KAM tori in 1D random discrete nonlinear Schrödinger model?

Johansson, Magnus; Kopidakis, Georgios; Aubry, S.

doi:10.1209/0295-5075/91/50001

Cited by 65 publications

(88 citation statements)

References 27 publications

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“…Published numerical data [17,18,19,20,21,22] (we refer to [16,23] for more original references and details of the theory) show that finite size initial wave packets i) stay localized if x ≪ d and display regular-like (i.e. quasiperiodic as in the KAM regime ) dynamics, which is numerically hardly distinguishable from very slow chaotic dynamics with subsequent spreading on unaccessible time scales); ii) spread subdiffusively if x ∼ d with the second moment of the wave packet m 2 ∼ t α and chaotic dynamics inside the wave packet core; segregate into a two component field with a selftrapped component, and a subdiffusively spreading part for x > ∆ .…”

Section: Disorder + Nonlinearitymentioning

confidence: 99%

Spreading, Nonergodicity, and Selftrapping: A Puzzle of Interacting Disordered Lattice Waves

Flach

2015

Springer Proceedings in Physics

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Localization of waves by disorder is a fundamental physical problem encompassing a diverse spectrum of theoretical, experimental and numerical studies in the context of metal-insulator transitions, the quantum Hall effect, light propagation in photonic crystals, and dynamics of ultra-cold atoms in optical arrays, to name just a few examples. Large intensity light can induce nonlinear response, ultracold atomic gases can be tuned into an interacting regime, which leads again to nonlinear wave equations on a mean field level. The interplay between disorder and nonlinearity, their localizing and delocalizing effects is currently an intriguing and challenging issue in the field of lattice waves. In particular it leads to the prediction and observation of two different regimes of destruction of Anderson localization -asymptotic weak chaos, and intermediate strong chaos, separated by a crossover condition on densities. On the other side approximate full quantum interacting many body treatments were recently used to predict and obtain a novel many body localization transition, and two distinct phases -a localization phase, and a delocalization phase, both again separated by some typical density scale. We will discuss selftrapping, nonergodicity and nonGibbsean phases which are typical for such discrete models with particle number conservation and their relation to the above crossover and transition physics. We will also discuss potential connections to quantum many body theories.

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Section: Disorder + Nonlinearitymentioning

confidence: 99%

Spreading, Nonergodicity, and Selftrapping: A Puzzle of Interacting Disordered Lattice Waves

Flach

2015

Springer Proceedings in Physics

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“…9,11,19,21 Currently, a greatly debatable problem is the long time behavior of wave packet spreading in disordered nonlinear lattices. Recently, it was conjectured 15,16 that chaotically spreading wave packets will asymptotically approach KAM torus-like structures in phase-space, while numerical simulations typically do not show any sign of slowing down of the spreading behavior. 13,14,29 Nevertheless, for particular disordered nonlinear models some numerical indications of a possible slowing down of spreading have been reported.…”

Section: A the Disordered Quartic Klein-gordon Modelmentioning

confidence: 99%

“…Thus, we suggest that the motion of this system will never approach a Kolmogorov-Arnold-Moser (KAM) regime of invariant tori as suggested by some authors. 15,16 This paper is organized as follows: In Sec. II, we outline the details of our study of the statistical distributions corresponding to weakly and strongly chaotic behaviors and Sec.…”

Section: Introductionmentioning

confidence: 99%

Complex statistics and diffusion in nonlinear disordered particle chains

Antonopoulos

Bountis

Skokos

et al. 2014

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We investigate dynamically and statistically diffusive motion in a Klein-Gordon particle chain in the presence of disorder. In particular, we examine a low energy (subdiffusive) and a higher energy (self-trapping) case and verify that subdiffusive spreading is always observed. We then carry out a statistical analysis of the motion in both cases in the sense of the Central Limit Theorem and present evidence of different chaos behaviors, for various groups of particles. Integrating the equations of motion for times as long as 10 9 , our probability distribution functions always tend to Gaussians and show that the dynamics does not relax onto a quasi-periodic KAM torus and that diffusion continues to spread chaotically for arbitrarily long times.

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“…Recent analytical and numerical studies demonstrate that nonlinearity breaks AL and leads to subdiffusive wave packet propagation, caused by nonintegrability, deterministic chaos, phase decoherence and a consequent loss of wave localization [10][11][12][13][14][15][16] . A positive measure of initially localized excitations gets delocalized for an arbitrarily small nonlinearity, tending to one above some threshold which may depend on the initial state 17 .…”

mentioning

confidence: 99%

Quantum chaotic subdiffusion in random potentials

2014

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Two interacting particles (TIP) in a disordered chain propagate beyond the single particle localization length ξ1 up to a scale ξ2 > ξ1. An initially strongly localized TIP state expands almost ballistically up to ξ1. The expansion of the TIP wave function beyond the distance ξ1 1 is governed by highly connected Fock states in the space of noninteracting eigenfunctions. The resulting dynamics is subdiffusive, and the second moment grows as m2 ∼ t 1/2 , precisely as in the strong chaos regime for corresponding nonlinear wave equations. This surprising outcome stems from the huge Fock connectivity and resulting quantum chaos. The TIP expansion finally slows down towards a complete halt -in contrast to the nonlinear case.PACS numbers: 63.20.Pw, 63.20.Ry, Anderson localization (AL), the absence of diffusion in linear lattice wave equations due to disorder 1 , is now broadly seen as a fundamental physical phenomenon manifested by light, sound, and matter waves 2,3 . Rigorous results state that in one dimension all single-particle (SP) states become exponentially localized at arbitrary weak disorder [4][5][6] . Going beyond the assumption of noninteracting particles or linear waves has proved to be extremely complex, and the current answers on the interplay between disorder and interactions remain controversial and debated.For quantum many body systems predictions range from no major effect of interactions 7 to the emergence of a finite-temperature AL transition already in dimension one 8 . Advance has been achieved within mean field approximations, which lead to nonlinear wave equations like the Gross-Pitaevsky equation 9 . Recent analytical and numerical studies demonstrate that nonlinearity breaks AL and leads to subdiffusive wave packet propagation, caused by nonintegrability, deterministic chaos, phase decoherence and a consequent loss of wave localization [10][11][12][13][14][15][16] . A positive measure of initially localized excitations gets delocalized for an arbitrarily small nonlinearity, tending to one above some threshold which may depend on the initial state 17 .Few interacting quantum particles may bridge the two extremes from above. In particular the case of two interacting particles (TIP) appears to be an interesting testing ground for any of the above statements. There is not much doubt that the TIP case also yields a finite localization length ξ 2 , similar to but potentially much larger than the single particle localization length ξ 1 . Most of the studies only debate whether and how the TIP localization length ξ 2 scales with ξ 1 18-23 . Paradoxically, almost nothing is known on the interaction induced wave packet dynamics beyond ξ 1 . Numerical experiments explored only the case of the strong disorder, limiting to small connectivity of TIP states in the relevant Hilbert space of the problem, to report, unsurprisingly, quite a trivial ballistic expansion up to ξ 1 followed by a quick saturation 24 . Here we report the first study of TIP wave packet dynamics much beyond the single particle localiza...

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KAM tori in 1D random discrete nonlinear Schrödinger model?

Cited by 65 publications

References 27 publications

Spreading, Nonergodicity, and Selftrapping: A Puzzle of Interacting Disordered Lattice Waves

Spreading, Nonergodicity, and Selftrapping: A Puzzle of Interacting Disordered Lattice Waves

Complex statistics and diffusion in nonlinear disordered particle chains

Quantum chaotic subdiffusion in random potentials

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