Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Relevance of space anisotropy in the critical behavior of m-axial
  Lifshitz points   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
${\mathbb R}^m$.

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The categories of this paper are: astro-ph
19
The categories of this paper are: eess
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The categories of this paper are: quant-ph
31
The categories of this paper are: ao-sci
37
The categories of this paper are: supr-con
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The categories of this paper are: q-fin
36
The categories of this paper are: cmp-lg
2
The categories of this paper are: math
10
The categories of this paper are: q-bio
32
The categories of this paper are: atom-ph
30
The categories of this paper are: q-alg
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The categories of this paper are: acc-phys
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The categories of this paper are: plasm-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: mtrl-th
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The categories of this paper are: adap-org