Dynamical Simulation of Integrable and Nonintegrable Models in the Heisenberg Picture

The numerical simulation of quantum many-body dynamics is typically limited by the linear growth of entanglement with time. Recently numerical studies have shown that for 1D Bethe-integrable models the simulation of local operators in the Heisenberg picture can be efficient. Using the spin-$1/2$ $XX$ chain as generic example of an integrable model that can be mapped to free fermions, we provide a simple explanation for this. We show furthermore that the same reduction of complexity applies to operators that have a high-temperature autocorrelation function which decays slower than exponential, i.e., with a power law. Thus efficient simulability may already be implied by a single conservation law as we will illustrate numerically for the spin-1 $XXZ$ model.

Figure 1

OSRE dynamics for the 40 site spin-$\frac{1}{2}$ $XXZ$ model for a split in the center. The legend gives initial operator and anisotropy in the order in which the arrow cuts the graphs. Dashed lines mark infinite index operators (see text). $\chi =1000$ is used in all cases and the numerical error is negligible on the time scale shown.

Figure 2

OSRE dynamics for the 40 site spin-1 $XXZ$ model for a split in the center. Dotted, dashed, and solid lines indicate simulations using $\chi =300$, $\chi =500$, and $\chi =1000$, respectively. The left panel features a logarithmic, the right panel a linear time scale. The curves show clear indication of the predicted long-time scaling. The insets show two corresponding ITAC curves for ${\widehat{\sigma }}_{20}^{+}$ (note the logarithmic vertical scaling).