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

Abstract

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.

Non-Gibbs states on a Bose-Hubbard lattice

“…Therefore uncorrelated disorder with finite variance will result in the same border between the Gibbs and the non-Gibbs regime given by equation (18). This result holds independent of the dimensionality of the lattice, and generalizes the previously obtained consideration of a one-dimensional disordered GP lattice [21].…”

“…Let us consider the Bose-Hubbard model with disorder by adding to the Hamiltonian (2) a disorder potential Ĥdis = i ǫ i â † i âi with random on-site energies ǫ i , obeying some probability density distribution (see, e.g., Ref. [21]). Their average value ǫ = lim M→∞ M −1 i ǫ i is assumed to be zero, while the variance σ ǫ is finite.…”

Statistical Mechanics of Systems with Negative Temperature

“…( 36), where the nature of the localization process is exclusively nonlinear. Recent studies on the interplay between nonlinearity and disorder can be found in [181][182][183].…”

On the Wave Turbulence Theory for the Nonlinear Schrödinger Equation with Random Potentials

“…The resolution of this question may shed lights on many nonlinear problems, such as the famous Fermi–Pasta–Ulam–Tsingou (FPUT) problem [ 12 , 13 ]. We also refer to [ 14 , 15 , 16 ] for recent numerical works on the microcanonical Gross–Pitaevskii (also known as the semiclassical Bose–Hubbard) lattice model dynamics.…”