Insulator/Chern-insulator transition in the Haldane model

We study the behavior of several physical properties of the Haldane model as the system undergoes its transition from the normal-insulator to the Chern-insulator phase. We find that the density matrix has exponential decay in both insulating phases, while having a power-law decay, more characteristic of a metallic system, precisely at the phase boundary. The total spread of the maximally localized Wannier functions is found to diverge in the Chern-insulator phase. However, its gauge-invariant part, related to the localization length of Resta and Sorella, is finite in both insulating phases and diverges as the phase boundary is approached. We also clarify how the usual algorithms for constructing Wannier functions break down as one crosses into the Chern-insulator region of the phase diagram.

Figure 1

(a) Four unit cells of the Haldane model. Solid (open) circles have site energy $-\Delta$ $(+\Delta )$. The first-nearest-neighbor hopping $t_1$ is real, while the second-nearest-neighbor hopping $t_2{e}^{i\phi }$ has a complex phase. Arrows indicate the direction of positive phase hopping. The Wigner-Seitz unit cell corresponds to the hexagon in the center of the plot. (b) First Brillouin zone of the Haldane model with high-symmetry points marked.

Figure 2

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 vertical line shows the range of parameters that we have chosen for all our calculations.

Figure 3

Band structure of the Haldane model along some high-symmetry lines for several values of $\Delta ∕t_2$ along the path marked in Fig. 2. The inset shows a magnification of the bands at $K$. Note that at ${(\Delta ∕t_2)}_{\mathrm{cr}}$ the dispersion is linear.

Figure 4

Energy vs wave vector $k_y$ for the Haldane model in a strip geometry 30 cells wide along the ${\mathbf{b}}_3$ direction and extending infinitely along ${\mathbf{b}}_2$ direction. For $\Delta ∕t_2<{(\Delta ∕t_2)}_{\mathrm{cr}}$ (bottom panel), chiral edge states are visible.

Figure 5

Lowest eigenvalue of the overlap matrix $S_{\alpha \beta }=⟨y_{\alpha }\mid y_{\beta }⟩$ as a function of the parameter $\Delta ∕t_2$ for different sample sizes $n\times n$. Note the logarithmic scale.

Figure 6

Plot of $s\left(\mathbf{k}\right)$ of Eq. (12) along some high-symmetry lines for several values of $\Delta ∕t_2$.

Figure 7

Density of $s\left(\mathbf{k}\right)$ values as a function of $\Delta ∕t_2$ obtained for periodic samples with a $300\times 300$ $\mathbf{k}$ mesh.

Figure 8

Gauge-independent part ${\Omega }_I$ and gauge-dependent part $\tilde{\Omega }$ of the spread functional for the Haldane model as a function of the $\mathbf{k}$-mesh density.

Figure 9

Logarithm of the density-matrix kernel $\mid {\xi }_{\alpha \beta }\left(R\widehat{\mathbf{x}}\right)\mid$ of the Haldane model for several values of $\Delta ∕t_2$. The linear asymptotic behavior indicates exponential decay except at ${(\Delta ∕t_2)}_{\mathrm{cr}}\approx 3.67$ (solid curves) where the decay is power law instead.