Analysis of the 3d massive renormalization group perturbative expansions: a delicate case

TL;DR

This paper critically examines the effectiveness of perturbative renormalization group methods at fixed dimension d=3 for various models, revealing issues with spurious fixed points and implications for phase transition nature in frustrated magnets.

Contribution

It provides a detailed analysis of resummation procedures in fixed-dimension RG, identifying spurious fixed points and clarifying the nature of phase transitions in frustrated magnetic systems.

Findings

Spurious fixed points persist after resummation in the O(N) model.

The stable fixed point in frustrated models is spurious for N<Nc.

Transitions in XY and Heisenberg frustrated magnets are of first order.

Abstract

The effectiveness of the perturbative renormalization group approach at fixed space dimension d in the theory of critical phenomena is analyzed. Three models are considered: the O(N) model, the cubic model and the antiferromagnetic model defined on the stacked triangular lattice. We consider all models at fixed d=3 and analyze the resummation procedures currently used to compute the critical exponents. We first show that, for the O(N) model, the resummation does not eliminate all non-physical (spurious) fixed points (FPs). Then the dependence of spurious as well as of the Wilson-Fisher FPs on the resummation parameters is carefully studied. The critical exponents at the Wilson-Fisher FP show a weak dependence on the resummation parameters. On the contrary, the exponents at the spurious FP as well as its very existence are strongly dependent on these parameters. For the cubic model, a new stable FP is found and its properties depend also strongly on the resummation parameters. It appears to be spurious, as expected. As for the frustrated models, there are two cases depending on the value of the number of spin components. When N is greater than a critical value Nc, the stable FP shows common characteristic with the Wilson-Fisher FP. On the contrary, for N<Nc, the results obtained at the stable FP are similar to those obtained at the spurious FPs of the O(N) and cubic models. We conclude from this analysis that the stable FP found for N<Nc in frustrated models is spurious. Since Nc>3, we conclude that the transitions for XY and Heisenberg frustrated magnets are of first order.