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Higher-order bulk-boundary correspondence for topological crystalline phases

Luka Trifunovic (Freie Universität Berlin)


It is less than a year ago that the concept of higher-order topological insulators and superconductors was mentioned for the first time in print (see Viewpoint: and the accompanying publications). Since then, the concept of higher-order topological phases has been recognized to be key to the description of the boundary phenomenology of topological crystalline insulators and superconductors.

Although higher-order boundary phenomenology can exist without a topologically nontrivial bulk band structure, the recent excitement stems from the observation that higher-order boundary states can also be the consequence of (and protected by) a nontrivial bulk topology if there are additional crystalline symmetries. In this picture, higher-order topological phases are not a new kind of topological phases, but they are a new type of boundary manifestation of nontrivial bulk topology.

In this talk, I will present 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. I will also present the results for the complete bulk and boundary classification of higher-order topological phases protected by an order-two (mirror, two-fold rotation, inversion) crystalline symmetry or antisymmetry (i.e. magnetic symmetry).