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The renormalization group

James Jeffrey Binney, N J Dowrick, Andrew J. Fisher, Mark E. J. Newman

Published Jun 11, 1992• 4 citations
Publisher

Abstract

Abstract We are part of the way towards understanding the Landau-Ginzburg model in the critical regime. We have been able to calculate two critical exponents, one of which describes the system at T = Tc and the other at T ≠ Tc. If we are prepared to accept the scaling laws for which we found experimental evidence in Chapter 1, and theoretical evidence in Chapter 5, we can stop there; all four remaining critical exponents follow. However, this is rather unsatisfactory. In Chapter 5 we saw that the scaling relations can be deduced from the properties of the critical fixed point that is reached by repeatedly reblocking and rescaling the system; it would be good to provide a similarly general derivation in the context of the Landau-Ginzburg model.

DOI
10.1093/oso/9780198513940.003.0011

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Physical review. B, Solid state•May 1, 1976•6 citations
Abstract We are part of the way towards understanding the Landau-Ginzburg model in the critical regime. We have been able to calculate two critical exponents, one of which describes the system at T = Tc and the other at T ≠ Tc. If we are prepared to accept the scaling laws for which we found experimental evidence in Chapter 1, and theoretical evidence in Chapter 5, we can stop there; all four remaining critical exponents follow. However, this is rather unsatisfactory. In Chapter 5 we saw that the scaling relations can be deduced from the properties of the critical fixed point that is reached by repeatedly reblocking and rescaling the system; it would be good to provide a similarly general derivation in the context of the Landau-Ginzburg model.
Abstract The successful calculation of critical exponents for continuous phase transitions is one of the main achievements of theoretical physics over the last quarter-century. This was achieved through the use of scaling and field-theoretic techniques which have since become standard equipment in many areas of physics, especially quantum field theory. This book provides a thorough introduction to these techniques. Continuous phase transitions are introduced, then the necessary statistical mechanics is summarized, followed by standard models, some exact solutions and techniques for numerical simulations. The real-space renormalization group and mean-field theory are then explained and illustrated. The final chapters cover the Landau-Ginzburg model, from physical motivation, through diagrammatic perturbation theory and renormalization to the renormalization group and the calculation of critical exponents above and below the critical temperature.
Abstract In Chapters 10 to 12 we have seen that the critical exponents of the Landau-Ginzburg model agree well with experiment (see Box 14.1 for a summary of results). This is good, but it is also slightly surprising. The Landau-Ginzburg Hamiltonian is one of the simplest possible non-Gaussian Hamiltonians, containing only one non-Gaussian term. Even the Ising model has an effective Hamiltonian which is much more complicated than this (see Appendix K). Yet not only do the critical exponents of the Landau-Ginzburg model agree with those of the Ising model, they agree with those of real physical systems too. This requires an explanation.
Abstract In Chapter 8 we gave an introduction to the formalism of the renormalization group. A renormalization transformation changes the scale of a system by a factor b. Systems at criticality flow to fixed points under the transformation and their critical exponents and scaling behaviour are determined by linearizing the recursion equations about the fixed point. The aim of this chapter is two-fold: firstly to illustrate these ideas by giving several examples of renormalization group transformations and secondly to demonstrate the ways in which they can be used to calculate critical properties. Our examples are divided into three classes. In Section 9.1, we shall consider a one-dimensional model where the recursion relations can be written down exactly. The drawback of this as a canonical example is that any critical behaviour is at zero temperature. The decimation transformation presented in one dimension is an example of a real-space transformation and the second task, in Section 9.2, is to show how these may be used approximately to treat models in higher dimensions. Thirdly, in Section 9.5, we shall mention the E-expansion where the renormalization takes place in momentum space. This enables an understanding of the crossover to mean-field behaviour. However, the formalism is somewhat involved and the details are beyond the scope of this book.
Using the Landau Hamiltonian for the description of second-order phase transitions, we give a proof of scaling for any continuous number of dimensions below four. The proof is based on a summation of diagrams having a power-law divergence and standard renormalization-group methods. The proof is constructive in that it leads to an unambiguous calculation for the critical exponents $\ensuremath{\eta}$ and $\ensuremath{\gamma}$. We present in this paper a detailed discussion of the proof; we also compare our method with the $\ensuremath{\epsilon}$ expansion leading to an interesting aspect of that theory: We find that the contribution to the critical exponents of order $\ensuremath{\epsilon}$ can be gotten without any calculation of diagrams. In this paper we have only made a lowest-order calculation in three dimensions. To this order we are of course unable to locate the relevant fixed point, but it leads to a relation between $\ensuremath{\eta}$ and $\ensuremath{\gamma}$---also to lowest order---which is such that if $\ensuremath{\gamma}$ is fixed to be 1.25, then $\ensuremath{\eta}$ turns out to be 0.12.