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cohomology

# Contents

## Idea

In solid state physics, the Kane-Mele invariant (Kane & Mele 05b) is an element in $\mathbb{Z}/2$ naturally assigned to time-reversal symmetric spinful insulators, whose non-triviality partially detects whether the insulator is in a non-trivial topological phase, hence whether it is a topological insulator.

Concretely, for 3d materials the Kane-Mele invariant is obtained from the Chern-Simons invariant of the Berry connection on the Brillouin torus (eg. KLW 16, Sec. 4.3).

Under the expected K-theory classification of topological phases of matter, time-reversal symmetric and spinful topological insulators are classified by the class of their valence bundle in KO-theory of degree 4 (hence in KQ-theory) equipped with the time-reversal involution $\tau$. The Kane-Mele invariant is a non-trivial group homomorphism from that K-theory group to $\mathbb{Z}/2$ (Bunk & Szabo 20, Prop. 2.25, Pro. 5.22(4)):

$K Q \big( \mathbb{T}^d, \tau \big) \xrightarrow{\text{KM invariant}} \mathbb{Z}/2 \,.$

## References

The idea of the Kane-Mele invariant originates in:

This was motivated by the observation that graphene appears as a topological insulator with a non-trivial MK-invariant when its spin-orbit coupling is resolved, due to a quantum spin Hall effect:

On the relation of the Kane-Mele invariant to the K-theory classification of topological phases of matter:

Last revised on May 25, 2022 at 12:50:26. See the history of this page for a list of all contributions to it.