Boundary critical behavior at m-axial Lifshitz points for a boundary plane parallel to the modulation axes

Abstract

The critical behavior of semi-infinite $d$-dimensional systems with $n$-component order parameter $\bmϕ$ and short-range interactions is investigated at an $m$-axial bulk Lifshitz point whose wave-vector instability is isotropic in an $m$-dimensional subspace of $\mathbb{R}^d$. The associated $m$ modulation axes are presumed to be parallel to the surface, where $0\le m\le d-1$. An appropriate semi-infinite $|\bmϕ|^4$ model representing the corresponding universality classes of surface critical behavior is introduced. It is shown that the usual O(n) symmetric boundary term $\propto \bmϕ^2$ of the Hamiltonian must be supplemented by one of the form $\mathringλ \sum_{α=1}^m(\partial\bmϕ/\partial x_α)^2$ involving a dimensionless (renormalized) coupling constant $λ$. The implied boundary conditions are given, and the general form of the field-theoretic renormalization of the model below the upper critical dimension $d^*(m)=4+{m}/{2}$ is clarified. Fixed points describing the ordinary, special, and extraordinary transitions are identified and shown to be located at a nontrivial value $λ^*$ if $ε\equiv d^*(m)-d>0$. The surface critical exponents of the ordinary transition are determined to second order in $ε$. Extrapolations of these $ε$ expansions yield values of these exponents for $d=3$ in good agreement with recent Monte Carlo results for the case of a uniaxial ($m=1$) Lifshitz point. The scaling dimension of the surface energy density is shown to be given exactly by $d+m (θ-1)$, where $θ=ν_{l4}/ν_{l2}$ is the anisotropy exponent.