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Chinese Journal of Physics

Historical Review on Analytic, Monte Carlo, and Renormalization Group Approaches to Critical Phenomena of Some Lattice Models

Chin‐Kun Hu

Published Feb 1, 2014• 46 citations
Publisher

Abstract

Analytic calculations, Monte Carlo (MC) simulations [N. Metropolis and S. Ulam, J. Am. Stat. Asso 44, 335 (1949)] and renormalization group (RG) theory [K. G. Wilson, Phys. Rev. B 4, 3174 and 3184 (1971)] are important methods of statistical physics to study phase transitions and critical phenomena. This paper gives a brief historical review on analytic, MC, and RG approaches to some lattice critical systems in memory of late Professors Shang-keng Ma (1940-1983) and Yu-Ming Shih (1942-2005), who played some key roles in early developments of statistical physics of critical phenomena in Taiwan. The paper first introduces some developments in the study of critical phenomena from late 19th century to early 70s in the 20th century when renormalization group theory was formulated. The paper then reviews some topics on analytic, Monte Carlo, and RG approach to some lattice models, including variational RG approach to the Ising model, slow relaxation of a spin glass model at low temperatures, iterative method for quantum spin models, percolation theory of critical phenomena of Ising-like models and hard-core particle model, histogram Monte Carlo method and histogram Monte Carlo RG method, universal finite-size scaling functions of lattice models, critical slowing down of the Ising model, finite-size corrections for critical lattice models, and self-organized critical systems. Such topics are closely related to the author's own research experience. Finally, the paper outlines some very recent developments, summarizes main results of this paper, and points out some interesting problems for further studies, e.g. why a biological system can maintain in a non-equilibrium state for a very long time.

DOI
10.6122/cjp.52.1

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Analytic calculations, Monte Carlo (MC) simulations [N. Metropolis and S. Ulam, J. Am. Stat. Asso 44, 335 (1949)] and renormalization group (RG) theory [K. G. Wilson, Phys. Rev. B 4, 3174 and 3184 (1971)] are important methods of statistical physics to study phase transitions and critical phenomena. This paper gives a brief historical review on analytic, MC, and RG approaches to some lattice critical systems in memory of late Professors Shang-keng Ma (1940-1983) and Yu-Ming Shih (1942-2005), who played some key roles in early developments of statistical physics of critical phenomena in Taiwan. The paper first introduces some developments in the study of critical phenomena from late 19th century to early 70s in the 20th century when renormalization group theory was formulated. The paper then reviews some topics on analytic, Monte Carlo, and RG approach to some lattice models, including variational RG approach to the Ising model, slow relaxation of a spin glass model at low temperatures, iterative method for quantum spin models, percolation theory of critical phenomena of Ising-like models and hard-core particle model, histogram Monte Carlo method and histogram Monte Carlo RG method, universal finite-size scaling functions of lattice models, critical slowing down of the Ising model, finite-size corrections for critical lattice models, and self-organized critical systems. Such topics are closely related to the author's own research experience. Finally, the paper outlines some very recent developments, summarizes main results of this paper, and points out some interesting problems for further studies, e.g. why a biological system can maintain in a non-equilibrium state for a very long time.
We review recent progress in the understanding of phase transitions and critical phenomena, obtained by means of numerical simulations of the early dynamic evolution of systems prepared at well-specified initial conditions. This field has seen exhaustive scientific research during the last decade, when the renormalization-group (RG) theoretical results obtained at the end of the 1980s were applied to the interpretation of dynamic Monte Carlo simulation results. While the original RG theory is restricted to critical phenomena under equilibrium conditions, numerical simulations have been applied to the study of far-from-equilibrium systems and irreversible phase transitions, the investigation of the behaviour of spinodal points close to first-order phase transitions and the understanding of the early-time evolution of self-organized criticality (SOC) systems when released far form the SOC regime. The present review intends to provide a comprehensive overview of recent applications in those fields, which can give the flavour of the main ideas, methods and results, and to discuss the directions for further studies. All of these numerical results pose new and interesting theoretical challenges that remain as open questions to be addressed by new research in the coming years.
We review the current state of knowledge of dynamic critical phenomena in simple magnetic models and present high statistics Monte Carlo data for critical slowing down in Ising and Potts models in two and three dimensions. By fitting the relaxation functions to multi-exponential decays at the critical temperature and applying finite size scaling we are able to extract very accurate estimates for the dynamical critical exponent z. Our results are compared with previous numerical estimates and with the predictions of renormalization group theory.
Based on the subgraph expansions for phase transition models, we present a histogram Monte Carlo method to calculate the free energy and other physical quantities for such models as continuous functions of the system parameter. The method does not have any critical slowing down. We then use physical quantities thus obtained in a percolation renormalization group method to calculate critical point, critical exponents, and thermodynamic order parameter. Our method gives quite accurate results and may be applied to many phase transition models.
The current state of Monte Carlo work on phase transitions and critical phenomena is reviewed. Both classical and quantum Monte Carlo results are discussed with emphasis on statistical lattice-model studies involving the highly accurate calculation of critical exponents. It is shown that finite-size scaling is an effective method whereby not only simple lattice models but also those of complex practical magnetic materials can be treated. For models allowing crossover phenomena, it is shown that both finite-size scaling theory and conventional power-law functions must be used in carrying out a Monte Carlo analysis.
Critical phenomena and Goldstone mode effects in spin models with the O(n) rotational symmetry are considered. Starting with Goldstone mode singularities in the XY and O(4) models, we briefly review various theoretical concepts, as well as state-of-the-art Monte Carlo simulation results. They support recent results of the GFD (grouping of Feynman diagrams) theory, stating that these singularities are described by certain nontrivial exponents, which differ from those predicted earlier by perturbative treatments. Furthermore, we present the recent Monte Carlo simulation results of the three-dimensional Ising model for lattices with linear sizes up to L = 1536, which are very large as compared to L ≤ 128 usually used in the finite-size scaling analysis. These results are obtained, using a parallel OpenMP implementation of the Wolff single-cluster algorithm. The finite-size scaling analysis of the critical exponent η, assuming the usually accepted correction-to-scaling exponent ω ≈ 0.8, shows that η is likely to be somewhat larger than the value 0.0335 ± 0.0025 of the perturbative renormalization group (RG) theory. Moreover, we have found that the actual data can be well described by different critical exponents: η = ω =1/8 and ν = 2/3, found within the GFD theory.
The renormalization-group (RG) method discussed in this chapter has assumed a pivotal role in the modern theory of critical phenomena. It attempts to relate the partition function of a given system to that of a “similar system” with decreased degrees of freedom through a process referred to as renormalization. Exactly how these degrees of freedom are removed from the system, what we mean by a “similar” system, and how successive systems are coupled to one another are essentially the questions we take up in the introductory treatment given in this chapter. The RG method is a topic with large scope and found widely disseminated in an extensive physics literature on the topic; however, it is seldom found in engineering journals. Our purpose here is to try and make sense of some basic ideas with the RG approach so that it is more accessible to this wider community. For this we often rely upon some prior exposes of the subject in more specialized settings [1, 2, 3, 5]. In its complete sense, the RG method has only been made to work, at least analytically, for a few simple statistical-mechanical models. But aside from these numerical results, many important and quite general insights about critical phenomena can be developed from studying this approach to the problem, especially the central role played by length scale as a factor in describing the phenomenology. These ideas have significantly enhanced our understanding of ideas like scale invariance, universality classes, relevant scaling fields (as opposed to irrelevant ones), Hamiltonian renormalization, and so on; these and related concepts lie at the center of modern discourse on the subject. The essential concepts of the approach can be well illustrated using the Ising system since, with this model, lattice spins are fixed in space, which makes the analytical work quite transparent. This approach, called real space renormalization, is the RG method studied in this chapter.