Bulk-interface correspondence from quantum distance in flat band systems

The key notion of topological analysis is the bulk-boundary correspondence, which provides us with direct access to the topological structure of a Bloch wave function via the existence of boundary or interface modes. While only the topology of the wave function has been considered relevant to boundary modes, we demonstrate that another geometric quantity, the so-called quantum distance, can also host a bulk-interface correspondence. We consider a generic class of two-dimensional flat band systems, where the flat band has a parabolic band-crossing with another dispersive band. While such flat bands are known to be topologically trivial, we show that the nonzero maximum quantum distance between the eigenstates of the flat band around the touching point guarantees the existence of boundary modes at the interfaces between two domains with different chemical potentials or different maximum quantum distances. Moreover, the maximum quantum distance can predict even the explicit form of the dispersion relation and decay length of the interface modes.