Abstract
We study the bulk-boundary correspondence for topological crystalline phases, where the crystalline symmetry is an order-two (anti)symmetry, unitary or antiunitary. We obtain a formulation of the bulk-boundary correspondence in terms of a subgroup sequence of the bulk classifying groups, which uniquely determines the topological classification of the boundary states. This formulation naturally includes higher-order topological phases as well as topologically nontrivial bulk systems without topologically protected boundary states. The complete bulk and boundary classification of higher-order topological phases with an additional order-two symmetry or antisymmetry is contained in this work.
4 More- Received 7 May 2018
DOI:https://doi.org/10.1103/PhysRevX.9.011012
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
A central paradigm for topological materials is the correspondence between boundary properties and internal behavior. In crystalline materials, this link exists for certain orientations of the crystal boundary. Recent work suggests that this constraint could be relaxed if one considers higher-order topological phases, where the dimension of the conducting boundary region is less than the dimension of the crystal surface. Examples include 1D modes along crystal edges or Majorana states at corners of topological superconductors. Here, we present a full classification of such higher-order topological phases for a subset of crystalline materials.
We mathematically analyze crystalline materials with one of three types of crystal symmetries: mirror reflection, inversion, or rotation by . Our classification of the topological phases takes the mathematical form of a “subgroup sequence,” where the position in the sequence contains information about the dimension of the anomalous boundary states. In particular, the obtained sequence shows that materials with certain symmetries cannot have topologically protected 1D modes or Majorana corner states on their boundaries, which helps narrow down the search for candidate materials.
Our results could spur applications of higher-order topological phases. Higher-order topological insulators may be used in scattering-free electron transport, whereas their superconducting counterparts could facilitate storage of quantum information using Majorana corner states.