Topological Insulators in Three Dimensions

We study three dimensional generalizations of the quantum spin Hall (QSH) effect. Unlike two dimensions, where the QSH effect is distinguished by a single $Z_2$ topological invariant, in three dimensions there are 4 invariants distinguishing 16 ``topological insulator'' phases. There are two general classes: weak (WTI) and strong (STI) topological insulators. The WTI states are equivalent to layered 2D QSH states, but are fragile because disorder continuously connects them to band insulators. The STI states are robust and have surface states that realize the 2+1 dimensional parity anomaly without fermion doubling, giving rise to a novel ``topological metal'' surface phase. We show that the $Z_2$ invariants can be easily determined for systems with inversion symmetry. This allows us to predict specific materials are STI's, including semiconducting alloy Bi$_{1-x}$ Sb$_x$ as well as $\alpha-$Sn and HgTe under uniaxial strain.\newline \newline 1. Liang Fu, C.L. Kane, E.J. Mele, cond-mat/0607699. \newline 2. Liang Fu, C.L. Kane, cond- mat/0611341.