Critical roughening of interfaces: A new class of renormalizable field theories

Abstract

A renormalizable field theory is developed for (multi)critical roughening of interacting interfaces in systems of dimension d < 3. There is an infinite hierarchy of universality classes that mirrors the series of multicritical points in Ising systems. The relevant operator algebra of these theories is built up by local scaling fields that are singular distributions of the basic field variable. Critical indices, e.g., the exponent a s of the specific heat, are obtained analytically in an e expansion. The extension of our results to d = 3 is discussed.PACS numbers: 64.60. Pr, 05.70.Jk, ll.10.Gh In recent years, much effort has been devoted to the study of low-dimensional manifolds such as domain boundaries, interfaces, polymers, or membranes. New experimental tools have been developed that make it possible to probe these objects in much detail. The structure of surfaces and interfaces can be studied on a microscopic scale using surface x-ray and neutron scattering, and even one-dimensional domain boundaries or surface steps can now be observed directly by atomic force microscopy.The statistical mechanics of these systems is governed by the interplay between intermolecular forces and fluctuations due to thermal excitation or quenched disorder. If these fluctuations are strong enough, a single manifold is in a scale-invariant rough state. Ising domain walls in two dimensions or in three dimensions above the roughening temperature are an example.In real systems, however, the position of a manifold may still be constrained by walls or defect planes, or by the presence of other manifolds, if they lead to an effective potential that is attractive over a microscopic range a and tends to zero at larger distances. For example, in the standard Ising model with a plane of weaker bonds, a domain wall is subject to an effective potential well; if the spin-spin interactions or the structure of the defect plane are more complex, the potential may have both attractive and repulsive parts within the range a. The discussion in the sequel includes such more general potentials.At low temperature or weak quenched disorder, the manifold is then localized to the position of lowest energy up to fluctuations of order a and is hence smooth on larger scales, while at high temperature or strong disorder, it is in a delocalized state with shape fluctuations on all scales up to the size of the system. The roughening, wetting, or unbinding transition separating these two regimes [1] may be of first or second order. In the latter case, the size of typical fluctuations diverges as the critical point is approached from below. Close to criticality, as they become large compared to a, the fluctuations wipe out microscopic details of the interactions, i.e., they renormalize the binding potential. Universal scaling behavior emerges, at least for sufficiently short-ranged potentials.So far, this type of transition has been studied by transfer matrix calculations for interfaces in twodimensional systems and by functional renormalization group methods ...

Critical behavior of interacting surfaces with tension

“…Local contacts have also been studied for 1-dimensional lines (or strings or directed walks) governed by line tension [27,28,29,30] and for tensionless membranes [30]. In the following sections, we will study these quantities for two and three interacting surfaces governed by tension.…”

Delocalization transitions of semiflexible manifolds

“…The perturbation series at u = 0, however, is factorizable according to (17) and one-loop renormalizable in exactly the same way as with the wall constraint. Hence, the (in D = 1) exact relations (15) and (16) still hold (with ǫ given by (23) and x Ω = 2d), resulting in ξ ∼ (T c − T ) 3/(2−4d) for 2/3 < d < 1 and x * Ω = 2 − 2d. This scaling dimension turning negative for d > 1 indicates that the transition becomes of first order; see the discussion and extensive numerics in [28].…”

“…1(b) but are confined to the region r i > 0. The most interesting application of (15) and (16) is the delocalization transition of a fluid membrane from a hard wall (D = 2, d = 1), where ξ ∼ (T c − T ) −1 (since the effective coupling is temperature-dependent) and x * Ω = 1. Not surprisingly, these one-loop results are in agreement with those from functional renormalization [23].…”

“…There exists a well known perturbative framework [8,9] to treat such interactions, which has been applied extensively to the problem of self-avoiding manifolds [10][11][12][13][14]. More complicated short-ranged [15] and long-ranged [16] interactions have been studied as well.…”

Dynamical selection of critical exponents

“…(iii) Another possibility is to consider short-ranged potentials V(u) δ(u) from the start. This leads to a renormalizable field theory, pioneered by David, Duplantier and Guitter [16,17], and further studied by several authors [18][19][20][21][22]. It is this approach that has been successful to tackle the renormalization of self-avoiding manifolds [23][24][25][26][27][28][29][30][31][32].…”