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Renormalization

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

Published Jun 11, 1992• 0 citations
Publisher

Abstract

Abstract In Chapter 7 we introduced the Landau-Ginzburg ‘metamoder of phase transitions. The model’s partition function takes the form of a functional integral which cannot be done exactly. So in the last chapter we expressed this integral as an infinite series of functional integrals that can be reduced to ordinary multi-dimensional integrals. To keep track of these integrals we associated with each a Feynman diagram and used these diagrams to classify the integrals by the number of integrations or ‘loops’ each involves.

DOI
10.1093/oso/9780198513940.003.0009

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Abstract In Chapter 7 we introduced the Landau-Ginzburg ‘metamoder of phase transitions. The model’s partition function takes the form of a functional integral which cannot be done exactly. So in the last chapter we expressed this integral as an infinite series of functional integrals that can be reduced to ordinary multi-dimensional integrals. To keep track of these integrals we associated with each a Feynman diagram and used these diagrams to classify the integrals by the number of integrations or ‘loops’ each involves.
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 For every physical model defined on a generic graph or factor graph, the Bethe M -layer construction allows building a different model for which the Bethe approximation is exact in the large M limit, and coincides with the original model for M = 1 . The 1 / M perturbative series is then expressed by a diagrammatic loop expansion in terms of so-called fat diagrams. Our motivation is to study some important second-order phase transitions that do exist on the Bethe lattice, but are either qualitatively different or absent in the corresponding fully connected case. In this case, the standard approach based on a perturbative expansion around the naive mean field theory (essentially a fully connected model) fails. On physical grounds, we expect that when the construction is applied to a lattice in finite dimension there is a small region of the external parameters, close to the Bethe critical point, where strong deviations from mean-field behavior will be observed. In this region, the 1 / M expansion for the corrections diverges, and can be the starting point for determining the correct non-mean-field critical exponents using renormalization group arguments. In the end, we will show that the critical series for the generic observable can be expressed as a sum of Feynman diagrams with the same numerical prefactors of field theories. However, the contribution of a given diagram is not evaluated by associating Gaussian propagators to its lines, as in field theories: one has to consider the graph as a portion of the original lattice, replacing the internal lines with appropriate one-dimensional chains, and attaching to the internal points the appropriate number of infinite-size Bethe trees to restore the correct local connectivity of the original model. The actual contribution of each (fat) diagram is the so-called line-connected observable, which also includes contributions from sub-diagrams with appropriate prefactors. In order to compute the corrections near to the critical point, Feynman diagrams (with their symmetry factors) can be read directly from the appropriate field-theoretical literature; the computation of momentum integrals is also quite similar; the extra work consists of computing the line-connected observable of the associated fat diagram in the limit of all lines becoming infinitely long.
This manuscript is a review of the main results obtained in a series of papers involving the present authors and their collaborator J.L. Cardy over the last two years. In our work we have developed and applied a new approach for the computation of the bi-partite entanglement entropy in massive 1+1-dimensional quantum field theories. In most of our work we have considered these theories to be also integrable. Our approach combines two main ingredients: the "replica trick" and form factors for integrable models and more generally for massive quantum field theory. Our basic idea for combining fruitfully these two ingredients is that of the branch-point twist field. By the replica trick we obtained an alternative way of expressing the entanglement entropy as a function of the correlation functions of branch-point twist fields. On the other hand, a generalisation of the form factor program has allowed us to study, and in integrable cases to obtain exact expressions for, form factors of such twist fields. By the usual decomposition of correlation functions in an infinite series involving form factors, we obtained exact results for the infrared behaviours of the bi-partite entanglement entropy, and studied both its infrared and ultraviolet behaviours for different kinds of models: with and without boundaries and backscattering, at and out of integrability.