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We study the emergence of quasiperiodic Bloch wave functions inquasicrystals, employing the one-dimensional Fibonacci model as a test case. Wefind that despite the fact that Bloch functions are not eigenfunctionsthemselves, superpositions of relatively small numbers of nearly degenerateeigenfunctions give rise to extended quasiperiodic Bloch functions. Thesefunctions possess the structure of earlier ancestors of the underlyingFibonacci potential, and it is often possible to obtain different ancestors asdifferent superpositions around the same energy. There exists an effectivecrystal momentum that characterizes these ancestors, which is determined by themean energy of the superimposed eigenfunctions, giving rise to an effectivedispersion curve. We also find that quasiperiodic Bloch functions do emerge aseigenfunctions when weak disorder is introduced into the otherwise perfectquasiperiodic potential. These theoretical results may explain a number ofexperimental observations, and may have practical consequences on emergingtheories of band topology and correlated electrons in quasicrystals.

Emergence of quasiperiodic Bloch wave functions in quasicrystals