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Phase coexistence of gradient Gibbs states

We consider the (scalar) gradient fields $η=(η_b)$--with $b$ denoting the nearest-neighbor edges in $\Z^2$--that are distributed according to the Gibbs measure proportional to $\texte^{-βH(η)}ν(\textdη)$. Here $H=\sum_bV(η_b)$ is the Hamiltonian, $V$ is a symmetric potential, $β>0$ is the inverse temperature, and $ν$ is the Lebesgue measure on the linear space defined by imposing the loop condition $η_{b_1}+η_{b_2}=η_{b_3}+η_{b_4}$ for each plaquette $(b_1,b_2,b_3,b_4)$ in $\Z^2$. For convex $V$, Funaki and Spohn have shown that ergodic infinite-volume Gibbs measures are characterized by their tilt. We describe a mechanism by which the gradient Gibbs measures with non-convex $V$ undergo a structural, order-disorder phase transition at some intermediate value of inverse temperature $β$. At the transition point, there are at least two distinct gradient measures with zero tilt, i.e., $E η_b=0$.