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

Diehl, H. W.; Rutkevich, S.; Gerwinski, A.

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

Authors

Diehl, H. W.

Rutkevich, S.

Gerwinski, A.

Description

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 $λ$) must be included in addition to the familiar ones $\proptoϕ^2$. Beyond one-loop order the infrared-stable fixed points describing the ordinary, special and extraordinary transitions in $d=4+\frac{m}{2}-ε$ dimensions (with $ε>0$) are located at $λ=λ^*=\Or(ε)$. At second order in $ε$, the surface critical exponents of both the ordinary and the special transitions start to deviate from their $m=0$ analogues. Results to order $ε^2$ are presented for the surface critical exponent $β_1^{\rm ord}$ of the ordinary transition. The scaling dimension of the surface energy density is shown to be given exactly by $d+m (θ-1)$, where $θ=ν_{l4}/ν_{l2}$ is the bulk anisotropy exponent.
Latex file using iop stylefiles, 2 eps files as gaphics included; 6pages in print; version 2: only cosmetic changes, to appear in J. Phys. A L