From the Hofstadter to the Fibonacci butterfly

Abstract

We show that the electronic spectrum of a tight-binding Hamiltonian defined in a quasiperiodic chain with an on-site potential given by a Fibonacci sequence, can be obtained as a superposition of Harper potentials. The electronic spectrum of the Harper equation is a fractal set, known as Hofstadter butterfly. Here we show that is possible to construct a similar butterfly for the Fibonacci potential just by adding harmonics to the Harper potential. As a result, the equations in reciprocal space for the Fibonacci case have the form of a chain with a long range interaction between Fourier components. Then we explore the transformation between both spectra, and specifically the origin of energy gaps due to the analytical calculation of the components in reciprocal space of the potentials. We also calculate some localization properties by finding the correlator of each potential.

7 pages, 4 figures

References in corpus (4)

• Noncollinear magnetic order in quasicrystals
• Hofstadter butterfly for a finite correlated system
• Phase Diagram for the Hofstadter butterfly and integer quantum Hall effect in three dimensions
• Electronic properties of the 1D Frenkel-Kontorova model

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• Chern and Majorana Modes of Quasiperiodic Systems
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• Majorana Zero-Energy Mode and Fractal Structure in Fibonacci-Kitaev Chain
• Spectral butterfly, mixed Dirac-Schrödinger fermion behavior and topological states in armchair uniaxial strained graphene
• Quantum Emulation of Extreme Non-equilibrium Phenomena with Trapped Atoms
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• Topological critical states and anomalous electronic transmittance in one dimensional quasicrystals
• Universal Route for the Emergence of Exceptional Points in PT-Symmetric Metamaterials with Unfolding Spectral Symmetries
• A higher-dimensional quasicrystalline approach to the Hofstadter and Fibonacci butterflies topological phase diagram and band conductance: symbolic sequences, Sturmian coding and self-similar rules at all magnetic fluxes
• Noise, delocalization, and quantum diffusion in one-dimensional tight-binding models
• Quantum superballistic diffusion in presence of white noise
• Critical region of topological trivial and nontrivial phases in interacting Kitaev chain with spatially varying potentials