Entanglement transitions in a boundary-driven open quantum many-body system

We introduce a numerical framework for integrating Markovian dynamics on tree tensor operator (TTO) ansatz states. This framework enables the simulation of both transient and steady-state regimes of systems governed by the Lindblad master equation, while preserving positivity of the density matrix and providing direct access to entanglement monotones. We demonstrate its capability to probe entanglement in open quantum many-body systems and to distinguish it from other correlations by studying a boundary-driven XXZ spin chain. Our analysis uncovers entanglement transitions driven by both the coupling to the environment and the anisotropy, revealing a striking connection between spatial entanglement scaling and spin-current.

💡 Research Summary

The authors present a novel numerical framework for simulating Markovian open‑quantum many‑body dynamics using the tree tensor operator (TTO) ansatz. A TTO represents a density matrix as two conjugate binary‑tree networks, P and P†, whose root tensor R encodes all bipartite entanglement between the left and right halves of the system. By evolving the Lindblad master equation with a second‑order Suzuki‑Trotter decomposition, the unitary part is treated with the time‑dependent variational principle (TDVP) while each local dissipative term is implemented as a Kraus channel directly attached to the corresponding site tensor. After each Kraus application, singular‑value decompositions compress the bond dimension χ and the Kraus dimension K back to preset maxima (χmax, Kmax), guaranteeing positivity and efficient access to entanglement monotones such as the entanglement of formation and the logarithmic negativity.

The method is applied to a paradigmatic boundary‑driven XXZ spin‑½ chain of length ℓ with open boundaries. The Hamiltonian HXXZ = Σj J (XjXj+1 + YjYj+1 + Δ ZjZj+1) is coupled at both ends to Markovian baths that inject spin up and down with rates set by a driving parameter μ=1 (maximal driving). This configuration is known to exhibit three transport regimes depending on the anisotropy Δ: ballistic (Δ<1), sub‑diffusive (Δ=1), and insulating (Δ>1). Starting from a fully polarized product state |Z−⟩, the authors quench the system‑bath coupling γ from zero to a finite value and monitor the ensuing dynamics.

Key observations include:

1. Current propagation – The spin current ⟨Jj⟩ spreads ballistically at early times, forming a clear light‑cone. The arrival of the current at the chain centre triggers simultaneous growth of the von Neumann entropies of the left half (SL), right half (SR), and the total system (S).

2. Entanglement vs. total correlations – The logarithmic negativity NL (a bona‑fide entanglement measure) and the mutual information IL:R = SL + SR – S are computed from the reduced density matrices. In the ballistic regime NL rises sharply together with IL:R and saturates at an extensive value, indicating volume‑law entanglement. In the sub‑diffusive regime IL:R continues to increase while NL remains almost constant and comparatively small, showing that most correlations are classical or mixed. In the insulating regime neither NL nor IL:R grows appreciably, reflecting strongly localized dynamics.

3. Steady‑state scaling – For large times the authors examine the scaling of NL with system size. In the ballistic case NL grows linearly with ℓ (volume law), whereas in the sub‑diffusive case both NL and IL:R display logarithmic growth, consistent with an area‑law‑like scaling. The insulating case shows no scaling, confirming the absence of long‑range entanglement.

4. Computational complexity – The required bond dimension χ and Kraus dimension K increase most dramatically in the ballistic regime, limiting simulations to intermediate times and moderate system sizes. The sub‑diffusive and insulating regimes are computationally cheaper, allowing longer simulations with moderate χ and K.

5. Relation to transport – The authors uncover a striking quantitative link between the spin current and the entanglement measures: the current’s dependence on γ mirrors that of NL in the ballistic regime, suggesting that the strength of the dissipative driving directly controls the amount of generated entanglement.

Overall, the paper bridges a methodological gap by providing a tensor‑network tool that preserves positivity, yields direct access to entanglement monotones, and can treat both transient and steady‑state regimes of open many‑body systems. The study of the boundary‑driven XXZ chain demonstrates that entanglement transitions are driven not only by the anisotropy Δ but also by the coupling to the environment, and that different transport regimes correspond to distinct entanglement scaling laws. These insights open avenues for engineering dissipative quantum devices that exploit boundary driving as a resource for entanglement generation and quantum information transport.