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We reconsider the phase diagram of a three-dimensional $\mathbb{Z}_2$topological insulator in the presence of short-ranged potential disorder withthe insight that non-perturbative rare states destabilize the noninteractingDirac semimetal critical point separating different topological phases. Basedon our numerical data on the density of states, conductivity, andwavefunctions, we argue that the putative Dirac semimetal line is destabilizedinto a diffusive metal phase of finite extent due to non-perturbative effectsof rare regions. We discuss the implications of these results for past andcurrent experiments on doped topological insulators.

Direct topological insulator transitions in three dimensions are destabilized by non-perturbative effects of disorder