A continuum of pure states in the Ising model on a halfplane

TL;DR

This paper characterizes the pure Gibbs states of the 2D Ising model on a half-plane with boundary conditions, revealing a continuum of states parametrized by interface angles, extending to the critical temperature.

Contribution

It provides a complete classification of pure states in the half-plane Ising model with Dobrushin boundary conditions at low temperatures, including new results at the critical temperature.

Findings

Exactly one pure state per interface angle at low temperature

Continuum of pure states parametrized by interface angles

Existence of at least one pure state for each angle at critical temperature

Abstract

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 $\theta\in\lbrack-\pi/2,+\pi/2]$; 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 $\theta$.