Interface, Surface Tension and Reentrant Pinning Transition in the 2D Ising Model

Abstract:

We develop a new way to look at the high-temperature representation of the Ising model up to the critical temperature and obtain a number of interesting consequences. In the two-dimensional case, it is possible to use these tools to prove results on phase-separation lines in the whole phase-coexistence regime, by way of a duality transformation. We illustrate the power of these techniques by studying an Ising model with a boundary magnetic field, in which a reentrant pinning transition takes place; more precisely we show that the typical configurations of the model can be described, at the macroscopic level, by interfaces which are solutions of the corresponding thermodynamic variational problem; this variational problem is solved explicitly. There exist values of the boundary magnetic field and temperatures 0<T ₁<T ₂<T _c such that the interface is not pinned for T<T ₁ or T>T ₂, but is pinned for T ₁<T<T ₂; we can also find values of the boundary magnetic field and temperatures 0<T ₁<T ₂<T ₃<T _c such that for T<T ₁ or T ₂<T<T ₃ the interface is pinned, while for T ₁<T<T ₂ or T>T ₃ it is not pinned. An important property of the surface tension which is used in this paper is the sharp triangle inequality about which we report some new results. The techniques used in this work are robust and can be used in a variety of different situations.