Order to Disorder in Quasiperiodic Systems

Four decades since the discovery of quasicrystals, the material properties arising from quasiperiodic microgeometry remain an active topic of theoretical intrigue and engineering utili ty. Here we introduce a class of Moiré type quasiperiodic media with novel macroscopic behavior. As the microgeometry of a Moiré system is tuned, the transport properties switch from those of ordered to randomly disordered materials in a fashion which parallels the Anderson localization transition, even though there are no scattering or interference effects at play. This transition is evident within an integral representation that applies broadly to the effective electrical, thermal, elastic and optical properties of two-phase composite media. This representation distills the relationship between microgeometry and bulk transport into the spectral characteristics of an operator, which is analogous to the Hamiltonian in quantum transport phenomena. Periodic media display sharp resonances, band gaps, and spatially extended eigenstates separated by "mobility edges" of localized states. As we tune system parameters to aperiodicity and disorder, eigenstates become more uniformly extended, and level repulsion increases as the spectral properties transition toward obeying universal Wigner-Dyson statistics.