Topologically Universal Spectral Hierarchies of Quasiperiodic Systems

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.