Interface Hamiltonians and bulk critical behavior

We examine the derivation and use of a short-ranged (d-1)-dimensional interface Hamiltonian to describe properties of a d-dimensional liquid-vapor or Ising system near the critical point. We argue that such a simplified description, which ignores bulk excitations (bubbles of the opposite phase) and multiple-valued interface configurations (overhangs) is valid only on length scales larger than the bulk correlation length B. Such excitations with wavelengths up to order B are essential for a correct description of the critical fluctuations, and preclude the use of an interface Hamiltonian to study bulk critical properties. This is explicitly demonstrated in d=2 by showing that bubbles and overhangs are relevant operators and we argue that this is true in any dimension. (However, these contributions do not necessarily affect the formal perturbation expansion about the degenerate case d=1, as carried out by Wallace and Zia.) This viewpoint is implicit in the physical picture Widom used to derive scaling laws relating interface and bulk critical properties. The long-wavelength fluctuations accurately described by an interface Hamiltonian produce a wandering of the interface, but this plays no important role in the critical behavior and can be reconciled with Widoms picture. We examine several modifications of the usual Ising model for which in certain limits a single-valued description becomes exact. Such models either exhibit no bulk critical behavior at all, even if the surface tension vanishes, or have critical properties in a different universality class from the usual Ising-model (liquid-vapor) critical point.