Energy levels and their correlations in quasicrystals

Anu Jagannathan (1) , Frédéric Piéchon (1)

1 LPS - Laboratoire de Physique des Solides

Quasicrystals can be considered, from the point of view of their electronic properties, as being intermediate between metals and insulators. In the theoretical studies we consider in this review, the quasicrystals are described by a pure hopping tight binding model on simple tilings. We focus on spectral properties, which we compare with those of other complex systems, in particular, the Anderson model of a disordered metal. We discuss “strong” and “weak” quasicrystals, which are described by different universal laws. We find similarities and universal behavior, but also significant differences between quasiperiodic models and models with disorder. Like weakly disordered metals, the quasicrystal can be described by the universal level statistics that can be derived from random matrix theory. These level statistics are only one aspect of the energy spectrum, whose very large fluctuations can also be described by a level spacing distribution that is log-normal. An analysis of spectral rigidity shows that electrons diffuse with a bigger exponent (super-diffusion) than in a disordered metal. Adding disorder attenuates the singular properties of the perfect quasicrystal, and leads to improved transport. Spectral properties are also used in computing conductances of such systems, and to attempt to resolve the experimental enigmas such as whether quasicrystals are intrinsically conductors, and if so, how conductances depend on the structure.

Mots clés: Physical Sciences

Anu Jagannathan, Frédéric Piéchon. Energy levels and their correlations in quasicrystals. Philosophical Magazine, 2008, 87 (16), pp.2389-2415. ⟨10.1080/14786430701196990⟩. ⟨hal-00513816⟩