# `Unhinging' the surfaces of higher-order topological insulators and superconductors

## TL;DR

This paper demonstrates that the hinge modes in 3D higher-order topological insulators and superconductors can be gapped with non-Abelian surface topological order, revealing new ways to engineer surface and hinge states.

## Contribution

It introduces a method to gap hinge modes in HOTIs and HOTSCs using non-Abelian topological order while preserving certain symmetries, a novel approach in topological matter.

## Key findings

- Hinge modes can be gapped with non-Abelian topological order.
- Surface topological order breaks time-reversal symmetry on one side.
- Patterned topological order can engineer new surface and hinge states.

## Abstract

We show that the chiral Dirac and Majorana hinge modes in three-dimensional higher-order topological insulators (HOTIs) and superconductors (HOTSCs) can be gapped while preserving the protecting $\mathsf{C}_{2n}\mathcal T$ symmetry upon the introduction of non-Abelian surface topological order. In both cases, the topological order on a single side surface breaks time reversal symmetry, but appears with its time-reversal conjugate on alternating sides in a $\mathsf{C}_{2n}\mathcal T$ preserving pattern. In the absence of the HOTI/HOTSC bulk, such a pattern necessarily involves gapless chiral modes on hinges between $\mathsf{C}_{2n}\mathcal T$-conjugate domains. However, using a combination of $K$-matrix and anyon condensation arguments, we show that on the boundary of a 3D HOTI/HOTSC these topological orders are fully gapped and hence `anomalous'. Our results suggest that new patterns of surface and hinge states can be engineered by selectively introducing topological order only on specific surfaces.