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 $\theta \in [-\pi \mathrm{/}2,+\pi \mathrm{/}2]$; here $\theta$ 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 result – in particular, there is at least one pure state for each $\theta$.