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 $\mathbf{\varphi }$ 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 $\mathbf{\varphi }$ 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, ${\sum }_{\alpha =1}^m({\partial }_{\alpha }^2\mathbf{\varphi }{)}^2$, are investigated in an $\epsilon$ expansion about the upper critical dimension $d^{\ast }(m)=4+m/2$. 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 ${\mathbb{R}}^m$.

arXiv commentRevtex4, 11 pages, to appear in PRB; v2: some additional references and minor changes

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