Bulk-LDOS Correspondence in Topological Insulators

Seeking the criterion for diagnosing topological phases in real materials has been one of the major tasks in topological physics. Currently, bulk-boundary correspondence based on spectral measurements of in gap topological boundary states and the fractional corner anomaly derived from the measurement of the fractional spectral charge are two main approaches to characterize topologically insulating phases. However, these two methods require a complete band-gap with either in-gap states or strict spatial symmetry of the overall sample which significantly limits their applications to more generalized cases. Here we propose and demonstrate an approach to link the non-trivial hierarchical bulk topology to the multidimensional partition of local-density of states (LDOS) respectively, denoted as the bulk-LDOS correspondence. Specifically, in a finite-size topologically nontrivial photonic crystal, we observe that the distribution of LDOS is divided into three partitioned regions of the sample - the two-dimensional interior bulk area (avoiding edge and corner areas), one-dimensional edge region (avoiding the corner area), and zero-dimensional corner sites. In contrast, the LDOS is distributed across the entire two-dimensional bulk area across the whole spectrum for the topologically trivial cases. Moreover, we present the universality of this criterion by validating this correspondence in both a higher-order topological insulator without a complete band gap and with disorders. Our findings provide a general way to distinguish topological insulators and unveil the unexplored features of topological directional band-gap materials without in-gap states.