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Physical review. B, Condensed matter

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

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

Published Jul 1, 1979• 17 citations

Abstract

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.

DOI: 10.1103/physrevb.20.243

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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.

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.

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.

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 dynamic Monte Carlo renormalization group is applied to the two-dimensional Kawasaki and three-dimensional Glauber models. Our best "matching" results for the Kawasaki model yield a value of

z=3.80

for the dynamical exponent, as compared with the exact value

z=3.75

. This provides strong confirmation for the validity of this method. The results for the Glauber model are less accurate, but our estimate of $z\ensuremath{\simeq}2.08$ is in reasonable agreement with the $\ensuremath{\epsilon}$ expansion.

In this work we study the Glauber dynamics of the one-dimensional Ising model with nearest-neighbor and next-nearest-neighbor interactions, for which an approximate solution of the magnetization per site is obtained. When the dynamical critical exponent z is investigated following the treatment of Cordery, Sarker, and Tobochnik [Phys. Rev. B 24, 5402 (1981)], our observation shows that its upper-bound value is the same as the known value, thus implying that z is independent of the range of the interaction. We also suggest a high-temperature expansion approximation which is then used to solve the two-dimensional Glauber dynamics governed by a master equation; this solution is compared with that of the decoupling method. The time-delayed correlation function is also calculated.