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Diagrammatic perturbation theory

James Jeffrey Binney, N J Dowrick, Andrew J. Fisher, Mark E. J. Newman

Published Jun 11, 1992• 1 citation
Publisher

Abstract

Abstract In the last chapter we introduced the Landau-Ginzburg model and calculated its partition function in a crude approximation known as Landau theory. In this chapter we develop a systematic method for calculating the Landau-Ginzburg partition function, the associated correlation functions, and the Helmholtz and Gibbs free energies. In subsequent chapters we shall use the techniques developed here to find the critical exponents for the Landau-Ginzburg model.

DOI
10.1093/oso/9780198513940.003.0008

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Abstract In the last chapter we introduced the Landau-Ginzburg model and calculated its partition function in a crude approximation known as Landau theory. In this chapter we develop a systematic method for calculating the Landau-Ginzburg partition function, the associated correlation functions, and the Helmholtz and Gibbs free energies. In subsequent chapters we shall use the techniques developed here to find the critical exponents for the Landau-Ginzburg model.
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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.
Abstract Actual computations of fixed points and eigenvalues usually involve approximations, often crude ones, except for a very limited number of simple cases such as the one-dimensional Ising model of the previous chapter. In real- and momentum-space renormalization group theory, there are no general prescriptions to systematically improve the degree of the approximation with a modest amount of effort. There are established methods to systematically improve precision, but they usually need a large amount of numerical calculations. The scope of the present chapter is modest as we limit ourselves to basic examples, including the epsilon expansion about the Gaussian fixed-point of the Landau-Ginzburg-Wilson model. Finally, the last section illustrates the extension of the renormalization group framework to study quantum phase transitions.