Atomic limit and inversion-symmetry indicators for topological superconductors

Symmetry indicators have proven to be extremely helpful in identifying topological crystalline insulators using symmetry-group representations of their Bloch states. An extension of this approach to superconducting systems requires defining an appropriate atomic limit for Bogoliubov-de-Gennes Hamiltonians. Here, we introduce such a notion of atomic limit and derive a ${\mathbb{Z}}_{2^d}$-valued symmetry indicator for inversion-symmetric superconductors in $d$ dimensions. This indicator allows for a refined topological classification including higher-order phases for systems in the superconducting symmetry classes D and DIII. We further elucidate their bulk-boundary correspondence using Dirac surface theories. Requiring only the normal-state band structure and the superconducting order-parameter symmetry as input, this indicator is well suited for a search of topological superconductors using first-principles calculations.

Figure 1

The BdG spectrum ${\varepsilon }_k^{\text{BdG}}$ obtained from a partially filled (empty) normal band ${\varepsilon }_k$ and an inversion-odd order parameter ${\mathrm{\Delta }}_k$. The inversion eigenvalues at TRIMs $k=\{0,\pi \}$ for two different inversion representations $\mathcal{I}={\tau }_z$ and $\mathcal{I}=-e^{ik}{\tau }_z$ are indicated by unboxed and boxed symbols $\pm$, respectively.

Figure 2

Construction of a 3D second-order TSC in class D from a first-order TSC in class DIII with a Majorana surface cone (a). Adding a TRS-breaking mass term respecting inversion symmetry, ${\tilde{M}}_{\mathit{r}}=-{\tilde{M}}_{-\mathit{r}}$, gaps out the surface except for an inversion-symmetric path, where a chiral Majorana mode remains (b).