Electronic Spectral and Wavefunction Properties of One-Dimensional Quasiperiodic Systems: A Scaling Approach

Hiramoto, Hisashi; Kohmoto, Mahito

doi:10.1142/s0217979292000153

Cited by 231 publications

(176 citation statements)

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“…and considering a sequence of rational numbers α n , that converges to an irrational α as n → ∞ (see for instance [27,28]). The sequence of approximants α n can be found by successive truncations of the continued-fraction expansion of α.…”

Section: Noninteracting Particlesmentioning

confidence: 99%

Effects of interaction on the diffusion of atomic matter waves in one-dimensional quasiperiodic potentials

2009

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We study the behaviour of an ultracold atomic gas of bosons in a bichromatic lattice, where the weaker lattice is used as a source of disorder. We numerically solve a discretized mean-field equation, which generalizes the one-dimensional Aubry-Andrè model for particles in a quasi-periodic potential by including the interaction between atoms. We compare the results for commensurate and incommensurate lattices. We investigate the role of the initial shape of the wavepacket as well as the interplay between two competing effects of the interaction, namely self-trapping and delocalization. Our calculations show that, if the condensate initially occupies a single lattice site, the dynamics of the interacting gas is dominated by self-trapping in a wide range of parameters, even for weak interaction. Conversely, if the diffusion starts from a Gaussian wavepacket, self-trapping is significantly suppressed and the destruction of localization by interaction is more easily observable.

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Section: Noninteracting Particlesmentioning

confidence: 99%

Effects of interaction on the diffusion of atomic matter waves in one-dimensional quasiperiodic potentials

2009

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“…For λ < 2 the quantum dynamics is similar to that of a free particle in a periodic potential. For λ = 2 the system undergoes a metal-insulator transition [5]. We note that the potential is strongly * Present address: Department of Physics, Kyoto University, Kitashirakawa, Sakyo-ku, Kyoto 606-8502, Japan † tezuka@scphys.kyoto-u.ac.jp correlated V (n)V (0) ∝ cos(2πωn) [7].…”

Section: Introductionmentioning

confidence: 99%

Stability of the superfluid state in a disordered one-dimensional ultracold fermionic gas

Tezuka

García-García

2010

Phys. Rev. A

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We study a 1D Fermi gas with attractive short range-interactions in a disordered potential by the density matrix renormalization group (DMRG) technique. Our results can be tested experimentally by using cold atom techniques. We identify a region of parameters for which disorder enhances the superfluid state. As disorder is further increased, global superfluidity eventually breaks down. However this transition seems to occur before the transition to the insulator state takes place. This suggests the existence of an intermediate metallic 'pseudogap' phase characterized by strong pairing but no quasi long-range-order.

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“…It should be pointed out that certain fractal and multifractal studies of some operators with the singularcontinuous spectrum (including some of the examples we discussed above) have been carried out by several authors [19,21]. While such studies are related to the above decomposition theory, the relations are generally far from trivial, and we believe that they are only partial.…”

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confidence: 80%

“…Remark.-There exists strong numerical evidence [19] that the spectrum of H l (as a set) has Hausdorff dimension strictly less than 1 (for every l fi 0), and this would imply that its spectrum must also be b singular (see below) for some b , 1.…”

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confidence: 99%

Dimensional Hausdorff Properties of Singular Continuous Spectra

Jitomirskaya

1996

Phys. Rev. Lett.

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We present an extension of the Gilbert-Pearson theory of subordinacy, which relates dimensional Hausdorff spectral properties of one-dimensional Schrödinger operators to the behavior of solutions of the corresponding Schrödinger equation. We use this theory to analyze these properties for several examples having the singular-continuous spectrum, including sparse barrier potentials, the almost Mathieu operator and the Fibonacci Hamiltonian.PACS numbers: 02.30. Sa, 71.23.An, 72.15.Rn Singular continuous spectra have been extensively studied recently. Our interest here is in the classification and decomposition of such spectra with respect to dimensional Hausdorff measures. The measure-theoretical aspect of this point of view goes back to Rogers and Taylor [1], and it has been studied recently within spectral theory by Last [2] and by del Rio et al. [3] who have shown that the singular-continuous spectrum which is produced by localized rank-one perturbations of Anderson-model Hamiltonians in the localized regime [4] must be purely zero dimensional-in the sense that the associated spectral measures are supported on a set of zero Hausdorff dimension.The main purpose of this paper is to report a general method for spectral analysis of one-dimensional Schrödinger operators from this point of view. It is a natural extension of the Gilbert-Pearson theory of subordinacy [5,6], and it allows us to analyze the dimensional Hausdorff properties for a number of examples with the singular-continuous spectrum. Below we describe the main ideas of our study and some of the main results. Mathematically complete proofs of these results will be given elsewhere [7].Most of our discussion will be restricted to onedimensional discrete (tight-binding) Schrödinger operators of the form ͑Hc͒ ͑n͒ c͑n 1 1͒ 1 c͑n 2 1͒ 1 V͑n͒c͑n͒ .We shall consider two kinds of such operators: "line" operators acting on ᐉ 2 ‫͒ޚ͑‬ ͑2`, n ,`͒, and "half-line" operators acting on ᐉ 2 ‫ޚ͑‬ 1 ͒ ͑n . 0͒, which are considered with a phase boundary condition of the formwhere 2p͞2 , u , p͞2. Before formulating our main result, which would require some definitions, we would like to describe some of its applications. We stress at this point that the dimensional Hausdorff properties which we study are those which are associated with the spectral measures of the corresponding operators. The spectra themselves, as sets, are closed sets, and their dimensions may be larger than those which are associated with the spectral measures. A description of the precise spectral-theoretic scheme which underlies our study is given below.We start with a somewhat artificial example of half-line operators with sparse barrier potentials. More specifically, we consider potentials which vanish for all n's outside a sparse (fastly growing) sequence of points ͕L n ͖ǹ 1 where jV ͑L n ͒j !`as n !`. Simon and Spencer [8] have shown that the Schrödinger operators corresponding to such potentials have no absolutely continuous spectrum, and Gordon [9] has shown that if the jV ͑L n ͒j's grow suffi...

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Electronic Spectral and Wavefunction Properties of One-Dimensional Quasiperiodic Systems: A Scaling Approach

Cited by 231 publications

References 0 publications

Effects of interaction on the diffusion of atomic matter waves in one-dimensional quasiperiodic potentials

Effects of interaction on the diffusion of atomic matter waves in one-dimensional quasiperiodic potentials

Stability of the superfluid state in a disordered one-dimensional ultracold fermionic gas

Dimensional Hausdorff Properties of Singular Continuous Spectra

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