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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.

Designing reconfigurable metasurfaces that can dynamically control scattered electromagnetic waves and work in the near-infrared (NIR) and optical regimes remains a challenging task, which is hindered by the static material property and fixed structures. Phase change materials (PCMs) can provide high contrast optical refractive indexes at high frequencies between amorphous and crystal states, therefore are promising as feasible materials for reconfigurable metasurfaces. Here, we propose a hybrid metasurface that can arbitrarily modulate the complex amplitude of incident light with uniform amplitude and full $2\pi$ phase coverage by utilizing composite concentric rings (CCRs) with different ratios of gold and PCMs. Our designed metasurface possesses a bi-functionality that is capable of splitting beams or generating vortex beams by thermal switching between metal and semiconductor states of vanadium oxide (VO2), respectively. It can be easily integrated into low loss photonic circuits with an ultra-small footprint. Our metadevice serves as a novel paradigm for active control of beams, which may open new opportunities for signal processing, memory storage, holography, and anti-counterfeiting.