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Physical review. B, Condensed matter

Real-space dynamic renormalization group. I. General formalism

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

Published Aug 1, 1980• 26 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.22.1263

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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.
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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.
The theory of the critical dynamics of the single spin-flip kinetic Ising model is examined. Various unproven assumptions and misconceptions are uncovered and discussed. In particular, we point out that the condition of detailed balance places important constraints on the renormalization-group analysis of this problem. The relationship among different fixed-point dynamical operators and dynamic universality classes is clarified. The role and convergence of perturbation-theory expansions which treat the bare coupling between cells as a small parameter in the real-space renormalization-group treatment are analyzed. We give reasons for believing that this approach cannot compete with methods using direct high-temperature expansions. We also comment on the usefulness of other methods in treating this problem.
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.
We present a detailed discussion of a novel dynamical renormalization group scheme: the Dynamically Driven Renormalization Group (DDRG). This is a general renormalization method developed for dynamical systems with non-equilibrium critical steady-state. The method is based on a real space renormalization scheme driven by a dynamical steady-state condition which acts as a feedback on the transformation equations. This approach has been applied to open non-linear systems such as self-organized critical phenomena, and it allows the analytical evaluation of scaling dimensions and critical exponents. Equilibrium models at the critical point can also be considered. The explicit application to some models and the corresponding results are discussed. Key words: Renormalization group; non-equilibrium steady states; driven dynamical systems; self-organized criticality. 1 Vespignani, Zapperi and Loreto 2 1