A Continuum of Pure States in the Ising Model on a Halfplane

We study the homogeneous nearest–neighbor Ising ferromagnet on the right half plane with a Dobrushin type boundary condition—say plus on the top part of the boundary and minus on the bottom. For sufficiently low temperature T, we completely characterize the pure (i.e., extremal) Gibbs states, as follows. There is exactly one for each angle θ∈ [- π/ 2 , + π/ 2] ; here θ specifies the asymptotic angle of the interface separating regions where the spin configuration looks like that of the plus (respectively, minus) full-plane state. Some of these conclusions are extended all the way to T= T_c by developing new Ising exact solution results—in particular, there is at least one pure state for each θ.