Quasiperiodic tilings in a magnetic field

We study the electronic properties of a two-dimensional quasiperiodic tiling, the isometric generalized Rauzy tiling, embedded in a magnetic field. Its energy spectrum is computed in a tight-binding approach using the recursion method. As for the square lattice, a rich band structure is obtained when the magnetic flux per elementary plaquette is varied. Then, we study the quantum dynamics of wave packets and discuss the influence of the magnetic field on the diffusion and spectral exponents. We show that contrary to the zero field case where the propagation is faster in a periodic structure than in a quasiperiodic one, the diffusion exponent may be larger in the Rauzy tiling than in the square lattice for some incommensurate fluxes. Finally, we consider a quasiperiodic superconducting wire network with the same geometry and we determine the critical temperature as a function of the magnetic field. A non-trivial cusp-like structure arises that reveals the real importance of the quasiperiodic order.