Mobility edges and multifractal properties in a one-dimensional system with three incommensurate frequencies

Almost all models for one-dimensional incommensurate systems studied up to now have two incommensurate frequencies, and the electronic spectra for the most studied models have shown no mobility edge (the Aubry model) or at the most one (the Soukoulis-Economou model). Influenced by the classical work of Newhouse, Ruelle, and Takens that developed quasiperiodic three-frequency flow to chaotic flow and by the recent work by Casati, Guarneri, and Shepelyansky on a kicked rotator with three incommensurate scales, we present here a one-dimensional model with three incommensurate frequencies generated from the concept of the spiral mean due to Kim and Ostlund. A way to analyze the structure of the electronic spectrum by rational approximants to the spiral mean with common denominators is indicated. For small (large) values of the strength of the on-site potential all eigenstates are extended (localized), but for intermediate values of the strength we find, as opposed to the above-mentioned models, up to four different mobility edges. In contrast to what in analogy with disordered systems is normally expected, the eigenstates in the central part of the spectrum are easier to localize than the eigenstates in some of the side bands, when the amplitude of the incommensurate potential is increased from zero. A multifractal analysis of the structure of the eigenstates is presented and shows similar properties as have been previously found for the Aubry model. In agreement with several other recent authors, we find some anomalies using the standard version of the thermodynamic approach of the multifractal formalism. The possibility of extended states having multifractal properties at large length scales is discussed.