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Physical review. B, Condensed matter

Dynamic correlations in the spin-conserving two-dimensional kinetic Ising model

Steven John Heilig, James H. Luscombe, Gene F. Mazenko, Enis Oḡuz, Oriol T. Valls

Published Jun 1, 1982• 20 citations
Publisher

Abstract

We present in this paper the application of the real-space dynamic renormalization-group method to a two-dimensional kinetic Ising model with a conserved order parameter. We obtain the diffusion coefficient as a function of the temperature in the conventional approximation. We show that our renormalization procedure automatically preserves the conservation laws upon iteration. We present recursion relations and results for the time-dependent correlation functions, with particular emphasis put on the evaluation of those dynamic correlations which are obtainable in field-emission experiments. The proper hydrodynamic behavior is obtained at long times.

DOI
10.1103/physrevb.25.7003

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Physical Review Letters•Aug 14, 1978•44 citations
We present in this paper the application of the real-space dynamic renormalization-group method to a two-dimensional kinetic Ising model with a conserved order parameter. We obtain the diffusion coefficient as a function of the temperature in the conventional approximation. We show that our renormalization procedure automatically preserves the conservation laws upon iteration. We present recursion relations and results for the time-dependent correlation functions, with particular emphasis put on the evaluation of those dynamic correlations which are obtainable in field-emission experiments. The proper hydrodynamic behavior is obtained at long times.
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.
The real-space dynamic renormalization group method developed in previous papers is applied to the kinetic Ising model defined on a square lattice. In particular we extend the formalism to the calculation of space- and time-dependent equilibrium averaged correlation functions. We find that conventional methods for implementing the real-space renormalization group via cumulant expansions in terms of the intercell coupling lead to correlation functions which decay algebraically in space at large distances in the disordered phase in qualitative disagreement with the known exponential decay. We indicate how one can develop new perturbation theory expansion methods which lead to the proper exponential decay at large distances and also lead to good quantitative results for other observable quantities like the magnetization, susceptibility, and single spin time autocorrelation function. As the result of a first-order calculation we obtain excellent results for the static critical exponents and a value of z=1.79z=1.79z=1.79 for the dynamic critical exponent. The critical exponents obtained from the correlation functions calculated using this method satisfy the proper static and dynamic scaling relations.
The critical dynamics of the two dimensional Kinetic Ising Model have been studied using Monte Carlo Real Space Renormalization Group techniques. Time dependent correlation functions were calculated for square lattices of 16 x 16 and 32 x 32 spins, using periodic boundary conditions wth 60,000 and 15,000 Monte Carlo steps per spin, respectively;The critical exponent z was estimated using a matching process for a variety of renormalization group operators. The estimates for z were surprisingly sensitive to small changes in the value of the renormalization group operator, with physically reasonable operators giving estimates for z less than the rigorous lower bound for z of z(,o) = 1.75. The estimates for z from a given renormalization group operator showed poor convergence properties. Attempts to estimate z from the scaling form of the correlation function were inconclusive, as were attempts to estimate the static exponent (eta);Possible explanations of these unexpected results are discussed. The most plausible source of the complex behavior seen in this work is the discrete nature of time in the Monte Carlo computer simulation, which generates phenomena not present in standard theoretical treatments of the Kinetic Ising Model.