Phase transition in the antiferromagnetic Ising model with external field

Consider the nearest-neighbor antiferromagnetic Ising model on $\mathbb{Z}^d$, $d\ge2$, at inverse temperature $β$ and in a uniform external field $h$. Let $β_c(d)$ be the critical inverse temperature of the zero-field Ising model. We prove that for every $β>β_c(d)$, the Gibbs state is not unique wh