nLab Chern insulator

Idea

Chern insulators

In solid state physics, a topological insulator which is not “protected by any symmetry” has a valence bundle $V$ over the Brillouin torus $\mathbb{R}^d$ which is a complex vector bundle unconstrained by any reality or other conditions, and hence whose class in complex K-theory $KU^0(\mathbb{T}^d)$ is thought to classify the topological phase of the “unprotected” topological insulator, according to the expected K-theory classification of topological phases of matter.

Now the Brillouin torus of a realistic material is of effective dimension $d \leq 3$ and over spaces of these dimensions the only characteristic class of a complex vector bundle is its first Chern class $c_1(V) \,\in\, H^2\big(\mathbb{T}^d; \, \mathbb{Z}\big)$. Specifically for effectively 2-dimensional materials, the first Chern number $c_1[V] = \int_{\mathbb{T}^2} c_1(V) \,\in\, \mathcal{C}$ coincides with this K-theory class:

For this reason such “symmetry un-protected” topological insulators are commonly known as Chern insulators and the first Chern number of their valence bundle is regarded as labeling their topological phase.

Haldane model & QAHE

While mathematically the case without any quantum symmetry is the simplest one, physically this typically has to be realized with more effort by starting with a symmetry-protected material (such as graphene) and then “breaking” its symmetries (either by manipulating the material in the lab or, quite often, just theoretically by adding more terms to the Hamiltonian of its theoretical model).

Therefore it is regarded as a seminal achievement when the Haldane model was found, which intrinsically breaks the symmetries of an idealized graphene-like model (by spin-orbit coupling) and which is generally regarded as the archetype of a (model for a) Chern insulator.

An extrinsic way to break the time-reversal symmetry of a material like graphene is to simply place it in an external magnetic field; in this case one sometimes speaks of a quantum Hall material, instead.

With reference to this relation, the intrinsic effect that breaks the time-reversal symmetry of some material to a Chern insulator-phase are also known as a quantum anomalous Hall effect (QAHE).

Chern semi-metals

If a Chern insulator is deformed (mentally at least) to a semi-metal with band gap closures over isolated nodal points $\{k_1, \cdots, k_N\}$ in the Brillouin torus, then the relevant domain space becomes the complement $\mathbb{T}^3 \setminus \{k_1, \cdots, k_N\}$, which admits non-trivial Chern numbers for each puncture, corresponding to the evaluation of $c_1(V)$ on any 2-sphere surrounding the $I$th nodal point. This now measures the topological protection of the semimetal phase (see eg. Mathai-Thiang 17b for a transparent account).

Some authors speak of “Chern semi-metals” to amplify this.

Related concepts

• topological insulator

• K-theory classification of topological phases of matter

References

General

• David Vanderbilt, Section 5.1 of: Berry Phases in Electronic Structure Theory – Electric Polarization, Orbital Magnetization and Topological Insulators, Cambridge University Press (2018) (doi:10.1017/9781316662205)

• Panagiotis Kotetes, Chapter 5 of: Topological Insulators, IOP Science 2019 (ISBN:978-1-68174-517-6)

• N. Regnault, B. Andrei Bernevig, Fractional Chern Insulator, Phys. Rev. X 1, 021014 (2011) (arXiv:1105.4867)

Models akin to the Haldane model:

• Doru Sticlet, Frédéric Piéchon, Jean-Noël Fuchs, Pavel Kalugin, Pascal Simon, Geometrical engineering of a two-bands Chern insulator in two dimensions with arbitrary topological index, Phys. Rev. B 85 165456 (2012) [arXiv:1201.6613, doi:10.1103/PhysRevB.85.165456]

• Marwa Mannaï, Jean-Noël Fuchs, Frédéric Piéchon, Sonia Haddad, Stacking-induced Chern insulator [arXiv:2208.02491]

Experimental realization

• Aizhu Wang, Xiaoming Zhang, Yuanping Feng, Mingwen Zhao,Chern Insulator and Chern Half-Metal States in the Two-Dimensional Spin-Gapless Semiconductor $Mn_2 \mathrm{C}_6\mathrm{S}_12$, J. Phys. Chem. Lett. 8 16 (2017) 3770–3775 $[$doi:10.1021/acs.jpclett.7b01187$]$