Topologically Universal Spectral Hierarchies of Quasiperiodic Systems

Abstract

Topological properties of energy spectra of general one-dimensional quasiperiodic systems, describing also Bloch electrons in magnetic fields, are studied for an infinity of irrational modulation frequencies corresponding to irrational numbers of flux quanta per unit cell. These frequencies include well-known ones considered in works on Fibonacci quasicrystals. It is shown that the spectrum for any such frequency exhibits a self-similar hierarchy of clusters characterized by universal (system-independent) values of Chern topological integers which are exactly determined. The cluster hierarchy provides a simple and systematic organization of all the spectral gaps, labeled by universal topological numbers which are exactly determinable, thus avoiding their numerical evaluation using rational approximants of the irrational frequency. These numbers give both the quantum Hall conductance of the system and the winding number of the edge-state energy traversing a gap as a Bloch quasimomentum is varied.

REXTEX4, 6 pages, 3 figures

References in corpus (3)

• Observation of Topological Phase Transitions in Photonic Quasicrystals
• Quantum Hall effect and the topological number in graphene
• Periodic Landau gauge and Quantum Hall effect in twisted bilayer graphene

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• Theory of quantum oscillations in quasicrystals: Quantizing spiral Fermi surfaces
• Topological quantum optical states in quasiperiodic cold atomic chains
• Topological Phase Transitions from Harper to Fibonacci Crystals
• Topological phonon polariton enhanced radiative heat transfer in bichromatic nanoparticle arrays mimicking Aubry-André-Harper model
• Floquet Systems with Hall Effect: Topological Properties and Phase Transitions
• Topological states in the Hofstadter model on a honeycomb lattice
• Edge modes in the Hofstadter model of interacting electrons
• Topological Properties of Adiabatically Varied Floquet Systems
• The quantum Hall effect at a weak magnetic field