Observation of Many-Body Localization in a One-Dimensional System with a Single-Particle Mobility Edge

We experimentally study many-body localization (MBL) with ultracold atoms in a weak one-dimensional quasiperiodic potential, which in the noninteracting limit exhibits an intermediate phase that is characterized by a mobility edge. We measure the time evolution of an initial charge density wave after a quench and analyze the corresponding relaxation exponents. We find clear signatures of MBL when the corresponding noninteracting model is deep in the localized phase. We also critically compare and contrast our results with those from a tight-binding Aubry-André model, which does not exhibit a single-particle intermediate phase, in order to identify signatures of a potential many-body intermediate phase.

Figure 1

Heuristic phase diagram of the GAA model: The noninteracting GAA model exhibits three phases (single-particle extended, SPIP, and single-particle localized), with the phase boundary denoted by $A$ and $B$. Here $\mathrm{\Delta }$ is the strength of the detuning lattice [Eq. (2)], while $U$ is the strength of the Hubbard on-site interactions [Eq. (4)]. The situation with finite interactions is unknown in theory, although a full MBL phase is believed to exist in the regime, where the corresponding noninteracting system is single-particle localized. Below the single-particle localization transition point $A$, interactions will lead to a thermal phase, where the ETH holds. The existence of an MBIP (marked in gray) is highly debated.

Figure 2

Time evolution of the imbalance: Measured imbalance time traces in the AA model [$V_p=8.0(1)E_r^p$] and the GAA model [$V_p=4.0(1)E_r^p$] at a fixed interaction strength $U/J_0=1$. Every data point is averaged over six different detuning phases $\varphi$, and error bars denote the standard error of the mean. The dashed lines are power-law fits to the experimental data. The solid lines are numerical simulations of the time traces in a system of $L=16$ sites [51] and the shaded regions indicate numerical uncertainties.

Figure 3

Power-law exponents: Measured relaxation exponents as a function of the detuning strength for the GAA model at $U/J_0=1$. The error bars denote the uncertainty of the fit. The blue shaded region shows the result of numerical simulations including fit uncertainties, while the brown shaded area indicates a regime of slow dynamics with finite relaxation exponents reminiscent of the slow dynamics observed in the interacting AA model [50]. The lower part of the figure represents the situation in the noninteracting system which exhibits an extended and a localized phase as well as a single-particle intermediate phase whose numerically predicted width [51] is represented by the gray shaded region.

Figure 4

Power-law exponents: Direct comparison of the relaxation exponents for both models and interaction strengths. Error bars denote the uncertainty of the fit. Solid lines are guides to the eye. The width of the corresponding SPIP for various lattice depths can be found in Ref. [51].