The critical behavior of semi-infinite d-dimensional systems with n-component order parameter φ 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 R d . The associated m modulation axes are presumed to be parallel to the surface, where 0 ≤ m ≤ d − 1. An appropriate semi-infinite |φ| 4 model representing the corresponding universality classes of surface critical behavior is introduced. It is shown that the usual O(n) symmetric boundary term ∝ φ 2 of the Hamiltonian must be supplemented by one of the formλ m α=1 (∂φ/∂xα) 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 ǫ ≡ d * (m) − d > 0. The surface critical exponents of the ordinary transition are determined to second order in ǫ. Extrapolations of these ǫ 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.

The critical behaviour of semi-infinite d-dimensional systems with short-range interactions and an O(n) invariant Hamiltonian is investigated at an m-axial Lifshitz point with an isotropic wave-vector instability in an m-dimensional subspace of R d parallel to the surface. Continuum |φ| 4 models representing the associated universality classes of surface critical behaviour are constructed. In the boundary parts of their Hamiltonians quadratic derivative terms (involving a dimensionless coupling constant λ) must be included in addition to the familiar ones ∝ φ 2 . Beyond one-loop order the infrared-stable fixed points describing the ordinary, special and extraordinary transitions in d = 4 + m 2 − ǫ dimensions (with ǫ > 0) are located at λ = λ * = O(ǫ). At second order in ǫ, the surface critical exponents of both the ordinary and the special transitions start to deviate from their m = 0 analogues. Results to order ǫ 2 are presented for the surface critical exponent β ord 1 of the ordinary transition. 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 bulk anisotropy exponent.

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