Dresden 2009 – scientific programme

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MM: Fachverband Metall- und Materialphysik

MM 40: Quasicrystals I

MM 40.4: Talk

Thursday, March 26, 2009, 11:00–11:15, IFW D

Scaling Behavior of the Participation Ratio in d-dimensional Quasiperiodic Models based on the Octonacci Sequence — •Stefanie Thiem and Michael Schreiber — Institut für Physik, Technische Universität Chemnitz, 09107 Chemnitz, Germany

The characteristics of quasicrystals are determined by the nature of their eigenstates. Studying the scaling behavior of the participation ratio p is a practicable way to obtain the localization properties of these wave functions. We investigate d-dimensional quasiperiodic models based on the octonacci sequence and prove that the scaling exponent is independent of the dimension for these models.

The eigenstates of the octonacci chain are obtained by numerical calculations for a tight-binding model. Higher dimensional eigenstates of the associated labyrinth tiling are constructed then by a product approach from the one-dimensional results, allowing the numerical consideration of large systems up to 10¹¹ sites. We give explicit construction rules for the energies E and wave functions Φ_r in d dimensions.

The participation ratio p= 1V [ ∑_r |Φ_r|⁴ ]⁻¹ is studied in one, two, and three dimensions. It is a known result that p scales for fractal states with p ∼ V^−γ (0 < γ < 1) in the number of sites V. We calculated the scaling exponent γ of the average participation ratio ⟨ p ⟩ over all eigenstates for different dimensions and various strengths of the coupling parameter v (0<v<1). These results suggest that γ(v) is independent of the dimension d. We also give a mathematical proof for the dimension independence of this scaling exponent using the product structure of the labyrinth tiling.