Faithful derivation of symmetry indicators: A case study for topological superconductors with time-reversal and inversion symmetries

Topological crystalline superconductors have attracted rapidly rising attention due to the possibility of higher-order phases, which support Majorana modes on boundaries in d - 2 or lower dimensions. However, although the classification and bulk topological invariants in such systems have been well studied, it is generally difficult to faithfully predict the boundary Majoranas from the band-structure information due to the lack of well-established bulk-boundary correspondence. Here we propose a protocol for deriving symmetry indicators that depend on a minimal set of necessary symmetry data of the bulk bands and can diagnose boundary features. Specifically, to obtain indicators manifesting clear bulk-boundary correspondence, we combine the topological crystal classification scheme in real space and a twisted equivariant K-group analysis in momentum space. The key step is to disentangle the generally mixed strong and weak indicators through a systematic basis-matching procedure between our real-space and momentum-space approaches. We demonstrate our protocol using an example of two-dimensional time-reversal odd-parity superconductors, where the inversion symmetry is known to protect a higher-order phase with corner Majoranas. Symmetry indicators derived from our protocol can be readily applied to an ab initio database and could fuel material predictions for strong and weak topological crystalline superconductors with various boundary features.

Automated summary

The study proposes a protocol for predicting boundary features using symmetry indicators, combining topological crystal classification in real space and equivariant K-group analysis in momentum space. By systematically matching bases, clear bulk-boundary correspondence indicators are obtained. The protocol is demonstrated using a two-dimensional time-reversal odd-parity superconductor example, showing its applicability for predicting materials with various boundary features in topological crystalline superconductors.