The approach of Chapter 11, in which we have kept a large momentum or short-distance cut-off, is sufficient for the needs of computing the various universal indices and functions that characterize criticality. All calculations were made in the limit in which the ratio ξ/a of the correlation length ξ to the lattice spacing a is large. In the previous chapters, this was done by keeping a fixed and increasing ξ . The renormalized theory relies on the opposite strategy: it deals directly with the scaling theory in which one takes the limit of a vanishingly small a . I have included here a chapter on the renormalized theory for several reasons: • There is no ultraviolet (or lattice) cut-off any more and, getting rid of one dimensionful parameter considerably simplifies the integrations, especially at higher orders. • All the critical exponents and scaling functions can be computed at T c , i.e., in a massless theory, even when one considers temperature-dependent laws, such as those concerning the correlation length, the magnetic susceptibility or the spontaneous magnetization. Integrals in a massless theory without cut-off are by far simpler, and they are often given by mere dimensional analysis. • However, the main reason is to understand the meaning of renormalizability from the viewpoint both of critical phenomena and of high-energy physics. This will be done in the following section.
The approach of Chapter 11, in which we have kept a large momentum or short-distance cut-off, is sufficient for the needs of computing the various universal indices and functions that characterize criticality. All calculations were made in the limit in which the ratio ξ/a of the correlation length ξ to the lattice spacing a is large. In the previous chapters, this was done by keeping a fixed and increasing ξ . The renormalized theory relies on the opposite strategy: it deals directly with the scaling theory in which one takes the limit of a vanishingly small a . I have included here a chapter on the renormalized theory for several reasons: • There is no ultraviolet (or lattice) cut-off any more and, getting rid of one dimensionful parameter considerably simplifies the integrations, especially at higher orders. • All the critical exponents and scaling functions can be computed at T c , i.e., in a massless theory, even when one considers temperature-dependent laws, such as those concerning the correlation length, the magnetic susceptibility or the spontaneous magnetization. Integrals in a massless theory without cut-off are by far simpler, and they are often given by mere dimensional analysis. • However, the main reason is to understand the meaning of renormalizability from the viewpoint both of critical phenomena and of high-energy physics. This will be done in the following section.
This book provides an elementary introduction to the notions of continuum limit and universality in statistical systems with a large number of degrees of freedom. The existence of a continuum limit requires the appearance of correlations at large distance, a situation that is encountered in second order phase transitions, near the critical temperature. In this context, the book emphasizes the role of gaussian distributions and their relations with the mean field approximation and Landau′s theory of critical phenomena. The book shows that quasi-gaussian or mean-field approximations cannot describe correctly phase transitions in three space dimensions. The book assigns this difficulty to the coupling of very different physical length scales, even though the systems we will consider have only local, that is, short range, interactions. To analyze the unusual situation, a new concept is required: the renormalization group, whose fixed points allow understanding the universality of physical properties at large distance, beyond mean-field theory. In the continuum limit, critical phenomena can be described by quantum field theories. In this framework, the renormalization group is directly related to the renormalization process; that is, the necessity to cancel the infinities that arise in straightforward formulations of the theory. The book discusses the renormalization group in the context of various relevant field theories. This leads to proofs of universality and to efficient tools for calculating universal quantities in a perturbative framework. Finally, the book constructs a general functional renormalization group, which can be used when perturbative methods are inadequate.
Abstract The approaches described so far in this book have given a broad phenomenological understanding of critical phenomena. However, although a substantial framework of results and connections has been built up, we have, as yet, no explanations for the following: What is needed is a theory, based on the physics of what is happening at the critical point. we argued in Chapter 1 that the special feature of criticality is that the correlation length is infinite and that the critical system is invariant on all length scales. The aim is to write down a (hopefully short, elegant, and comprehensible) mathematical theory which embodies this physics and explains all the observations listed above. A useful theory will also allow the calculation of critical exponents and transition temperatures, if not exactly, then within an accurate and well-controlled approximation scheme.
Abstract In Chapter 15, the scaling behaviour of correlation functions at criticality, T = Tc, has been derived. This chapter is devoted to the critical domain, where the correlation length is large with respect to the microscopic scale, but finite. In dimensions d < 4, above Tc, the property of strong scaling is derived: in the critical domain above Tc, all correlation functions, after rescaling, can be expressed in terms of universal correlation functions, in which the scale of distance is provided by the correlation length. However, because the correlation length is singular at Tc, in this formalism, the critical temperature cannot be crossed. Alternatively, one can expand correlation functions in formal power series of the deviation (T −Tc) from the critical temperature, in presence of a magnetic field. The sum of the expansion satisfies renormalization group (RG) equations also valid for T < Tc and in a magnetic field, from which follow scaling properties in the whole critical domain. The universal two-point function can be expanded when T approaches Tc, using the short-distance expansion (SDE). A few terms of the ϵ expansion (ϵ is the deviation from dimension 4) of a few universal quantities are reported. Calculations at fixed dimension and summation of perturbative expansions are described. The conformal bootstrap based on the SDE and conformal invariance at the infrared (IR) fixed point provides an alternative method to determine critical exponents.
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.
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.
Wilson's momentum-shell transformation with the concomitant expansion around the upper critical dimension is defined. The basic notions of relevant and irrelevant couplings, renormalization flow, and fixed points are introduced. The origins of scaling and of universality are explained, and corrections to mean-field values of critical exponents are computed. The field theoretic renormalization group is briefly discussed and used to calculate the anomalous dimension. Idea We found that the direct perturbation theory in the Ginzburg–Landau–Wilson theory breaks down below the upper critical dimension because the perturbation parameter grows with the correlation length, and so becomes arbitrarily large as the critical point is approached. If the system were finite, on the other hand, the correlation length would be bound by its size, and perturbation theory could succeed. Singular thermodynamic behavior near the critical point comes from the thermodynamic limit, or, more precisely, from those modes that have arbitrary low energies in an infinitely large system. This is called the infrared singularity . This observation suggests the following strategy to avoid the problem of direct perturbation theory. First, note that mass m only provides the energy scale in Eq. (2.36). It is practical to rescale it out by absorbing it into the chemical potential, rede- fined as 2 m μ/ħ 2 → μ, and into the interaction coupling, as 2 m λ/ħ 2 → λ. Similarly, near the critical point, the temperature T ≈ T c may be replaced by the critical temperature, and then eliminated by rescaling the action as 2 m κ B T c S /ħ 2 → S.