Electron Levels in a One-Dimensional Random Lattice

Frisch, H.; Lloyd, Seth

doi:10.1103/physrev.120.1175

Cited by 223 publications

(134 citation statements)

References 5 publications

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“…In order to build up an estimate for the trace T , we need to know how many such edge states contribute to (4.8). This number can be estimated by making use of anomalous band-edge scaling, which has been investigated first in continuum models with a white-noise potential [32,33], and then at the band edges of tight-binding models [34,35,36]. The main result of the latter works can be expressed in the following compact form.…”

Section: Weak-disorder Localized Regimementioning

confidence: 99%

“…This perturbative estimate breaks down near the band edges (E → ±2, i.e., q → 0 and π), where the numerator vanishes. Right at the band edges, eigenstates are actually more strongly localized, as their localization length only diverges as w −2/3 [32,33,34,35,36]. This anomalous band-edge scaling will play a key role in the following (see section 5).…”

Section: Particle In a Random Potential: Generalitiesmentioning

confidence: 99%

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An investigation of equilibration in small quantum systems: the example of a particle in a 1D random potential

Luck¹

2016

J. Phys. A: Math. Theor.

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Abstract. We investigate the equilibration of a small isolated quantum system by means of its matrix of asymptotic transition probabilities in a preferential basis. The trace of this matrix is shown to measure the degree of equilibration of the system launched from a typical state, from the standpoint of the chosen basis. This approach is substantiated by an in-depth study of the example of a tightbinding particle in one dimension. In the regime of free ballistic propagation, the above trace saturates to a finite limit, testifying good equilibration. In the presence of a random potential, the trace grows linearly with the system size, testifying poor equilibration in the insulating regime induced by Anderson localization. In the weak-disorder situation of most interest, a universal finite-size scaling law describes the crossover between the ballistic and localized regimes. The associated crossover exponent 2/3 is dictated by the anomalous band-edge scaling characterizing the most localized energy eigenstates.

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Section: Weak-disorder Localized Regimementioning

confidence: 99%

Section: Particle In a Random Potential: Generalitiesmentioning

confidence: 99%

An investigation of equilibration in small quantum systems: the example of a particle in a 1D random potential

Luck¹

2016

J. Phys. A: Math. Theor.

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“…This simple picture, first provided by Jona-Lasinio [134] allows recovering the exponential tail obtained by several methods in refs. [99,124,15,157,130] 22 . The distribution of the length ℓ is a Poisson law (see [105] or appendix of ref.…”

Section: Random Schrödinger Hamiltonianmentioning

confidence: 99%

Functionals of Brownian motion, localization and metric graphs

Comtet¹,

Desbois²,

Texier³

2005

J. Phys. A: Math. Gen.

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We review several results related to the problem of a quantum particle in a random environment.In an introductory part, we recall how several functionals of the Brownian motion arise in the study of electronic transport in weakly disordered metals (weak localization).Two aspects of the physics of the one-dimensional strong localization are reviewed : some properties of the scattering by a random potential (time delay distribution) and a study of the spectrum of a random potential on a bounded domain (the extreme value statistics of the eigenvalues).Then we mention several results concerning the diffusion on graphs, and more generally the spectral properties of the Schrödinger operator on graphs. The interest of spectral determinants as generating functions characterizing the diffusion on graphs is illustrated.Finally, we consider a two-dimensional model of a charged particle coupled to the random magnetic field due to magnetic vortices. We recall the connection between spectral properties of this model and winding functionals of the planar Brownian motion.

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“…These phenomena involving disorder can be conveniently described using 1D tight-binding models ͑see, e.g., Refs. [16][17][18]20, and 21͒. For a long time, it was believed that localized and extended states in 1D potentials did not coexist and 1D systems could not display complex dynamic features such as the metal-insulator transition.…”

Section: Introductionmentioning

confidence: 99%

Controlled terahertz frequency response and transparency of Josephson chains and superconducting multilayers

et al. 2007

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A fundamental property of wave propagation is Anderson localization, which affects the transfer of information, energy, mass, and charge in disordered media. This localization can manifest itself via, e.g., the metal-insulator transition. We exactly map the behavior of a quantum particle moving in a potential with correlated disorder to the sub-terahertz wave propagation in either Josephson chains or superconducting multilayers. When the Josephson junction parameters vary randomly, the sub-THz electromagnetic waves cannot propagate through these Josephson structures due to localization. For parameter variations with long-range correlations, we predict sharp transitions from transparent to reflective frequency regions for Josephson plasma waves. With appropriate choices of the correlation function, frequency windows with targeted or designed transparencies for THz or sub-THz electromagnetic waves could be achieved. This could be useful for tailoring the electromagnetic wave spectrum of Josephson arrays within the THz frequency range, which is important for applications in physics, astronomy, chemistry, biology, and medicine.

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Electron Levels in a One-Dimensional Random Lattice

Abstract: Let the potential of a one-dimensional scalar particle be F(ic) = FoS_ 00 00 5(^-Xj), -<*> , constitute an ergodic stationary Markov process. The stationary density T(z) of the (ZJ) satisfies a first-order linear d… Show more

Cited by 223 publications

References 5 publications

An investigation of equilibration in small quantum systems: the example of a particle in a 1D random potential

An investigation of equilibration in small quantum systems: the example of a particle in a 1D random potential

Functionals of Brownian motion, localization and metric graphs

Controlled terahertz frequency response and transparency of Josephson chains and superconducting multilayers

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