Higher-order topological insulators

TLDR

Three‑dimensional topological (crystalline) insulators have insulating bulks but conduct surface states protected by time‑reversal or spatial symmetries. The authors extend this concept to higher‑order topological insulators that lack gapless surface states yet host protected gapless hinge states. They identify two symmetry classes—chiral higher‑order topological insulators protected by time‑reversal and four‑fold rotation, and helical ones protected by time‑reversal and mirror symmetries—and provide the corresponding topological invariants. The hinge states are chiral modes in the $\mathbb{Z}_2$‑classified case and Kramers‑paired modes in the $\mathbb{Z}$‑classified case; SnTe, surface‑modified Bi$_2$TeI, BiSe, and BiTe are shown to be helical higher‑order topological insulators, and an experimental setup to detect the hinge states is proposed.

Abstract

Three-dimensional topological (crystalline) insulators are materials with an insulating bulk, but conducting surface states which are topologically protected by time-reversal (or spatial) symmetries. Here, we extend the notion of three-dimensional topological insulators to systems that host no gapless surface states, but exhibit topologically protected gapless hinge states. Their topological character is protected by spatio-temporal symmetries, of which we present two cases: (1) Chiral higher-order topological insulators protected by the combination of time-reversal and a four-fold rotation symmetry. Their hinge states are chiral modes and the bulk topology is $\mathbb{Z}_2$-classified. (2) Helical higher-order topological insulators protected by time-reversal and mirror symmetries. Their hinge states come in Kramers pairs and the bulk topology is $\mathbb{Z}$-classified. We provide the topological invariants for both cases. Furthermore we show that SnTe as well as surface-modified Bi$_2$TeI, BiSe, and BiTe are helical higher-order topological insulators and propose a realistic experimental setup to detect the hinge states.