Perturbative renormalization of the Ginzburg-Landau model revisited

TL;DR

This paper revisits the perturbative renormalization of the Ginzburg-Landau model using Feynman diagrams, deriving RG flow equations up to epsilon^3 order, and discusses the consistency and limitations of the epsilon-expansion method.

Contribution

It provides an exact calculation of vertices up to epsilon^3 order and analyzes the RG flow and correlation functions, highlighting inconsistencies in the perturbative approach.

Findings

RG flow converges to a fixed point on the critical surface

Inconsistency found in the epsilon-expansion of the correlation function

Modified approach with fixed dimensionality discussed

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 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.