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Localized waves in disordered one-dimensional materials have been studied fordecades, including white-noise and correlated disorder, as well asquasi-periodic disorder. How these wave phenomena relate to those incrystalline (periodic ordered) materials---arguably the better understoodsetting---has been a mystery ever since Anderson discovered disorder-inducedlocalization. Nonetheless, together these revolutionized materials science andtechnology and led to new physics far beyond the solid state. We introduce abroad family of structurally complex materials---chaotic crystals---thatinterpolate between these organizational extremes---systematically spanningperiodic structures and random disorder. Within the family one can tune thedegree of disorder to sweep through an intermediate structurally disorderedregion between two periodic lattices. This reveals new transport andlocalization phenomena reflected in a rich array of energy-dependentlocalization degree and density of states. In particular, strong localizationis observed even with a very low degree of disorder. Moreover, markedlyenhanced localization and delocalization coexist in a very narrow range ofenergies. Most notably, beyond the simply smoothed bands found in previousdisorder studies, islands of transport emerge in band gaps and sharp bandboundaries persist in the presence of substantial disorder. Finally, the familyof materials comes with rather direct specifications of how to assemble therequisite material organizations.

Islands in the Gap: Intertwined Transport and Localization in Structurally Complex Materials