Mapping topological order in coordinate space

The organization of the electrons in the ground state is classified by means of topological invariants, defined as global properties of the wave function. Here we address the Chern number of a two-dimensional insulator and we show that the corresponding topological order can be mapped by means of a “topological marker,” defined in $\mathbf{r}$ space, and which may vary in different regions of the same sample. Notably, this applies equally well to periodic and open boundary conditions. Simulations over a model Hamiltonian validate our theory.

Figure 1

Chern number of the bottom band of the Haldane model as a function of the parameters $\phi$ and $\Delta /t_2$ ($t_1=1,t_2=1/3$). The points marked with letters (a)–(e) in this phase diagram are relevant for the subsequent discussion and figures. In order to avoid special features, the $\phi$ parameter is not a multiple of $\pi /4$.

Figure 2

A typical flake, with 2550 sites, showing the honeycomb lattice of the Haldane model (Ref.19). The 50 sites on the horizontal line will be used in all the subsequent one-dimensional plots. Black and gray circles indicate nonequivalent sites (with on-site energies $\pm \Delta$).

Figure 3

Local Chern number (top) and site occupancy (bottom) for the 50 sites along the line shown in Fig. 2. Left-hand panel: Point (b) in the phase diagram, Fig. 1. Right-hand panel: Point (c). Notice the different scales.

Figure 4

Local Chern number for a few points on the line $\Delta /t_2=3.67$, i.e., on the (b)–(c) segment in Fig. 1, close to the transition from $C=0$ to $C=1$. The exact transition occurs at $\phi /\pi =0.25$; our five plots correspond (bottom to top) to $\phi /\pi =0.17,0.25,0.27,0.29,0.33.$

Figure 5

Local Chern number (top) and site occupancy (bottom) for disordered systems (see text). Left-hand panel: Disordered system along the line (a)–(b) in the phase diagram, Fig. 1. Right-hand panel: Line (c)–(d). Notice the different scales.

Figure 6

Local Chern number (top) and site occupancy (bottom) across heterojunctions. Left-hand panel: Hamiltonian parameters as in (a) and in (b) for the left- and the right-hand halves of the sample, respectively. Right-hand panel: Parameters as in (e) and in (c) for left- and the right-hand halves of the sample.