Dynamics of pattern-loaded fermions in bichromatic optical lattices

Motivated by experiments in Munich [M. Schreiber et al., Science 349, 842 (2015).], we study the dynamics of interacting fermions initially prepared in charge density wave states in one-dimensional bichromatic optical lattices. The experiment sees a marked lack of thermalization, which has been taken as evidence for an interacting generalization of Anderson localization, dubbed “many-body localization.” We model the experiments using an interacting Aubry-Andre model and develop a computationally efficient low-density cluster expansion to calculate the even-odd density imbalance as a function of interaction strength and potential strength. Our calculations agree with the experimental results and shed light on the phenomena. We also explore a two-dimensional generalization. The cluster expansion method we develop should have broad applicability to similar problems in nonequilibrium quantum physics.

Figure 1

Imbalance $I=\frac{N_{\mathrm{odd}}-N_{\mathrm{even}}}{N_{\mathrm{odd}}+N_{\mathrm{even}}}$ vs time $t$, measured in units of the nearest-neighbor hopping strength $J$ for fermions in an incommensurate superlattice of strength $\mathrm{\Delta }$. $N_{\mathrm{odd}\left(\mathrm{even}\right)}$ is the number of fermions on odd (even) sites. The inset shows the geometry. At time $t=0,I=1$. The dark (blue) curves show the result of keeping the first two terms in the cluster expansion in Eq. (6) for 20 sites. The light (orange) curve shows the result of including three-particle terms in the cluster expansion. Red dots correspond to a time-dependent DMRG simulation. Here $\mathrm{\Delta }=3J,U=3J$, the superlattice period ${\beta }^{-1}={\left(0.721\right)}^{-1}$, and the superlattice phase $\varphi =0$. The density is $\epsilon =0.2$ in the top graph and $\epsilon =0.5$ in the bottom graph.

Figure 2

Long-time density imbalance $I_{\infty }$ as a function of interaction strength $U/J$ for a 1D lattice with 20 sites at density $\epsilon =0.2$. The superlattice period is ${\beta }^{-1}={\left(0.721\right)}^{-1}$ in units of the lattice spacing. The different curves correspond to different superlattice strengths: $\mathrm{\Delta }/J=2,3,4,5$ (from bottom to top). The inset shows $I$ as a function of superlattice strength for $U/J=0$.

Figure 3

Long-time density imbalance $I_{\infty }$ as a function of interaction strength $U/J$ for a 1D lattice at superlattice strength $\mathrm{\Delta }/J=3$. The different curves show calculations using a cluster expansion on a 20-site lattice with different densities $\eta$ of doublons in the ensemble of initial states: The bottom (blue), middle (orange), and top (green) curves correspond to a ratio of doublons to particles of $\eta /\epsilon =0,0.23,0.5$, respectively. The blue and orange points are experimental measurements for a small doublon fraction $\left(\eta /\epsilon \approx 0.08\right)$ and larger doublon fraction $\left(\eta /\epsilon \approx 0.5\right)$, from Fig. 6 of Ref. [26].

Figure 4

Density imbalance $I$ averaged over time from $t=5/J$ to $t=10/J$ as a function of interaction strength $U/J$ for a 2D lattice with $10\times 10$ sites at superlattice strength $\mathrm{\Delta }/J=3.0$ and density $\epsilon =0.2$. The superlattice potential is only one dimensional: $V(i_x,i_y)=\mathrm{\Delta }cos(2\pi \beta i_x+\varphi )$. $J_y/J=0,0.1,1$ for the top, middle, and bottom (blue, orange, green) curves, respectively. The inset shows a diagram of the setup.