<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi mathvariant="double-struck">Z</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> topological invariants from Green function's diagonal zeros

We investigate the relationship between the analytical properties of the Green's function and ${\mathbb{Z}}_{2}$ topological insulators, focusing on three-dimensional inversion-symmetric systems. We show that the diagonal zeros of the Green's function in the orbital basis provide a direct and visual way to calculate the strong and weak ${\mathbb{Z}}_{2}$ topological invariants. We introduce the surface of crossings of diagonal zeros in the Brillouin zone, and show that it separates time-reversal invariant momenta (TRIMs) of opposite parity in two-band models, enabling the visual computation of the ${\mathbb{Z}}_{2}$ invariants by counting the relevant TRIMs on either side. In three-band systems, a similar property holds in every case except when a trivial band is added in the band gap of a nontrivial two-band system, reminiscent of the band topology of fragile topological insulators.