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Physical Review A

Feynman - graph calculation of the (0, l ) critical exponents to order ε2

A. Houghton, Franz Wegner

Published Jul 1, 1974• 22 citations
Publisher

Abstract

It is shown that the critical indices corresponding to (0, $l$ perturbations can be calculated by a Feynman-diagram method. At order ${\ensuremath{\epsilon}}^{2}$, the results are consistent with the special cases $l=1,2, \mathrm{and} 4$ obtained previously. There is also general agreement to order $\ensuremath{\epsilon}$ with the exponents calculated from Wilson's recurrence relation.

DOI
10.1103/physreva.10.435

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It is shown that the critical indices corresponding to (0, lll perturbations can be calculated by a Feynman-diagram method. At order \ensuremathϵ2{\ensuremath{\epsilon}}^{2}\ensuremathϵ2, the results are consistent with the special cases l=1,2,and4l=1,2, \mathrm{and} 4l=1,2,and4 obtained previously. There is also general agreement to order \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ with the exponents calculated from Wilson's recurrence relation.
We use field theory to calculate the critical exponents ${\ensuremath{\eta}}_{l2},$ ${\ensuremath{\eta}}_{l4},$ ${\ensuremath{\nu}}_{l2},$ and ${\ensuremath{\nu}}_{l4}$ for the Lifshitz point (LP) with m=2m=2m=2 and m=6m=6m=6. We were motivated by an old controversy on the order ${\ensuremath{\epsilon}}^{2}$ corrections for the exponents of this multicritical point. In a previous paper, we studied the renormalization of the theory that describes the LP and derived expressions for the exponents and scaling relations that hold to all orders in perturbation. In this work we concentrate on the calculation of the Feynman diagrams that are involved in the determination of the exponents and present our results for the exponents to order ${\ensuremath{\epsilon}}^{2}.$ The evaluation of these diagrams, due to a free propagator that contains a quartic term in the momentum and is not rotationally invariant, requires special techniques that are presented in detail in the appendixes.
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.
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 relaxational models introduced in a previous treatment of critical dynamics are studied in detail using renormalization-group methods. The earlier results are justified by an analysis to all order in $\ensuremath{\epsilon}=4\ensuremath{-}d$, where ddd is the dimensionality. The diagrammatic formalism of the full dynamic renormalization group is presented, and applied to the earlier models. A generalization of Wilson's Feynman-graph expansion method is used to calculate the exponents to second order in $\ensuremath{\epsilon}$. In model CCC, where a nonconserved order parameter is coupled to a conserved energy field, ambiguities were found in the earlier recursion-relation treatment for 2<n<42<n<42<n<4, $d\ensuremath{\approx}4$ (nnn is the number of components of the order parameter). These ambiguities are discussed in the present work, but are not fully resolved.
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.