We study time dynamics of 1D disordered Heisenberg spin-1/2 chain focusing on a regime of large system sizes and a long time evolution. This regime is relevant for observation of manybody localization (MBL), a phenomenon that is expected to freeze the dynamics of the system and prevent it from reaching thermal equilibrium. Performing extensive numerical simulations of the imbalance, a quantity often employed in the experimental studies of MBL, we show that the regime of a slow power-law decay of imbalance persists to disorder strengths exceeding by at least a factor of 2 the current estimates of the critical disorder strength for MBL. Even though we investigate time evolution up to few thousands tunneling times, we observe no signs of the saturation of imbalance that would suggest freezing of system dynamics and provide a smoking gun evidence of MBL. We demonstrate that the situation is qualitatively different when the disorder is replaced by a quasiperiodic potential. In this case, we observe an emergence of a pattern of oscillations of the imbalance that is stable with respect to changes in the system size. This suggests that the dynamics of quasiperiodic systems remain fully local at the longest time scales we reach provided that the quasiperiodic potential is sufficiently strong. Our study identifies challenges in an unequivocal experimental observation of the phenomenon of MBL.
The datasets described below were used to produce all figures in the linked arXiv paper.
Figure 1. Interactions induce a slow decay of the imbalance I(t) that persists to long times. Jupyter notebook creating plots of Fig. 1 is fig1.ipynb; it employs data stored in directory ./fig1/
Figure 2. Time evolution of density correlation function C(t) in disordered XXZ model. Jupyter notebook creating plots of Fig. 2 is fig2.ipynb; it employs data stored in directory ./fig2/
Figure 3. Comparison of the time evolution for noninteracting system between exact propagation and TDVP approximate algorithm (L/2 fermions for the system size L = 50 at disorder strength W = 10). Jupyter notebook creating plots of Fig. 3 is fig3.ipynb; it employs data stored in directory ./fig3/
Figure 4. Time evolution of imbalance I(t) for systems of size L = 50, 100, 200 at disorder strength W = 8. Jupyter notebook creating plots of Fig. 4 is fig4.ipynb; it employs data stored in directory ./fig4/
Figure 5. Time evolution of imbalance for W = 10. Jupyter notebook creating plots of Fig. 5 is fig5.ipynb; it employs data stored in directory ./fig5/ Figure 6. Time evolution of imbalance for W = 10 in extended time interval. Jupyter notebook creating plots of Fig. 6 is fig6.ipynb; it employs data stored in directory ./fig6/
Figure 7. Time evolution of entanglement entropy for L = 50 and W = 10. Jupyter notebook creating plots of Fig. 7 is fig7.ipynb; it employs data sets stored in directory ./fig7/
Figure 8. Time evolution of imbalance I(t) for quasi-periodic potential, for small systems. Jupyter notebook creating plots of Fig. 8 is fig8.ipynb; it employs data sets stored in directory ./fig8/
Figure 9. Time evolution of imbalance I(t) for quasi-periodic potential for WQP = 4, 5. Jupyter notebook creating plots of Fig. 9 is fig9.ipynb; it employs data sets stored in directory ./fig9/
Figure 10. Time evolution of entanglement entropy for quasi-periodic potential with WQP = 5. Jupyter notebook creating plots of Fig. 10 is fig10.ipynb; it employs data sets stored in directory ./fig10/
Figure 11. Comparison of the imbalance I(t) (averaged over times [t − 10, t + 10]) for system size L = 200 and disorder strength W = 10 obtained with TEBD and TDVP algorithms. Jupyter notebook creating plots of Fig. 11 is fig11.ipynb; it employs data sets stored in directory ./fig11/
Figure 12. Comparison of the imbalance I(t) (averaged over times [t − 10, t + 10]) for system size L = 50 and disorder strength W = 8 obtained with TEBD and TDVP algorithms. Jupyter notebook creating plots of Fig. 12 is fig12.ipynb; it employs data sets stored in directory ./fig12/
Figure 13. The imbalance I(t) (averaged over times [t − 10, t + 10]) for system size L and disorder strength W obtained with TDVP algorithm. Jupyter notebook creating plots of Fig. 13 is fig13.ipynb; it employs data sets stored in directory ./fig13/
Figure 14. Convergence of the TDVP algorithm. Jupyter notebook creating plots of Fig. 14 is fig14.ipynb; it employs data sets stored in directory ./fig14/
Figure 15. Comparison of the values of the exponent β governing the decay of the imbalance I(t) for various bond dimensions. Jupyter notebook creating plots of Fig. 15 is fig15.ipynb; it employs data sets stored in directory ./fig15/
Figure 16. The long-time exponent β. Jupyter notebook creating plots of Fig. 16 is fig16.ipynb; it employs data sets stored in directory ./fig16/
Figure 17. Persistent oscillations for QP potential. Jupyter notebook creating plots of Fig. 17 is fig17.ipynb; it employs data sets stored in directory ./fig17/
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