nLab Kane-Mele invariant

Idea

In solid state physics, the Kane-Mele invariant (after Kane & Mele 05b) or Fu-Kane-Mele invariant (after Fu, Kane & Mele 2007) 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)):

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

Related concepts

• Haldane model

References

The idea originates in:

• Charles Kane, Eugene Mele, $Z_2$ Topological Order and the Quantum Spin Hall Effect, Phys. Rev. Lett. 95 (2005) 146802 (doi:10.1103/PhysRevLett.95.146802)

• Liang Fu, Charles L. Kane: Time Reversal Polarization and a $\mathbb{Z}_2$ Adiabatic Spin Pump, Phys. Rev. B 74 (2006) 195312 [arXiv:cond-mat/0606336, doi:10.1103/PhysRevB.74.195312]

• Liang Fu, Charles L. Kane, Eugene J. Mele: Topological Insulators in Three Dimensions, Phys. Rev. Lett. 98 (2007) 106803 [arXiv:cond-mat/0607699, arXiv:10.1103/PhysRevLett.98.106803]

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:

• Charles Kane, Eugene Mele, Quantum Spin Hall Effect in Graphene, Phys. Rev. Lett. 95 226801 (2005) (arXiv:cond-mat/0411737, doi:10.1103/PhysRevLett.95.226801)

Discussion relating to a previously defined FKMM invariant:

• Giuseppe De Nittis, Kiyonori Gomi: Classification of “Quaternionic” Bloch-bundles: Topological Quantum Systems of type AII, Commun. Math. Phys. 339 (2015) 1–55 [arXiv:1404.5804, doi:10.1007/s00220-015-2390-0]

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

• Ralph M. Kaufmann, Dan Li, Birgit Wehefritz-Kaufmann, Notes on topological insulators, Rev. Math. Phys. 28 10 (2016) 1630003 (arXiv:1501.02874, doi:10.1142/S0129055X1630003X)

• Severin Bunk, Richard Szabo, Topological Insulators and the Kane-Mele Invariant: Obstruction and Localisation Theory, Reviews in Mathematical Physics 32 06 (2020) 2050017 (arXiv:1712.02991, doi:10.1142/S0129055X20500178)