Phase transition in the antiferromagnetic Ising model with external field

Consider the nearest-neighbor antiferromagnetic Ising model on ${\mathbb{Z}}^d$, $d\ge 2$, at inverse temperature $\beta$ and in a uniform external field $h$. Let ${\beta }_c(d)$ be the critical inverse temperature of the zero-field Ising model. We prove that for every $\beta >{\beta }_c(d)$, the Gibbs state is not unique whenever $\mathrm{\mid }h\mathrm{\mid }$ is sufficiently small. This extends a previous result of Dobrushin (1968) from sufficiently large $\beta$ to every $\beta >{\beta }_c(d)$.