Spectra and gap amplification for systems with two widely different incommensurate periodicities

We derive analytically the spectrum for the Schrödinger equation for quasiperiodic systems with two length scales: one large ‘‘macroscopic’’ scale [e.g., a cos(2πx/λ)] and one small ‘‘microscopic’’ scale [e.g., v cos(2πx)]. The phase diagram includes regimes with exponentially narrow gaps due to the slowly varying potential, regimes where the rapidly varying potential amplifies these narrow gaps, and regimes with exponentially narrow ‘‘Landau bands.’’ The full ‘‘devil’s-staircase’’ spectrum with gaps at wave vectors q=mπ+nπ/λ develops in a hierarchical manner as a increases. The results apply to systems with superlattices, to celestial orbits with two periodic perturbations, to systems with slowly varying lattice distortions, and, in particular, to quasi-one-dimensional magnets such as bis(tetramethyltetraselenafulvalene) perchlorate [(TMTSF${)}_2$ ${\mathrm{ClO}}_4$] in magnetic fields, where our findings may provide insight into the experimentally observed cascade of phase transitions.