Aizenman-Wehr argument for a class of disordered gradient models

We consider random gradient fields with disorder where the interaction potential $V_e$ on an edge $e$ can be expressed as $e^{-V_e(s)} = \int \rho(\mathrm{d}\kappa)\, e^{-\kappa \xi_e} e^{-\frac{\kappa s^2}{2}}$. Here $\rho$ denotes a measure with compact support in $(0,\infty)$ and $\xi_e\in\mathbb{R}$ a nontrivial edge dependent disorder. We show that in dimension $d=2$ there is a unique shift covariant disordered gradient Gibbs measure such that the annealed measure is ergodic and has zero tilt. This shows that the phase transitions known to occur for this class of potential do not persist to the disordered setting. The proof relies on the connection of the gradient Gibbs measures to a random conductance model with compact state space, to which the well known Aizenman-Wehr argument applies.