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 \(\bm{\phi}\) 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 \(\mathbb{R}^d\). The associated \(m\) modulation axes are presumed to be parallel to the surface, where \(0\le m\le d-1\). An appropriate semi-infinite \(|\bm{\phi}|^4\) model representing the corresponding universality classes of surface critical behavior is introduced. It is shown that the usual O(n) symmetric boundary term \(\propto \bm{\phi}^2\) of the Hamiltonian must be supplemented by one of the form \(\mathring{\lambda} \sum_{\alpha=1}^m(\partial\bm{\phi}/\partial x_\alpha)^2\) 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 \(d^*(m)=4+{m}/{2}\) is clarified. Fixed points describing the ordinary, special, and extraordinary transitions are identified and shown to be located at a nontrivial value \(\lambda^*\) if \(\epsilon\equiv d^*(m)-d>0\). 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 \(d=3\) 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 \(d+m (\theta-1)\), where \(\theta=\nu_{l4}/\nu_{l2}\) is the anisotropy exponent.