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7 Wave functions on a Fibonacci lattice

The properties of the wave functions for one-dimensional quasiperiodic discrete Schrodinger equations have been thoroughly studied by Kohmoto and others [13,14,15]. In quasiperiodic systems, the wave functions exhibit three possibilities:
1) localized wave functions (dense-point spectrum);
2) extended wave functions (absolutely continuous spectrum);
3) critical wave functions (singular continuous spectrum).

Wave functions of this last type are further classified into two types: selfsimilar and non-self-similar. It has been shown that a wave function which corresponds to a cycle of the renormalization group map is self-similar. On the other hand, a wave function which corresponds to a bounded chaotic orbit is not self-similar.

For several forms of the potential $V(x),$ the properties of the wave functions and the spectra have been thoroughly investigated [16,17]. The model defined by

\begin{displaymath}
V(x) = \lambda \cos (2\pi x),
\end{displaymath} (7.1)

which is known as the Harper model,1 undergoes a metal-insulator transition at $\lambda = 2.$ That is, for $\lambda
< 2,$ all the states are extended, while for $\lambda
> 2,$ all the states are localized. At $\lambda = 2,$ all the states are critical.

In the model defined by

\begin{displaymath}
V(x) = \lambda \tan (2\pi x),
\end{displaymath} (7.2)

all the states are localized for any value of $\lambda.$

An important model is the Fibonacci model, defined by $\frac{{\sqrt 5-1}}{2}=\frac{1}{\phi}$ (the reciprocal value of the Golden Mean) and

\begin{displaymath}
V(x) = \left\{
\begin{array}{cl}
-\lambda & m-1/\phi<x\le m \\
\lambda & m<xm+1-1/\phi, \\
\end{array} \right.
\end{displaymath}

where $m$ is an integer. In this model, all the states are critical, irrespective of the value of $\lambda.$


next up previous
Next: 8 Behavior of a Up: SPINADEL Previous: 6 Density states for
2003-06-05