  Localized waves in disordered one-dimensional materials have been studied for
decades, including white-noise and correlated disorder, as well as
quasi-periodic disorder. How these wave phenomena relate to those in
crystalline (periodic ordered) materials---arguably the better understood
setting---has been a mystery ever since Anderson discovered disorder-induced
localization. Nonetheless, together these revolutionized materials science and
technology and led to new physics far beyond the solid state. We introduce a
broad family of structurally complex materials---chaotic crystals---that
interpolate between these organizational extremes---systematically spanning
periodic structures and random disorder. Within the family one can tune the
degree of disorder to sweep through an intermediate structurally disordered
region between two periodic lattices. This reveals new transport and
localization phenomena reflected in a rich array of energy-dependent
localization degree and density of states. In particular, strong localization
is observed even with a very low degree of disorder. Moreover, markedly
enhanced localization and delocalization coexist in a very narrow range of
energies. Most notably, beyond the simply smoothed bands found in previous
disorder studies, islands of transport emerge in band gaps and sharp band
boundaries persist in the presence of substantial disorder. Finally, the family
of materials comes with rather direct specifications of how to assemble the
requisite material organizations.
