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

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