Subdiffusion in a one-dimensional Anderson insulator with random dephasing: Finite-size scaling, Griffiths effects, and possible implications for many-body localization

Abstract

We study transport in the boundary-driven XX spin chain with onsite disorder and randomly positioned onsite dephasing, observing a transition from diffusive to subdiffusive spin transport below a critical density of sites with dephasing. We then present an exactly solvable semiclassical model of conductors and insulators, which exhibits both diffusive and subdiffusive phases, and qualitatively reproduces the results of the quantum system. The subdiffusion in these models is a consequence of rare insulating regions, and they therefore exhibit, in a more controlled setup, the physics of "Griffiths effects" which have been conjectured to cause the subdiffusive transport observed in interacting many-body localizable systems. For the quantum model we show that finitesize effects come from the interplay of three characteristic lengths: one associated with disorder (the localization length), one with dephasing, and the third with the percolation problem defining large, rare, insulating regions. The latter grows logarithmically with system size, and we conjecture that this may be the reason why the heavy-tailed distributions typical of Griffiths effects have not been observed in subdiffusive interacting systems.

Diffusion and thermalization in a boundary-driven dephasing model

“…( 11) is polynomial in resources, the stationary solution is obtained exactly by solving ∂ t C = 0. This is given by [20,21]…”

A constructive theory of the numerically accessible many-body localized to thermal crossover

“…The exponent θ 0 in (56) vanishes as |ξ| → ∞ in the RM. The strongest low-frequency divergence [S(ω)] is however not ∼ 1/ω (indeed, as noted in [2] such a strong divergence would violate an elementary sum rule) because the exponential term in ( 53) modifies the exponent.…”

“…1a. The exponent θ characterising the low-frequency divergence of [S(ω)] in region II jumps at the transition (56).…”

“…Phenomenological models suggest that the transition has Kosterlitz-Thouless-type scaling [45][46][47], and occurs when the localised phase is destabilised by rare thermal regions which seed "thermalisation avalanches" [48][49][50][51][52][53][54]. Numerical studies, which are limited to small systems, do not find any evidence of rare thermal regions [55,56], but are known to be plagued by unexplained finite-size effects [57][58][59][60]. The absence of a theory of the finite-size crossover leaves unclear which features of the numerical data may survive in the thermodynamic limit, and has led Refs.…”

Localization of a mobile impurity interacting with an Anderson insulator

“…In order to quantify the density fluctuations of the clean boson, we average the absolute value of the deviation of its density from the mean value, stood as resulting from an Anderson insulator globally coupled to a weak, non-Markovian, local noise. While it is known [44][45][46][47][48] that Anderson localization is unstable with respect to global noise, a recent work [49] has demonstrated that coupling of an Anderson insulator to a local Markovian white noise leads to a logarithmically slow particle transport and entanglement growth. Our dynamics differs from that of Ref.…”