2003/08/31 by H. W. Diehl, A. Gerwinski, S. Rutkevich
Physics and Astronomy · #cond-mat.stat-mech #cond-mat.soft
paper · pdf · doi:10.1103/physrevb.68.224428
published as Phys. Rev. B 68, 224428 (2003) (30 pages) · revtex4, 31 pages with eps-files for figures, uses texdraw to generate some graphs; to appear in PRB; v2: some references and additional remarks added, labeling in figure 1 and some typos corrected
arxiv created 2003/10/27 · arxiv updated 2009/12/01
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 ℝd. The associated m modulation axes are presumed to be parallel to the surface, where 0≤ m≤ 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 ∝ \bmϕ2 of the Hamiltonian must be supplemented by one of the form \mathringλ ∑α=1m(∂\bmϕ/∂ 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.