About the relevance of the fixed dimension perturbative approach to frustrated magnets in two and three dimensions

TL;DR

This paper critically examines the reliability of perturbative renormalization group methods for describing the critical behavior of frustrated magnets in two and three dimensions, highlighting significant limitations and potential spurious results.

Contribution

It demonstrates that existing high-loop perturbative results are unreliable for frustrated magnets, questioning the validity of fixed points and critical exponents derived from these methods.

Findings

Critical exponents in 3D show strong dependence on resummation parameters.

In 2D, apparent convergence leads to incorrect critical exponent values.

Fixed points identified in perturbative series are likely spurious.

Abstract

We show that the critical behaviour of two- and three-dimensional frustrated magnets cannot reliably be described from the known five- and six-loops perturbative renormalization group results. Our conclusions are based on a careful re-analysis of the resummed perturbative series obtained within the zero momentum massive scheme. In three dimensions, the critical exponents for XY and Heisenberg spins display strong dependences on the parameters of the resummation procedure and on the loop order. This behaviour strongly suggests that the fixed points found are in fact spurious. In two dimensions, we find, as in the O(N) case, that there is apparent convergence of the critical exponents but towards erroneous values. As a consequence, the interesting question of the description of the crossover/transition induced by Z2 topological defects in two-dimensional frustrated Heisenberg spins remains open.