The electronic spectrum of a quasiperiodic potential: From the Hofstadter butterfly to the Fibonacci chain

We show that an electronic tight-binding Hamiltonian, defined in a quasiperiodic chain with an on-site potential given by a Fibonacci sequence, can be obtained using a superposition of Harper potentials. Since the spectrum of the Harper equation as a function of the magnetic flux is a fractal set, known as the Hofstadter butterfly, we follow the transformation of the butterfly to a new one that contains the Fibonacci potential and related approximants. As a result, the equation in reciprocal space for the Fibonacci case has the form of a chain with long range interaction between Fourier components. Then, the structure of the resulting spectrum is analyzed by calculating the components in reciprocal space of the related potentials. As an application, the correlator of each potential and some localization properties are obtained.