Metals, Insulators, and Beyond in Quasi-Periodic Systems with Power-Law Hoppings

Recent analytic methods are applied to single-frequency quasi-periodic systems with power-law hoppings to generate an exact phase diagram. Taking advantage of Avila's Global Theory, we show the existence of a metallic phase and produce sharp bounds in the phase diagram. We then introduce self-dual power-law hopping models as a perturbation to the single-frequency operators to show the existence of an insulating phase both in the self-dual models and the single-frequency operators. This insulating phase survives via an exact perturbative expansion to the metallic transition at the cost of countably many gaps closings in the spectrum, generating a hierarchy of mixed spectra, consistent with existing numerical works.