Topology behind topological insulators
Koushik Ray, Siddhartha Sen

TL;DR
This paper uses topological $K$-theory to explain why topological insulators have conducting surface states, linking their properties to advanced mathematical calculations involving fiber bundles, index theorems, and spin-orbit interactions.
Contribution
It provides a novel topological $K$-group calculation framework for understanding surface states in topological insulators, connecting mathematical topology with physical phenomena.
Findings
Surface states are gap-less and conducting due to topological reasons.
$K$-group calculations over tori can predict the existence of these states.
The approach is applicable to condensed matter systems with periodic lattices.
Abstract
In this paper topological -group calculations for fiber bundles with structure group 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 -group calculations. The existence of gap-less surface points is established with the help of an additional topological property of the -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 -groups over tori require some special topological tools that are are not widely known.…
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Taxonomy
TopicsTopological Materials and Phenomena · Algebraic structures and combinatorial models · Noncommutative and Quantum Gravity Theories
