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We derive an efficient formula for Z$_2$ topological invariantscharacterizing the quantum spin Hall effect. It is defined in a latticeBrillouin zone, which enables us to implement numerical calculations forrealistic models even in three dimensions. Based on this, we study the quantumspin Hall effect in Bi and Sb in quasi-two and three dimensions using atight-binding model.Comment: 5 pages, 5 figure

Quantum Spin Hall Effect in Three Dimensional Materials: Lattice Computation of Z2 Topological Invariants and Its Application to Bi and Sb

Quantum Spin Hall Effect in Three Dimensional Materials: Lattice Computation of Z₂ Topological Invariants and Its Application to Bi and Sb