Surface critical behavior in the presence of linear or cubic weak surface fields

The effects of weak symmetry-breaking surface fields on the surface critical behavior of semi-infinite systems with a scalar order parameter are investigated near a bulk critical point. Previous field-theoretic analyses in 4-ε dimensions are generalized by incorporating an additional surface term ${\mathit{w}}_0$ ${\mathrm{\phi }}_{\mathit{s}}^3$ in the Hamiltonian. The mapping of microscopic models of the lattice-gas type on a continuum field theory is reconsidered, and we conclude that the cubic surface field ${\mathit{w}}_0$ generically does not vanish both in the case of an Ising ferromagnet in an applied magnetic field H (where ${\mathit{w}}_0$∼H as H→0) as well as in the case of critical adsorption of fluids onto a wall or an interface. Explicit results of a perturbative renormalization-group analysis are given, which show that a fixed point with nonvanishing cubic surface field ${\mathit{w}}^{\mathrm{*}}$=O(√ε ) exists but that at order ε the usual ${\mathit{w}}^{\mathrm{*}}$=0 fixed point is stable in the w direction. The associated correction-to-scaling exponent is ${\mathrm{\omega }}_{\mathit{w}}$=ε+O(${\mathrm{\epsilon }}^2$). It is shown that the ${\mathrm{\phi }}_{\mathit{s}}^3$ term leads to a mixing of surface enhancement and surface field in the leading odd relevant surface scaling field. Furthermore, it is proven beyond perturbation theory that a (redundant) surface operator exists, that corresponds to a similar mixing in the leading even surface scaling field.