On antiferromagnetic regimes in the Ashkin-Teller model

The Ashkin-Teller model can be represented by a pair $(\tau,\tau')$ of Ising spin configurations with coupling constants $J$ and $J'$ for each, and $U$ for their product. We study this representation on the integer lattice $\mathbb{Z}^d$ for $d\geq 2$. We confirm the presence of a partial antiferromagnetic phase in the isotropic case ($J=J'$) when $-U>0$ is sufficiently large and $J=J'>0$ is sufficiently small, by means of a graphical representation. In this phase, $\tau$ is disordered, admitting exponential decay of correlations, while the product $\tau\tau'$ is antiferromagnetically ordered, which is to say that correlations are bounded away from zero but alternate in sign. No correlation inequalities are available in this part of the phase diagram. In the planar case $d=2$, we construct a coupling with the six-vertex model and show, in analogy to the first result, that the corresponding height function is localised, although with antiferromagnetically ordered heights on one class of vertices of the graph. We then return to $d\geq 2$ and consider a part of the phase diagram where $U<0$ but where correlation inequalities still apply. Using the OSSS inequality, we proceed to establish a subcritical sharpness statement along suitable curves covering this part, circumventing the difficulty of the lack of general monotonicity properties in the parameters. We then address the isotropic case and provide indications of monotonicity.