Clustering resonance effects in the electronic energy spectrum of tridiagonal Fibonacci quasicrystals

Abstract

In this work, we show that the fundamental structure of the electronic energy spectrum of binary Fibonacci quasicrystals can be decomposed in terms of two main contributions, stemming from two related characteristic symmetries. The algebraic approach, we introduce allows us for a unified and systematic description of the energy spectrum finer structure details in terms of block matrices commutators properties, within the framework of a renormalization approach based on transfer matrices. Close analytical expressions can be explicitly obtained by exploiting some algebraic properties of these blocks commutators. In particular, the overall main features of the electronic energy spectrum structure are related to the roots of a series of polynomials of the form pFjfalse(Efalse), where the subscript denotes the degree of the polynomial, which is given by a Fibonacci number Fj. These polynomials, in turn, are derived in a systematic way from commutators involving progressively longer palindromic fundamental blocks containing two types of basic renormalized matrices. In this way, we can classify the resonance energies defining the different fragmentation patterns of the energy spectrum on the basis of purely algebraic criteria. The transmission coefficient of these resonant states is always bounded below, although their related Landauer conductance values may range from highly conductive to highly resistive ones, depending on the system length. The obtained results significantly contribute to gain a better understanding of the symmetry principles governing the fragmentation process leading to the characteristic nested structure of the energy spectrum.

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

“…The connection between local resonators and quasibands in quasiperiodic setups has been commented on in Ref. 18,20, and 91. For the Thue-Morse sequence, a similar analysis has been achieved in Ref.…”

“…The corresponding eigenstates generally neither extend homogeneously across the system like Bloch states in regular crystals, nor do they decay exponentially like in disordered systems, and are therefore dubbed "critical" 1,[12][13][14][15] . In specific cases, quasibands have been shown to originate from the localization of different eigenstates on similar repeated substructures in the system yielding similar eigenenergies [16][17][18][19][20] . The formation of quasibands typically becomes less distinct with increasing spatial complexity, which in turn can be classified by the structure's spatial Fourier transform-accordingly altering from point-like to singular continuous to absolutely continuous 1,[21][22][23] .…”

Real Space Theory for Electron and Phonon Transport in Aperiodic Lattices via Renormalization

“…The renormalization technique was also used for the study of localization [51][52][53], electronic spectra of GaAs/Ga x Al 1−x As superlattices [54], and arrays of quantum dot [55], as well as for a unified transport theory of phonon [56], photon [57], and fermionic atom [58] based on the tight-binding model. On the other hand, by means of RSRM, the fine structure of energy spectra [59] and electronic transport in Hubbard Fibonacci chains [60,61] were investigated, and a new universality class was found in spin-one-half Heisenberg quasiperiodic chains [62].…”

Chemical Bonding and Physical Properties in Quasicrystals and Their Related Approximant Phases: Known Facts and Current Perspectives

“…This polyvalent transmission characteristic of electronic states in solids endowed with self-similar invariance symmetry is at the root of the so-called critical nature of these wave functions, which belong to fractal-like energy (vibration) spectra in the ideal case. Thus, critical electronic states embrace a diverse set of wave functions exhibiting a broad palette of possible diffusivity values, ranging from highly-conductive transparent states to highly-resistive, almost localized ones [23,24].…”