Electronic energy spectra and wave functions on the square Fibonacci tiling

Abstract

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 behaviours, 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. Background and motivationWe continue our initial studies [1] of the off-diagonal tight-binding hamiltonian on the square Fibonacci tiling [2], in order to gain a better quantitative understanding of the nature of the transitions between different spectral behaviours in this 2-dimensional (2D) quasicrystal. We also consider more carefully the transition of the spectrum from singular-continuous to absolutely continuous, in going from one to two dimensions, and the implications of this transition on the possible emergence of extended Bloch wave functions.The square Fibonacci tiling is constructed by taking two identical grids-each consisting of an infinite set of lines whose inter-line spacings follow the well-known Fibonacci sequence of short (S ) and long (L) distances-and superimposing them at a 90 angle. This construction can be generalized, of course, to any quasiperiodic sequence as well as to higher dimensions. On this tiling we define an off-diagonal tight-binding hamiltonian with zero on-site energy, and hopping amplitudes: t for vertices connected by short (S ) edges; 1 for vertices connected by long (L) edges; and zero for vertices that are not connected by edges.The 2-dimensional Schro¨dinger equation for this model is given by t nþ1 Éðn þ 1, mÞ þ t n Éðn À 1, mÞ þ t mþ1 Éðn, m þ 1Þ þ t m Éðn, m À 1Þ ¼ EÉðn, mÞ, ð1Þ

Mixed spectral regimes for square Fibonacci Hamiltonians

“…More precisely, we study the square of the extensively studied Fibonacci Hamiltonian and its generalizations (compare [7,11,12,23]). The reason for this choice is three-fold: (1) the one-dimensional versions have been extensively studied as prototypical examples of one-dimensional quasicrystals and the square models present a natural next step towards honest models of higherdimensional quasicrystals; (2) the square models have also been considered in a physical setting [17,[19][20][21], and while numerical studies have appeared, analytical results are scarce (to the best of our knowledge, the only comprehensive work in this direction to date is [11]); (3) while the more commonly accepted models of two-dimensional quasicrystals (such as those based on the Penrose tiling) are currently out of reach, the square models are amenable to a rigorous analysis using tools which are presently available. Our techniques are based on spectral theory, smooth dynamical systems, and geometric measure theory.…”

“…Moreover, like the general model, the square model is physically motivated (e.g. [17,[19][20][21]). However, even this simplified case presents substantial challenges, and some of the resulting problems are of independent mathematical interest (e.g.…”

Electronic energy spectra of square and cubic Fibonacci quasicrystals

“…3 we use direct numerical calculation of the 2d and 3d spectra of Fibonacci approximants of finite order to extrapolate for the behavior in the quasiperiodic limit. In an earlier paper [3] we studied only two of the transitions,…”

“…Among the open questions is a lack of understanding of the dependence of electronic properties-such as the nature of electronic wave functions, their energy spectra, and the nature of electronic transport-on the dimension of the quasicrystal. In an attempt to bridge some of this gap, we [2,3] have been studying the spectrum and electronic wave functions of an off-diagonal tight-binding hamiltonian on the separable n-dimensional Fibonacci quasicrystals 1 [4]. The advantage of using such separable models, despite the fact that they do not occur in nature, is the ability to obtain exact results in one, two, and three dimensions, and compare them directly to each other.…”

“…This allows one to use the known solutions for the 1d problem [7][8][9][10][11][12][13][14] in order to construct the solutions in two and higher dimensions (as was done for similar models in the past [15][16][17][18][19][20][21]). Two-dimensional eigenfunctions can therefore be expressed as Cartesian products of the 1d eigenfunctions [3], and the corresponding 2d eigenvalues are given by pairwise sums of the 1d eigenvalues.The 1d spectrum for the N th order Fibonacci approximant is composed of F N bands, whereis the N th Fibonacci number, starting with F 0 = F 1 = 1. The edges of each such band correspond to either periodic or antiperiodic boundary conditions.…”