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CRC Press eBooks

Critical Phenomena Near Four Dimensions

Nigel D. Goldenfeld

Published Mar 8, 2018• 0 citations
Publisher

Abstract

This chapter discusses a very popular method of studying critical phenomena analytically: the so-called ϵ-expansion. It describes the renormalisation group calculation of the critical exponents in the complete Landau theory, near d = 4. The chapter outlines the strategy of the calculation of 0, to prepare the reader for the algebraic horrors to follow. R. P. Feynman developed useful shorthand for the terms, which is to write down diagrams indicating how different fields are paired up. There is one-to-one correspondence between each Feynman diagram or Feynman graph and a term in the perturbation series. This correspondence is not unique: there are many ways to write down diagrams for the terms, and people have their own preferences. Once one has figured out this correspondence, then one only has to be able to draw all the appropriate diagrams to see what one should compute.

DOI
10.1201/9780429493492-12

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This chapter discusses a very popular method of studying critical phenomena analytically: the so-called ϵ-expansion. It describes the renormalisation group calculation of the critical exponents in the complete Landau theory, near d = 4. The chapter outlines the strategy of the calculation of 0, to prepare the reader for the algebraic horrors to follow. R. P. Feynman developed useful shorthand for the terms, which is to write down diagrams indicating how different fields are paired up. There is one-to-one correspondence between each Feynman diagram or Feynman graph and a term in the perturbation series. This correspondence is not unique: there are many ways to write down diagrams for the terms, and people have their own preferences. Once one has figured out this correspondence, then one only has to be able to draw all the appropriate diagrams to see what one should compute.
The application of renormalisation group recursion relations to the calculation of critical exponents in d dimensions to second order in epsilon =4-d is discussed. It is shown that in the limit of a large scale factor b, the fixed point value, u* of the four-spin coupling constant is correctly given to second order by a renormalisation group equation in which all irrelevant variables, mass renormalisation graphs, and mass terms in the propagators are ignored. Related approximations in the calculation of the critical exponents are justified. The unexpected equality to order epsilon 2 of u* and u 0c , the Feynman graph coupling constant, is discussed.
Different perturbation theory treatments of the Ginzburg-Landau phase transition model are discussed. This includes a criticism of the perturbative renormalization group (RG) approach and a proposal of a novel method providing critical exponents consistent with the known exact solutions in two dimensions. The usual perturbation theory is reorganized by appropriate grouping of Feynman diagrams of φ4 model with O(n) symmetry. As a result, equations for calculation of the two-point correlation function are obtained which allow to predict possible exact values of critical exponents in two and three dimensions by proving relevant scaling properties of the asymptotic solution at (and near) the criticality. The new values of critical exponents are discussed and compared to the results of numerical simulations and experiments.
Different perturbation theory treatments of the Ginzburg-Landau phase transition model are discussed. This includes a criticism of the perturbative renormalization group (RG) approach and a proposal of a novel method providing critical exponents consistent with the known exact solutions in two dimensions. The usual perturbation theory is reorganized by appropriate grouping of Feynman diagrams of φ4 model with O(n) symmetry. As a result, equations for calculation of the two-point correlation function are obtained which allow to predict possible exact values of critical exponents in two and three dimensions by proving relevant scaling properties of the asymptotic solution at (and near) the criticality. The new values of critical exponents are discussed and compared to the results of numerical simulations and experiments.
Pertinent Concepts and Ideas in the Theory of Critical Phenomena Formulation of the Problem of Phase Transitions in Terms of Functional Integrals Functional Integrals in Quantum Field Theory Perturbation Theory and Feynman Graphs Vertex Functions and Symmetry Breaking Expansions in the Number of Loops and Components Renormalization The Renormalization Group and Scaling the Critical Region The Computation of the Critical Exponents Beyond Leading Scaling Universality Revisited Critical Behavior with Several Couplings Crossover Phenomena Critical Phenomena near Two Dimensions Real Space Methods Finite Size Scaling Monte Carlo Methods. Numerical Field Theory.
Abstract Chapter 6 describes how the perturbative renormalization group (RG) of quantum field theory has made it possible to derive universal properties of continuous macroscopic phase transitions. The RG initially emerged as a consequence of the necessity to cancel infinities that appear in the perturbative expansion (the renormalization procedure) and the possibility of defining the parameters of the renormalized theory at different momentum scales. Although the field theory RG is now understood to be an asymptotic form, it has made it possible to confirm the Wilson–Fisher fixed point and led to an understanding of universality for a large class of critical phenomena. In the framework of dimensional continuation, zeros of RG beta functions, which correspond to Wilson–Fisher’s fixed points, have made it possible to recover Wilson–Fisher’s epsilon expansion, proving scaling relations and calculating critical exponents. Series summation methods have then been used to generate precise values of exponents.
In the previous chapter, we developed the perturbative formulation of the Landau–Ginzburg theory and introduced Feynman diagrams as a systematic way to compute corrections to the Gaussian (mean-field) results. Mean-field theory provides a qualitatively correct picture of phase transitions but fails quantitatively: it neglects fluctuations, leading to incorrect values of the critical exponents, especially in low dimensions ( https://www.w3.org/1998/Math/MathML" display="inline"> D < 4 ). To obtain accurate exponents, one must incorporate fluctuations perturbatively and study how couplings evolve with scale.