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

Abstract

The critical behavior of $d$-dimensional systems with $n$-component order parameter $\bmϕ$ 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 $\bmϕ$ and fourth order in the derivatives $\partial_α$ with respect to the Cartesian coordinates $x_α$ of ${\mathbb R}^m$ is presented. For the specific case of systems with cubic anisotropy, the effects of having an additional term, $\sum_{α=1}^m(\partial_α^2\bmϕ)^2$, are investigated in an $ε$ expansion about the upper critical dimension $d^{*}(m)=4+m/2$. Its associated crossover exponent is computed to order $ε^2$ and found to be positive, so that it is a \emph{relevant} perturbation on a model isotropic in ${\mathbb R}^m$.