Topological insulators with weak and strong indices

三维拓扑绝缘体有弱拓扑指数和强拓扑指数。强拓扑绝缘体具有拓扑保护的表面态，具有奇数个狄拉克锥。在弱拓扑绝缘体中，线位错可能带有拓扑保护的螺旋态（取决于线位错的伯格斯矢量）。这两个特征同时出现在拓扑绝缘体中，它们具有非平凡的强{\em}和弱{\emand}指数。在平板几何中，位错线支撑的螺旋态可以连接顶面和底面的表面态。初步结果表明，在适当的条件下，这种耦合导致开放的间隙中的表面态谱。在这个建议中，我们提出了一个更完整的理解这个观察可以达到的方向，以及如何位错线诱导的耦合之间的表面状态的顶部和底部表面可以用来转移非平凡的性质的电子状态从一个表面到另一个。我们在这里讨论的显着的物理是特定的强拓扑绝缘体与非平凡的弱指数，一个条件是满足少数已知的拓扑绝缘体材料，包括某些铋锑化合物。对于具有平凡弱指数的强拓扑绝缘体，其适用于大多数目前已知的拓扑绝缘体材料，位错线不携带受保护的螺旋状态，使得这里描述的现象不适用。因此，具有非平凡弱拓扑指数和非平凡强拓扑指数的拓扑绝缘体必须被认为是它们自己的一个特殊类别，与纯粹的弱拓扑绝缘体或强拓扑绝缘体分开。这个提议解决了使这一类分开的物理学：位错线处的受保护螺旋态和受保护表面态的组合。

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.