Real-Space Renormalization Group Approach to the Kinetic Ising Model

Abstract

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 v and dynamic exponent z are given by 1/v=0.99 and z=2.23 from the present second order calculation. These results are quite reasonable. § 1. IntroductionThe renormalization group approach first introduced by Wilson!) to study critical phenomena has turned out a very useful and powerful method in this field. 2 ) This approach is classified roughly into two categories, namely, the momentum-space and real-space approaches. The latter one first proposed by Niemeijer and van Leeuwen,3) in contrast with the former, deals with the microscopic Hamiltonian directly in real space and it can be used in any dimension. In fact, the real-space approach has given very successful results concerning the static critical properties and the global thermodynamic properties of the twodimensional Ising-like spin systems. 4) Therefore, it is of great interest to generalize the real-space renormalization group method to the dynamic case. In recent years, many attempts 5H7 ) at this generalization have been made in the kinetic Ising model. 18)-20)There are two problems in the generalization of the real-space renormalization group approach to the dynamic case. The first one is how to deal with the memory effect accompanied with the coarse graining procedure. The second one is how to construct a consistent perturbation theory.Concerning the memory effect, one possible approach is the convolutionlesstype approach, in which a time-convolutionless master equation for the coarse grained system is constructed. The coarse grained time-evolution operator thus obtained has the time dependence, which corresponds to the memory effect. We usually take the long-time limit to make the time-evolution operator time-inde-*) Present address: