English

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

Statistical Mechanics 2007-05-23 v2 Soft Condensed Matter

Abstract

The critical behavior of semi-infinite dd-dimensional systems with nn-component order parameter ϕ\bm{\phi} and short-range interactions is investigated at an mm-axial bulk Lifshitz point whose wave-vector instability is isotropic in an mm-dimensional subspace of Rd\mathbb{R}^d. The associated mm modulation axes are presumed to be parallel to the surface, where 0md10\le m\le d-1. An appropriate semi-infinite ϕ4|\bm{\phi}|^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\propto \bm{\phi}^2 of the Hamiltonian must be supplemented by one of the form λ˚α=1m(ϕ/xα)2\mathring{\lambda} \sum_{\alpha=1}^m(\partial\bm{\phi}/\partial x_\alpha)^2 involving a dimensionless (renormalized) coupling constant λ\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(m)=4+m/2d^*(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\lambda^* if ϵd(m)d>0\epsilon\equiv d^*(m)-d>0. The surface critical exponents of the ordinary transition are determined to second order in ϵ\epsilon. Extrapolations of these ϵ\epsilon expansions yield values of these exponents for d=3d=3 in good agreement with recent Monte Carlo results for the case of a uniaxial (m=1m=1) Lifshitz point. The scaling dimension of the surface energy density is shown to be given exactly by d+m(θ1)d+m (\theta-1), where θ=νl4/νl2\theta=\nu_{l4}/\nu_{l2} is the anisotropy exponent.

Keywords

Cite

@article{arxiv.cond-mat/0308483,
  title  = {Boundary critical behavior at m-axial Lifshitz points for a boundary plane parallel to the modulation axes},
  author = {H. W. Diehl and A. Gerwinski and S. Rutkevich},
  journal= {arXiv preprint arXiv:cond-mat/0308483},
  year   = {2007}
}

Comments

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