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 all
ow 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.
We present the exact solution for the many-body wavefunction of a one-dimensional mixture of bosons and spin-polarized fermions with equal masses and infinitely strong repulsive interactions under external confinement. Such a model displays a large d
egeneracy of the ground state. Using a generalized Bose-Fermi mapping we find the solution for the whole set of ground-state wavefunctions of the degenerate manifold and we characterize them according to group-symmetry considerations. We find that the density profile and the momentum distribution depends on the symmetry of the solution. By combining the wavefunctions of the degenerate manifold with suitable symmetry and guided by the strong-coupling form of the Bethe-Ansatz solution for the homogeneous system we propose an analytic expression for the many-body wavefunction of the inhomogeneous system which well describes the ground state at finite, large and equal interactions strengths, as validated by numerical simulations.
