Diagnosis for Band Topology under Crystal Symmetries Based on Bulk Measurements

Topological gapped crystalline materials are a class of topological materials characterized by invariants protected by spatial symmetries, allowing for the emergence of robust boundary states. The band topology of gapped materials with crystal symmetries can be theoretically labeled by symmetry-indicator invariants, such as the Fu-Kane parity criterion for inversion-symmetric topological gapped materials. Here, we propose an approach to measuring these invariants and diagnosing topological gapped crystalline materials, which is experimentally validated in two-dimensional photonic crystals with finite size. Notably, this diagnosis approach relies exclusively on the measured field distribution within the bulk region, without necessitating the energy band diagram possessing complete band gaps, even in the absence of in-gap states in real materials. Our study provides an approach for measuring symmetry-indicator invariants and paves the way for experimentally diagnosing various gapped topological phases originating from crystalline symmetries.