Dynamical properties of two-dimensional quasicrystals

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Abstract
Electron and phonon spectra, densities and integrated densities of states, and dynamic-response functions are investigated for a variety of two-dimensional (2D) quasicrystals. The dynamical properties of a Fibonacci chain, extended both periodically and quasiperiodically into a second dimension, are obtained using an exact convolution relationship involving only one-dimensional quantities. For a particular on-site model we find two successive transitions in the spectrum of the quasiperiodically extended Fibonacci chain (2D Fibonacci quasilattice) as a function of coupling strength: from finite to infinite band number and from finite to zero total bandwidth. The periodically extended Fibonacci chain (Fibonacci superlattice) always has a finite band number and nonzero total bandwidth. Furthermore, in each case we find the plateaux in the integrated density of states to follow a gap-labeling rule, as is the case in one dimension. We also present for the first time a study of the spectrum of a particular 2D Penrose lattice generated via the Robinson tiling approach. Our study of the spectrum is made by diagonalizing the matrices of finite-size samples, our results indicating that the gap number is always finite, irrespective of the coupling strengths. Plots of the integrated electronic and vibrational densities of states indicate a similar gap-labeling rule as was found for the 2D Fibonacci superlattice and quasilattice. Surface plots of the full wave-vector and frequency-dependent response functions (dynamic-structure factor) are given for both the 2D Fibonacci and Penrose lattices, and show very rich structure. That for the Fibonacci quasilattice can be interpreted by comparison with an exact analytic expression obtained in the equal-coupling limit, which indicates that the response is strongly peaked along well-defined curves in the frequency-wave vector plane, unlike that for the Penrose lattice, which shows no apparent regularity, except in the long-wavelength limit.

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