Quantum spin Hall effect in three dimensional materials: Lattice computation of Z${}_2$ topological invariants and its application to Bi and Sb

We derive an efficient formula for Z${}_2$ topological invariants characterizing the quantum spin Hall effect. It is defined in a lattice Brillouin zone, which enables us to implement numerical calculations for realistic models even in three dimensions. Based on this, we study the quantum spin Hall effect in Bi and Sb in quasi-two and three dimensions using a tight-binding model.