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.

Figures (7)

• Figure 1: Osculatory (left) and intersecting (right) band functions for Kramer pairs
• Figure 2: Fiber Bundles: trivial (left) and non-trivial (right)
• Figure 3: Pictorial presentation of the exact sequence (\ref{['esq']})
• Figure 4: Cartesian product of circles: two-torus
• Figure 5: Wedge sum of two circles

Theorems & Definitions (12)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Theorem 2.1
• Theorem 2.2: Equivalence Theorem
• proof
• Lemma 2.4: Calculating Lemma
• Theorem 5.1: NashNash2
• Definition 5.1
• Lemma 5.1