Energy spectra, wave functions, and quantum diffusion for quasiperiodic systems

Yuan, H. Q.; Grimm, U.; Repetowicz, P. and Schreiber, M. (2000). Energy spectra, wave functions, and quantum diffusion for quasiperiodic systems. Physical Review B, 62(23) pp. 15569–15578.



We study energy spectra, eigenstates, and quantum diffusion for one- and two-dimensional quasiperiodic tight-binding models. As for our one-dimensional model system we choose the silver mean or “octonacci” chain. The two-dimensional labyrinth tiling, which is related to the octagonal tiling, is derived from a product of two octonacci chains. This makes it possible to treat rather large systems numerically. For the octonacci chain, one finds singular continuous energy spectra and critical eigenstates, which is the typical behavior for one-dimensional Schrödinger operators based on substitution sequences. The energy spectra for the labyrinth tiling can, depending on the strength of the quasiperiodic modulation, be either bandlike or fractal-like. However, the eigenstates are multifractal. The temporal spreading of a wave packet is described in terms of the autocorrelation function C(t) and the mean-square displacement d(t). In all cases, we observe power laws C(t)∼t-δ and d(t)∼tβ. For the octonacci chain, 0<δ<1, whereas for the labyrinth tiling a crossover is observed from δ=1 to 0<δ<1 with increasing modulation strength. Corresponding to the multifractal eigenstates, we obtain anomalous diffusion with 0<β<1 for both systems. Moreover, we find that the behavior of C(t) and d(t) is independent of the shape and the location of the initial wave packet. We use our results to check several relations between the diffusion exponent β and the fractal dimensions of energy spectra and eigenstates that were proposed in the literature.

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