Fate of higher-order topological corner modes in a mirror-symmetric photonic lattice without chiral symmetry

Abstract Higher-order topological states exhibit topologically protected lower dimensional boundary states such as corner states in contrast to the conventional first-order topological states. In a crystalline, the bulk topology information of higher-order topological states is encoded in the Bloch wave functions. It is usually deemed that the bulk-corner correspondence always holds as long as Bloch wave functions remain unchanged. In this paper we show that the corner modes can delocalize into the bulk if the chiral symmetry is broken while both the Bloch wave functions and the topological number of each energy band remain invariant. We take two typical higher-order topological insulators consisting of 2D Wannier-type second-order topological insulator and qaudrupole insulator as examples. With involved longer-range hopping that break the chiral symmetry but preserve the mirror symmetry, the Bloch wave functions do not change and the topological number for each band remains invariant. However, the topological corner modes delocalize into the bulk if the strength of the chiral symmetry breaking terms, namely the longer-range hopping amplitude, exceeds a critical value. This work provides a new insight into the bulk-boundary correspondence of higher-order topological states.