  Topological Insulators are a novel state of matter where spectral bands are
characterized by quantized topological invariants. This unique quantized
non-local property commonly manifests through exotic bulk phenomena and
corresponding robust boundary effects. In our work, we report a new type of
topological insulator exhibiting spectral bands with non-quantized topological
properties, but with a quantization that arises in a corresponding system where
the square of the Hamiltonian is taken. We provide a thorough theoretical
analysis as well as an experimental demonstration based on photonic
Aharonov-Bohm cages to highlight the bulk and boundary properties of this
neophyte state of matter.
