Topological origin of edge states in two-dimensional inversion-symmetric insulators and semimetals

Symmetries play an essential role in identifying and characterizing topological states of matter. Here, we classify topologically two-dimensional (2D) insulators and semimetals with vanishing spin-orbit coupling using time-reversal ($\mathcal{T}$) and inversion ($\mathcal{I}$) symmetry. This allows us to link the presence of edge states in $\mathcal{I}$ and $\mathcal{T}$ symmetric 2D insulators, which are topologically trivial according to the Altland-Zirnbauer table, to a $\mathbb{Z}_2$ topological invariant. This invariant is directly related to the quantization of the Zak phase. It also predicts the generic presence of edge states in Dirac semimetals, in the absence of chiral symmetry. We then apply our findings to bilayer black phosphorus and show the occurrence of a gate-induced topological phase transition, where the $\mathbb{Z}_2$ invariant changes.