Quantum chaotic subdiffusion in random potentials

Abstract

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...

Nonlinear Lattice Waves in Random Potentials

“…It is instructive to mention that the same overlap integrals play a crucial role when estimating the localization length of two interacting particles (e.g. within a Bose-Hubbard chain) with onsite disorder [34,19,35] and are the main reason for the absence of any consensus on the scaling properties of this localization length. This is mainly due to the strong correlations between eigenvectors of states residing in the same localization volume but having sufficiently well separated eigenvalues.…”

Interaction-induced weakening of localization in few-particle disordered Heisenberg chains

“…Third, for N > 2 such conclusions are less obvious. The long time scale relevant for our dynamics, as typical for disordered problems 15,16,44,57 , systematically increases with N and we do not observe a saturation of σ(t) at the time scales depicted in Fig. 1.…”

“…Refs. [41][42][43][44][45][46]. The dependence of λ on the interaction strength for bosonic models appears to be under dispute.…”

Quantum subdiffusion with two- and three-body interactions

“…[15]), which we also verified. The system size is varied within the limits N = 1000...15000 (2IP) and N = 200...1000 (3IP), whereas the integration time reaches 10 6 for 2IP (outperforming [23] by an order of magnitude) and 10 4.5 for 3IP. The averaging is done over 30 disorder realizations.…”

“…Second, in disordered lattices, it was found that 2IP produce self-sustained subdiffusive propagation beyond the single particle localization length, provided that the disorder is weak and ξ 2 ξ 1 [23]. This regime was associated with quantum chaos and high effective connectivity of states due to interaction [17,23,24].…”