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In our understanding of solids, the formation of highly spatially coherentelectronic states, fundamental to command the quantum behavior of materials,relies on the existence of discrete translational symmetry of the crystallinelattice. In contrast, in the absence of long-range order, as in the case ofnon-crystalline materials, the electronic states are localized and electroniccoherence does not develop. This brings forward the fundamental questionwhether long range order is necessary condition to establish coherence andstructured momentum-dependent electronic state, and how to characterize it inthe presence of short-range order. Here we study Bi$_2$Se$_3$, a material thatexists in its crystalline form with long range order, in amorphous form, withshort and medium range order, and in its nanocrystalline form, with reducedshort range order. By using angle resolved photoemission spectroscopy todirectly access the electronic states in a momentum resolved manner, we revealthat, even in the absence of long-range order, a well-defined real-space lengthscale is sufficient to produce dispersive band structures. Moreover, we observefor the first time a repeated Fermi surface structure of duplicated annuli,reminiscent of Brillouin zone-like repetitions. These results, together withour simulations using amorphous Hamiltonians, reveal that the typical momentumscale where coherence occurs is the inverse average nearest-neighbor distance,the direct fingerprint of the local order of the underlying atomic structure.These results, not only lead the way to a new understanding of electroniccoherence in solids, but also open the way to the realization of novelmomentum-dependent quantum phenomena such as momentum pairing and spin-orbitcoupling, in a much broader class of materials than the currently studied ones,lacking long range crystalline translational symmetry.

Establishing Coherent Momentum-Space Electronic States in Locally Ordered Materials