Abstract
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.
Highlights
Quasicrystals are alloys that show long range order but possess symmetries that prevent them to be crystals
We develop a very straightforward method to calculate the energy-band dispersion for quasicrystals as a function of the extended Brillouin zones
Our strategy allows to do without the rough approximation exploited usually in the literature to assume that the first Brillouin zone can be considered as the fundamental one
Summary
Quasicrystals are alloys that show long range order but possess symmetries that prevent them to be crystals. In this paper we introduce a very efficient method that allow us to calculate the band-energy dispersion for quasicrystals as a function of the (effective) extended BZs and we apply it for the case of a Penrose-tiled quasicrystal This explicit relation between the spectrum and the quasi-momenta allows to pass from the gap labelling to the Brillouin zone labelling. The importance of this step is both conceptual and pratical since the extended BZs are more accessible than the IDOS and can be measured in optics [3], with polaritons [12] or using matter waves [13, 14].
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