Dynamics of an Electron in Quasiperiodic Systems. I. Fibonacci Model

Hiramoto, Hisashi; Abe, Shuji

doi:10.1143/jpsj.57.230

Cited by 74 publications

(68 citation statements)

References 28 publications

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“…More generally, one may expect that the ratio of potential terms to hopping terms is increased by these ABCs with twist. This would naturally explain the observed results, as for increasing V 0 ͞t 0 the Harper model makes a transition from metal to insulator [3], the Anderson model has a decreasing localization length [1], and the energy spectrum of the Fibonacci model has a decreasing fractal dimension [19]. Further insight can be obtained from a perturbative treatment for l 0.…”

Section: Two Subsequent Abcs With Twist [See Fig 2(a)] Can Be Modelementioning

confidence: 65%

Avoided Band Crossings: Tuning Metal-Insulator Transitions in Chaotic Systems

1998

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We study transverse spin in a sub-wavelength metal-dielectric-metal (MDM) sphere when the MDM sphere exhibits avoided crossing due to hybridization of the surface plasmon with the Mie localized plasmon. We show that the change in the absorptive and dipersive character near the crossing can have significant effect on the transverse spin. An enhancement in the transverse spin is shown to be possible associated with the transparency (suppression of extinction) of the MDM sphere. The effect is attributed to the highly structured field emerging as a consequence of competition of the electric and magnetic modes. Light-matter interaction in the strong coupling regime resulting in vacuum-field Rabi split-tings has been one of the central themes in cavity quantum electrodynamics [1-5]. The coupling occurs when the dispersion branches of the uncoupled systems cross and results in the avoided crossing phenomenon and normal mode splitting. The resulting physics can have interesting implications for various applications ranging from fast and slow light [6, 7] and optical sensing [8] to counting and sizing of nanoparticles [9]. The avoided crossing phenomena have been observed in a variety of optical as well as condensed matter systems. These include planar or spherical plasmonic or guided wave structures, metamaterial cavities, in photonic crystal fibers [8, 10-12], etc. Contrary to the belief that the split modes can be resolved only with high-finesse optical modes, recent focus on systems with substantial losses (eg. plasmonic nanocavities, and leaky cavities) demonstrated the avoided crossing phenomena [13-15] in such systems. Our interest in the strongly coupled systems is motivated by the fact that there is a significant change in the dispersive and absorptive properties near the avoided crossing caused by mode mixing and exchange [16-18]. This could result in a highly structured field which is essential for observing yet another important recent discovery, namely, the transverse spin [19] in optical systems [20-22]. It is well known that the linear momentum (P) carried by light waves can be decomposed into orbital (P o) and spin (P s) parts [23, 24]. The orbital momentum (P o) is the so-called canon-ical momentum of light which is responsible for the energy transport and radiation pressure. The spin momentum, on the other hand, has long been known as a 'virtual' entity which does not transport energy or exerts pressure on the dipolar particle and is only responsible for generating the spin angular momentum (SAM) (P s ≈ ∇ × S) [19]. However, recent studies have shown that

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Section: Two Subsequent Abcs With Twist [See Fig 2(a)] Can Be Modelementioning

confidence: 65%

Avoided Band Crossings: Tuning Metal-Insulator Transitions in Chaotic Systems

1998

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“…This was rigorously established by Sütő for the Fibonacci case [69,70] and by Bellissard, Iochum, Scoppola, and Testard [11] and Damanik, Killip, and Lenz [20] in the general Sturmian case. Abe and Hiramoto studied the transport exponents for the Fibonacci model numerically [1,44]. They found that they are decreasing in λ and behave like (20) α…”

Section: Upper Bounds In Quantum Dynamics 805mentioning

confidence: 99%

Upper bounds in quantum dynamics

Damanik

Tcheremchantsev²

2006

J. Amer. Math. Soc.

138

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We develop a general method to bound the spreading of an entire wavepacket under Schrödinger dynamics from above. This method derives upper bounds on time-averaged moments of the position operator from lower bounds on norms of transfer matrices at complex energies. This general result is applied to the Fibonacci operator. We find that at sufficiently large coupling, all transport exponents take values strictly between zero and one. This is the first rigorous result on anomalous transport. For quasi-periodic potentials associated with trigonometric polynomials, we prove that all lower transport exponents and, under a weak assumption on the frequency, all upper transport exponents vanish for all phases if the Lyapunov exponent is uniformly bounded away from zero. By a well-known result of Herman, this assumption always holds at sufficiently large coupling. For the particular case of the almost Mathieu operator, our result applies for coupling greater than two.

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“…In particular, the anomalous diffusion of wave packets in quasiperiodic systems has attracted wide interest. [14][15][16][17][18][19][20][21][22][23] Multifractal eigenstates -neither extended over the system, nor exponentially localized -exist at the metal-insulator transition of the Anderson model of localization. 24,25 In tight-binding models of quasicrystals, this kind of eigenstates has also been revealed.…”

Section: Introductionmentioning

confidence: 99%

“…or the mean square displacement [18][19][20]27 d(t) = n |r n − r n 0 | 2 |Ψ n (t)| 2 1/2 (1.2) where Ψ n (t) is the amplitude of the wavefunction at time t at the nth site which is located at the position r n in space. Apparently, C(t) is the time-averaged probability of a wave packet staying at the initial site at time t, and d(t) determines the spreading of the width of a wave packet.…”

Section: Introductionmentioning

confidence: 99%

Energy spectra, wave functions, and quantum diffusion for quasiperiodic systems

et al. 2000

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We study energy spectra, eigenstates and quantum diffusion for one-and two-dimensional quasiperiodic tight-binding models. As our one-dimensional model system we choose the silver mean or "octonacci" chain. The twodimensional labyrinth tiling, which is related to the octagonal tiling, is derived from a product of two octonacci chains. This makes it possible to treat rather large systems numerically. For the octonacci chain, one finds singular continuous energy spectra and critical eigenstates which is the typical behaviour for one-dimensional Schrödinger operators based on substitution sequences. The energy spectra for the labyrinth tiling can, depending on the strength of the quasiperiodic modulation, be either band-like or fractallike. However, the eigenstates are multifractal. The temporal spreading of a wavepacket is described in terms of the autocorrelation function C(t) and the mean square displacement d(t). In all cases, we observe power laws C(t) ∼ t −δ and d(t) ∼ t β . For the octonacci chain, 0 < δ < 1, whereas for the labyrinth tiling a crossover is observed from δ = 1 to 0 < δ < 1 with increasing modulation strength. Corresponding to the multifractal eigenstates, we obtain anomalous diffusion with 0 < β < 1 for both systems. Moreover, we find that the behaviour of C(t) and d(t) is independent of the shape and the location of the initial wavepacket. We use our results to check several relations between the diffusion exponent β and the fractal dimensions of energy spectra and eigenstates that were proposed in the literature.

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Dynamics of an Electron in Quasiperiodic Systems. I. Fibonacci Model

Cited by 74 publications

References 28 publications

Avoided Band Crossings: Tuning Metal-Insulator Transitions in Chaotic Systems

Avoided Band Crossings: Tuning Metal-Insulator Transitions in Chaotic Systems

Upper bounds in quantum dynamics

Energy spectra, wave functions, and quantum diffusion for quasiperiodic systems

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