Relevance of space anisotropy in the critical behavior of m-axial Lifshitz points

TL;DR

This paper investigates how space anisotropy influences the critical behavior at m-axial Lifshitz points using renormalization group methods, revealing that certain anisotropic terms are relevant perturbations.

Contribution

It introduces a framework for analyzing anisotropic effects at Lifshitz points and computes the crossover exponent for cubic anisotropy to second order in epsilon expansion.

Findings

Crossover exponent for cubic anisotropy is positive.

Anisotropic terms are relevant perturbations.

Framework for general operators at Lifshitz points.

Abstract

The critical behavior of $d$-dimensional systems with $n$-component order parameter $\bm{\phi}$ is studied at an $m$-axial Lifshitz point where a wave-vector instability occurs in an $m$-dimensional subspace ${\mathbb R}^m$ ($m{>}1$). Field theoretic renormalization group techniques are exploited to examine the effects of terms in the Hamiltonian that break the rotational symmetry of the Euclidean group ${\mathbb E}(m)$. The framework for considering general operators of second order in $\bm{\phi}$ and fourth order in the derivatives $\partial_\alpha $ with respect to the Cartesian coordinates $x_\alpha $ of ${\mathbb R}^m$ is presented. For the specific case of systems with cubic anisotropy, the effects of having an additional term, $\sum_{\alpha=1}^m(\partial_\alpha^2\bm{\phi})^2$, are investigated in an $\epsilon$ expansion about the upper critical dimension $d^{*}(m)=4+m/2$. Its associated crossover exponent is computed to order $\epsilon^2$ and found to be positive, so that it is a \emph{relevant} perturbation on a model isotropic in ${\mathbb R}^m$.