Smooth gauge for topological insulators

We develop a technique for constructing Bloch-like functions for 2D ${\mathbb{Z}}_2$ insulators (i.e., quantum spin-Hall insulators) that are smooth functions of $\mathbf{k}$ on the entire Brillouin-zone torus. As the initial step, the occupied subspace of the insulator is decomposed into a direct sum of two “Chern bands,” that is, topologically nontrivial subspaces with opposite Chern numbers. This decomposition remains robust independent of underlying symmetries or specific model features. Starting with the Chern bands obtained in this way, we construct a topologically nontrivial unitary transformation that rotates the occupied subspace into a direct sum of topologically trivial subspaces, thus facilitating a Wannier construction. The procedure is validated and illustrated by applying it to the Kane-Mele model.

Figure 1

Brillouin zone in 2D represented as (a) a torus and (b) a cylinder. We choose the gauge discontinuity to be distributed along the cross-sectional cut of the torus that maps onto the end loops of the cylinder at $k_y=\pm \pi$.

Figure 2

(a) BZ in $k$ space. (b) States are parallel transported along $k_y=0$, but are not periodic. (c) Periodicity is restored at $k_y=0$. (d) Parallel transport of states at all $k_x$ from $k_y=0$ to $k_y=\pm \pi$. (e) Periodicity is restored at $k_x=0$, and, hence, $k_x=2\pi$, but not at other $k_x$.

Figure 3

Trajectory of $V_{11}$ in the complex plane as $k_x$ runs across the BZ, before (red dashed line) and after (solid black line) the global ${\mathcal{U}}_0$ rotation that minimizes ${\mathcal{V}}_{\mathrm{OD}}$ for a ${\mathbb{Z}}_2$-odd insulator. In neither case is the graph exactly a unit circle (dotted line), but $V_{11}\left(0\right)=1$ and $V_{11}\left(\pi \right)=-1$.