Comment on ``Renormalization-group picture of the Lifshitz critical behavior''

TL;DR

This paper critiques a recent renormalization-group analysis of Lifshitz critical behavior, highlighting fundamental theoretical flaws and incorrect critical exponent calculations, thus questioning its validity.

Contribution

It identifies severe deficiencies in Leite's RG approach to Lifshitz criticality, emphasizing the need for a more consistent theoretical framework.

Findings

Leite's approach is not ultraviolet finite.

It misses the nontrivial anisotropy exponent .

Results for critical exponents are incorrect.

Abstract

We show that the recent renormalization-group analysis of Lifshitz critical behavior presented by Leite [Phys. Rev. B {\bf 67}, 104415 (2003)] suffers from a number of severe deficiencies. In particular, we show that his approach does not give an ultraviolet finite renormalized theory, is plagued by inconsistencies, misses the existence of a nontrivial anisotropy exponent $\theta\ne 1/2$, and therefore yields incorrect hyperscaling relations. His $\epsilon$-expansion results to order $\epsilon^2$ for the critical exponents of $m$-axial Lifshitz points are incorrect both in the anisotropic ($0<m<d$) and the isotropic cases ($m=d$). The inherent inconsistencies and the lack of a sound basis of the approach makes its results unacceptable even if they are interpreted in the sense of approximations.