‘Unhinging’ the surfaces of higher-order topological insulators and superconductors

We show that the chiral Dirac and Majorana hinge modes in three-dimensional higher-ordertopological insulators (HOTIs) and superconductors (HOTSCs) can be gapped while preservingthe protectingC2nTsymmetry upon the introduction of non-Abelian surface topological order.In both cases, the topological order on a single side surface breaks time reversal symmetry, butappears with its time-reversal conjugate on alternating sides in aC2nTpreserving pattern. In theabsence of the HOTI/HOTSC bulk, such a pattern necessarily involves gapless chiral modes onhinges betweenC2nT-conjugate domains. However, using a combination ofK-matrix and anyoncondensation arguments, we show that on the boundary of a 3D HOTI/HOTSC these topologicalorders are fully gapped and hence ‘anomalous’. Our results suggest that new patterns of surface andhinge states can be engineered by selectively introducing topological order only on specific surfaces