Exponential decay of fractional moments in the $H^{2\mathrm{\mid }2}$ model on ${\mathbb{Z}}^2$

For the $H^{2\mathrm{\mid }2}$ model on ${\mathbb{Z}}^2$, we prove that, at energy zero, fractional moment of Green functions, $\mathbb{E}[G(x,y{)}^s]$ decays exponentially in the distance between $x$ and $y$, for every constant edge weight $W>0$ and every $0<s<1\mathrm{/}2$. Here $G$ is the infinite-volume Green function at energy zero of the associated random Schrödinger operator. The main ingredient is a quantitative lower bound on the derivative of $\mathbb{E}[e^{u_x\mathrm{/}2}]$ with respect to the edge weights, where $u$ is the pinned effective field.