Electronic Energy Spectra of Square and Cubic Fibonacci Quasicrystals

Abstract

Understanding the electronic properties of quasicrystals, in particular the dependence of these properties on dimension, is among the interesting open problems in the field of quasicrystals. We investigate an off-diagonal tight-binding Hamiltonian on the separable square and cubic Fibonacci quasicrystals. We use the well-studied singular-continuous energy spectrum of the 1-dimensional Fibonacci quasicrystal to obtain exact results regarding the transitions between different spectral behaviors of the square and cubic quasicrystals. We use analytical results for the addition of 1d spectra to obtain bounds on the range in which the higher-dimensional spectra contain an absolutely continuous component. We also perform a direct numerical study of the spectra, obtaining good results for the square Fibonacci quasicrystal, and rough estimates for the cubic Fibonacci quasicrystal.

References in corpus (1)

• Electronic energy spectra and wave functions on the square Fibonacci tiling

Cited by in corpus (12)

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• Wave Functions, Quantum Diffusion, and Scaling Exponents in Golden-Mean Quasiperiodic Tilings
• Emergence of quasiperiodic Bloch wave functions in quasicrystals
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• Observation of Log-Periodic Oscillations in the Quantum Dynamics of Electrons on the One-Dimensional Fibonacci Quasicrystal
• The Fibonacci quasicrystal: case study of hidden dimensions and multifractality
• Hexagonal and trigonal quasiperiodic tilings
• Inflation versus projection sets in aperiodic systems: The role of the window in averaging and diffraction
• Triangular and dice quasicrystals modulated by generic 1D aperiodic sequences
• Missing link between the 2D Quantum Hall problem and 1D quasicrystals
• The Spectrum and the Spectral Type of the Off-Diagonal Fibonacci Operator
• Modulated honeycomb lattices and their magnetic properties