Dynamical simulation of integrable and non-integrable models in the Heisenberg picture

TL;DR

This paper explains why simulating local operators in the Heisenberg picture is efficient for certain integrable quantum models, due to slow growth of operator-space entanglement, and extends this understanding to models with power-law decaying correlations.

Contribution

It provides a simple explanation for efficient simulation of integrable models in the Heisenberg picture and shows this applies to models with slow-decaying correlations, supported by numerical examples.

Findings

Operator-space entanglement grows logarithmically in integrable models.

High-temperature autocorrelation functions with power-law decay also exhibit reduced complexity.

Efficient simulation is possible with just a single conservation law, like magnetization.

Abstract

The numerical simulation of quantum many-body dynamics is typically limited by the linear growth of entanglement with time. Recently numerical studies have shown, however, that for 1D Bethe-integrable models the simulation of local operators in the Heisenberg picture can be efficient as the corresponding operator-space entanglement grows only logarithmically. Using the spin-1/2 XX chain as generic example of an integrabel model that can be mapped to free particles, we here provide a simple explanation for this. We show furthermore that the same reduction of complexity applies to operators that have a high-temperature auto correlation function which decays slower than exponential, i.e., with a power law. This is amongst others the case for models where the Blombergen-De Gennes conjecture of high-temperature diffusive dynamics holds. Thus efficient simulability may already be implied by a single conservation law (like that of total magnetization), as we will illustrate numerically for the spin-1 XXZ model.