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 ${ \mathbb R}^m$.
Categories: cond-mat.stat-mech hep-th