  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.
