Brillouin zone labelling for quasicrystals

Abstract

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

13 pages, 5 figures

References in corpus (6)

• Direct observation of Dirac cones and a flatband in a honeycomb lattice for polaritons
• Orbital superfluidity in the $P$-band of a bipartite optical square lattice
• Fractal energy spectrum of a polariton gas in a Fibonacci quasi-periodic potential
• Pure Point Dynamical and Diffraction Spectra
• Complete Band Gaps in 2D Photonic Quasicrystals
• Quantum criticality in disordered bosonic optical lattices

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• Landau levels in quasicrystals
• Exploiting Anyonic Behavior of Quasicrystals for Topological Quantum Computing
• Equivalence classes of Fibonacci lattices and their similarity properties
• Weyl superconductivity and quasiperiodic Majorana arcs in quasicrystals