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

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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 Rᵈ. 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 ∝ ϕ² of the Hamiltonian must be supplemented by one of the form λ ∑α₌₁ᵐ(∂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 ε≡ 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=νₗ₄/νₗ₂ is the anisotropy exponent.