Measuring topological invariants from generalized edge states in polaritonic quasicrystals

We investigate the topological properties of Fibonacci quasicrystals using cavity polaritons. Composite structures made of the concatenation of two Fibonacci sequences allow one to investigate generalized edge states forming in the gaps of the fractal energy spectrum. We employ these generalized edge states to determine the topological invariants of the quasicrystal. When varying a structural degree of freedom (phason) of the Fibonacci sequence, the edge states spectrally traverse the gaps, while their spatial symmetry switches: The periodicity of this spectral and spatial evolution yields direct measurements of the gap topological numbers. The topological invariants that we determine coincide with those assigned by the gap-labeling theorem, illustrating the direct connection between the fractal and topological properties of Fibonacci quasicrystals.