Automated construction of maximally localized Wannier functions for bands with nontrivial topology

We show that an optimized projection functions method can automatically construct maximally localized Wannier functions even for bands with nontrivial topology. We demonstrate this method on a tight-binding model of a two-dimensional $\mathbb{Z}_2$ topological insulator, on a three-dimensional strong $\mathbb{Z}_2$ topological insulator, as well as on first-principles density functional theory calculated valence states of Bi$_2$Se$_3$. In all cases, the resulting Wannier functions contain large imaginary components and are more extended than those in the topologically trivial phase.