Chapter 25 established the scaling behaviour of correlation functions at criticality, T = Tc . This chapter studies the critical domain which is defined by the property that the correlation length is large with respect to the microscopic scale, but finite. Using results proven in Chapter 10, it first demonstrates strong scaling above Tc : 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. The first part of the chapter is restricted to Ising-like systems. It then generalizes the results to N-component order parameters in Section 26.6. In Section 26.7, shows how to expand the universal two-point function when T approaches Tc , using the short distance expansion. Finally, the appendix discusses the energy correlation function when the specific heat exponent vanishes.
Chapter 25 established the scaling behaviour of correlation functions at criticality, T = Tc . This chapter studies the critical domain which is defined by the property that the correlation length is large with respect to the microscopic scale, but finite. Using results proven in Chapter 10, it first demonstrates strong scaling above Tc : 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. The first part of the chapter is restricted to Ising-like systems. It then generalizes the results to N-component order parameters in Section 26.6. In Section 26.7, shows how to expand the universal two-point function when T approaches Tc , using the short distance expansion. Finally, the appendix discusses the energy correlation function when the specific heat exponent vanishes.
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.
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.
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.
We write the singular parts of the free energy and correlation functions in terms of universal quantities, identical for all members of the same symmetry class, and in terms of two system‐dependent scale‐factors. Besides the universality and homogeneity of the functions, we also assume that the free energy in a volume defined by the correlation length is a universal constant. We can then write the critical singularities of properties derived from these functions in terms of universal quantities and the two scale factors. This two scale‐factor universality has important experimental consequences because once any two properties have been measured for a system the two scale‐factors may be determined via comparison with some reference system in the same symmetry class. Then the remaining properties of the system are predicted by our formalism using the two scale‐factors and the known properties of the reference system. Both our extra assumption and the predictions of two‐scale factor universality are exact for the d = 2 Ising model and agree satisfactorily with results for d = 3 systems.
"Critical point phenomena in magnetic systems are studied with the aid of the scaling laws. These laws relate the critical indices which describe the behavior of thermodynamic and correlation functions near T. Kadanoff's derivation of the scaling 1aw is reviewed. An experimental survey of ferromagnetic and antiferromagnetic data near T is presented. Susceptibility, correlation length, magnectization, specific heat and a magnetic equation of state are discussed. It is found that the ""critical region"" can extend as far as [equation] although frequently it is restricted to E < 10-2 The critical indices S, y, v, ~ and ~, are close to the three dimensional ISing model values. Values for 0, y' and ~ are not known as conclusively. Specific heat behavior is discussed in terms -~ of a three-parameter fit, namely C = aE + b. Varying b allows one to estimate a possible range of a's. The experimental information is consistent with the scaling law relations 2 - alpha = 2 - alpha' = dV = d gamma / (2-pi) = gamma + 2B = B(d+1) The scaling law theory is extended to predict the temperature, field and spatial dependence of various spin correlation functions. These predictions are checked by analytic calculations for the two dimensional Ising model in zero field. Results are presented for the energy density-energy density and energy density-spin correlation functions in the limit E « 1, R » 1, but the product ER arbitrary. Results are also presented for a group of two, three and four spin correlations at cT. In each case the scaling law predictions are verified."