Surface critical behaviour at m-axial Lifshitz points: continuum models, boundary conditions and two-loop renormalization group results

The critical behaviour of semi-infinite \(d\)-dimensional systems with short-range interactions and an O(n) invariant Hamiltonian is investigated at an \(m\)-axial Lifshitz point with an isotropic wave-vector instability in an \(m\)-dimensional subspace of \(\mathbb{R}^d\) parallel to the surface. Continuum \(|\bphi|^4\) models representing the associated universality classes of surface critical behaviour are constructed. In the boundary parts of their Hamiltonians quadratic derivative terms (involving a dimensionless coupling constant \(\lambda\)) must be included in addition to the familiar ones \(\propto\phi^2\). Beyond one-loop order the infrared-stable fixed points describing the ordinary, special and extraordinary transitions in \(d=4+\frac{m}{2}-\epsilon\) dimensions (with \(\epsilon>0\)) are located at \(\lambda=\lambda^*=\Or(\epsilon)\). At second order in \(\epsilon\), the surface critical exponents of both the ordinary and the special transitions start to deviate from their \(m=0\) analogues. Results to order \(\epsilon^2\) are presented for the surface critical exponent \(\beta_1^{\rm ord}\) of the ordinary transition. 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 bulk anisotropy exponent.