Chern Insulator

A Chern insulator is a class of materials that exhibit an anomalous quantum Hall effect, a quantized Hall conductance in the absence of a magnetic field.

An early model of such a material was introduced by Haldane. He proposed a tight-binding model where flux quanta were introduced “by hand” through a complex second nearest neighbor hopping term. This term breaks time-reversal symmetry, and allows the system to exhibit a ground state with a non-zero Chern number. The model is

$$\begin{array}{rl}H=& \Delta \sum _i(-{)}^i{c}_i^{†}{c}_i-t_1\sum _{⟨i,j⟩}(c_i^{†}{c}_j+h.c.)+t_2\sum _{⟨⟨i,j⟩⟩}(i{c}_i^{†}{c}_j+h.c.)\end{array}$$

The global invariance of the Chern number introduces an equivalence class of insulators characterized by the Chern number of their ground state. Insulators whose occupied states have a net $C\ne 0$ have become known as “Chern insulators”, while insulators with $C=0$ are called trivial. This model became the first example of a Chern insulator, launching the systematic investigation of topological materials.

The manifestation of the Chern number in 2D is that the Berry phase winds along BZ in one k direction ($k_x$) as a function of the other ($k_y$). Another observable consequence is that there are gapless surface states creating a chiral edge channel around the sample.