Bott periodicity for the topological classification of gapped states of matter with reflection symmetry

Using a dimensional reduction scheme based on scattering theory, we show that the classification tables for topological insulators and superconductors with reflection symmetry can be organized in two period-2 and four period-8 cycles, similar to the Bott periodicity found for topological insulators and superconductors without spatial symmetries. With the help of the dimensional reduction scheme, the classification in arbitrary dimensions $d\ge 1$ can be obtained from the classification in one dimension, for which we present a derivation based on relative homotopy groups and exact sequences to classify one-dimensional insulators and superconductors with reflection symmetry. The resulting classification is fully consistent with a comprehensive classification obtained recently by Shiozaki and Sato [Phys. Rev. B 90, 165114 (2014)]. The use of a scattering-matrix-inspired method allows us to address the second descendant ${\mathbb{Z}}_2$ phase, for which the topological nontrivial phase was previously reported to be vulnerable to perturbations that break translation symmetry.

Figure 1

Schematic picture of a $d$-dimensional gapped insulator occupying the half space (blue) with twisted boundary conditions applied along $(d-1)$-dimension (black line), coupled to an ideal lead (red) with a $(d-1)$-dimensional cross section. The reflection matrix $r_d\left({\mathbf{k}}_{\perp }\right)$ relates the amplitudes of outgoing and incoming modes in the lead.

Figure 2

Schematic illustration of the spaces ${\mathcal{H}}_0$ and ${\mathcal{M}}_0$. The solid dot indicates the trivial element. The thick curve shows a path in ${\mathcal{H}}_0$ that starts at the trivial element and ends in ${\mathcal{M}}_0$. Equivalence classes of such paths form the relative homotopy group ${\pi }_1({\mathcal{H}}_0,{\mathcal{M}}_0)$.

Figure 3

(a) Schematic illustration of the two nonequivalent positions for the reflection plane (solid vertical lines) and the corresponding choice of the unit cell (yellow box). (b) Two nonequivalent ways of doubling unit cell. The first row depicts the unit cell before the doubling; the second row shows the doubled unit cell where the reflection axis has been moved by half of the lattice period; the third row gives the choice of the unit cell that is consistent with original reflection axis. (c) Schematic illustration of a two-dimensional insulator, attached to an ideal lead of finite length. The insulator has reflection matrix $r$, and the lead is terminated with reflection matrix $r^{\prime }$. This setup is used to obtain the boundary state from the scattering approach.

Figure 4

Schematic illustration of the transport setup we consider. We consider two leads (red) that run along the $y$ direction, attached to the system (black). Twisted periodic boundary conditions (corresponding to the wave number $k_x$) are applied in the $x$ direction, with unit-cell size $M$. The whole system (together with the leads) is reflection symmetric with reflection plane parallel to the $y$ axis. Each site (point) contains eight orbitals. The system size $L$ in the $y$ direction is chosen to be large enough that $r\left(k_x\right)$ is unitary.

Figure 5

The dependence of topological invariant (39) on the unit-cell size $M$. The solid line is for $m=3$ (topologically trivial), whereas the dashed line is for $m=1$ (topologically nontrivial). The remaining parameters take the values $L=20$, $\overline{\mathrm{\Delta }}=0.1$, and $\delta =0.03$.

Figure 6

(a) Dispersion over half of the Brillouin zone $k_x\in [0,\pi ]$ for the system depicted in Fig. 4. The inset shows a closeup of the spectra that contains two Dirac-like dispersions, the red (blue) states being localized on the top (bottom) edge of the sample. (b) Phases $\mathrm{\Phi }$ of the eigenvalues of the unitary operator $r\left(k_x\right)r^{\prime }\left(k_x\right)$, where $r\left(k_x\right)$ is the reflection matrix from the upper lead. The inset show that the position of the red Dirac cone matches the position where the phase is zero. The remaining parameters take the values $L=20$, $M=2$, $m=1$, $\overline{\mathrm{\Delta }}=0.1$, and $\delta =0.03$.

Figure 7

Phases $\mathrm{\Phi }$ of the eigenvalues of the unitary operator $r\left(k_x\right)r^{\prime }\left(k_x\right)$, where $r\left(k_x\right)$ is the reflection matrix from the system in the trivial phase. The remaining parameters take the values $L=20$, $M=2$, $m=3$, $\overline{\mathrm{\Delta }}=0.1$, and $\delta =0.03$.