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Physical review. B, Condensed matter

Critical slowing down of energylike perturbation in the Glauber-Ising model

Y Achiam

Published Jan 1, 1979• 17 citations
Publisher

Abstract

A real-space renormalization-group analysis of Glauber's equation of motion for a two-dimensional Ising system on a triangular lattice is carried out up to the second order of the cumulant approximations. The study of the relaxation of the energy gives the dynamic exponent ${\ensuremath{\Delta}}_{E}=2.09$.

DOI
10.1103/physrevb.19.376

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Physical Review Letters•Jul 10, 1978•73 citations
A real-space renormalization-group analysis of Glauber's equation of motion for a two-dimensional Ising system on a triangular lattice is carried out up to the second order of the cumulant approximations. The study of the relaxation of the energy gives the dynamic exponent \ensuremathΔE=2.09{\ensuremath{\Delta}}_{E}=2.09\ensuremathΔE​=2.09.
A previously developed real-space renormalization-group approach to critical dynamics is examined. A correction to the formalism has to be introduced. The need for this correction is demonstrated through an exactly soluble one-dimensional Ising model. The new version imposes additional serious limitations on the usefulness of the method within standard approximation schemes. The second-order cumulant approximation of the two-dimensional kinetic Ising model on a triangular lattice is used to demonstrate these limitations.
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=1.70z=1.70z=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=2.22z=2.22z=2.22, together with a critical coupling Kc=0.251{K}_{c}=0.251Kc​=0.251 and a thermal critical index $\ensuremath{\nu}=0.950$.
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 real-space renormalization-group technique to treat dynamic critical phenomena in two-dimensional Glauber-Ising models. The invariant form of the time-dependent probability distribution is found within linear response theory and an estimate of the dynamic exponent zzz is given.