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

Abstract

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.

Figures (4)

• Figure 1: Tree tensor operator (TTO): a positive and loopless tensor network ansatz for density operators $\rho$. (a) The TTO consists of two branches, $\mathcal{P}$ and $\mathcal{P}^\dagger$, linked through the Kraus dimension $K$, such that $\rho = \mathcal{P} \mathcal{P}^\dagger = \sum_{k=1}^{K} p_k \ket{\Psi_k}\bra{\Psi_k}$. The root tensor $\mathcal{R}$, whose size is determined by $K$ and the bond dimension $\chi$, encodes all information on entanglement between the system's halves $L$ and $R$. Computations of entanglement monotones, such as entanglement of formation eof_benett and logarithmic negativity log_neg_plenio, thus only involve $\mathcal{R}$. (b) The Lindblad master equation \ref{['eq:lindblad_master_loc']} is approximated by successively applying Eq. \ref{['eq:time_evo_trott']}, whose unitary part is implemented via the time-dependent variational principle. The dissipative part is carried out via local Kraus channels, which are realized by connecting the Kraus tensor $\mathcal{K}_j$ to site $j$ and contracting it with its adjoint in the upper branch, as shown for $j=1$. (c) Contracting $\mathcal{K}_j$ with its connecting tensor in the TTO results in a four-leg tensor, whose additional leg is moved toward $\mathcal{R}$ via consecutive singular value decompositions (SVD), see (d). (e) After applying the Kraus channel, the bond dimension and the Kraus dimension are compressed via SVDs on the involved tensors.
• Figure 2: (a) Pictorial sketch of the boundary-driven spin-chain. The XXZ model from Eq. \ref{['eq:ham_xxz']} is coupled at its boundaries to Markovian environments. The latter induce a current flowing through the system, resulting in a non-equilibrium stationary state. (b) Spatio-temporal profile of the spin-current $\left<\mathcal{J}_j\right>$, following a quench with initial state $\ket{Z-}$ for the different transport regimes, i.e., $\Delta=1/2,1,3/2$. System size is $\ell=32$ and $\gamma/J=1$. Horizontal dashed lines mark the arrival of the spin-current at the system's center.
• Figure 3: Dynamics following a quench with initial state $\ket{Z-}$ for the different transport regimes. System size is $\ell=32$ and $\gamma/J=1$. (a) Von Neumann entropy of the left half, $S_L$, the right half, $S_R$, and total system $S$. (b) Logarithmic negativity $N_L$ and mutual information $I_{L:R}$, quantifying the entanglement and the total correlation between left and right halves respectively. Vertical dashed lines mark the arrival of the spin-current at the system's center.
• Figure 4: Entanglement scaling across regimes: (a) Dynamics of logarithmic negativity $N_L$ in the ballistic regime ($\Delta = 1/2$) for various system sizes $\ell$ at $\gamma/J = 1$. (b) $N_L$ and mutual information $I_{L:R}$ at $t^* = 10J$ for $\ell = 32$, $\Delta = 1/2$, and varying $\gamma$. Solid line shows analytical spin-current in the stationary state as $\ell \rightarrow \infty$prosen_ex_mpo_xxz2, exhibiting similar $\gamma$-dependence. (c) Scaling of $N_L$ and $I_{L:R}$ in the stationary state of the subdiffusive regime ($\Delta = 1$). (d) Logarithmic negativity $N_{j,j+1}$ of neighboring pairs of sites against the rest of the system in the stationary state of (c) for $\ell=128$.