Higher-Order Topological Insulators

Three-dimensional topological (crystalline) insulators are materials with an insulating bulk but conducting surface states that are topologically protected by time-reversal (or spatial) symmetries. 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 spatiotemporal symmetries of which we present two cases: (i) Chiral higher-order topological insulators protected by the combination of time-reversal and a fourfold rotation symmetry. Their hinge states are chiral modes, and the bulk topology is Z(2)-classified. (ii) Helical higher-order topological insulators protected by time-reversal and mirror symmetries. Their hinge states come in Kramers pairs, and the bulk topology is Z-classified. We provide the topological invariants for both cases. Furthermore, we show that SnTe as well as surface-modified Bi(2)Tel, BiSe, and BiTe are helical higher-order topological insulators and propose a realistic experimental setup to detect the hinge states.F.S. and T.N. acknowledge support from the Swiss National Science Foundation (grant number: 200021_169061) and from the European Union's Horizon 2020 research and innovation program (ERC-StG-Neupert-757867-PARATOP). A.M.C. wishes to thank the Aspen Center for Physics, which is supported by NSF grant PHY-1066293, for hosting during some stages of this work. M.G.V. was supported by FIS2016-75862-P national projects of the Spanish Ministry of Economy and Competitiveness. B.A.B. acknowledges support for the analytic work from the Department of Energy (de-sc0016239), Simons Investigator Award, the Packard Foundation, and the Schmidt Fund for Innovative Research. The computational part of the Princeton work was performed under NSF Early-Concept Grants for Exploratory Research grant DMR-1643312, ONR-N00014-14-1-0330, ARO MURI W911NF-12-1-0461, and NSF-MRSEC DMR-1420541