Wilson’s Renormalization Scheme
At present there is a diversity of formulations for the Renormalization Group (RG) approach to the theory of critical behavior. Analysis of different methods can be found in the review articles by Fisher (1974), Bresin et al (1976), Di Castro and Jona-Lasinio (1976), Wegner (1976) and in the books by Ma (1976), Patashinskii and Pokrovskii (1979), Amit (1984), Baker (1990), Zinn-Justin (1989), and others. From the variety of different formulations one needs to conceive only two disparate approaches in order to be able to grasp any other approach. The first approach, which has been successfully applied to the theory of phase transitions in the critical region, is Wilson’s RG formulation (1971). In this approach the scaling hypothesis in Kadanoff’s formulation (see Chapter 1) has been exploited to the fullest extent. Kadanoff (1966) substantially developed the intuitive idea of the scaling invariance of thermodynamic functions in the vicinity of the critical point. He showed that this invariance can be explained on the basis of a simple assumption. The essence of this assumption can be formulated as follows. The experimentally-seen scaling invariance is a direct consequence of some coarsened “dynamical” invariance of the Ginzburg-Landau functional in the critical region. The latter in turn is an intuitively-logical consequence of the existence of a single characteristic length — correlation radius ξ. In reality, this is a very restrictive condition and this condition is the origin of invariance of the Ginzburg-Landau functional under very unusual transformations at the critical point.
KeywordsPartition Function Renormalization Group Critical Exponent Critical Behavior Renormalization Group Equation
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