Effective interface Hamiltonians for short-range critical wetting

The derivation of effective interface Hamiltonians on the basis of an underlying noncritical bulk order-parameter theory is critically examined for use in studying critical wetting transitions in (d=3)-dimensional systems with short-range forces. A crossing constraint on the interfacial profile is used to define the fluctuating interface location l(y) and exact general expressions are obtained for the effective wall-interface potential W[l(y)] and for the wall-modified interfacial stiffness {Σ} ̃ \~{}{}[l(y)] in terms of a constrained planar order-parameter profile. Previous discussions in the literature are shown to be inadequate. Explicit formulas for W and {Σ} ̃ \~{}{} are obtained when the bulk thermodynamic potential for the wetting-layer phase is purely parabolic. Novel terms varying as le^-jκl (j=2,3, . . .) appear in the decay of {Σ} ̃ \~{}{}(l) to the free-interface limit {Σ}̃ \~_∞; here, 1/{κ}==ξ_β is the true correlation length of the bulk wetting-layer phase. General nonparabolic bulk potentials are analyzed perturbatively, leading to terms in W(l) decaying as wⱼₖ{l}ᵏe^-jκl for 0{≤}k{≤}j=1,2, . . .. An alternative, generalized adsorption definition for l(y) can be solved exactly for a cphi⁴ bulk potential and yields closely similar results for W(l). On approach to critical wetting at T=TcW the important coefficients wⱼₖ for k{≥}1 vanish rapidly with {}T-TcW{}; hence previous renormalization-group (RG) treatments of critical wetting remain essentially unchanged. However, these treatments neglect the variation of {Σ} ̃ \~{}{}(l) with l which, under RG flow, is seen here to destabilize wetting criticality; further analyses reported elsewhere, show that first-order transitions then arise in many cases.