The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian

Damanik, David; Embree, Mark; Gorodetski, Anton; Tcheremchantsev, Serguei

doi:10.1007/s00220-008-0451-3

Cited by 63 publications

(93 citation statements)

References 35 publications

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“…We need to consider only symmetric solutions, since they include the lowest frequency branch, corresponding to the lowest transverse quasi-mode, k y,0 (x) = π w(x) . An infinite hierarchy of coupled differential equations for ψ m (x) is obtained by substituting the expansion (4) into the wave equation (2) and subsequently integrating over y with the weight 2 w(x) cos (k y,m (x) y). Neglecting the coupling to the higher quasi-modes, leads to the following approximate equation for the lowest quasi-mode:…”

Section: Derivation Of the Expression Of The Effective Potential Givementioning

confidence: 99%

“…We observe a quantitative agreement between experiments and the calculated modes and density of states. In particular, we evidence features of a fractal energy spectrum, arXiv:1311.3453v2 [cond-mat.quant-gas] 24 Feb 2014 namely gaps densely distributed and an integrated density of states (IDOS) reflecting the existence of a discrete scaling symmetry as expressed by (2).…”

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confidence: 95%

“…The notion of spectral distribution has been deepened in the wake of quasicrystals discovery and it led to a classification of energy spectra into absolutely continuous, pure point and singular continuous spectral distributions [1]. The latter class proved to be surprisingly rich and it encompasses a broad range of potentials, such as quasi-periodic potentials which have been thoroughly studied [2,3].An interesting quasi-periodic potential can be designed using a Fibonacci sequence. The corresponding singular continuous energy spectrum has a fractal structure of the Cantor set type [4][5][6][7], and it displays self-similarity i.e., a symmetry under a discrete scaling transformation.…”

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confidence: 99%

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Fractal Energy Spectrum of a Polariton Gas in a Fibonacci Quasiperiodic Potential

et al. 2014

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We report on the study of a polariton gas confined in a quasi-periodic one dimensional cavity, described by a Fibonacci sequence. Imaging the polariton modes both in real and reciprocal space, we observe features characteristic of their fractal energy spectrum such as the opening of mini-gaps obeying the gap labeling theorem and log-periodic oscillations of the integrated density of states. These observations are accurately reproduced solving an effective 1D Schrödinger equation, illustrating the potential of cavity polaritons as a quantum simulator in complex topological geometries.PACS numbers: 71.36.+c,78.55.Cr, 05.45.Df, 61.43.Hv, 71.23.Ft Free quantum particles or waves propagating in a spatially varying potential present modifications of their spectral density, which depend on the symmetry of this potential. The richness of spectral distributions in constrained geometries has long been recognized. The case of a periodic potential described by means of the Bloch theorem is a significant example. The notion of spectral distribution has been deepened in the wake of quasicrystals discovery and it led to a classification of energy spectra into absolutely continuous, pure point and singular continuous spectral distributions [1]. The latter class proved to be surprisingly rich and it encompasses a broad range of potentials, such as quasi-periodic potentials which have been thoroughly studied [2,3].An interesting quasi-periodic potential can be designed using a Fibonacci sequence. The corresponding singular continuous energy spectrum has a fractal structure of the Cantor set type [4][5][6][7], and it displays self-similarity i.e., a symmetry under a discrete scaling transformation. Denoting ρ(ε) the relevant density of states (DOS) in ε (either energy or frequency), a discrete scaling symmetry about a particular value ε u is expressed by the propertywhere µ (ε) = ε −∞ ρ (ε ) dε is the integrated density of states (IDOS), or density measure, and α and β are scaling parameters which usually, depend on ε u . Defining a shifted IDOS by N εu (ε) ≡ µ(ε) − µ (ε u ), the general solution of (1) can be written as [8]where γ = ln α ln β is the local (ε u -dependent) scaling exponent and F(z) is a periodic function of period unity, whose (non-universal) form depends on the problem at hand. Generally, the exponent γ takes values between zero and unity, so that the density ρ (ε) is a singular function. Such scaling properties of a fractal spectrum are expected to modify the behavior of physical quantities [8]. Recently studied examples include thermodynamic properties of photons [9], random walks [10], quantum diffusion of wave packets [11] and spontaneous emission triggered by a fractal vacuum [12]. The diffusion of a wave packet in a quasi-periodic medium is predicted to be neither diffusive, nor ballistic but to present a behavior characterized by non-universal exponents and a logperiodic modulation of its time dynamics. Experimental demonstration of these specific properties of quasiperiodic structures is still missing ...

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Section: Derivation Of the Expression Of The Effective Potential Givementioning

confidence: 99%

mentioning

confidence: 95%

mentioning

confidence: 99%

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Fractal Energy Spectrum of a Polariton Gas in a Fibonacci Quasiperiodic Potential

et al. 2014

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“…Damanik et al [14,23] recently used the second moment of the position operator for the diagonal tight-binding Hamiltonian, to show that far from the periodic limit, or very close to it, the dynamics of wave-packets is independent of the initial site. However, since we are interested as a function of T , which is displayed in Fig.…”

Section: Quantum Dynamics Of Electronic Wave-packetsmentioning

confidence: 99%

Observation of log-periodic oscillations in the quantum dynamics of electrons on the one-dimensional Fibonacci quasicrystal

Lifshitz

Mandel

2011

Philosophical Magazine

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We revisit the question of quantum dynamics of electrons on the off-diagonal Fibonacci tight-binding model. We find that typical dynamical quantities, such as the probability of an electron to remain in its original position as a function of time, display log-periodic oscillations on top of the leading-order power-law decay. These periodic oscillations with the logarithm of time are similar to the oscillations that are known to exist with the logarithm of temperature in the specific heat of Fibonacci electrons, yet they offer new possibilities for the experimental observation of this unique phenomenon. Fibonacci ElectronsThe Fibonacci sequence of Long (L) and Short (S) intervals on the 1-dimensional line-generated by the simple substitution rules L → LS and S → L-is a favorite textbook model for demonstrating the peculiar nature of electrons in quasicrystals [1][2][3]. The wave functions of Fibonacci electrons are neither extended nor exponentially-localized, but rather decay algebraically; the spectrum of energies is neither absolutely-continuous nor discrete, but rather singular-continuous, like a Cantor set; and the quantum dynamics is anomalous. In recent years we have studied how these three electronic properties change as the dimension of the Fibonacci quasicrystal increases to two and three [4][5][6][7][8], by constructing square and cubic versions of the Fibonacci quasicrystal [9].The 1-dimensional off-diagonal Fibonacci tight-binding model is constructed by associating a unit hopping amplitude between sites connected by a Long interval, and a hopping amplitude T > 1 between sites connected by a Short interval, while assuming equal on-site energies that are taken to be zero. The resulting tight-binding Schrödinger equation, on an F N -site model, is 1

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“…Le comportement asymptotique pour une grande constante de couplage de la dimension du spectre d'un opérateur de Schrödinger discret dont le potentiel est une suite sturmienne associée au nombre d'or vient d'être obtenu par Damanik et al (2007). Dans cette Note, nous donnons une démonstration plus simple de ce résultat et l'étendons au cas d'un potentiel sturmien associé à une fréquence irrationnelle dont les quotient partiels de sa décomposition en fraction continue sont bornés.…”

Section: Résuméunclassified

Dimension of the spectrum of one-dimensional discrete Schrödinger operators with Sturmian potentials

Liu

Peyrière

Zhou

2007

Comptes Rendus. Mathématique

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The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian

Cited by 63 publications

References 35 publications

Fractal Energy Spectrum of a Polariton Gas in a Fibonacci Quasiperiodic Potential

Fractal Energy Spectrum of a Polariton Gas in a Fibonacci Quasiperiodic Potential

Observation of log-periodic oscillations in the quantum dynamics of electrons on the one-dimensional Fibonacci quasicrystal

Dimension of the spectrum of one-dimensional discrete Schrödinger operators with Sturmian potentials

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