Anomalous Quantum Diffusion and Conductivity of Quasicrystals

A phenomenological diffusion law $L(t)\propto t^{\beta}$, where L(t) measures the spreading of a wave-packet in a time t, is assumed for perfect quasicrystals. We show that it affects their conductivity with striking differences compared to the case of periodic metals. In the absence of defects the dissipative part of conductivity is non zero even at low frequencies contrary to the case of crystals. Also if $\beta<1/2$ the d.c. conductivity increases when disorder increases and the so-called Drude peak, characteristic of metals, is replaced by a dip . Experimental results are briefly discussed.