Equivalent expression of ${\mathbb{Z}}_2$ topological invariant for band insulators using the non-Abelian Berry connection

We introduce an expression for the ${\mathbb{Z}}_2$ topological invariant of band insulators using the non-Abelian Berry connection. Our expression can identify the topological nature of a general band insulator without any of the gauge-fixing problems that plague the concrete implementation of previous invariants. This expression can be derived from the “partner switching” of the Wannier function center during time-reversal pumping and is thus equivalent to the $Z_2$ topological invariant proposed by Kane and Mele. Using our expression, we have recalculated the $Z_2$ topological index for several topological insulator material systems and obtained consistent results with the previous studies.

Figure 1

Schematic plots of the Wannier center curves: (a) for the trivial case, the Wannier center winding the cylinder zero times; (b) the Wannier center winding the cylinder one time; (c) the Wannier center winding the cylinder twice, and the cross point labeled by $P$ is not protected by time-reversal symmetry and it is usually eliminated by some perturbation terms; and (d) the Wannier center winding the cylinder three times, which is topologically equal to the case in (b).

Figure 2

Wannier centers for the BHZ model. (a) For the quantum spinn Hall phase ($A=-13.68$ eV$$ Å, $B=-16.9$ eV$$ Å${}^2$, $C=-0.0263$ eV, $D=-0.514$ eV  Å${}^2$, and $M=-2.058$ eV, $\Delta$ = 1.20 eV), the Wannier center crosses the reference line (red dashed line) once (odd number of times). (b) For the normal insulating phase ($A=-14.48$ eV   Å, $B=-18.0$ eV$$ Å${}^2$, $C=-0.018$ eV, $D=-0.594$ eV$$ Å${}^2$, $M=2.766$ eV, and $\Delta$ $=1.20$ eV), the Wannier center crosses the reference line (red dashed line) zero (even number of) times.

Figure 3

The evolution lines of Wannier centers for HgTe [(a and (b)] and CdTe [(c) and (d)]. For the HgTe system, the evolution lines cross the reference line an odd number of times in the $k_z=0$ plane and an even number of times in the $k_z=\pi$ plane, indicating HgTe is a strong topological insulator. For CdTe, the evolution lines cross the reference line zero times for both $k_z=0$ and $\pi$ planes, indicating CdTe is a normal insulator.

Figure 4

The evolution lines of Wannier centers for the Bi${}_2$Se${}_3$ system without [(a) and (b)] and with SOC [(c) and (d)]. If we turn off SOC the system is a normal insulator and the evolution lines never cross the reference line for both $k_z=0$ and $\pi$ planes as shown in (a) and (b), indicating the system to be a normal insulator. When the SOC is turned on the system is in a strong topological insulator phase, and the evolution lines cross the reference line an odd number of times in $k_z=0$ and an even number of times in the $k_z=\pi$ plane as shown in (c) and (d), indicating the system is topologically nontrivial.

Figure 5

The evolution lines of Wannier centers for the 1QL and 2QL Bi${}_2$Te${}_3$ slabs. (a) The 1QL Bi${}_2$Te${}_3$ slab is in normal insulator phase. (b) The 2QL Bi${}_2$Te${}_3$ slab is in topological insulator phase.

Figure 6

The evolution lines of Wannier centers for 2D bilayer Bi (a) and Sb (b) systems, indicating the bilayer Bi is topologically nontrivial but Sb is topologically trivial.

Figure 7

The evolution lines of Wannier centers for graphene in the (a) QSH phase ${\lambda }_{\upsilon }=0.1t$ and (b) the normal insulating phase ${\lambda }_{\upsilon }=0.4t$. In both cases ${\lambda }_{\text{so}}=0.06t$ and ${\lambda }_R=0.05t$.

Figure 8

(a) Definition of the half BZ and the paths to define Wilson loops. (b) Schematic picture of eigenvalues ${\phi }_n\left(k_y\right)$ for a two-band model. ${\phi }_1\left(k_y\right)$ has winding number 0 and ${\phi }_{\overline{1}}\left(k_y\right)$ has winding number 1, when ${\phi }_1\left(0\right)$ and ${\phi }_1\left(\pi \right)$ are chosen to be in $[0,2\pi )$.