Phase diagram of the Ashkin-Teller model

The Ashkin-Teller model is a pair of interacting Ising models and has two parameters: $J$ is a coupling constant in the Ising models and $U$ describes the strength of the interaction between them. In the ferromagnetic case $J,U>0$ on the square lattice, we establish a complete phase diagram conjectured in physics in 1970s (by Kadanoff and Wegner, Wu and Lin, Baxter and others): when $J<U$, the transitions for the Ising spins and their products occur at two distinct curves that are dual to each other; when $J\geq U$, both transitions occur at the self-dual curve. All transitions are shown to be sharp using the OSSS inequality. We use a finite-criterion argument and continuity to extend the result of Peled and the third author \cite{GlaPel19} from a self-dual point to its neighborhood. Our proofs go through the random-cluster representation of the Ashkin-Teller model introduced by Chayes-Machta and Pfister-Velenik and we rely on couplings to FK-percolation.