Higher-order topological insulators with double band-inversions

Symmetry indicators and band-inversion play pivotal roles in determining the topology of bulk bandstructures. Here, we discuss a class of higher-order topological insulators that do not accommodate symmetry indicators, but continue to support a notion of band-inversion. We identify a general principle for constructing multi-band models of such insulators with an even number of band-inversions, and propose suitable topological invariants. We also discuss the surface states they support and establish a bulk-boundary correspondence.