Extended Data Fig. 10: Finite-size effects from the chain length.
From: Spin transport in a tunable Heisenberg model realized with ultracold atoms
a, Contrast c(t) obtained after a weighted average over all different chain lengths between L = 0 and 44a (shown in b), for Δ = 0 and λ = 10.4a. The averaged dynamics (orange, yellow, blue) shows almost no dependence on the phase θ, in contrast to the dynamics determined from a single chain length L = 40a (Extended Data Fig. 9a). Also overlaid are the contrasts for a fixed chain length (L = 40a) averaged over all initial phases 0 ≤ θ < 2π (black solid line), and averaged over only the two phases θ = 0 and π/2 (black dashed line). The close agreement implies that averaging over either chain lengths or phases suppresses the dependence on initial or boundary conditions. b, A cut through the spherical Mott insulator with diameter Lmax = 44a (as in the experiment) illustrates the distribution of different chain lengths (oriented along the z direction). Averaging the local magnetization Sz over the x and y directions provides a 1D magnetization profile (bottom panel), which is an average over all chains. c, The number of chains with length L is given by (π/2)(L/a). The total number of chains is πLmax/(2a)2 ≈ 1,500. d, The number of atoms in chains with length L is given by (π/2)(L/a)2. The contribution of each chain to the imaging signal is proportional to the atom number in the chain, and so the relevant average over chain lengths is weighted by the atom number and is ⟨L⟩ = (3/4)Lmax = 33a.
