Boundary critical behaviour at m-axial Lifshitz points of semi-infinite systems with a surface plane perpendicular to a modulation axis

Semi-infinite \(d\)-dimensional systems with an \(m\)-axial bulk Lifshitz point are considered whose (\(d-1\))-dimensional surface hyper-plane is oriented perpendicular to one of the \(m\) modulation axes. An \(n\)-component \(\phi^4\) field theory describing the bulk and boundary critical behaviour when (i) the Hamiltonian can be taken to have O(n) symmetry and (ii) spatial anisotropies breaking its Euclidean symmetry in the \(m\)-dimensional coordinate subspace of potential modulation directions may be ignored is investigated. The long-distance behaviour at the ordinary surface transition is mapped onto a field theory with the boundary conditions that both the order parameter \(\bm{\phi}\) and its normal derivative \(\partial_n\bm{\phi}\) vanish at the surface plane. The boundary-operator expansion is utilized to study the short-distance behaviour of \(\bm{\phi}\) near the surface. Its leading contribution is found to be controlled by the boundary operator \(\partial_n^2\bm{\phi}\). The field theory is renormalized for dimensions \(d\) below the upper critical dimension \(d^*(m)=4+m/2\), with a corresponding surface source term \(\propto \partial_n^2\bm{\phi}\) added. The anomalous dimension of this boundary operator is computed to first order in \(\epsilon=d^*-d\). The result is used in conjunction with scaling laws to estimate the value of the single independent surface critical exponent \(\beta_{\mathrm{L}1}^{(\mathrm{ord},\perp)}\) for \(d=3\). Our estimate for the case \(m=n=1\) of a uniaxial Lifshitz point in Ising systems is in reasonable agreement with published Monte Carlo results.