Topological insulators I: Theory

We present a new topological class of insulating Hamiltonians, which we dub boundary-obstructed delicate topological insulators (BODTIs). These are defined as Hamiltonians whose bulk energy bands appear to be topologically trivial, but whose Wannier bands exhibit a delicate topological obstruction. BODTIs are boundary obstructed; this means they are deformable to atomic limit in the absence of boundaries, but not in the presence of suitably chosen boundary terminations. If the boundary terminations are sharp, BODTIs exhibit in-gap states reminiscent of higher-order topological insulators. Topological transition from a BODTI to the trivial phase occurs through a surface-only gap closing.

We formulate a general principle for constructing $d$-dimensional BODTIs by layering a pair of alternating $(d-1)$-dimensional delicate topological insulators (DTIs), i.e., of models characterized by delicate topology of energy bands. Recalling the paradigm model of two-dimensional DTI, called returning Thouless pump (RTP) insulator, we apply the layer construction to introduce a paradigm model of a three-dimensional BODTI. We call this model the layered RTP insulator. We find that by virtue of the layer construction, the anomalous $\pi$ Berry phase associated with edges of 2D RTP insulator becomes elevated to an anomalous $\pi$ Berry phase associated with hinges of the 3D layered RTP insulator. We illustrate the general principle for obtaining BODTIs through the layer construction by applying the technique to other examples of DTIs, including a one-dimensional mirror-symmetric chain and a two-dimensional dartboard Chern insulator.