Atomic clusters, local isomorphism, and recurrently localized states in quasicrystals

Perfect icosahedral quasicrystals have a structural skeleton based on hierarchical selfsimilar packing of atomic clusters. The related inflation rules constrain both composition and atomic valences to have strictly defined values. Stability of the skeleton requires that bonding electrons are recurrently localized at sites forming selfsimilar isomorphic subsets of the structure and are distributed into `magic' cluster states. The corresponding eigenstates show power law dependences in space which generate a hopping mechanism for conductivity and unexpected physical properties.