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Cambridge University Press eBooks

Scaling transformations and the XY-model

C. Itzykson, Jean-Michel Drouffe

Published Sep 29, 1989• 0 citations
Publisher

Abstract

Condensed media show a large variety of critical phenomena, ranging from critical opalescence at the end point of the liquid–gas coexistence curve, the Curie transition of ferromagnetic materials, to the superfluid transition of helium, the behaviour of solutions of polymers, the conductivity of random media, … To this list should be added systems with local symmetries, such as those suggested by particle physics in order to understand quark confinement. Surprisingly all these phenomena can be classified into a few universality classes, characterized by specific large distance behaviour with the same critical exponents. In this chapter, we sketch the methods and ideas of the renormalization procedure. We illustrate the concepts with simple approximations in the language of classical spin models in the first section, treating in more detail the XY -model as an example in the second section. Scaling laws. Real space renormalization Homogeneity and scale invariance The discussion given in previous chapters suggests that, close to a continuous transition, critical systems exhibit universal properties. Correlations at large distance are not sensitive to the details of microscopic interactions. Their behaviour is described by a specific dimensional analysis governed by some essential characteristics of the system, such as the dimension of space, the nature of the order parameter and the underlying symmetries. The mean field approximation gave a first idea of a simple critical behaviour. Fluctuations have only a quantitative effect in dimension greater than the upper critical one.

DOI
10.1017/cbo9780511622779.005

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Condensed media show a large variety of critical phenomena, ranging from critical opalescence at the end point of the liquid–gas coexistence curve, the Curie transition of ferromagnetic materials, to the superfluid transition of helium, the behaviour of solutions of polymers, the conductivity of random media, … To this list should be added systems with local symmetries, such as those suggested by particle physics in order to understand quark confinement. Surprisingly all these phenomena can be classified into a few universality classes, characterized by specific large distance behaviour with the same critical exponents. In this chapter, we sketch the methods and ideas of the renormalization procedure. We illustrate the concepts with simple approximations in the language of classical spin models in the first section, treating in more detail the XY -model as an example in the second section. Scaling laws. Real space renormalization Homogeneity and scale invariance The discussion given in previous chapters suggests that, close to a continuous transition, critical systems exhibit universal properties. Correlations at large distance are not sensitive to the details of microscopic interactions. Their behaviour is described by a specific dimensional analysis governed by some essential characteristics of the system, such as the dimension of space, the nature of the order parameter and the underlying symmetries. The mean field approximation gave a first idea of a simple critical behaviour. Fluctuations have only a quantitative effect in dimension greater than the upper critical one.
To set the stage for our subsequent thorough discussion of dynamic critical phenomena, we first review the theoretical description of second-order equilibrium phase transitions. (Readers already well acquainted with this material may readily move on to Chapter 2.) To this end, we compare the critical exponents following from the van-der-Waals equation of state for weakly interacting gases with the results from the Curie–Weiss mean-field approximation for the ferromagnetic Ising model. We then provide a unifying description in terms of Landau–Ginzburg theory, i.e., a long-wavelength expansion of the effective free energy with respect to the order parameter. The Gaussian model is analyzed, and a quantitative criterion is established that defines the circumstances when non-linear fluctuations need to be taken into account properly. Thereby we identify d c = 4 as the upper critical dimension for generic continuous phase transitions in thermal equilibrium. The most characteristic feature of a critical point turns out to be the divergence of the correlation length that renders microscopic details oblivious. As a consequence, not only the correlation functions, but remarkably the thermodynamics as well of a critical system are governed by an emergent unusual symmetry: scale invariance. A simple scaling ansatz is capable of linking different critical exponents; as an application, we introduce the basic elements of finite-size scaling. Finally, a brief sketch of Wilson's momentum shell renormalization group method is presented, intended as a pedagogical preview of the fundamental RG ideas. Exploiting the scale invariance properties at the critical point, the scaling forms of the free energy and the order parameter correlation function are derived.
Abstract This first chapter is an elementary introduction to the models and methods that will be used later. The ferromagnetic (Curie) transition serves as our prototype for all second-order phase transitions, and serves also to introduce the notions of broken symmetry and of order parameter. The basic model of ferromagnetism, namely the Ising model, is described in Section 1.2, and solved for an elementary case. The reason why this model is so important is that it has an exact solution in two dimensions, which exhibits spontaneous magnetization. Section 1.3 is devoted to a first approach to mean-field methods, which remain amongst the most frequently exploited approximations even today. They will lead us to define a first set of critical exponents; the listing of such exponents will then be completed in Section 1.4, once correlation functions have been defined. Finally, in Section 1.5 a qualitative description of critical phenomena will enable us, in a first approximation, to define the concept of scale invariance and its relation to the behaviour of the theory under dilatations.
As discussed in the previous chapter, the phenomenological theory of critical phenomena finds a sound basis in the logical sequence of universality, scaling, relevant and irrelevant variables. The difficult problem of a large number of degrees of freedom strongly correlated within the coherence distance, which diverges at criticality, is thereby, in principle, reduced to the determination of the homogeneous form of the relevant response functions and of the evaluation of a few critical indices related to the relevant parameters, whereas the details specifying each system do not matter. In order to give a microscopic foundation to universality and to the scaling theory, an exact transformation is needed, which changes into a scaling transformation asymptotically near the critical point. This is the renormalization group (RG) approach, recently presented in a huge number of reviewpapers and books.Historically, the first comprehensive overview was given by the sixth volume of the series edited by Domb and Green (1976) and by the classic books by Patashinskij and Pokrovskij (1979) and by Ma (1976). Along with scaling theory, the field-theoretic renormalization group (RG) approach (Gell- Mann and Low (1954); Bogoliubov and Shirkov (1959); Bonch-Bruevich and Tyablikov (1962)) was introduced in critical phenomena (Di Castro and Jona-Lasinio (1969)) by showing how the RG equations generalize the universality relations in the sense that they relate one model system to another by varying the coupling and suitably rescaling the other variables and the correlation functions. At the same time a detailed analysis of the pertubative diagrammatic structure of the correlation functions together with the use of Ward identities yielded a renormalized theory (Migdal (1969); Polyakov (1968, 1969)). By iterating the RG transformation the coupling disappears from the equations near to the critical point, the homogeneous form of the static order-parameter correlation function is obtained and a microscopic definition of the critical indices is given. The scaling picture then acquires a theoretical basis. Wilson (Wilson (1971a, b); Wilson and Kogut (1974)) gave a great contribution to the understanding of the physics underlying the RG procedure and to the explicit calculation of the critical indices. He proposed a simple mathematical realization of the other very physical idea of universality due to Kadanoff of grouping together degrees of freedom on the scale s associated with larger and larger cells.
Overview As discussed in Chapter 16 , when we approach the critical temperature of a phase coexistence line, several fascinating things start to happen which together have come to be known as “critical phenomena.” It is observed that the spatial correlation between relevant order parameters like density, magnetization, etc. becomes infinitely large. This is reflected in the divergence of inverse susceptibility of the respective order parameters. Landau’s theory explains this in terms of the flattening of the free energy surface but fails to unravel many of the other mysteries of critical phenomena that have kept scientists engaged now for about a century. Critical phenomena are characterized by certain singular properties, like the universality of the critical exponents. Several aspects of the critical phenomena are discussed in this chapter with an aim to maintain continuity with earlier chapters. A simple theory (like that of Landau) cannot explain many of the observed features and this problem remained unsolved for a long time until K. G. Wilson could develop a sophisticated theory that explained almost everything. At the core of Wilson’s method is the renormalization group technique. Physically, this technique employs the expected invariance of the Hamiltonian due to a transformation that removes local interactions and replaces them by effective longer range interactions. Because of the existence of long range correlations, the transformation is expected to converge to a fixed point. The procedure is quite mathematical and is the only robust method to obtain the critical exponents. Here we discuss the method briefly, as an introduction to the subject.
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.
Abstract This chapter is devoted to a brief review of general properties of phase transitions in macroscopic physics and, in particular in lattice models. Some of these lattice models actually appear as lattice regularizations of Euclidean (imaginary time) quantum physics theory (QFT). Most of the transitions considered in this work have the following character: spins on the lattice, or macroscopic particles in the continuum, interact through short-range forces, assumed, for simplicity, to decay exponentially. For simple systems, it is possible to find a local observable, called order parameter, whose expectation values depend on the phase in the several phase region, for example, the spin in ferromagnetic systems. In the disordered phase, the connected two-point function decreases exponentially at large distance, at a rate characterized by the correlation length (the inverse of the smallest physical mass in particle physics). In continuous transitions, the correlation length diverges at the critical temperature. Within the mean-field approximation (consistent with Landau's theory of critical phenomena), it can be shown that the singular behaviour of thermodynamic quantities at the critical temperature is universal. These properties can also be reproduced by calculating correlation functions with a perturbed Gaussian measure. It is then shown that the leading corrections to the mean-field approximation, in Ising-like systems, diverge at the critical temperature for dimensions smaller than or equal to 444.