Characterizing weak topological properties: Berry phase point of view

We propose classification schemes for characterizing two-dimensional topological phases with nontrivial weak indices. Here, weak implies that the Chern number in the corresponding phase is trivial, while the system shows edge states along specific boundaries. As concrete examples, we analyze different versions of the so-called Wilson-Dirac model with (i) anisotropic Wilson terms, (ii) next-nearest-neighbor hopping terms, and (iii) a superlattice generalization of the model, here in the tight-binding implementation. For types (i) and (ii) a graphic classification of strong properties is successfully generalized for classifying weak properties. As for type (iii), weak properties are attributed to quantized Berry phase $\pi$ along a Wilson loop.

Figure 1

The anisotropic Wilson-Dirac model. (a) Phase diagram determined by numerical estimation of the bulk Chern number $\mathcal{N}$. $b_x=1,t=1$. $\mathcal{N}=+1 \left(\mathcal{N}=-1\right)$ regions in blue (red). (b) Phase boundaries determined by (i) the existence vs absence of edge modes, and (ii) closing of the bulk energy gap at the four symmetric points: ${\mathit{k}}_D=(0,0),(\pi ,0),(0,\pi )$, and $(\pi ,\pi )$. For each phase boundary, we have checked that the two conditions (i) and (ii) coincide. The bulk band indices (see Table 1) are also indicated in the figure. (c), (d) Graphic representation of the closed surface $\mathcal{R}\left[T^2\right]$ and the closed loops $\mathcal{R}[{\mathcal{C}}_{k_y}\left[0\right]]$ (blue), $\mathcal{R}[{\mathcal{C}}_{k_y}\left[\pi \right]]$ (green), $\mathcal{R}[{\mathcal{C}}_{k_x}\left[0\right]]$ (red), $\mathcal{R}[{\mathcal{C}}_{k_x}\left[\pi \right]]$ (orange). [(c): case of $\mathrm{SCI}{+}_1$, (d): case of WCI-$y$] (see Sec. 3). (e)–(h) Edge (and bulk) spectra in different topological phases: $\mathrm{SCI}{+}_1$ [(e), (g)], WCI-$y$ [(f), (h)], and in two different types of ribbon geometries: $x$-oriented [(e), (f)], $y$-oriented [(g), (h)].

Figure 2

NNN hopping model. (a) Phase diagram determined by numerical estimation of the Chern number $\mathcal{N}$. Parameter regions corresponding to different $\mathcal{N}$ are specified by different colors; respectively, in blue $\left(\mathcal{N}=1\right)$, red $\left(\mathcal{N}=-1\right)$, green $\left(\mathcal{N}=2\right)$, orange $\left(\mathcal{N}=-2\right)$, and white $\left(\mathcal{N}=0\right)$. $b=1,t=1$. (b) Phase boundaries, and the bulk band indices. (c) Graphic representation of the closed surface $\mathcal{R}\left[T^2\right]$ and the closed loops $\mathcal{R}[{\mathcal{C}}_{k_y}\left[0\right]]$ (blue), $\mathcal{R}[{\mathcal{C}}_{k_y}\left[\pi \right]]$ (green), $\mathcal{R}[{\mathcal{C}}_{k_x}\left[0\right]]$ (red), $\mathcal{R}[{\mathcal{C}}_{k_x}\left[\pi \right]]$ (orange) in the ${\mathrm{SCI}}_{\mathcal{N}=2}$ phase. $m=9,b_{xy}=1.5$. (d) The edge spectrum in the ${\mathrm{SCI}}_{\mathcal{N}=2}$ phase. Here, the ribbon is extended along the $x$ axis, while the same spectrum is obtained for a ribbon along the $y$ axis.

Figure 3

Phase diagram of the superlattice model determined by numerical estimation of the Chern number $\mathcal{N}$. Parameter regions represented by different colors correspond to $\mathcal{N}=1$ (blue), $\mathcal{N}=-1$ (red), and $\mathcal{N}=0$ (white). (a), (b): case of the $\mathrm{ABAB}...$-like pattern. (c), (d): $\mathrm{AABAAB}...$-type case. $b=1,t=1$ [(a), (c)], $b=1,t=4$ [(b), (d)].

Figure 4

Energy spectrum in the ribbon geometry. The superlattice is $\mathrm{AABAAB}...$ type. The system is in the WCI-$x$ phase in (a), (b), while it is in the WCI-$y$ phase in (c), (d). The ribbon is extended along $x$ axis in (a), (c), while the same is along the $y$ axis in (b), (d).

Figure 5

Phase boundaries between and bulk band indices in different topological phases. (a) and (b): cases of the $\mathrm{ABAB}...$-type pattern. (c) and (d): cases of the $\mathrm{AABAAB}...$-type pattern. Other parameters are set the same as in the corresponding panel in Fig. 3. The phase boundaries are determined by tracing the zeros of $detH$ at the four symmetric $\mathit{k}$ points in the BZ (see the main text). Different colors correspond to zeros of $detH$ at $(0,0)$ (black), $(\pi ,0)$ (blue), $(0,\pi )$ (green), and $(\pi ,\pi )$ (red).

Figure 6

Evolution of the Wilson loop $W[{\mathcal{C}}_{k_y}\left[k_x\right]]$ and $W[{\mathcal{C}}_{k_x}\left[k_y\right]]$. (a), (c), (e), (g) represent $W[{\mathcal{C}}_{k_y}\left[k_x\right]]$, while the remaining panels (b), (d), (f), (h) are for $W[{\mathcal{C}}_{k_x}\left[k_y\right]]$. Case of the $\mathrm{AABAAB}...$-type superlattice. (a), (b) represent the strong case with $\mathcal{N}=+1$ ($\mathrm{SCI}{+}_1$ phase), while the remaining panels are for the weak and ordinary phases. (c), (d) are for the WCI-$x$ phase; edge states appear in the ribbon oriented in the $x$ direction. Similarly, (e), (f) are for WCI-$y$ phase. Finally, (g), (h) are for the $\mathrm{OI}+$ phase. $b=1,t=1$.