Analytical Results for Scaling Properties of the Spectrum of the Fibonacci Chain

Abstract

We solve the approximate renormalisation group found by Qiu Niu and Franco Typeset using REVT E X 1

Quantum chaos in small quantum networks

“…The chaotic sequence of Fibonacci-type can be interpreted as a temporal random (chaotic) analogue to 1-dimensional chaotic potentials in real space [19,20].…”

Analytical investigation of the specific heat for the Cantor energy spectrum

“…More specifically, the Molecular Beam Epitaxy technique has produced and driven a multiplication of possible such structures (Fibonacci, Thue-Morse, double-period sequences; other possibilities could be Cantor sets, prime numbers, etc.). The behavior of a variety of particles and quasi-particles (electrons, photons, plasmon-polaritons, magnons) in quasi-periodic sequences has been and is currently being studied [3][4][5][6][7][8][9][10][11]. Now, there is a common feature which can be considered as the basic signature of such structures, and this is a fractal energy spectrum.…”

Fractal Property of Band Branching in Fibonacci Mesoscopic Rings

“…Using the renormalization group method, the energy spectrum of the Fibonacci chain with the on-site model is proved as three main bands and each band split into three subbands in a hierarchical structure. [9][10][11] According to the energy spectrum, the existence of pure critical or both critical and extended states in the on-site Fibonacci chain have been proposed. 13) As the mixed Fibonacci structure is considered, the coexistence of the extended, critical and localized states may occur in the chains.…”

“…An absence of localization may also occur if the long-range correlation is introduced in the disorder chains. 20) For the one-dimensional quasiperiodic chains, branching of the energy spectra [9][10][11][12] has been applied to investigate the localization and delocalization of electron behavior. All eigenstates are localized, extended, and critical as the energy spectrum in the system are purely absolutely continuous, purely a dense point, and purely singular continuous, respectively.…”

“…6) Recently, localizationdelocalization of quantum transport in a quasiperiodic array has received intensive interest since the discovery of quasicrystals. [7][8][9][10][11][12][13][14][15][16] For the one-dimensional array system, the tight-binding model (TBM) is one of the simplest ways to examine the behavior of quantum transport. Based on TBM, the diffusion of electrons vanishing in one-dimensional chains with sitediagonal disorder has been proven by Anderson.…”