Hidden multi-topological phases mediated by constrained inter-cell coupling

Abstract The discovery of topological phases of matter and topological boundary states had a tremendous impact on condensed matter physics, photonics, and material sciences, where topological phases are defined via energy bands, described by the topological band theory. However, there are topological materials that cannot be described by this theory, which support non-trivial boundary states but are little-known and largely unexplored. Here, we uncover a new class of topological phases—termed "multi-topological phase" (MTPs)—arising from constrained inter-cell coupling in lattice systems, and experimentally demonstrate them in a photonic platform. The MTP features multiple sets of boundary states, where each set is associated with one distinct topological invariant. Unlike conventional topological phases, the MTP cannot be identified via the original band structure, being a "hidden" topological phase, where the phase transition can occur without band-gap closing. We present typical examples of MTPs in both one- and two-dimensional structures, as well as in indirectly gapped Chern insulators, beyond the regime where the conventional bulk-boundary correspondence predicts the existence of boundary states. Furthermore, we directly observe the MTPs in the first two examples using laser-written photonic lattices. Our work offers a new design strategy for topological materials, paving the way for future exploration and applications in photonics.