Abstract
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 the real space and a twisted equivariant K group analysis in the 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 ab initio database and could fuel material predictions for strong and weak topological crystalline superconductors with various boundary features.
Highlights
Topological crystalline superconductors (TCSCs) [1,2,3,4,5,6,7,8,9,10,11,12,13,14] have attracted rapidly rising attention since certain crystalline symmetries can enlarge the classifications and protect new types of topological superconductors
By systematically writing down symmetry-allowed mass terms that couple the two 2D topological superconductor in AZ class DIII (2DTSC), we find that there exists no mass term that can fully gap out the Majorana edge modes, and the mass terms with the least level of spatial modulation still leave two leftover Majorana Kramers pairs related by inversion
After performing the basis-matching procedure for the above topological crystal states, which are the generators for the entire topological superconductor group CT SC (G), we have shown that the map φ takes the form of Eq (70)
Summary
Topological crystalline superconductors (TCSCs) [1,2,3,4,5,6,7,8,9,10,11,12,13,14] have attracted rapidly rising attention since certain crystalline symmetries can enlarge the classifications and protect new types of topological superconductors. Given that weak and strong phases carry different boundary features, symmetry indicators suffering from such mixtures fail to serve as boundary diagnostics and have limited application to actual material searches Such an issue has been raised in prior work for a certain case [17], but a protocol that can be systematically generalized to general space groups is still absent. By bridging the topological crystal approach with this K-theory analysis, we provide a systematic protocol to identify the canonical bases for the symmetry indicator group, in which each indicator is purely strong or purely weak and corresponds to unambiguous boundary types. These examples include models with momentum-independent and -dependent inversion operators, as well as both minimal models and ab initio-based models for monolayer WTe2
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have