Higher-order topological insulators and superconductors protected by inversion symmetry

We study surface states of topological crystalline insulators and superconductors protected by inversion symmetry. These fall into the category of “higher-order” topological insulators and superconductors which possess surface states that propagate along one-dimensional curves (hinges) or are localized at some points (corners) on the surface. We provide a complete classification of inversion-protected higher-order topological insulators and superconductors in any spatial dimension for the 10 symmetry classes by means of a layer construction. We discuss possible physical realizations of such states starting with a time-reversal-invariant topological insulator (class AII) in three dimensions or a time-reversal-invariant topological superconductor (class DIII) in two or three dimensions. The former exhibits one-dimensional chiral or helical modes propagating along opposite edges, whereas the latter hosts Majorana zero modes localized to two opposite corners. Being protected by inversion, such states are not pinned to a specific pair of edges or corners, thus offering the possibility of controlling their location by applying inversion-symmetric perturbations such as magnetic field.

Figure 1

Surface states of inversion-protected higher-order TI/TSC in two and three dimensions.

Figure 2

The upper panel illustrates how two chiral modes can be removed without breaking inversion symmetry. The lower panel illustrates how a pair of 0D topological defects (each defects consists of two zero modes at two inversion related points with opposite chirality) can be removed by moving states of opposite chirality towards each other and annihilating them. As a result, the classification of inversion-protected higher-order TIs/TSCs is always ${\mathbb{Z}}_2$ regardless of the classification of the defect.

Figure 3

Illustration of the pattern of surface states obtained by applying a magnetic field to a cubic sample of a time-reversal-symmetric 3D TI. The four possible patterns correspond to the four possible values $(\mathrm{sgn}tan{\varphi }_{xy},\mathrm{sgn}tan{\varphi }_{xz})=(\pm ,\pm )$, where ${\varphi }_{xy}$ and ${\varphi }_{xz}$ correspond to the angle between the $x$ axis and the projection of the field on the $x-y$ and the $x-z$ planes, respectively.

Figure 4

Schematic illustration of the construction of a third-order TSC with two corner states in 3D. The first stage involves combining two (first-order) 3D time-reversal-symmetric TSCs (class DIII) with opposite topological index $\nu =\pm 1$ leading to a second-order time-reversal-symmetric TSC (class DIII) with a helical 1D mode. The second stage involves applying a TRS-breaking perturbation leading to a third-order TSC with broken TRS (class D) with a pair of Majorana zero modes at two antipodal points.

Figure 5

Schematic illustration of the layer construction. Here, we can construct a second-order TI in 3D starting with a 2D strong (first order) TI and adjoining it with 2D layers and their images under inversion. We choose here to do it such that the resulting system has translational symmetry along the stacking direction, thus realizing a 3D weak TI. Breaking the translational symmetry while keeping inversion leads to a second-order TI with a single 1D mode which lives on some inversion-symmetric curve on the surface.