Time-dependent variational principle in matrix-product state manifolds: Pitfalls and potential

We study the applicability of the time-dependent variational principle in matrix-product state manifolds for the long time description of quantum interacting systems. By studying integrable and nonintegrable systems for which the long time dynamics are known we demonstrate that convergence of long time observables is subtle and needs to be examined carefully. Remarkably, for the disordered nonintegrable system we consider the long time dynamics are in good agreement with the rigorously obtained short time behavior and with previous obtained numerically exact results, suggesting that at least in this case, the apparent convergence of this approach is reliable. Our study indicates that, while great care must be exercised in establishing the convergence of the method, it may still be asymptotically accurate for a class of disordered nonintegrable quantum systems.

Figure 1

Clean XX model ($\mathrm{\Delta }=0,W=0$). Upper panel: MSD as a function of time for various bond dimensions (32, 64, 128) averaged over 200–500 realizations of initial spin configurations and disorder. More intense shades represent larger bond dimensions and shaded areas indicate the standard deviation of the observables obtained using a bootstrap procedure. The black solid line is an exact solution, obtained numerically. The inset shows the log-log scale of the main panel with the black dotted line corresponding to diffusion. Lower panel: Time-dependent diffusion constant $D\left(t\right)$. The dashed black line on both plots represents the convergence time, $t_{*}$.

Figure 2

Same as Fig. (1) but for the disordered XX model ($\mathrm{\Delta }=0,W=1$) for 100 realizations of initial spin configurations and disorder.

Figure 3

Same as Fig. (1) but for clean XX ladder of length $L=50$ with isotropic coupling between the rungs. The results were obtained by averaging 100 realizations of initial spin configurations. The black dotted line in the bottom panel represents the previously reported diffusion constant [20].

Figure 4

Same as Fig. (1) but for disordered XXZ model in the subdiffusive regime ($\mathrm{\Delta }=1,W=1.5$) for 200 realizations of initial spin configurations and disorder.