Our community has a deep and sophisticated understanding of phase transitions and their universal scaling functions. We outline and advocate an ambitious program to use this understanding as an anchor for describing the surrounding phases. We explain how to use normal form theory to write universal scaling functions in systems where the renormalization-group flows cannot be linearized. We use the 2d Ising model to demonstrate how to calculate high-precision implementations of universal scaling functions, and how to extend them into a complete description of the surrounding phases. We discuss prospects and challenges involved into extending these early successes to the many other systems where the RG has successfully described emergent scale invariance, making them invaluable tools for engineers, biologists, and social scientists studying complex systems.
Our community has a deep and sophisticated understanding of phase transitions and their universal scaling functions. We outline and advocate an ambitious program to use this understanding as an anchor for describing the surrounding phases. We explain how to use normal form theory to write universal scaling functions in systems where the renormalization-group flows cannot be linearized. We use the 2d Ising model to demonstrate how to calculate high-precision implementations of universal scaling functions, and how to extend them into a complete description of the surrounding phases. We discuss prospects and challenges involved into extending these early successes to the many other systems where the RG has successfully described emergent scale invariance, making them invaluable tools for engineers, biologists, and social scientists studying complex systems.
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.
Abstract This chapter analyzes systems with emergent scale invariance -- fractal, self-similar behavior -- by developing the renormalization group. The renormalization group is an amazing abstraction. It describes the flow of the laws governing the system as one coarse-grains -- blurring out the short-distance or short-time details. In the huge space of possible systems (experimental and theoretical), a fixed point of the renormalization group will be the same after blurring and shrinking -- implying emergent scale invariance, a fractal self-similarity. The points which flow into the fixed point share its properties under rescaling -- implying universality, with behavior shared by theory and a wide variety of different experimental systems. The renormalization group also predicts universal power laws and universal functions, describing all behavior on long length and/or time scales. Exercises explore applications to the Ising model, the onset of lasing, superconductors, the onset of chaos, percolation, crackling noise and avalanches, earthquakes, random walks and diffusion, chemical reaction rate theory, and extreme value statistics.
The method of the renormalization group (RG) provides theorists with powerful and efficient tools to explore interacting theories, often in regimes where perturbation theory fails. Motivating our discussion with two introductory examples drawn from a classical and a quantum theory, we will first become acquainted with the RG as a concept whereby nonlinear theories can be analyzed beyond the level of plain perturbation theory. With this background, we will then proceed to discuss the idea and practice of RG methods in more rigorous and general terms, introducing the notion of scaling, dimensional analysis, and the connection to the general theory of phase transitions and critical phenomena. Finally, to conclude this chapter, we will visit a number of concrete implementations of the RG program introduced and exemplified on a number of canonical applications. In Chapter 5 φ 4 -theory was introduced as an archetypal model of interacting continuum theories. Motivated by the existence of nonlinearities inherent in the model, a full perturbative scheme was developed, namely Wick contractions and their diagrammatic implementation. However, from a critical perspective, one may say that such perturbative approaches present only a limited understanding. Firstly, the validity of the φ 4 -action as a useful model theory was left unjustified, i.e. the φ 4 -continuum description was obtained as a gradient expansion of, in that case, a d -dimensional Ising model. But what controls the validity of the low-order expansion? Indeed, the same question could be applied to any one of the many continuum approximations we have performed throughout the first chapters of the text.
If we study the history of science we see produced two phenomena which are, so to speak, each the inverse of the other. Sometimes it is simplicity which is hidden under what is apparently complex; sometimes, on the contrary, it is simplicity which is apparent, and which conceals extremely complex realities. Henri Poincaré Devoting only a short chapter to the renormalization group (RG) is a challenge owing to the wealth of both fundamental concepts and computational tools associated with this term. The RG is indeed encountered in many different domains of theoretical physics, ranging from quantum electrodynamics to second-order phase transitions to fractal growth and diffusion processes – one should rather speak of renormalization group S ! We refer to Brown (1993) and Fisher (1998) for an historical account, to Goldenfeld (1992) and Lesne (1998) and references therein for an overview of the domains of application and variants of RGs. We here emphasize that the RG is a way, maybe the most successful and with no doubt the most systematic and constructive, to derive effective low-dimensional descriptions that capture large-scale and/or long-time behavior . The RG can be extended far beyond the specific scope of critical phenomena, to an iterated multiscale approach allowing the construction of robust and minimal macroscopic models describing the universal large-scale features and asymptotics of a complex system. This generalized viewpoint brings out the close logical and even technical connections that bridge, within a unified framework, perturbative RG for singular series expansions, spin-block RG and momentum-shell RG for critical phenomena, RG for the asymptotic analysis of differential and partial differential equations, and probabilistic RG for the derivation of statistical laws and limit theorems.
Renormalization group theory is a powerful and intriguing technique with a wide range of applications. One of the main successes of renormalization group theory is the description of continuous phase transitions and the development of scaling theory. Most courses on phase transitions focus on scaling and critical exponents, while less attention is paid to universality, renormalization group flow, and the existence of a unique fixed point, which are the ultimate reasons why scaling theory is so effective in describing continuous phase transitions. We use a combination of Monte Carlo simulations and real space renormalization group theory to determine the renormalization group flow and to show the existence of a universal fixed point in the context of the ferromagnetic Ising model.