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Perturbative renormalization of the Ginzburg-Landau model revisited

J. Kaupuÿzs

Published Sep 30, 2006• 0 citations
OpenAlex

Abstract

The perturbative renormalization of the Ginzburg–Landau model is reconsidered based on the Feynman diagram technique. We derive renormalization group (RG) flow equations, exactly calculating all vertices appearing in the perturbative renormalization of the ' 4 model up to the 3 order of the -expansion. In this case, the ' 2 , ' 4 , ' 6 , and ' 8 vertices appear. All these are relevant in the sense that they do not vanish at the fixed point. We have tested the expected basic properties of the RG flow, such as the semigroup property. Under repeated RG transformation Rs, appropriately represented RG flow on the critical surface converges to certain s–independent fixed point. The Fourier–transformed two–point correlation function G(k) has been considered. Although the -expansion of X(k) = 1=G(k) is well defined on the critical surface, we have revealed a fundamental inconsistency of the perturbative method with the exact rescaling of X(k), represented as an expansion in powers of k at k ! 0. We discuss also certain paradox, related to the -expansion of X(k) above the critical point. Apart from the –expansion, we have tested also a modified approach, where the ' 4 coupling constant u is the expansion parameter at a fixed spatial dimensionality d. Our tests point to some internal inconsistency of such a method.

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The perturbative renormalization of the Ginzburg–Landau model is reconsidered based on the Feynman diagram technique. We derive renormalization group (RG) flow equations, exactly calculating all vertices appearing in the perturbative renormalization of the ' 4 model up to the 3 order of the -expansion. In this case, the ' 2 , ' 4 , ' 6 , and ' 8 vertices appear. All these are relevant in the sense that they do not vanish at the fixed point. We have tested the expected basic properties of the RG flow, such as the semigroup property. Under repeated RG transformation Rs, appropriately represented RG flow on the critical surface converges to certain s–independent fixed point. The Fourier–transformed two–point correlation function G(k) has been considered. Although the -expansion of X(k) = 1=G(k) is well defined on the critical surface, we have revealed a fundamental inconsistency of the perturbative method with the exact rescaling of X(k), represented as an expansion in powers of k at k ! 0. We discuss also certain paradox, related to the -expansion of X(k) above the critical point. Apart from the –expansion, we have tested also a modified approach, where the ' 4 coupling constant u is the expansion parameter at a fixed spatial dimensionality d. Our tests point to some internal inconsistency of such a method.
The perturbative renormalization of the Ginzburg-Landau model is reconsidered based on the Feynman diagram technique. We derive renormalization group (RG) flow equations, exactly calculating all vertices appearing in the perturbative renormalization of the phi^4 model up to the epsilon^3 order of the epsilon-expansion. In this case, the phi^2, phi^4, phi^6, and phi^8 vertices appear. All these vertices are relevant. We have tested the expected basic properties of the RG flow, such as the semigroup property. Under repeated RG transformation R_s, appropriately represented RG flow on the critical surface converges to certain s-independent fixed point. The Fourier-transformed two-point correlation function G(k) has been considered. Although the epsilon-expansion of X(k)=1/G(k) is well defined on the critical surface, we have revealed an inconsistency of the perturbative method with the exact rescaling of X(k), represented as an expansion in powers of k at k --> 0. We have discussed also some aspects of the perturbative renormalization of the two-point correlation function slightly above the critical point. Apart from the epsilon-expansion, we have tested and briefly discussed also a modified approach, where the phi^4 coupling constant u is the expansion parameter at a fixed spatial dimensionality d.
The perturbative renormalization group (RG) treatment of the Ginzburg–Landau model is reconsidered based on the Feynman diagram technique. We derive RG flow equations, exactly calculating all vertices appearing in the perturbative RG transformation of the φ4 model up to the ε3 order of the ε-expansion. The Fourier-transformed two-point correlation function G(k) has been considered. Although the ε-expansion of X(k) = 1/G(k) is well defined on the critical surface, we have revealed an inconsistency with the exact rescaling of X(k), represented as an expansion in powers of k at k →0. This new result can serve as a basis to challenge the correctness of the ε-expansion-based perturbative RG method.
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.
A renormalization-group scheme is developed for the three-dimensional O(2N)O(2N)O(2N)-symmetric Ginzburg-Landau-Wilson model, which is consistent with the use of a 1/N1/N1/N expansion as a systematic method of approximation. It is motivated by an application to the critical properties of superconductors, reported in a separate paper. Within this scheme, the infrared stable fixed point controlling critical behavior appears at z=0,z=0,z=0, where $z={\ensuremath{\lambda}}^{\ensuremath{-}1}$ is the inverse of the quartic coupling constant, and an efficient renormalization procedure consists of the minimal subtraction of ultraviolet divergences at z=0.z=0.z=0. This scheme is implemented at next-to-leading order, and the standard results for critical exponents calculated by other means are recovered. An apparently novel result of this nonperturbative method of approximation is that corrections to scaling (or confluent singularities) do not, as in perturbative analyses, appear as simple power series in the variable ${y=zt}^{\ensuremath{\omega}\ensuremath{\nu}}.$ At least in three dimensions, the power series are modified by powers of lny.\mathrm{ln}y.lny.
We include spontaneous symmetry breaking in the functional renormalization group equations for the irreducible vertices of Ginzburg–Landau theories by augmenting these equations by a flow equation for the order parameter, which is determined from the requirement that at each renormalization group (RG) step the vertex with one external leg vanishes identically. Using this strategy, we propose a simple truncation of the coupled RG flow equations for the vertices in the broken symmetry phase of the Ising universality class in D dimensions. Our truncation yields the full momentum dependence of the self-energy Σ( k ) and interpolates between lowest-order perturbation theory at large momenta k and the critical scaling regime for small k . Close to the critical point, our method yields the self-energy in the scaling form Σ( k ) = k c 2 σ − (| k |ξ,| k |/ k c ), where ξ is the order parameter correlation length, k c is the Ginzburg scale, and σ − ( x , y ) is a dimensionless two-parameter scaling function for the broken symmetry phase which we calculate explicitly within our truncation.