Interfaces and wetting transition on the half plane. Exact results from field theory

Abstract

We consider the scaling limit of a generic ferromagnetic system with a continuous phase transition, on the half plane with boundary conditions leading to the equilibrium of two different phases below criticality. We use general properties of low energy two-dimensional field theory to determine exact asymptotics of the magnetization profile perperdicularly to the boundary, to show the presence of an interface with endpoints pinned to the boundary, and to determine its passage probability. The midpoint average distance of the interface from the boundary grows as the square root of the distance between the endpoints, unless the reflection amplitude of the bulk excitations on the boundary possesses a stable bound state pole. The contact angle of the phenomenological wetting theory is exactly related to the location of this pole. Results available from the lattice solution of the Ising model are recovered as a particular case.

Fields, particles and universality in two dimensions

“…[120][121][122]). In this and in the next subsection we show how field theory provides general and exact results for universal properties in the twodimensional case [123,124]. We start from the half-plane with the boundary located at x = 0 and consider the scaling limit below T c in which two or more phases coexist; as usual the kinks K ab provide the elementary excitations in the particle description.…”

“…On the other hand, considering first the case of the half-plane (α = 0), we saw in Section 8.1 that in some range of the boundary parameters the particle may form with the boundary a stable bound state with energy e B ′ specified by (123), i.e. smaller than the energy e B + m cosh θ of the unbound boundary-particle system.…”

“…18). When this state is present it dominates the spectral decomposition of (127) at large mR and produces an order parameter equal to ⟨σ ⟩ a for mx ≫ 1, no matter how large R is [123]. We know from (105) that m is the interfacial tension between the phases a and b; on the other hand, e B corresponds to the interfacial tension between the boundary and phase b, and e ′ B that between the excited boundary and phase a (Fig.…”

“…18). It follows that (123) amounts to the condition which in the phenomenological theory of boundary wetting (see e.g. [122]) determines the equilibrium of a drop of phase b between the boundary and phase a.…”

Long range correlations generated by phase separation. Exact results from field theory

“…This has allowed, in the first place, to determine the order parameter profiles 2 (one-point functions) and to derive from them the properties of the interfacial region, including the deviations from the simple curve picture [9,10]. The theory has also been extended to interfaces at boundaries [14][15][16] and to interface localization [17].…”