Perturbation theory for the nonlinear Schrödinger equation with a random potential

Fishman, Shmuel; Krivolapov, Yevgeny; Soffer, Avy

doi:10.1088/0951-7715/22/12/004

Cited by 41 publications

(76 citation statements)

References 65 publications

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“…The question of whether or not localization is stable in the presence of interactions was first considered by Fleishman and Anderson [4], who concluded that short-ranged interactions cannot destabilize the insulating phase. A similar and still open question also exists for Bose-Einstein condensates, treated in the framework of the time-dependent Gross-Pitaevskii (or nonlinear Schrödinger) equation [5]. In this case numerics, as well as analytical arguments, suggest a temporally sub-diffusive or even logarithmic thermalization behavior for not very strong interactions [5].…”

mentioning

confidence: 92%

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confidence: 92%

“…A similar and still open question also exists for Bose-Einstein condensates, treated in the framework of the time-dependent Gross-Pitaevskii (or nonlinear Schrödinger) equation [5]. In this case numerics, as well as analytical arguments, suggest a temporally sub-diffusive or even logarithmic thermalization behavior for not very strong interactions [5].Using a diagrammatic approach, Basko et al argued that for a general class of isolated, disordered and interacting systems, a many-body mobility edge exists, similarly to the Anderson mobility edge in a threedimensional non-interacting system [1]. Namely, a critical energy separates "insulating" and "metallic" eigenstates, which can be distinguished by evaluating the spatial correlations of any local operator.…”

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confidence: 96%

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Dynamics of many-body localization

2014

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Following the field theoretic approach of Basko et al., Ann. Phys. 321, 1126(2006, we study in detail the real-time dynamics of a system expected to exhibit many-body localization. In particular, for time scales inaccessible to exact methods, we demonstrate that within the second-Born approximation that the temporal decay of the density-density correlation function is non-exponential and is consistent with a finite value for t → ∞, as expected in a non-ergodic state. This behavior persists over a wide range of disorder and interaction strengths. We discuss the implications of our findings with respect to dynamical phase boundaries based both on exact diagonalization studies and as well as those established by the methods of Ref. 1.It has been known for more than 50 years that noninteracting particles in a one-dimensional disordered system exhibit Anderson localization [2], namely the exponential suppression of transport. While a localized system is non-ergodic and thus does not thermalize, coupling the system to other degrees of freedom with a continuous spectrum, such as a heat bath, allows thermalization to occur via processes such as variable-range hopping [3]. For an isolated many-body system, only interactions between the particles may lead to thermalization. The question of whether or not localization is stable in the presence of interactions was first considered by Fleishman and Anderson [4], who concluded that short-ranged interactions cannot destabilize the insulating phase. A similar and still open question also exists for Bose-Einstein condensates, treated in the framework of the time-dependent Gross-Pitaevskii (or nonlinear Schrödinger) equation [5]. In this case numerics, as well as analytical arguments, suggest a temporally sub-diffusive or even logarithmic thermalization behavior for not very strong interactions [5].Using a diagrammatic approach, Basko et al. argued that for a general class of isolated, disordered and interacting systems, a many-body mobility edge exists, similarly to the Anderson mobility edge in a threedimensional non-interacting system [1]. Namely, a critical energy separates "insulating" and "metallic" eigenstates, which can be distinguished by evaluating the spatial correlations of any local operator. "Metallic" eigenstates will have non-vanishing or slowly decaying correlations, while "insulating" states will have exponentially decaying correlations. By changing the energy (or the micro canonical temperature) of the system across the mobilityedge, the system will undergo an insulator-metal transition. Similar to the Anderson transition, the many-body localization (MBL) transition is a dynamical and not a thermodynamic phenomena [1]. However, the MBL transition is also not a conventional quantum phase transition since the critical energy, which depends on the parameters of the system, may be very far from the ground state. In fact, for systems of bounded energy density (e.g., finite number of states per site), Oganesyan and Huse suggested that the transition will persist up t...

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Dynamics of many-body localization

2014

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“…We consider here the one-dimensional random discrete nonlinear Schrödinger (DNLS) system (see, e.g., [9][10][11][12][13][14]), iψ n = ( n + χ|ψ n | 2 )ψ n − C(ψ n+1 + ψ n−1 ),…”

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confidence: 99%

KAM tori in 1D random discrete nonlinear Schrödinger model?

Johansson

Kopidakis

Aubry

2010

EPL

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PACS 05.45.-a -Nonlinear dynamics and chaos PACS 45.05.+x -General theory of classical mechanics of discrete systems PACS 42.25.Dd -Wave propagation in random mediaAbstract. -We suggest that KAM theory could be extended for certain infinite-dimensional systems with purely discrete linear spectrum. We provide empirical arguments for the existence of square summable infinite-dimensional invariant tori in the random discrete nonlinear Schrödinger equation, appearing with a finite probability for a given initial condition with sufficiently small norm. Numerical support for the existence of a fat Cantor set of initial conditions generating almost-periodic oscillations is obtained by analyzing (i) sets of recurrent trajectories over successively larger time scales, and (ii) finite-time Lyapunov exponents. The norm region where such KAM-like tori may exist shrinks to zero when the disorder strength goes to zero and the localization length diverges.

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“…From a practical perspective, it can yield some insight on the nonlinear Anderson model via multi-state resonance phenomenon, [9].…”

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confidence: 99%

Eigenvalue counting inequalities, with applications to Schrödinger operators

Elgart¹,

Schmidt²

2015

J. Spectr. Theory

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ABSTRACT. We derive a sufficient condition for a Hermitian N × N matrix A to have at least m eigenvalues (counting multiplicities) in the interval (−ǫ, ǫ). This condition is expressed in terms of the existence of a principal (N − 2m) × (N − 2m) submatrix of A whose Schur complement in A has at least m eigenvalues in the interval (−Kǫ, Kǫ), with an explicit constant K.We apply this result to a random Schrödinger operator H ω , obtaining a criterion that allows us to control the probability of having m closely lying eigenvalues for H ω -a result known as an m-level Wegner estimate. We demonstrate its usefulness by verifying the input condition of our criterion for some physical models. These include the Anderson model and random block operators that arise in the Bogoliubov-de Gennes theory of dirty superconductors.

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Perturbation theory for the nonlinear Schrödinger equation with a random potential

Cited by 41 publications

References 65 publications

Dynamics of many-body localization

Dynamics of many-body localization

KAM tori in 1D random discrete nonlinear Schrödinger model?

Eigenvalue counting inequalities, with applications to Schrödinger operators

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