Based on ideas of Kadanoff and Wilson, we develop a lattice- (cutoff-) independent formulation of a theory of critical behavior. We are able to construct arbitrary fluctuating variables and their critical indices. As an illustrative example we consider the energy-density correlation function. The fluctuating variables of the critical theory obey the Kadanoff short-distance algebra.
Based on ideas of Kadanoff and Wilson, we develop a lattice- (cutoff-) independent formulation of a theory of critical behavior. We are able to construct arbitrary fluctuating variables and their critical indices. As an illustrative example we consider the energy-density correlation function. The fluctuating variables of the critical theory obey the Kadanoff short-distance algebra.
We recall that, in the study of critical phenomena, the discrete nature of the lattice (finite-cutoff effects) necessitates considering an infinite set of couplings in Wilson's Hamiltonian. This is particularly important for quantities, like the specific heat, which crucially depend on all scales of fluctuations between the lattice spacing and the correlation length and not only on large scales. As a consequence, by taking a precocious infinite-cutoff limit in the renormalization-group equations, one reaches erroneous results.
The primary goal of modern theories of critical phenomena is the evaluation of critical exponents from first principles, implying the need to get beyond mean field theories which predict the same exponents independent of the nature of the system. This chapter focuses on a new approach emerged in the 1960s based on two ideas—scaling and renormalization. It presents the physical picture developed by L. P. Kadanoff that provides a conceptual basis for scaling and which has become part of the standard language of critical phenomena. Renormalization provides a way to calculate the free energy not involving a direct evaluation of the partition function—it's a new paradigm in statistical mechanics. The chapter analyzes a simplified, approximate set of recursion relations valid at high temperature for a model with nearest and next-nearest-neighbor interactions. In 1971 K. G. Wilson published two articles marking the introduction of the renormalization group to the study of critical phenomena.
Continuous field models were originally systematically elaborated (and still are) in the context of particle physics. Most relevant to the study of critical phenomena is the derivation of a renormalization group flow, characterized by a set of Callan–Symanzik equations for Green functions, with regular coefficients derived from perturbation theory. This is supplemented by the idea due to Fisher and Wilson of an expansion in powers of the deviation from the strict renormalizability dimension, i.e. four in the case of the ϕ 4 model, but can also be presumed to work directly in the physical dimension three (and even possibly two). We devote this chapter to a general presentation and a survey of some applications. An appendix gives a short introduction to multicritical phenomena. The Lagrangian and dimensional analysis Introduction We want to investigate universal critical properties. A discrete lattice model appears as a regularizing intermediate stage, which allows a precise meaning to be given to the functional integrals of the field theory. Following the scheme suggested by the mean field approximation, one is tempted to start directly from a continuous model, in which the lattice is replaced by a d -dimensional continuous Euclidean space, and also, even for models with a discrete symmetry, the dynamical variables are replaced by continuous fields. From the original formulation, we just retain the idea of a cutoff factor ∧ large with respect to all momenta, which ensures that some possibly divergent expressions remain finite.
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 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.
ficld theory. 111 adclition to W~lson's r.olorii~alizotioti qr-oup method, whicl ~ is basctf on the idea of eliminating short range fluctuations, we discuss the renor~nali~ation method of quant~~m ficld theory which. in the present context. we call rc~ptrrtrt~rc~trizolion (in order to avoid confusion). A reparametrization which is of particular interest In the theory of critical phenomena is the one which leads to cc.cllin ~ c~qrrtrtioris. We derive new scaling equations which remain free of infl-arcd divergoices in two and three dimensions. Our method allows 11s to givc:I rather compact and unified discussion of Kadanoffs scaling laws and the related concept of global.sc.irlitrg /ic,ld.s, as well as the scale invariant correlation f~~nclions. Revemos nqui o trnbalho dc Kadanoff, Wilson c Wegner, na lingnagem da tcoria euclidiana de campus. AlCm do mCtodo do qrupo tlc rc~normtrliztr(~ilo dc Wilson, qi~c sc baseia na idCia de eliminar as flutuac;i,es dc curto nlcancc, discutimos o metodo de rcnormali/a~ao da teoria clu5ntica de campos o quai, no prcsente contexto, cienomin;lmos r-c.l)trrcinretriz(~~~(ii>.;I firn dc cvllar confi~siTo. UmLr repara~nci~-izi~qao dc especial inlcl-e\\se, na tcoria dos kn6rncnos criticos, i. a quc conduz a c,yrrtr(.cics (I(, c,.\\c,trltr. Obtcmos novas