We study in this paper the EM scheme for a family of well-posed critical SDEs with the drift $-x\log(1+|x|)$ and $α$-stable noises. Specifically, we find that when the SDE is driven by a rotationally symmetric $α$-stable processes with $α=2$ (i.e. Brownian motion), the EM scheme is bounded in the $L^2$ sense uniformly w.r.t. the time. In contrast, if the SDE is driven by a rotationally symmetric $α$-stable process with $α\in (0,2)$, all the $β$-th moments, with $β\in (0,α)$, of the EM scheme blow up. This demonstrates a phase transition phenomenon as $α\uparrow 2$. We verify our results by simulations.