Entanglement transition in unitary system-bath dynamics

TL;DR

The paper investigates entanglement transitions in open quantum dynamics by comparing trajectory unravelings of GKSL evolution with fully unitary system–bath dynamics in a 2D lattice of free fermions. It shows a transition from logarithmic to area-law entanglement scaling across a left–right bipartition as the system–bath coupling $\\gamma$ increases, observable in the unitary model through the steady-state values of the logarithmic fermionic negativity $E$, mutual information $I$, and the connected correlation weight $\\mathcal{C}$. Remarkably, the transition is carried by bath–bath correlations even though the system reaches a trivial infinite-temperature state when traced alone, and finite-size scaling near the critical point $\\gamma_c \\approx 0.13J$ yields critical exponents $\\nu \\approx 1.26$ and $\\zeta \\approx 0$. The work highlights that entanglement transitions can be witnessed without post-selection in fully unitary system–bath dynamics and suggests experimental routes via quantum simulators, with potential extensions to interacting baths where non-Gaussian entanglement could emerge.

Abstract

The evolution of a system coupled to baths is commonly described by a master equation that, in the long-time limit, yields a steady-state density matrix. However, when the same evolution is unraveled into quantum trajectories, it is possible to observe a transition in the scaling of entanglement within the system as the system-bath coupling increases - a phenomenon that is invisible in the trajectory-averaged reduced density matrix of the system. Here, we go beyond the paradigm of trajectories from master equations and explore whether a qualitatively analogous entanglement-scaling transition emerges in the unitary evolution of the combined system-bath setup. We investigate the scaling of entanglement in a unitary quantum setup composed of a 2D lattice of free fermions, where each site is coupled to a fermionic bath. Varying the system-bath coupling reveals a transition from logarithmic-law to area-law scaling, visible in the logarithmic fermionic negativity, mutual information, and also in the correlations. This occurs while the system's steady-state properties are trivial, highlighting that the signatures of these different scalings are within the bath-bath correlations.

Figures (8)

• Figure 1: (a) A $2$D lattice of free fermions with open boundary conditions is coupled to external baths. After tracing out the baths, the dissipative dynamics of the system can be approximated by the GKSL master equation. For further analysis, we partition the system into the left and right partitions, each consisting of $N/2$ system sites. (b) We can also consider a unitary evolution, in which the environment is modeled as $M$ discrete local bath modes coupled to each system site. The left and right partitions each contain $N/2 \times (M+1)$ system and bath sites, and the degrees of freedom of the baths are retained.
• Figure 2: The trajectory-averaged steady-state (a) bipartite von Neumann entropy $\overline{\mathcal{S}}$ and (b) bipartite logarithmic fermionic negativity $\overline{\mathcal{E}}$ as a function of $L$. In the inset of (a), we plot $\overline{\mathcal{S}}$ against $L\log{L}$ for $\gamma=0.1J$ and $0.5J$. The filled symbols are numerical results, and the dashed lines are the linear best-fit lines based on all data points. The error bars are smaller than the symbol sizes and thus not shown.
• Figure 3: The unitary-evolved steady-state (a) bipartite logarithmic fermionic negativity $\mathcal{E}$, (b) mutual information $\mathcal{I}$, and (c) the connected correlation weight between the left and the right partitions $\mathcal{C}$ as a function of $L$. In the insets, we plot the same quantities against $L\log{L}$ for $\gamma=0.1J$. The filled symbols are numerical results, and the dashed lines are linear best-fit lines based on all data points.
• Figure 4: (a) The steady-state density-density correlation as a function of the distance $\mathcal{D}(d)$ for different $\gamma$ values. The system size is $N=676$. (b) $c(\gamma, L)$ obtained by fitting $\mathcal{C}(\gamma, x) = c(\gamma, L)x \ln{(x)} + b(\gamma, L)x$ within the range $x \in [8, L]$. The vertical dashed line marks the estimated size-invariant phase transition point $\gamma_c$. (c) Finite size scaling analysis of $c(\gamma, L)$. The data show a good collapse with exponent $\nu = 1.26 \pm 0.48$, $\zeta = 0.00 \pm 0.01$, and $\gamma_c = (0.13 \pm 0.01)J$. Subplots (b, c) share the same legends.
• Figure S1: The dynamics of $C^{\text{SS}}$ in the unitary-evolved $C_0$. (a) $\tilde{C}_{n,n}^{\text{SS}}$ quantifies the diagonal deviation from the infinite-temperature state. (b) $\tilde{C}_{n,n^\prime}^{\text{SS}}$ quantifies the off-diagonal deviation from the infinite-temperature state. (c) $\tilde{C}^{\text{SS}, \text{ME}}_{n, n^\prime}$ quantifies the difference between $C^{\text{SS}}$ and $C^{\text{ME}}$. In the steady-state, all three quantities are expected to diminish to $0$. The system size is $N=36$.