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Due to the absence of periodic length scale, electronic states and theirtopological properties in quasicrystals have been barely understood. Here, wefocus on one dimensional quasicrystal and reveal that their electronic criticalstates are topologically robust. Based on tiling space cohomology, we exemplifythe case of one dimensional aperiodic tilings especially Fibonacci quasicrystaland prove the existence of topological critical states at zero energy.Furthermore, we also show exotic electronic transmittance behavior near suchtopological critical states. Within the perturbative regime, we discuss lack oftranslational symmetries and presence of topological critical states lead tounconventional scaling behavior in transmittance. Considering both analyticanalysis and numerics, electronic transmittance is computed in cases where thesystem is placed in air or is connected by semi-infinite periodic leads.Finally, we also discuss generalization of our analysis to other quasicrystals.Our findings open a new class of topological quantum states which solely existin quasicrystals due to exotic tiling patterns in the absence of periodiclength scale, and their anomalous electronic transport properties applicable tomany experiments.

Topological critical states and anomalous electronic transmittance in one dimensional quasicrystals