We study the Schrödinger equation for an off-diagonal tight-binding hamiltonian, as well as the equations of motion for out-of-plane vibrations on the separable square Fibonacci quasicrystal. We discuss the nature of the spectra and wave functions of the solutions.
Details are given of the theory of magnetic symmetry in quasicrystals, which has previously only been outlined. A practical formalism is developed for the enumeration of spin point groups and spin space groups, and for the calculation of selection rules for neutron scattering experiments. The formalism is demonstrated using the simple, yet non-trivial, example of magnetically ordered octagonal quasicrystals in two dimensions. In a companion paper [Even-Dar Mandel & Lifshitz (2004). Acta Cryst. A60, 179-194], complete results are provided for octagonal quasicrystals in three dimensions.
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
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. Background and MotivationAs we celebrate the Silver Jubilee of the 1982 discovery of quasicrystals [1], and highlight the achievements of the past two and a half decades of research on quasicrystals, we are reminded that there still remains a disturbing gap in our understanding of their electronic properties. 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. Here we focus on the energy spectra of the 2-dimensional (2d) and 3-dimensional (3d) Fibonacci quasicrystals to obtain a quantitative understanding of the nature of the transitions between different spectral behaviors in these crystals, as their dimension increases from 1 up to 3. In particular, we consider the transitions between different regimes in the spectrum, taking into account the existence of a regime in which the spectrum contains both singular continuous 1 and absolutely continuous components. These different behaviors of the higher-dimensional spectra are expected to reflect on the physical extent of the electronic wave functions, as well as on the dynamics of electronic wave packets, and are therefore of great importance in unraveling the electronic properties of quasicrystals in general.Recall [2] that the off-diagonal tight-binding model assumes equal on-site energies (taken to be zero), and hopping that is restricted along tile edges, with amplitude 1 for long (L) edges and T for short (S) edges, where we take T ≥ 1. The Schrödinger equation for the square Fibonacci quasicrystal in 2d (with obvious extensions to higher dimensions) is then given bywhere Ψ...
Based on Monte Carlo calculations, multipolar ordering on the Penrose tiling, relevant for two-dimensional molecular adsorbates on quasicrystalline surfaces and for nanomagnetic arrays, has been analyzed. These initial investigations are restricted to multipolar rotors of rank one through four - described by spherical harmonics Ylm with l=1...4 and restricted to m=0 - positioned on the vertices of the rhombic Penrose tiling. At first sight, the ground states of odd-parity multipoles seem to exhibit long-range multipolar order, indicated by the appearance of a superstructure in the form of the decagonal Hexagon-Boat-Star tiling, in agreement with previous investigations of dipolar systems. Yet careful analysis establishes that long-range multipolar order is absent in all cases investigated here, and only short-range order exists. This result should be taken as a warning for any future analysis of order in either real or simulated arrangements of multipoles on quasiperiodic templates
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