Topological insulators with weak and strong indices

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Topological insulators in three dimensions have weak and strong topological indices. Strong topological insulators have topologically protected surface states with an odd number of Dirac cones. In weak topological insulators line dislocations may carry topologically protected helical states (depending on the Burgers vector of the line dislocation). These two characteristics appear simultaneously in topological insulators that have nontrivial strong {\em and} weak indices.In a slab geometry, the helical states supported by dislocation lines may connect the surface states of the top and bottom surfaces. Preliminary results show that under suitable conditions this coupling leads to the opening of a gap in the surface state spectrum. In this proposal we present directions in which a more complete understanding of this observation can be reached, and how the dislocation line-induced coupling between surface states of top and bottom surfaces can be used to transfer nontrivial properties of electronic states from one surface to the other. The remarkable physics we discuss is here specific for strong topological insulators with nontrivial weak indices, a condition that is met in a small number of known topological insulator materials, including certain BiSb compounds. For strong topological insulators with trivial weak indices, which applies to most presently known topological insulator materials, dislocation lines to not carry protected helical states, so that the phenomena described here do not apply. For this reason, topological insulators with nontrivial weak {\em and} strong topological indices must be considered a special class of their own, separate from the purely weak or strong topological insulators. This proposal addresses the physics that sets this class apart: The combination of protected helical states at dislocation lines and protected surface states.