Crystalline topological phases without symmetry indicators

Topological insulating phases of matter are quantum phases that cannot be deformed to atomic insulators. They generically feature protected anomalous surface states, which can be used for applications in spintronics and quantum computation.

Recently, two new types of topological phases protected by crystalline symmetries have been discovered. The first are higher-order topological insulators, featuring conducting states on the hinges [1], recently observed in bismuth [2]. The other are fragile topological states, which become trivial when occupying additional bands. Twisted bilayer graphene may be a realization of a fragile phase, which may help to explain superconductivity in this system [3].

Usually, crystalline topology can be detected by computing symmetry indicators. However, there exist topological phases which cannot be detected in this way, and we present the first classification of such states [4]. We show these crystalline invariants are also well-defined in quantum spin Hall systems and use this to construct a hybrid-order weak TI [5], which is both a weak and higher-order TI.

[1] Kooi et al. Phys. Rev. B 98, 245102
[2] Schindler et al. Nature Physics 14, 918–924 (2018)
[3] Zou et al. Phys. Rev. B 98, 085435
[4] Kooi et al. arXiv:1906.08695
[5] Kooi et al. arXiv:1908.00879