Abstract
Being Wannierizable is not the end of the story for topological insulators. We introduce a family of topological insulators that would be considered trivial in the paradigm set by the tenfold way, topological quantum chemistry, and the method of symmetry-based indicators. Despite having a symmetric, exponentially localized Wannier representation, each Wannier function cannot be completely localized to a single primitive unit cell in the bulk. Such multicellular topology is shown to be neither stable nor fragile, but delicate; i.e., the topology can be nullified by adding trivial bands to either valence or conduction band.
Highlights
Introduction.—Two themes have indelibly shaped the paradigm of topological insulators (TIs) and couched how topological properties are discussed, modeled, and measured
The strongest form of stability is the notion of stable equivalence introduced by K theory [8,9,10,11,12], where the bulk or surface topological invariant of a valence subspace is immune to addition of trivial bands
A distinct notion that we introduce here is delicate topology, where the topological property can be nullified by adding trivial bands to either valence or conduction subspace [Fig. 1(b)]
Summary
Aleksandra Nelson ,1,* Titus Neupert ,1 Tomáš Bzdušek ,2,1 and A. Exponentially localized Wannier representation, each Wannier function cannot be completely localized to a single primitive unit cell in the bulk Such multicellular topology is shown to be neither stable nor fragile, but delicate; i.e., the topology can be nullified by adding trivial bands to either valence or conduction band. All stably equivalent and fragile TIs present an obstruction to a WF representation It has been further argued through equivariant vector bundle theory that such Wannier obstructions represent a robust property of a valence subspace summed with an arbitrary conduction subspace [19], and such obstruction cannot exist for delicate topological insulators. We pick one representative eigenvalue from each ladder and define their sum (modulo integer) to be the (charge) polarization Pðk⊥Þ, in accordance with the geometric theory of polarization [31,32]
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