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arXiv:cond-mat/0610468 · Journal of Physics A Mathematical and Theoretical

A non-perturbative approach to critical dynamics

Léonie Canet, Hugues Chaté

Published Feb 14, 2007• 64 citations
arXivHTMLPDF

Abstract

This paper is devoted to a non-perturbative renormalization group (NPRG) analysis of Model A, which stands as a paradigm for the study of critical dynamics. The NPRG formalism has appeared as a valuable theoretical tool to investigate non-equilibrium critical phenomena, yet the simplest—and nontrivial—models for critical dynamics have never been studied using NPRG techniques. In this paper we focus on Model A taking this opportunity to provide a pedagogical introduction to NPRG methods for dynamical problems in statistical physics. The dynamical exponent z is computed in d = 3 and d = 2 and is found to be in close agreement with results from other methods.

DOI
10.1088/1751-8113/40/9/002

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This paper is devoted to a non-perturbative renormalization group (NPRG) analysis of Model A, which stands as a paradigm for the study of critical dynamics. The NPRG formalism has appeared as a valuable theoretical tool to investigate non-equilibrium critical phenomena, yet the simplest—and nontrivial—models for critical dynamics have never been studied using NPRG techniques. In this paper we focus on Model A taking this opportunity to provide a pedagogical introduction to NPRG methods for dynamical problems in statistical physics. The dynamical exponent z is computed in d = 3 and d = 2 and is found to be in close agreement with results from other methods.
In this study we focus on the non-equilibrium phase transition whose asymptotic critical behavior is governed by the dynamic isotropic percolation universality class. In order to quantitatively characterize universal properties associated with this class, we employ a field-theoretic formulation of the model augmented with perturbative renormalization group technique. Utilizing such approach enables us to systematically compute the critical exponents in higher orders of perturbation theory. In particular, we perform an analysis to the three-loop precision. To this order we determine the renormalization constants and present relations for critical exponents in terms of critical dimensions. Finally, we also present numerically estimated values of critical exponents obtained from perturbation theory.
We propose an implementation of the nonperturbative renormalization group (NPRG) which applies to lattice models. Contrary to the usual NPRG approach where the initial condition of the RG flow is the mean-field solution, the lattice NPRG uses the (local) limit of decoupled sites as the (initial) reference system. In the long-distance limit, it is equivalent to the usual NPRG formulation and therefore yields identical results for the critical properties. We discuss both a lattice field theory defined on a d-dimensional hypercubic lattice and classical spin models. The simplest approximation, the local potential approximation, is sufficient to obtain the critical temperature and the magnetization of the three-dimensional Ising, XY, and Heisenberg models to an accuracy on the order of 1%. We show how the local potential approximation can be improved to include a nonzero anomalous dimension η and discuss the Berezinskii-Kosterlitz-Thouless transition of the two-dimensional XY model on a square lattice.
In this central and essential chapter, we develop the dynamic renormalization group approach to time-dependent critical phenomena. Again, we base our exposition on the simple O ( n )-symmetric relaxational models A and B; the generalization to other dynamical systems is straightforward. We begin with an analysis of the infrared and ultraviolet singularities appearing in the dynamic perturbation expansion. Although we are ultimately interested in the infrared critical region, we first take care of the ultraviolet divergences. Below and at the critical dimension d c = 4, only a finite set of Feynman diagrams carries ultraviolet singularities, which we evaluate by means of the dimensional regularization prescription, and then eliminate via multiplicative as well as additive renormalization (the latter takes into account the fluctuation-induced shift of the critical temperature). The renormalization group equation then permits us to explore the ensuing scaling behavior of the correlation and vertex functions of the renormalized theory upon varying the arbitrary renormalization scale. Of fundamental importance is the identification of an infrared-stable renormalization group fixed point, which describes scale invariance and hence allows the derivation of the critical power laws in the infrared limit from the renormalization constants determined in the ultraviolet regime. This program is explicitly carried through for the relaxational models A and B. For model B, we derive a scaling relation connecting the dynamic exponent to the Fisher exponent η . The critical exponents ν , η , and z are computed to first non-trivial order in an ∊ expansion around the upper critical dimension d c = 4.
The non-perturbative renormalization group (NPRG), in its modern form, constitutes an efficient framework to investigate the physics of systems whose long-distance behavior is dominated by strong fluctuations that are out of reach of perturbative approaches. We present here the basic principles underlying the NPRG and illustrate its power in the context of two longstanding problems of condensed matter and soft matter physics: the nature of the phase transition occuring in frustrated magnets in three dimensions and the phase diagram of polymerized phantom membranes.
Introducing a unified framework for describing and understanding complex interacting systems common in physics, chemistry, biology, ecology, and the social sciences, this comprehensive overview of dynamic critical phenomena covers the description of systems at thermal equilibrium, quantum systems, and non-equilibrium systems. Powerful mathematical techniques for dealing with complex dynamic systems are carefully introduced, including field-theoretic tools and the perturbative dynamical renormalization group approach, rapidly building up a mathematical toolbox of relevant skills. Heuristic and qualitative arguments outlining the essential theory behind each type of system are introduced at the start of each chapter, alongside real-world numerical and experimental data, firmly linking new mathematical techniques to their practical applications. Each chapter is supported by carefully tailored problems for solution, and comprehensive suggestions for further reading, making this an excellent introduction to critical dynamics for graduate students and researchers across many disciplines within physical and life sciences.