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

The critical behavior of semi-infinite d-dimensional systems with n-component order parameter $\mathit{\phi }$ 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 $|\mathit{\phi }{|}^4$ model representing the corresponding universality classes of surface critical behavior is introduced. It is shown that the usual $O\left(n\right)\mathrm{}$ symmetric boundary term $\propto {\mathit{\phi }}^2$ of the Hamiltonian must be supplemented by one of the form $\mathring{\lambda }{\sum }_{\alpha =1\mathrm{}}^m(\partial \mathit{\phi }/\partial x_{\alpha }{)}^2$ involving a dimensionless (renormalized) coupling constant $\lambda .$ The implied boundary conditions are given, and the general form of the field-theoretic renormalization of the model below the upper critical dimension $d^{*}\mathrm{}\left(m\right)=4+m/2\mathrm{}$ is clarified. Fixed points describing the ordinary, special, and extraordinary transitions are identified and shown to be located at a nontrivial value ${\lambda }^{*}$ if $\epsilon \equiv d^{*}\mathrm{}\left(m\right)\mathrm{}-d>\mathrm{}0.\mathrm{}$ The surface critical exponents of the ordinary transition are determined to second order in $\epsilon .$ Extrapolations of these $\epsilon$ expansions yield values of these exponents for $d=3\mathrm{}$ in good agreement with recent Monte Carlo results for the case of a uniaxial $\mathrm{}(m=1\mathrm{})\mathrm{}$ Lifshitz point. The scaling dimension of the surface energy density is shown to be given exactly by $d+m\left(\theta -1\right),$ where $\theta ={\nu }_{l4\mathrm{}}/{\nu }_{l2\mathrm{}}$ is the anisotropy exponent.