摘要： Topological insulators have to date seen a variety of manifestations. All available realizations of topological insulators, however, share a common feature: their spectral bands are attributed with a nonlocal topological index that is quantized. In this work, we report a new type of insulator exhibiting spectral bands with nonquantized indices, yet robust boundary states. We provide a theoretical analysis based on the quantization of the indices in the corresponding system where the square of the Hamiltonian is taken and exemplify the general paradigm using photonic Aharonov-Bohm cages. Taking the square-root of the Klein-Gordon Hamiltonian has opened up an entirely new realm of physics: the resulting Dirac operator provided a relativistic quantum description of massive spin-1/2 fermionic particles, thus disclosing the fine-structure spectra of atoms or the anomalous Zeeman effect. Along a similar line, we consider a three-band model that can have nonquantized topological indices, but with a spectral symmetry that gives rise to quantized topological invariants when the square of the Hamiltonian is taken. This relation generates the topological characterization of the square-root model and allows us to establish a topological bulk-boundary correspondence with states that can be used as in-gap, protected and controllable qubits. Notably, whereas taking the square of the Dirac operator leads to the topologically trivial Klein-Gordon model, in our case, the squared model maps to an effective Su-Schrieffer-Heeger (SSH) model in its nontrivial phase. Optical settings prove to be ideally suited for realizing various topological phenomena. In this vein, we employ photonic waveguide lattices with effective negative hopping amplitudes to realize a 3-band quasi-1D chain made of photonic Aharonov-Bohm cages that exemplifies our general description. In summary, we predict and demonstrate the physics of a square-root topological insulator, using a photonic platform. Specifically, we show that the AB cages have in-gap states, above bands possessing nonquantized Zak’s phases. Furthermore, we find that these states are robust, both in energy and localization, against symmetry-preserving perturbation and disorder. We show that this robustness stems from the corresponding system where the square of the Hamiltonian is taken and where the bands exhibit quantized topological indices with associated in-gap boundary states.