Abstract
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.
Highlights
Systems without periodicity are studied in various contexts such as condensed matter physics, optics, and mathematics [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]
We first consider transmittance of the zero energy state. Both air boundary or semi-infinite periodic lead boundary share the similar behavior in this case, so without loss of generality we focus on the former case. [The only difference between the former and the latter cases may be interchanging between h(n) and f (n) which will be discussed in Sec
When the Fibonacci quasicrystal is sandwiched by two semi-infinite conducting leads, Fig. 10 represents the log-log plot of half-bandwidth 1/2 as a function of systematic constant ρ
Summary
Systems without periodicity are studied in various contexts such as condensed matter physics, optics, and mathematics [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]. If two given tilings are congruent up to 1 large region through translation on one tiling as amount of less than , we say that their distance is in abstract tiling space [1,13] Based on this kind of metric space for tilings itself, one can consider the pattern dependent topologies. Physical quantities which strongly depend on such PE cohomology or cochains, are considered as very robust concept [1,13,40,45,49,51] Such kind of discussion highly promoted understanding of quasicrystal or general aperiodic Hamiltonian that lack of periodicity but depend on the pattern of the system. We apply such pattern dependent topologies to the electronic system of the Fibonacci quasicrystal and discuss the topological critical state at zero energy and relevant electronic transports.
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