Progress of Theoretical Physics

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

Masuo Suzuki, Kiyoshi Sogo, Ikuo Matsuba, H. Ikeda, Terumi Chikama, Hiroshi Takano

Published Mar 1, 1979• 28 citations

Abstract

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.

DOI: 10.1143/ptp.61.864

Related papers

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

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Physical review. B, Condensed matterJul 1, 197917 citations

Real-Space Renormalization Group Approach to the Kinetic Ising Model

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Masuo Suzuki

Progress of Theoretical PhysicsNov 1, 197311 citations

Progress of Theoretical Physics

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

Masuo Suzuki, Kiyoshi Sogo, Ikuo Matsuba, H. Ikeda, Terumi Chikama, Hiroshi Takano

Published Mar 1, 1979• 28 citations

Publisher

Abstract

DOI: 10.1143/ptp.61.864

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.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 matterJul 1, 197917 citations

Real-Space Renormalization Group Approach to the Kinetic Ising Model

Hiroshi Takano, Masuo Suzuki

A systematic real-space dynamic renormalization group method is proposed and it is applied to the kinetic Ising model on a triangular lattice. The coarse grained master equation is constructed by using the memory function formalism and the Markov approximation. The high temperature series renormalization group method proposed by Betts, Cuthiell and Plischke is extended for the present purpose to make a perturbation expansion in a self-consistent way with respect to all orders of interactions. Terms nonlocal in space appear in the coarse grained time-evolution operator in our approach, because of the memory effect accompanied with the coarse graining procedure. It is shown within the second order calculation that these nonlocal terms are irrelevant around the nontrivial fixed point. The values of the static exponent ν and dynamic exponent z are given by 1/ν = 0.99 and z = 2.23 from the present second order calculation. These results are quite reasonable.A systematic real-space dynamic renormalization group method is proposed and it is applied to the kinetic Ising model on a triangular lattice. The coarse grained master equation is constructed by using the memory function formalism and the Markov approximation. The high temperature series renormalization group method proposed by Betts, Cuthiell and Plischke is extended for the present purpose to make a perturbation expansion in a self-consistent way with respect to all orders of interactions. Terms nonlocal in space appear in the coarse grained time-evolution operator in our approach, because of the memory effect accompanied with the coarse graining procedure. It is shown within the second order calculation that these nonlocal terms are irrelevant around the nontrivial fixed point. The values of the static exponent ν and dynamic exponent z are given by 1/ν = 0.99 and z = 2.23 from the present second order calculation. These results are quite reasonable.

Progress of Theoretical PhysicsMay 1, 198211 citations

Real-space dynamic renormalization group: Systematics for T>~Tc

Gene F. Mazenko, Oriol T. Valls

In this paper we systematize the real-space dynamic-renormalization-group method we developed elsewhere. We treat the two-dimensional kinetic Ising model with relaxational dynamics defined on a square lattice. We show how one can set up a systematic perturbation-theory expansion which treats the coupling between cells of spins as a small parameter. The novel feature of the expansion is that it provides a method for determining the appropriate effective interaction between spins in a set of uncoupled cells. By obtaining a good zeroth-order approximation for a set of uncoupled cells, one finds that the effective interaction between cells is reduced. Similar ideas are also applied to the dynamics of the problem and we indicate how one can obtain a good zeroth-order approximation in terms of the dynamics of uncoupled cells. We couple these perturbation theory ideas to the real-space dynamic-renormalization-group method and carry out explicit calculations of a variety of observable quantities including the magnetic susceptibility, specific heat, static structure factor, time-dependent spin-autocorrelation function, and the dynamic structure factor. In every case we obtain excellent results in keeping with the scaling behavior near the phase transition and the high-temperature limit.In this paper we systematize the real-space dynamic-renormalization-group method we developed elsewhere. We treat the two-dimensional kinetic Ising model with relaxational dynamics defined on a square lattice. We show how one can set up a systematic perturbation-theory expansion which treats the coupling between cells of spins as a small parameter. The novel feature of the expansion is that it provides a method for determining the appropriate effective interaction between spins in a set of uncoupled cells. By obtaining a good zeroth-order approximation for a set of uncoupled cells, one finds that the effective interaction between cells is reduced. Similar ideas are also applied to the dynamics of the problem and we indicate how one can obtain a good zeroth-order approximation in terms of the dynamics of uncoupled cells. We couple these perturbation theory ideas to the real-space dynamic-renormalization-group method and carry out explicit calculations of a variety of observable quantities including the magnetic susceptibility, specific heat, static structure factor, time-dependent spin-autocorrelation function, and the dynamic structure factor. In every case we obtain excellent results in keeping with the scaling behavior near the phase transition and the high-temperature limit.

Physical review. B, Condensed matterAug 1, 198119 citations

Real-space dynamic renormalization group. II. Simple examples

Gene F. Mazenko, Michael J. Nolan, Oriol T. Valls

The real-space renormalization-group method for the dynamics of critical phenomena introduced in the previous paper is applied to two calculations within the two-dimensional kinetic Ising model on a triangular lattice. In both cases a cumulant expansion in the intercell couplings is performed. In the first calculation are found the most general first-order recursion relations coupling neighboring cells. From these recursion relations is extracted the dynamic critical exponent $z=1.70$ . While this is a reasonable first-order result, the results for the statics are not as satisfactory. In the second calculation it is shown how the terms that are generated at second order are important in determining the static results. We obtain at second order $z=2.22$ , together with a critical coupling ${K}_{c}=0.251$ and a thermal critical index $\ensuremath{\nu}=0.950$.The real-space renormalization-group method for the dynamics of critical phenomena introduced in the previous paper is applied to two calculations within the two-dimensional kinetic Ising model on a triangular lattice. In both cases a cumulant expansion in the intercell couplings is performed. In the first calculation are found the most general first-order recursion relations coupling neighboring cells. From these recursion relations is extracted the dynamic critical exponent $z=1.70$. While this is a reasonable first-order result, the results for the statics are not as satisfactory. In the second calculation it is shown how the terms that are generated at second order are important in determining the static results. We obtain at second order $z=2.22$, together with a critical coupling ${K}_{c}=0.251$ and a thermal critical index $\ensuremath{\nu}=0.950$.

Physical review. B, Condensed matterAug 1, 198019 citations

Real-space dynamic renormalization group. III. Calculation of correlation functions

Gene F. Mazenko, J. E. Hirsch, Michael J. Nolan +1 more

The real-space dynamic renormalization group method developed in previous papers is applied to the kinetic Ising model defined on a square lattice. In particular we extend the formalism to the calculation of space- and time-dependent equilibrium averaged correlation functions. We find that conventional methods for implementing the real-space renormalization group via cumulant expansions in terms of the intercell coupling lead to correlation functions which decay algebraically in space at large distances in the disordered phase in qualitative disagreement with the known exponential decay. We indicate how one can develop new perturbation theory expansion methods which lead to the proper exponential decay at large distances and also lead to good quantitative results for other observable quantities like the magnetization, susceptibility, and single spin time autocorrelation function. As the result of a first-order calculation we obtain excellent results for the static critical exponents and a value of $z=1.79$ for the dynamic critical exponent. The critical exponents obtained from the correlation functions calculated using this method satisfy the proper static and dynamic scaling relations.The real-space dynamic renormalization group method developed in previous papers is applied to the kinetic Ising model defined on a square lattice. In particular we extend the formalism to the calculation of space- and time-dependent equilibrium averaged correlation functions. We find that conventional methods for implementing the real-space renormalization group via cumulant expansions in terms of the intercell coupling lead to correlation functions which decay algebraically in space at large distances in the disordered phase in qualitative disagreement with the known exponential decay. We indicate how one can develop new perturbation theory expansion methods which lead to the proper exponential decay at large distances and also lead to good quantitative results for other observable quantities like the magnetization, susceptibility, and single spin time autocorrelation function. As the result of a first-order calculation we obtain excellent results for the static critical exponents and a value of $z=1.79$ for the dynamic critical exponent. The critical exponents obtained from the correlation functions calculated using this method satisfy the proper static and dynamic scaling relations.

Physical review. B, Condensed matterFeb 1, 198117 citations

Master-Equation Approach to Dynamic Critical Phenomena Using Wilson's -Expansion

Masuo Suzuki

Progress of Theoretical Physics Vol. 51 No. 4 (1974) pp. 1257–1259 On the Derivation of the Dynamic Scaling Law by Means of Kadanoff’s Cell Analysis and Coarse Graining of Time Masao Suzuki Progress of Theoretical Physics Vol. 51 No. 6 (1974) pp. 1712–1730 Renormalization Group Theory of Dynamic Critical Phenomena Yoshiki Kuramoto Progress of Theoretical Physics Vol. 52 No. 2 (1974) pp. 722–725 Perturbational Renormalization Group Approach to Critical Dynamics in Stochastic Models Masuo Suzuki and Fumihiko Tanaka Progress of Theoretical Physics Vol. 52 No. 3 (1974) pp. 871–885 Critical Relaxation of a Non-Linear Fokker-Planck Equation Hideo Yahata Progress of Theoretical Physics Vol. 53 No. 1 (1975) pp. 97–110 On the Critical Dynamics in Bose Systems and the Time-Dependent Ginzburg-Landau Model Masuo Suzuki Progress of Theoretical Physics Vol. 58 No. 4 (1977) pp. 1142–1150 Static and Dynamic Finite-Size Scaling Theory Based on the Renormalization Group Approach Masuo SuzukiProgress of Theoretical Physics Vol. 51 No. 4 (1974) pp. 1257–1259 On the Derivation of the Dynamic Scaling Law by Means of Kadanoff’s Cell Analysis and Coarse Graining of Time Masao Suzuki Progress of Theoretical Physics Vol. 51 No. 6 (1974) pp. 1712–1730 Renormalization Group Theory of Dynamic Critical Phenomena Yoshiki Kuramoto Progress of Theoretical Physics Vol. 52 No. 2 (1974) pp. 722–725 Perturbational Renormalization Group Approach to Critical Dynamics in Stochastic Models Masuo Suzuki and Fumihiko Tanaka Progress of Theoretical Physics Vol. 52 No. 3 (1974) pp. 871–885 Critical Relaxation of a Non-Linear Fokker-Planck Equation Hideo Yahata Progress of Theoretical Physics Vol. 53 No. 1 (1975) pp. 97–110 On the Critical Dynamics in Bose Systems and the Time-Dependent Ginzburg-Landau Model Masuo Suzuki Progress of Theoretical Physics Vol. 58 No. 4 (1977) pp. 1142–1150 Static and Dynamic Finite-Size Scaling Theory Based on the Renormalization Group Approach Masuo Suzuki

Progress of Theoretical PhysicsNov 1, 197311 citations

{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.

z=1.70

. While this is a reasonable first-order result, the results for the statics are not as satisfactory. In the second calculation it is shown how the terms that are generated at second order are important in determining the static results. We obtain at second order

z=2.22

, together with a critical coupling

{K}_{c}=0.251

and a thermal critical index $\ensuremath{\nu}=0.950$.

z=1.79

for the dynamic critical exponent. The critical exponents obtained from the correlation functions calculated using this method satisfy the proper static and dynamic scaling relations.