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Reports on Progress in Physics

The renormalisation group approach to scaling in physics

D. J. Wallace, Royce K. P. Zia

Published Jan 1, 1978• 60 citations
Publisher

Abstract

The auth ors review the application of the renormalisation group to many-body problems, with particular emphasis on the scaling behaviour which may arise. An introductory section discusses the difficulties of calculating the many-body effects near a magnetic phase transition, and shows by a simple example how the renormalisation group may be used to predict scaling behaviour. A general discussion of scaling and universality follows, with a review of techniques for calculation in lattice models, and an outline of the approaches in field theory. The range of applications of these techniques is discussed.

DOI
10.1088/0034-4885/41/1/001

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The auth ors review the application of the renormalisation group to many-body problems, with particular emphasis on the scaling behaviour which may arise. An introductory section discusses the difficulties of calculating the many-body effects near a magnetic phase transition, and shows by a simple example how the renormalisation group may be used to predict scaling behaviour. A general discussion of scaling and universality follows, with a review of techniques for calculation in lattice models, and an outline of the approaches in field theory. The range of applications of these techniques is discussed.
These lectures describe the physics of phase transitions and critical phenomena. After introducing the thermodynamics of phase transitions, these general considerations are made explicit by presenting statistical mechanical calculations on a lattice gas model in the mean-field approximation. Then the Landau theory of phase transitions is presented. The form of free energy in the Landau theory is motivated by an explicit mean field calculation for the Ising model. This is followed by a discussion of spatial correlations and the breakdown of the mean field theory due to increase of spatial fluctuations near the critical point. This calculation also brings out the role of dimension of the system for the critical behaviour. The basic ideas of Renormalization Group (RG) are then introduced and illustrated by a simple RG calculation for the one-dimensional Ising model. Real-space RG calculations are then applied to the two-dimensional Ising model on the triangular lattice, which, in fact, brings forth several key features of the RG technique. A discussion of more general aspects of the RG formalism, such as scaling for spatial correlation functions, relevant and irrelevant couplings, universality, upper critical dimension, etc. are introduced.
This text provides a thoroughly modern graduate-level introduction to the theory of critical behaviour. Beginning with a brief review of phase transitions in simple systems and of mean field theory, the text then goes on to introduce the core ideas of the renormalization group. Following chapters cover phase diagrams, fixed points, cross-over behaviour, finite-size scaling, perturbative renormalization methods, low-dimensional systems, surface critical behaviour, random systems, percolation, polymer statistics, critical dynamics and conformal symmetry. The book closes with an appendix on Gaussian integration, a selected bibliography, and a detailed index. Many problems are included. The emphasis throughout is on providing an elementary and intuitive approach. In particular, the perturbative method introduced leads, among other applications, to a simple derivation of the epsilon expansion in which all the actual calculations (at least to lowest order) reduce to simple counting, avoiding the need for Feynman diagrams.
Studies of first-order phase transitions through the use of the exact renormalization group are reviewed. In the first part the emphasis is on universal aspects: We discuss the universal critical behaviour near weakly first-order phase transitions for a three-dimensional model of two coupled scalar fields – the cubic anisotropy model. In the second part we review the application of the exact renormalization group to the calculation of bubble-nucleation rates. More specifically, we concentrate on the pre-exponential factor. We discuss the reliability of homogeneous nucleation theory that employs a saddle-point expansion around the critical bubble for the calculation of the nucleation rate.
The renormalization group approach to the theory of critical behavior is reviewed at an introductory level with emphasis on magnetic systems. Among recent results reported are the dependence of critical exponents above Tc{T}_{c}Tc​ on dimensionality $d=4\ensuremath{-}\ensuremath{\epsilon}$; on the symmetry index or number of spin components, nnn; on the range and anisotropy of exchange couplings; and on dipolar interactions and lattice anisotropies, in ferro- and antiferromagnets. Calculations of the scaling functions for the equation of state and critical scattering are summarized.
The renormalization group (RG) method developed by Ken Wilson more than four decades ago has revolutionized the way we think about problems involving a broad range of energy scales such as phase transitions, turbulence, continuum limits and bifurcations in dynamical systems. The Theme Issue provides articles reviewing recent progress made using the RG method in atomic, condensed matter, nuclear and particle physics. In the following, we introduce these articles in a way that emphasizes common themes and the universal aspects of the method.
Quantum phase transitions are strongly relevant in a number of fields, ranging from condensed matter to cold atom physics and quantum field theory. This book, now in its second edition, approaches the problem of quantum phase transitions from a new and unifying perspective. Topics addressed include the concepts of scale and time invariance and their significance for quantum criticality, as well as brand new chapters on superfluid and superconductor quantum critical points, and quantum first order transitions. The renormalisation group in real and momentum space is also established as the proper language to describe the behaviour of systems close to a quantum phase transition. These phenomena introduce a number of theoretical challenges which are of major importance for driving new experiments. Being strongly motivated and oriented towards understanding experimental results, this is an excellent text for graduates, as well as theorists, experimentalists and those with an interest in quantum criticality.