Self-dual quasiperiodic systems with power-law hopping

We introduce and explore a family of self-dual models of single-particle motion in quasiperiodic potentials, with hopping amplitudes that fall off as a power law with exponent p. These models are generalizations of the familiar Aubry-Andr\'e model. For large enough p, their static properties are similar to those of the Aubry-Andr\'e model, although the low-frequency conductivity in the localized phase is sensitive to p. For p2.1 the Aubry-Andr\'e localization transition splits into three transitions; two distinct intermediate regimes with both localized and delocalized states appear near the self-dual point of the Aubry-Andr\'e model. In the intermediate regimes, the density of states is singular continuous in much of the spectrum, and is approximately self-similar: states form narrow energy bands, which are divided into yet narrower subbands; we find no clear sign of a mobility edge. When $p<1$, localized states are not stable in random potentials; in the present model, however, tightly localized states are present for relatively large systems. We discuss the frequency-dependence and strong sample-to-sample fluctuations of the low-frequency optical conductivity, although a suitably generalized version of Mott's law is recovered when the power law is slowly decaying. We present evidence that many of these features persist in models that are away from self-duality.