6 Density states for a two-dimensional quasiperiodic Penrose lattice

The problem of the quantitative analysis of the electronic states on a two-dimensional quasiperiodic Penrose lattice was solved by Kohmoto and others [11,12]. The approach consists in defining a ``hopping Hamiltonian" for independent electrons on a two-dimensional quasiperiodic Penrose lattice like the one depicted in Fig. 1. The hopping matrix elements are defined as being the same for all pairs of sites connected by a bond and zero otherwise.

Fig. 1. Penrose lattice. The center of the pattern is the center of the configuration, and the ten tiles at the center are the ``seed" from which the pattern was grown.

The time-independent Schrodinger equation for the energy eigenfunction $\Psi(x,y)$ with energy $E$ is

$$E\Psi (x,y) = \sum \Psi (x',y') = H\Psi,$$ (6.1)

where $H$ is the ``Hamiltonian" and the summation is taken over all sites $(x', y')$ connected to site $(x,y)$ by a bond. Electron spins are ignored in this model.

As in the one-dimensional case, equation (6.1) cannot be exactly solved. Therefore, to study the spectrum of the Hamiltonian, a renormalization group technique is introduced. Since the lattice sites appear at the vertices of the rhombus-shaped tiles, the lattice can be divided into two sublattices $A$ and $B$ such that the electron only hops from an $A$ site to a $B$ site, or back. In fact, we need only diagonalize a matrix half the size of the Hamiltonian.

The seed placed at the origin is obviously invariant under the dihedral group $D_5$ about the origin, so we may classify the states by their behavior under a rotation by $2\pi /5.$ Since $r^5 = 1$, we have $r = e^{2\pi in/5},$ where $n=-2,-1,0, 1,2$ and $i=\sqrt{-1}.$

The following results for the density of states were found by Kohmoto and Sutherland [11]:

• There is a central peak of zero width at $E=0$--about 10% of the total number of states.
• These states are strictly localized, and there is a sharp cut-off at a maximum and minimum energy $\pm E_1.$
• The remainder of the states lie in two bands, symmetric about $E = 0,$ separated from the localized states by a finite gap $E_0.$

For a large but finite lattice, inflated $s$ times, it is found that

$$E_1 \approx \phi^3-\phi^{-2(s-1)}+\ldots$$

where $\phi$ is the Golden Mean and again powers of it appear.