A Universal Approach to Hydrodynamics in Higher Dimensions via Tensor Networks

Introduction

Understanding how microscopic quantum systems give rise to macroscopic fluid behavior is a central challenge in many‑body physics. While one‑dimensional models can often be tackled with tensor‑network methods, higher‑dimensional systems quickly become intractable because entanglement spreads across a larger volume and the natural geometry of the lattice no longer matches the one‑dimensional structure of matrix product states (MPS). A recent preprint proposes a practical workaround: adapting the Density Matrix Truncation (DMT) algorithm so that it retains the geometrically local information needed for hydrodynamics, even when the underlying simulation uses a 1D MPS representation.

Extending DMT beyond one dimension

The core idea of DMT is to truncate the density matrix after each short time step, discarding high‑energy components that contribute little to long‑time transport. In one dimension this approach already yields accurate diffusion constants at high temperature. The authors generalize the method by:

1. Embedding higher‑dimensional lattices into a 1D chain while preserving nearest‑neighbor connectivity through a carefully chosen ordering (e.g., a snake‑like sweep). This ordering keeps the dominant local couplings adjacent in the MPS, limiting the growth of entanglement.
2. Applying a locality‑preserving truncation that respects the original lattice geometry. Instead of truncating purely on bond dimension, the algorithm monitors operators that span the physical distance on the lattice and retains those that dominate hydrodynamic currents.
3. Iterating short‑time evolutions with a Trotter decomposition that respects the higher‑dimensional interaction pattern, ensuring that errors do not accumulate in a way that would obscure transport coefficients.

These steps allow the simulation to remain computationally feasible up to three dimensions, a regime where conventional tensor‑network approaches typically fail.

Benchmarking transport and crossover dynamics

Using the extended DMT, the authors compute high‑temperature diffusion constants for several lattice models in two and three dimensions. The results match analytical high‑temperature expansions and, where available, quantum Monte Carlo data, confirming that the method captures the correct late‑time behavior. In addition to steady‑state transport, the study demonstrates that the algorithm can resolve the slow crossover from integrable to chaotic dynamics when a weak integrability‑breaking perturbation is introduced. This crossover appears as a gradual change in the decay exponent of correlation functions, which the DMT captures without requiring prohibitively large bond dimensions.

Practical implications and future directions

The ability to extract quantitative hydrodynamic coefficients from microscopic Hamiltonians in higher dimensions opens several avenues:

• Benchmarking quantum simulators: Experimental platforms such as cold‑atom arrays or Rydberg‑atom processors can now be compared against reliable classical predictions for diffusion and viscosity at temperatures where exact diagonalization is impossible.
• Exploring anomalous transport: By tuning interactions or disorder, researchers can investigate regimes where standard diffusion breaks down, using DMT as a diagnostic tool.
• Hybrid algorithms: The locality‑preserving truncation strategy could be combined with other tensor‑network architectures (e.g., projected entangled‑pair states) to further push the dimensional frontier.

Conclusion

By rethinking how to preserve geometric locality within a one‑dimensional tensor‑network framework, the authors provide a scalable route to simulate high‑temperature hydrodynamics in two and three dimensions. Their extended DMT algorithm delivers accurate diffusion constants and captures subtle crossover dynamics, offering a valuable bridge between microscopic quantum models and emergent fluid behavior. The work suggests that classical computation can still play a pivotal role in validating and guiding experiments on complex quantum many‑body systems.