The Fibonacci quasicrystal: Case study of hidden dimensions and multifractality

The one-dimensional Fibonacci chain is a toy model central to theoretical studies of the physics of electronic states in quasiperiodic structures. This review surveys the state of the art of methods to study the energy spectra and states of Fibonacci chain models, including exact solutions, renormalization group, perturbation theory, and numerical analysis. Results highlight distinctive properties of Fibonacci chain systems, including projection from higher dimensions, nontrivial topological properties, multifractal states, and hyperuniformity. Questions of current interest include the effects of disorder and interactions and experimental realizations in electronic, cold atom, phononic, and photonic systems.