[Paper Review] Dissipation-assisted operator evolution method for capturing hydrodynamic transport

Tibor Rakovszky, Curt von Keyserlingk+1 more|arXiv (Cornell University)|Apr 10, 2020

Theoretical and Computational Physics20 citations

TL;DR

The paper introduces the dissipation-assisted operator evolution (DAOE) method, a tensor network-based approach that uses artificial dissipation to suppress non-local operator entanglement during Heisenberg picture evolution, enabling long-time simulation of hydrodynamic transport. By gradually reducing dissipation strength and extrapolating to zero dissipation, the method achieves high-precision estimation of spin and energy diffusion constants in strongly interacting quantum systems, overcoming finite-size and time limitations of prior methods.

ABSTRACT

We introduce the dissipation-assisted operator evolution (DAOE) method for calculating transport properties of strongly interacting lattice systems in the high temperature regime. DAOE is based on evolving observables in the Heisenberg picture, and applying an artificial dissipation that reduces the weight on non-local operators. We represent the observable as a matrix product operator, and show that the dissipation leads to a decay of operator entanglement, allowing us to capture the dynamics to long times. We test this scheme by calculating spin and energy diffusion constants in a variety of physical models. By gradually weakening the dissipation, we are able to consistently extrapolate our results to the case of zero dissipation, thus estimating the physical diffusion constant with high precision.

Motivation & Objective

• To develop a numerically controlled method for computing transport coefficients in strongly interacting quantum systems beyond finite-size and short-time limitations.
• To address the challenge of simulating long-time dynamics of conserved operator evolution in the Heisenberg picture using tensor network techniques.
• To enable precise estimation of diffusion constants by introducing artificial dissipation that suppresses non-local operator growth while preserving physical transport features.
• To demonstrate convergence of results to the physical limit (zero dissipation) across diverse models, including Hamiltonian and Floquet circuits.
• To validate the method on spin-1/2 and spin-1 chains, showing robustness and consistency in non-integrable regimes.

Proposed method

• The method evolves local observables in the Heisenberg picture using a non-unitary evolution that includes an artificial dissipation superoperator.
• The dissipation selectively reduces the weight of operators with large spatial support, promoting a compact matrix product operator (MPO) representation.
• The Liouvillian evolution is approximated via time-evolving block decimation (TEBD) on a doubled Hilbert space, where operators are mapped to matrix product states (MPS).
• A dissipator $\mathcal{D}_{\ell_*,\gamma}$ is applied periodically, suppressing operators with support beyond a cutoff $\ell_*$, with strength $\gamma$.
• The method uses a systematic extrapolation to zero dissipation ($\gamma \to 0$) to extract the physical diffusion constant.
• The approach is applied to spin and energy diffusion in one-dimensional models, including XXZ chains and Floquet circuits with $S^z$ conservation.

Experimental results

Research questions

• RQ1 Can artificial dissipation be used to stabilize long-time operator evolution in tensor network simulations of quantum transport?
• RQ2 Does the DAOE method enable accurate extrapolation to the physical diffusion constant in the limit of vanishing dissipation?
• RQ3 How does the method perform in non-integrable models, particularly near integrable points?
• RQ4 Can the method capture both diffusive and ballistic contributions to operator spreading in hydrodynamic regimes?
• RQ5 What is the role of the support cutoff $\ell_*$ and dissipation strength $\gamma$ in achieving convergence of transport coefficients?

Key findings

• The DAOE method successfully captures long-time dynamics of conserved operator evolution by suppressing non-local contributions via artificial dissipation.
• Extrapolation to zero dissipation yields consistent and precise estimates of spin and energy diffusion constants across multiple models.
• For spin-1/2 Floquet circuits, non-monotonic behavior in diffusion constants at small $\ell_*$ suggests sensitivity to nearby integrable points, while $\ell_* \geq 3$ shows stable convergence.
• In spin-1 Floquet circuits, the method shows no strong non-monotonicities, indicating that the anomalous behavior in spin-1/2 is due to proximity to integrability rather than the Floquet structure.
• The method confirms that operator entanglement growth is tamed by dissipation, enabling long-time simulations with controlled bond dimensions.
• The weight of operators with support $\ell$ at time $t$ scales as $(Dt)^{-3/2}$, consistent with diffusive spreading and ballistic front propagation.