Topology behind topological insulators

Abstract

In this paper topological $K$-group calculations for fiber bundles with structure group $SO(3)$ over tori are carried out to explain why topological insulators have special conducting points on their surface but are bulk insulators. It is shown that these special points are gap-less and conducting for topological reasons and follow from the $K$-group calculations. The existence of gap-less surface points is established with the help of an additional topological property of the $K$-groups which relates them to the index theorem of an operator. The index theorem relates zeros of operators to topology. For the topological insulator the relevant operator is a Dirac operator, that emerges in the problem because the system has strong spin-orbit interactions and time-reversal invariance. Calculating $K$-groups over tori require some special topological tools that are are not widely known. These are explained. We then show that the actual calculation of $K$-groups over tori becomes straightforward once a few topological results are in place. Since condensed matter systems with periodic lattices, are always bundles over tori the procedures described is of general interest.