Classification of topological insulators and superconductors in the presence of reflection symmetry

We discuss a topological classification of insulators and superconductors in the presence of both (nonspatial) discrete symmetries in the Altland-Zirnbauer classification and spatial reflection symmetry in any spatial dimensions. By using the structure of bulk Dirac Hamiltonians of minimal matrix dimensions and explicit constructions of topological invariants, we provide the complete classification, which still has the same dimensional periodicities with the original Altland-Zirnbauer classification. The classification of reflection-symmetry-protected topological insulators and superconductors depends crucially on the way reflection symmetry operation is realized. When a boundary is introduced, which is reflected into itself, these nontrivial topological insulators and superconductors support gapless modes localized at the boundary.

Figure 1

For topological insulators and superconductors that are protected by reflection symmetry, the correspondence between gapless surface states and bulk topology holds only when the surface that reflects to itself is chosen. The figure shows that nontrivial bulk topology guarantees gapless states in the self-reflected surfaces. Furthermore, gapped states in the non-self-reflected surfaces does not imply trivial bulk topology.

Figure 2

(a) The energy spectrum for the $y$ edge states near $E=0$ as a function of a translational symmetry breaking term ($c$) and (b) the entanglement spectrum around the entanglement eigenvalue 1/2 as a function of $c$. We consider that the open boundary condition is the $y$ discretion. The parameters are set to $m=-1$, $\delta =\arcsin \left(0.1\pi \right)$, and $\eta =0.05\pi =0.157$. As $c>0.157$, the edge states start being gapped and the entanglement midgap states begin to move out from $0.5$. However, as $c<0.157$, $E_{\min }$ is nonzero and the entanglement eigenvalue is not $0.5$ due to the finite size effect in the numerical stimulation.