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Universality, renormalization and critical phenomena

Lawrence S. Schulman

Published Jan 18, 2022• 0 citations
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Abstract

Abstract The Ising model of ferromagnetism (defined in Chapter 5) is studied in three ways: mean field (which is incorrect), transfer matrix, and renormalization. Renormalization has the advantage of ignoring details and breaks the world into universality classes, each of which behaves in the same way in the neighborhood of the critical point. Spatial correlations are an example and are computed in two ways, which agree.

DOI
10.1093/oso/9780198861881.003.0008

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Abstract The Ising model of ferromagnetism (defined in Chapter 5) is studied in three ways: mean field (which is incorrect), transfer matrix, and renormalization. Renormalization has the advantage of ignoring details and breaks the world into universality classes, each of which behaves in the same way in the neighborhood of the critical point. Spatial correlations are an example and are computed in two ways, which agree.
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Abstract The order-disorder transition and ferromagnetism have their origins in quite different manifestations of atomic interactions and are exhibited in quite different kinds of material. But, as shown in chapter 9, both can be described by the Ising model with considerable success. In fact, there are other phenomena that are different from either magnetism or order-disorder that can also be described by similar models. These include adsorption of atoms or molecules on surfaces, absorption of gases in solids, and the attachment of impurities to dislocation cores. A simple description of such phenomena is provided by the lattice gas model, and this model is easily shown to be equivalent to the Ising model. The fact that the same model can describe such varied systems implies that there are some important similarities among them that do not depend on the specific nature of the systems or the interaction energies involved. This, in fact, turns out to be the case and is most clearly apparent in the neighborhood of critical points. The statistical mechanics of the lattice gas is described in appendix 7, and specific applications are given in chapter 12. Here, the model and its connection to the Ising model are described.
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.
Abstract Chapter 8 introduces the key ideas of the renormalization group, including how they provide a theoretical scheme and a proper language to face critical phenomena. It covers the scaling transformations of a system and their implementations in the space of the coupling constants and reducing the degrees of freedom. From this analysis, the reader is led to the important notion of relevant, irrelevant and marginal operators and then to the universality of the critical phenomena. Furthermore, the chapter also covers (as regards the RG) transformation laws, effective Hamiltonians, the Gaussian model, the Ising model, operators of quantum field theory, universal ratios, critical exponents and β-functions.
Abstract Mean-field theory is usually taken as a first step toward understanding critical phenomena, providing an overview that reveals qualitative behaviour of physical quantities. However, it is necessary to proceed beyond the mean-field theory to better understand the situation, both qualitatively and quantitatively, when fluctuations play vital roles leading to exponents that cannot be explained by dimensional analysis, thus introducing anomalous dimensions. The present chapter explains the basic concepts of the renormalization group and scaling theory, which allow us to analyze critical phenomena with fluctuations systematically taken into account. The essential step in a renormalization group calculation consists of establishing recursion relations between the parameters defining the Hamiltonian of the system. These recursion or renormalization group equations define a flow with well-defined fixed points. Details other than the values of the relevant operators have no influence on the critical exponents and this represents universality.
This chapter reviews a few properties, from the point of view of phase transitions, of simple ferromagnetic lattice models. In systems with finite range interactions a transfer matrix can be defined. It first examines its properties in a finite volume. In the infinite volume limit, low and high temperature considerations provide convincing evidence of the existence of phase transitions in Ising-like systems. The notion of order parameter is related to cluster properties in the low temperature broken phase. It is shown in a simple example that phase transitions indeed correspond to breaking of ergodicity. The chapter extends the analysis to ferromagnetic systems with continuous symmetries. The appendix contains a brief discussion of quenched disorder.