Brillouin zone labelling for quasicrystals

We propose a scheme to determine the energy-band dispersion of quasicrystals that does not require any periodic approximation and directly provides the correct structure of the extended Brillouin zones. From the gap labelling viewpoint, this allows us to transpose the measurement of the integrated density of states with that of the effective Brillouin zone areas, which are uniquely determined by the position of the Bragg peaks. Moreover, we show that the Bragg vectors can be determined by stability analysis of the law of recurrence used to generate the quasicrystal. Our analysis of gap labelling in the quasi-momentum space opens the way to experimental proof of gap labelling itself within the framework of optics experiments, polaritons, or with ultracold atoms.