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Physical review. B, Condensed matter

Real-space dynamic renormalization group: Systematics T

Gene F. Mazenko, Oriol T. Valls

Published Jul 1, 1982• 5 citations
Publisher

Abstract

We extend our previous analysis of the two-dimensional kinetic Ising model using the real-space dynamic-renormalization-group (RSDRG) method to the ordered phase. This extension involves a discussion of spontaneous symmetry breaking within the context of a perturbative treatment using uncoupled cells as a zeroth-order approximation. This leads us to consider the introduction of an effective local magnetic field which simulates the average local order and necessitates the generalization of the RSDRG approach to the case of a finite magnetic field. We explain the role of the local order and proceed to calculate various observables. We note that our results match smoothly with our results for $T>~{T}_{c}$.

DOI
10.1103/physrevb.26.389

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We extend our previous analysis of the two-dimensional kinetic Ising model using the real-space dynamic-renormalization-group (RSDRG) method to the ordered phase. This extension involves a discussion of spontaneous symmetry breaking within the context of a perturbative treatment using uncoupled cells as a zeroth-order approximation. This leads us to consider the introduction of an effective local magnetic field which simulates the average local order and necessitates the generalization of the RSDRG approach to the case of a finite magnetic field. We explain the role of the local order and proceed to calculate various observables. We note that our results match smoothly with our results for T> TcT>~{T}_{c}T> Tc​.
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 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 $\ensuremath{\nu}=0.950$.
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.
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.
This paper represents some general aspects of real-space renormalization group approach to critical dynamics. The main idea is to combine Kadanoff's cell analysis with coarse graining of time corresponding to Markoffian approximation, which has been proposed by Suzuki to derive the dynamic scaling law. In order to realize this idea explicitly, the Migdal approximation is extended to the two-dimensional kinetic Ising model, to obtain z=1.96 for the dynamical critical exponent z. This result agrees very well with the value z=2.0±0.05 obtained by Yahata and Suzuki in a high temperature expansion method. Other new methods for real-space RGA are also proposed.