Z_2 Topological Insulators in Ultracold Atomic Gases

References

I Supplementary Material

Shortcuts are possible in the presence of additional symmetries. For example, for 2D systems where the $z$-component of the spin is conserved, evaluating $\nu$ reduces to the calculation of the Chern number $C$ from the wavefunctions of one of the spin components. The result is simply

This result is used to establish the quantum spin Hall phase of Eq. (7) of the main text for $\delta=0$.

In the main text, we extensively use another shortcut, which is available if the system has inversion symmetry. This leads to a radical simplification in both 2D and 3D. Before citing the result of Ref. FuKaneinv, we briefly summarize some basic facts about the eigenstates of inversion symmetric, time-reversal invariant, lattice periodic Hamiltonians. Inversion symmetry means that the Hamiltonian $\hat{H}({\bm{r}})$ is invariant under $\hat{P}:{\bm{r}}\rightarrow-{\bm{r}}$. The Bloch Hamiltonian is

and it acts on lattice periodic functions (Bloch functions). We denote the reciprocal lattice vectors by ${\bm{\kappa}}_{j}$. At a generic ${\bm{k}}$, unlike $\hat{H}({\bm{r}})$, the Bloch Hamiltonian (12) is not invariant under $\hat{P}$ nor under time-reversal $\hat{\theta}$, since both map ${\bm{k}}\rightarrow-{\bm{k}}$. The momenta ${\bm{\Gamma}}_{\{n_{j}\}}=\sum_{j}n_{j}{\bm{\kappa}}_{j}/2$ ($n_{j}=0,1$), however, are special because ${\bm{\Gamma}}_{\{n_{j}\}}$ and $-{\bm{\Gamma}}_{\{n_{j}\}}$ are equivalent wavevectors. This means that $\hat{H}({\bm{r}})_{\mathbf{\Gamma}_{\{n_{j}\}}}$ is invariant under both generalized time-reversal and inversion operators: $\hat{\theta}_{\{n_{j}\}}=\exp(-i2{\bm{\Gamma}}_{\{n_{j}\}}{\bm{r}})\hat{\theta}$, and $\hat{P}_{\{n_{j}\}}=\exp(-i2{\bm{\Gamma}}_{\{n_{j}\}}{\bm{r}})\hat{P}$, where the exponentials take $H({\bm{r}})_{-{\bm{\Gamma}}_{\{n_{j}\}}}$ back to $H({\bm{r}})_{{\bm{\Gamma}}_{\{n_{j}\}}}$ [cf. Eq. (12)]. Note that the operators $\hat{\theta}_{\{n_{j}\}}$ and $\hat{P}_{\{n_{j}\}}$ map lattice periodic functions to lattice periodic functions.

Because of the generalized inversion symmetry, the eigenstates $u({\bm{r}})_{{\bm{\Gamma}}_{\{n_{j}\}}}$ of $H({\bm{r}})_{{\bm{\Gamma}}_{\{n_{j}\}}}$ can be labeled by inversion eigenvalues $\xi_{\{n_{j}\}}=\pm 1$,

It is straightforward to show that the corresponding eigenstates $\psi({\bm{r}})_{{\bm{\Gamma}}_{\{n_{j}\}}}=\exp(i{\bm{\Gamma}}_{\{n_{j}\}}{\bm{r}})u({\bm{r}})_{{\bm{\Gamma}}_{\{n_{j}\}}}$ of the Hamiltonian $\hat{H}({\bm{r}})$ are eigenstates of $\hat{P}$ (and not $\hat{P}_{\{n_{j}\}}$) with the same eigenvalue $\xi_{\{n_{j}\}}$.

The symmetry under $\hat{\theta}_{\{n_{j}\}}$ means that the energies at ${\bm{\Gamma}}_{\{n_{j}\}}$ (i.e., the eigenvalues of $\hat{H}({\bm{r}})_{{\bm{\Gamma}}_{\{n_{j}\}}}$) come in Kramers degenerate pairs. The Kramers pair of a Bloch eigenstate $u({\bm{r}})_{{\bm{\Gamma}}_{\{n_{j}\}}}$ is $\hat{\theta}_{\{n_{j}\}}u({\bm{r}})_{{\bm{\Gamma}}_{\{n_{j}\}}}$. Because the symmetries $\hat{\theta}_{\{n_{j}\}}$ and $\hat{P}_{\{n_{j}\}}$ commute, $u({\bm{r}})_{{\bm{\Gamma}}_{\{n_{j}\}}}$ and $\hat{\theta}_{\{n_{j}\}}u({\bm{r}})_{{\bm{\Gamma}}_{\{n_{j}\}}}$ share the same inversion eigenvalue. The $\alpha$-th Kramers doublet at ${\bm{\Gamma}}_{\{n_{j}\}}$ can thus be characterized by a single inversion eigenvalue $\xi^{(\alpha)}_{\{n_{j}\}}$.

We are now ready to state the result of Ref. FuKaneinv: Fu and Kane showed that the inversion eigenvalues corresponding to the filled bands straightforwardly determine whether an insulator is in a topological phase. The system is a topological insulator ($\nu=1$) if

where $\alpha$ runs over the Kramers doublets of the filled bands. Note that the invariant (14) is even simpler than the Chern number, since it does not require the integration of a (possibly complicated) function over the Brillouin zone; its sole inputs are the parity eigenvalues at a few discrete points in momentum space.