Theoretical Analysis of a Non-Quantized Square-Root Topological Insulator using Photonic Aharonov-Bohm Cages

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.