Physica A 177 (1991) 495-504 North-Holland Sum rules for interface Hamiltonians Lev V. Mikheev and John D. Weeks Institute for Physical,Science and Technology, University of Maryland, College Park, MD 20742, USA We derive exact identities for a general class of interface Hamiltonians describing wetting tran- sitions that relate force fluctuations to interface fluctuations and the surface tension. These sum rules are analogous to the Triezenbeg, Zwanzig and Wertheim formulas for the surface tension derived using the theory of inhomogeneous fluids. Relations between these two approaches are discussed. Our sum rules are consistent with the scaling theory of wetting transitions. 1. Introduction There are two basic strategies used to study properties of an inhomogeneous d- dimensional system containing interfaces and boundaries or walls. The most straight- forward approach in principle is to use the modern theory of inhomogeneous fluids ~1 [ 2 ] to determine thermodynamic properties and density correlation functions for the full d-dimensional system as a function of an external field ¢ and the interparticle interactions. In practice, approximations are often made whose errors are difficult to assess. Thus it is of particular interest to study certain exact sum rules: equilibrium identities relating a macroscopic thermodynamic property such as the surface tension to integrals over correlation functions. Relations between exponents describing inter- face phase transitions can often be derived by combining a scaling ansatz for the cor- relation functions along with an appropriate sum rule. Henderson [2,3] has shown the utility of such an approach for a variety of problems, including in particular wet- ting transitions [ 3 ], which will be our main focus in this article. A second, more phenomenological but often conceptually simpler, approach is to study the properties of a d' =- d- 1 construct, the interface Hamiltonian #2 [ 5--8 ]. At wavelengths long compared to the bulk correlation length, the important remaining degrees of freedom can often be described in terms of interfaces, their effective inter- actions and their fluctuations, as described by the appropriate interface Hamiltonian. This viewpoint is very general, and forms the basis for renormalization group studies of interfacial phenomena [4,6-8]. However, the relationship between the effective ~ For a general review, see ref. [ 1]. ~z For a general review, see ref. [4]. 0378-4371/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)