Local symmetry theory of resonator structures for the real-space control of edge states in binary aperiodic chains

Abstract

We propose a real-space approach explaining and controlling the occurrence of edge-localized gap states between the spectral quasibands of binary tight binding chains with deterministic aperiodic long-range order. The framework is applied to the Fibonacci, Thue-Morse and Rudin-Shapiro chains, representing different structural classes. Our approach is based on an analysis of the eigenstates at weak inter-site coupling, where they are shown to generically localize on locally reflection-symmetric substructures which we call local resonators. A perturbation theoretical treatment demonstrates the local symmetries of the eigenstates. Depending on the degree of spatial complexity of the chain, the proposed local resonator picture can be used to predict the occurrence of gap-edge states even for stronger couplings. Moreover, we connect the localization behavior of a given eigenstate to its energy, thus providing a quantitative connection between the real-space structure of the chain and its eigenvalue spectrum. This allows for a deeper understanding, based on local symmetries, of how the energy spectra of binary chains are formed. The insights gained allow for a systematic analysis of aperiodic binary chains and offers a pathway to control structurally induced edge states.

Transfer efficiency enhancement and eigenstate properties in locally symmetric disordered finite chains

“…As shown in Ref. 69 (and in a recent extension to discrete models 71 ), the position where the eigenstate will actually localize is then determined by the minimal spectral distance…”

Effects of Disorder in the Fibonacci Quasicrystal

“…7(a) offered very little disruption, even though it had no locally symmetric region surrounding it. In recent work, the on-site model of the Fibonacci chain has been studied using a local resonator mode framework [18]. It would be interesting to find an analog version of this framework in the hopping model, which offers an intuitive understanding of the effect of impurities on local symmetries.…”

“…The two models have been studied extensively in Refs. [12,18,22], where the renormalization scheme [12] was applied to understand the multifractal properties of the model [16,22]. The on-site model was also studied in Ref.…”

“…The on-site model was also studied in Ref. [18] through the perspective of local symmetries, where a systematic way to control the edge modes of a finite chain was devised.…”

“…Together with the conumbering scheme [15], it offered an insightful way to characterize the wavefunctions in terms of their renormalization paths [16]. Nowadays, new insights are still being provided and interestingly, it has been shown that this system also possesses topological characteristics [16][17][18][19][20].…”