinetic Ising model on fractals as well as to the time-dependent renormalization-group method is discussed.
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.