The critical behaviour of $d$-dimensional semi-infinite systems with $n$-component order parameter $\bm{ \phi}$ is studied at an $m$-axial bulk Lifshitz point whose wave-vector instability is isotropic in an $m$-dimensional subspace of $\mathbb{ R}^d$. Field-theoretic renormalization group methods are utilised to examine the special surface transition in the case where the $m$ potential modulation axes, with $0\leq m\leq d-1$, are parallel to the surface. The resulting scaling laws for the surface critical indices are given. The surface critical exponent $\eta_\|^{ \rm sp}$, the surface crossover exponent $\Phi$ and related ones are determined to first order in $\epsilon=4+\case{ m}{ 2}-d$. Unlike the bulk critical exponents and the surface critical exponents of the ordinary transition, $\Phi$ is $m$-dependent already at first order in $\epsilon$. The $\Or(\epsilon)$ term of $\eta_\|^{ \rm sp}$ is found to vanish, which implies that the difference of $\beta_1^{ \rm sp}$ and the bulk exponent $\beta$ is of order $\epsilon^2$.
Categories: cond-mat.stat-mech cond-mat.soft hep-th