  Due to the absence of periodic length scale, electronic states and their
topological properties in quasicrystals have been barely understood. Here, we
focus on one dimensional quasicrystal and reveal that their electronic critical
states are topologically robust. Based on tiling space cohomology, we exemplify
the case of one dimensional aperiodic tilings especially Fibonacci quasicrystal
and prove the existence of topological critical states at zero energy.
Furthermore, we also show exotic electronic transmittance behavior near such
topological critical states. Within the perturbative regime, we discuss lack of
translational symmetries and presence of topological critical states lead to
unconventional scaling behavior in transmittance. Considering both analytic
analysis and numerics, electronic transmittance is computed in cases where the
system 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 exist
in quasicrystals due to exotic tiling patterns in the absence of periodic
length scale, and their anomalous electronic transport properties applicable to
many experiments.
