Boundary critical behaviour at $m$-axial Lifshitz points: the special transition for the case of a surface plane parallel to the modulation axes

Abstract

The critical behaviour of $d$-dimensional semi-infinite systems with $n$-component order parameter $\bmϕ$ 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 $η_\|^{\rm sp}$, the surface crossover exponent $Φ$ and related ones are determined to first order in $ε=4+\case{m}{2}-d$. Unlike the bulk critical exponents and the surface critical exponents of the ordinary transition, $Φ$ is $m$-dependent already at first order in $ε$. The $\Or(ε)$ term of $η_\|^{\rm sp}$ is found to vanish, which implies that the difference of $β_1^{\rm sp}$ and the bulk exponent $β$ is of order $ε^2$.