Figure 2

The unit vector , of Eq. [4], plotted across the honeycomb lattice Brillouin zone.

Due to the periodic boundary conditions of the Brillouin zone, the unit vector must wrap the sphere an integer number of times, which gives the Chern number. In all three figures, t′ = 0.1t. In (a), Δ = 0.2t and so the system is a trivial insulator. The unit vector clearly never visits the north pole, but wraps and then un-wraps the lower hemisphere. In (b), Δ = 0, ϕ = π/2ϕ₀ and θ = π/2, so the system is a Chern insulator. In (c) is shown the Haldane model. In the later two cases, it can be seen that the vector visits both the north and south poles once only, giving a Chern number of 1. The bottom figure shows the representation of the unit vector in pseudo-spin space, together with the corresponding colour coding of the σ_z component of each vector.