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

Abstract

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 d parallel to the surface. Continuum |ϕ|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 λ) must be included in addition to the familiar ones ∝ϕ2. Beyond one-loop order the infrared-stable fixed points describing the ordinary, special and extraordinary transitions in d = 4 + m/2 − dimensions (with > 0) are located at λ = λ* = O(). At second order in , the surface critical exponents of both the ordinary and the special transitions start to deviate from their m = 0 analogues. Results to order 2 are presented for the surface critical exponent βord1 of the ordinary transition. 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 bulk anisotropy exponent.