Critical wave functions and a Cantor-set spectrum of a one-dimensional quasicrystal model (original) (raw)
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Journal of physics. Condensed matter : an Institute of Physics journal, 2015
We demonstrate, by explicit construction, that a single band tight binding Hamiltonian defined on a class of deterministic fractals of the b = 3N Sierpinski type can give rise to an infinity of dispersionless, flat-band like states which can be worked out analytically using the scale invariance of the underlying lattice. The states are localized over clusters of increasing sizes, displaying the existence of a multitude of localization areas. The onset of localization can, in principle, be 'delayed' in space by an appropriate choice of the energy of the electron. A uniform magnetic field threading the elementary plaquettes of the network is shown to destroy this staggered localization and generate absolutely continuous sub-bands in the energy spectrum of these non-translationally invariant networks.
Physica D: Nonlinear Phenomena, 1999
We study quantum chaos in open dynamical systems and show that it is characterized by quantum fractal eigenstates located on the underlying classical strange repeller. The states with longest life times typically reveal a scars structure on the classical fractal set.
Electronic spectra of quasi-regular Fibonacci systems: Analysis of simple 1D models
Microelectronics Journal, 2005
The electronic spectra of quasi-regular systems grown following the Fibonacci sequence are investigated via simple one-dimensional tight-binding, one-band models. Different models are considered and the influence of the model parameters and the number of atoms entering the different blocks on the electronic spectrum are discussed.
Physica A: Statistical Mechanics and its Applications, 1999
We present here a detailed multifractal scaling study for the electronic transmission resonances with the system size for an infinitely large one dimensional perfect and imperfect quasiperiodic system represented by a sequence of δ-function potentials. The electronic transmission resonances in the energy minibands manifest more and more fragmented nature of the transmittance with the change of system sizes. We claim that when a small perturbation is randomly present at a few number of lattice sites, the nature of electronic states will change and this can be understood by studying the electronic transmittance with the change of system size. We report the different critical states manifested in the size variation of the transmittance corresponding to the resonant energies for both perfect and imperfect cases through multifractal scaling study for few of these resonances.
EPL (Europhysics Letters), 2016
It is shown that, an entire class of off-diagonally disordered linear lattices composed of two basic building blocks and described within a tight binding model can be tailored to generate absolutely continuous energy bands. It can be achieved if linear atomic clusters of an appropriate size are side-coupled to a suitable subset of sites in the backbone, and if the nearest neighbor hopping integrals, in the backbone and in the side-coupled cluster bear a certain ratio. We work out the precise relationship between the number of atoms in one of the building blocks in the backbone, and that in the side-attachment. In addition, we also evaluate the definite correlation between the numerical values of the hopping integrals at different subsections of the chain, that can convert an otherwise point spectrum (or, a singular continuous one for deterministically disordered lattices) with exponentially (or power law) localized eigenfunctions to an absolutely continuous spectrum comprising one or more bands (subbands) populated by extended, totally transparent eigenstates. The results, which are analytically exact, put forward a non-trivial variation of the Anderson localization [P. W. Anderson, Phys. Rev. 109, 1492 (1958)], pointing towards its unusual sensitivity to the numerical values of the system parameters and, go well beyond the other related models such as the Random Dimer Model (RDM) [Dunlap et al., Phys. Rev. Lett. 65, 88 (1990)].
Electronic energy spectra and wave functions on the square Fibonacci tiling
Philosophical Magazine, 2006
We study the electronic energy spectra and wave functions on the square Fibonacci tiling, using an off-diagonal tight-binding model, in order to determine the exact nature of the transitions between different spectral behaviors, as well as the scaling of the total bandwidth as it becomes finite. The macroscopic degeneracy of certain energy values in the spectrum is invoked as a possible mechanism for the emergence of extended electronic Bloch wave functions as the dimension changes from one to two.
Electronic energy spectra of square and cubic Fibonacci quasicrystals
Philosophical Magazine, 2008
Understanding the electronic properties of quasicrystals, in particular the dependence of these properties on dimension, is among the interesting open problems in the field of quasicrystals. We investigate an off-diagonal tight-binding hamiltonian on the separable square and cubic Fibonacci quasicrystals. We use the well-studied singular-continuous energy spectrum of the 1-dimensional Fibonacci quasicrystal to obtain exact results regarding the transitions between different spectral behaviors of the square and cubic quasicrystals. We use analytical results for the addition of 1d spectra to obtain bounds on the range in which the higher-dimensional spectra contain an absolutely continuous component. We also perform a direct numerical study of the spectra, obtaining good results for the square Fibonacci quasicrystal, and rough estimates for the cubic Fibonacci quasicrystal.
Fractal character of wave functions in one-dimensional incommensurate systems
Physical review. B, Condensed matter, 1986
The electronic wave functions of simple one-dimensional systems with a modulation potential incommensurate with that of the underlying lattice are determined by a direct diagonalization method. The existence of the metal-insulator transition is also obtained by a renormalization-group method. Numerical evidence for a fractal character of the wave functions is obtained and the fractal dimensionality D is calculated as a function of the strength of the modulation potential Vo. At the critical point Vo-2t, we find that D =0.80+0. 15. The wave functions can also be characterized by the localization length i, and the amplitude correlation length g.
This article presents a novel Hamiltonian architecture based on vertex types and empires for demonstrating the emergence of aperiodic order (quasicrystal growth) in one dimension by a suitable prescription for breaking translation symmetry. At the outset, the paper presents different algorithmic, geometrical, and algebraic methods of constructing empires of vertex configurations of a given quasi-lattice. These empires have non-local scope and form the building blocks of the new lattice model. This model is tested via Monte Carlo simulations beginning with randomly arranged N tiles. The simulations clearly establish the Fibonacci configuration, which is a one dimensional quasicrystal of length N , as the final relaxed state of the system. The Hamiltonian is promoted to a matrix operator form by performing dyadic tensor products of pairs of interacting empire vectors followed by a summation over all permissible configurations. A spectral analysis of the Hamiltonian matrix is performed and a theoretical method is presented to find the exact solution of the attractor configuration that is given by the Fibonacci chain as predicted by the simulations. Finally, a precise theoretical explanation is provided which shows that the Fibonacci chain is the most probable ground state. The proposed Hamiltonian is a one dimensional model of quasicrystal growth.
Multifractal properties of critical eigenstates in two-dimensional systems with symplectic symmetry
Journal of Physics: Condensed Matter, 1995
The multifractal properties of electronic eigenstates at the metal-insulator transition of a two-dimensional disordered tight-binding model with spin-orbit interaction are investigated numerically. The correlation dimensions of the spectral measure D 2 and of the fractal eigenstate D 2 are calculated and shown to be related by D 2 = 2 D 2. The exponent η = 0.35 ± 0.05 describing the energy correlations of the critical eigenstates is found to satisfy the relation η = 2 − D 2 .