Wannier center sheets in topological insulators

We argue that various kinds of topological insulators (TIs) can be insightfully characterized by an inspection of the charge centers of the hybrid Wannier functions, defined as the orbitals obtained by carrying out a Wannier transform on the Bloch functions in one dimension while leaving them Bloch-like in the other two. From this procedure, one can obtain the Wannier charge centers (WCCs) and plot them in the two-dimensional projected Brillouin zone. We show that these WCC sheets contain the same kind of topological information as is carried in the surface energy bands, with the crucial advantage that the topological properties of the bulk can be deduced from bulk calculations alone. The distinct topological behaviors of these WCC sheets in trivial, Chern, weak, strong, and crystalline TIs are first illustrated by calculating them for simple tight-binding models. We then present the results of first-principles calculations of the WCC sheets in the trivial insulator ${\mathrm{Sb}}_2$Se${}_3$, the weak TI KHgSb, and the strong TI ${\mathrm{Bi}}_2$Se${}_3$, confirming the ability of this approach to distinguish between different topological behaviors in an advantageous way.

Figure 1

(a) Flow of Wannier charge centers along $\widehat{z}$ vs $k_y$ for a 2D Chern insulator. (b) Flow of surface energy bands vs $k_y$ for a 2D Chern insulator. (c), (d) Same, but for a 2D $Z_2$-odd (quantum spin Hall) insulator. Dashed lines are arbitrary reference positions in (a) and (c), or Fermi energies in (b) and (d).

Figure 2

Surface energy bands and WCC sheets for the TR-broken Chern insulator model. (a)–(c) Surface normal and WCCs along $\widehat{z}$ vs $(k_x,k_y)$. (d)–(f) Surface normal and WCCs along $\widehat{y}$ vs $(k_x,k_z)$. Surface states for a 24-layer slab in (a) and (d); WCCs around 2D BZ boundary in (b) and (e); WCCs in 2D BZ in (c) and (f). Dashed and solid surface states in (d) reside on the top and bottom of the (010) slab, respectively. The WCC sheets and surface bands wind by one unit in the $k_y$-$k_z$ plane, but not in the $k_x$-$k_y$ plane.

Figure 3

Surface energy bands (15-layer slab) and WCC sheets for the TR-invariant FKM model in the trivial phase ($\alpha =2.5$). (a)–(c) Surface normal and WCCs along $\widehat{z}$ vs $(k_x,k_y)$. (d)–(f) Surface normal and WCCs along $\widehat{y}$ vs $(k_x,k_z)$. The WCC sheets and surface bands show a trivial behavior in all directions.

Figure 4

First-principles WCC sheets along $\widehat{z}$ for topologically trivial Sb${}_2$Se${}_3$, plotted on (a) the boundary and (b) the interior of the 2D quarter BZ. The WCCs show trivial behavior as expected.

Figure 5

Surface energy bands (15-layer slab) and WCC sheets for the TR-invariant FKM model in the weak topological phase ($\alpha =-1$). (a)–(c) Surface normal and WCCs along $\widehat{z}$ vs $(k_x,k_y)$. (d)–(f) Surface normal and WCCs along $\widehat{y}$ vs $(k_x,k_z)$. Only the $(k_x,k_y)$ TRI faces at $k_z=0$ and $\pi$ are $Z_2$ odd.

Figure 6

First-principles WCC sheets for the weak TI KHgSb. (a), (b) Along $\widehat{z}$. (c), (d) Along $\widehat{y}$. Only the $(k_x,k_y)$ TRI faces at $k_z=0$ and $\pi$ are $Z_2$ odd.

Figure 7

Surface energy bands (15-layer slab) and WCC sheets for the TR-invariant FKM model in the strong topological phase $(\alpha =1)$. (a)–(c) Surface normal and WCCs along $\widehat{z}$ vs $(k_x,k_y)$. (d)–(f) Surface normal and WCCs along $\widehat{y}$ vs $(k_x,k_z)$. The $(k_x,k_y)$ TRI face at $k_z=\pi$, the $(k_x,k_z)$ TRI face at $k_y=0$, and the $(k_y,k_z)$ TRI face at $k_x=0$ are $Z_2$ odd.

Figure 8

First-principles WCC sheets for the strong TI Bi${}_2$Se${}_3$, plotted on (a) the boundary and (b) the interior of the 2D quarter BZ. The WCC sheets on parallel TRI faces (e.g., at $k_x=0$ and $\pi$) show opposite topological behavior.

Figure 9

Surface energy bands (24-layer slab) and WCC sheets for tight-binding model of a crystalline TI. (a)–(c) Surface normal and WCCs along $\widehat{z}$ vs $(k_x,k_y)$. (d)–(f) Surface normal and WCCs along $\widehat{y}$ vs $(k_x,k_z)$. Dashed and solid surface states in (a) reside on the top and bottom of the (001) slab, respectively. Quadratic band touching and cross linking in panels (a)–(c) signal the crystalline topological phase.