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Physical Review Letters

Application of the Real-Space Renormalization Group to Dynamic Critical Phenomena

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

Published Aug 14, 1978• 44 citations
Publisher

Abstract

We present a general approach for applying real-space renormalization-group methods to dynamic critical phenomena. In particular, we discuss the two-dimensional kinetic Ising model treating the interaction between blocks of spins as a small parameter.
DOI
10.1103/physrevlett.41.500

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Physical Review Letters•Aug 23, 1976•275 citations
A general scheme is presented for applying real-space renormalization-group methods to dynamic critical phenomena. The method is based on mapping the time evolution operator for the initial variables onto a new operator acting on the block variables. The general formalism is discussed in the context of the kinetic Ising model. It is shown that all non-Markovian effects in the new operator can be eliminated if the renormalization-group transformation is taken to be the solution of a generalized eigenvalue equation. We show how to solve this equation perturbatively. The resulting transformation is such that the dynamics of the block spins is given by the slowest dynamic modes of the initial spins within a block.
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.
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 $\ensuremath{\nu}=0.950$.
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.