Connecting higher-order topological insulators to lower-dimensional topological insulators

In recent years, the role of crystal symmetries in enriching the variety of TIs have been actively investigated. Higher-order TIs are a new type of topological crystalline insulators that exhibit gapless boundary states whose dimensionality is lower than those on the surface of conventional TIs. In this paper, relying on a concrete tight-binding model, we show that higher-order TIs can be smoothly connected to conventional TIs in a lower dimension without the bulk-gap closing or symmetry breaking. Our result supports the understanding of higher- order TIs as a stacking of lower-dimensional TIs in a way respecting all the crystalline symmetry.