Operator entanglement in $\mathrm{SU}(2)$-symmetric dissipative quantum many-body dynamics

TL;DR

The paper investigates operator entanglement dynamics in an open quantum many-body system with SU(2) symmetry under Lindblad dissipation. Using an MPDO representation and iTEBD evolution, it shows that after an initial rise and fall, the operator entanglement grows logarithmically in time: $S_{\mathrm{op}}(t) \sim \eta \log_{2}(tJ) + S_{0}$, with a dissipation-dependent prefactor $\eta$. The symmetry-resolved analysis reveals that both the Gaussian distribution of sector probabilities (variance $\delta^2 \sim (tJ)^{2\alpha}$) and nontrivial symmetry-sector entanglement contribute to the late-time growth, and the SU(2) results can be understood from its U(1) subsymmetry. Breaking SU(2) to U(1) still yields late-time logarithmic growth for certain initial states and dissipation types, indicating a potentially generic feature of open quantum dynamics with U(1) symmetry. Overall, the work deepens understanding of how symmetries shape operator entanglement and the classical simulability of dissipative quantum systems.

Abstract

The presence of symmetries can lead to nontrivial dynamics of operator entanglement in open quantum many-body systems, which characterizes the cost of an matrix product density operator (MPDO) representation of the density matrix in the tensor-network methods and provides a measure for the corresponding classical simulability. One example is the $\mathrm{U}(1)$-symmetric open quantum systems with dephasing, in which the operator entanglement increases logarithmically at late times instead of being suppressed by the dephasing. Here we numerically study the far-from-equilibrium dynamics of operator entanglement in a dissipative quantum many-body system with the more complicated $\mathrm{SU}(2)$ symmetry and dissipations beyond dephasing. We show that after the initial rise and fall, the operator entanglement also increases again in a logarithmic manner at late times in the $\mathrm{SU}(2)$-symmetric case. We find that this behavior can be fully understood from the corresponding $\mathrm{U}(1)$ subsymmetry by considering the symmetry-resolved operator entanglement. But unlike the $\mathrm{U}(1)$-symmetric case with dephasing, both the classical Shannon entropy associated with the probabilities for the half system being in different symmetry sectors and the corresponding symmetry-resolved operator entanglement have nontrivial contributions to the late time logarithmic growth of operator entanglement. Our results show evidence that the logarithmic growth of operator entanglement at long times is a generic behavior of dissipative quantum many-body dynamics with $\mathrm{U}(1)$ as the symmetry or subsymmetry and for more broad dissipations beyond dephasing. By breaking the $\mathrm{SU}(2)$ symmetry of our quantum many-body dynamics to $\mathrm{U}(1)$, we also show that the latter property is valid even for open quantum systems with only $\mathrm{U}(1)$ symmetry.

Figures (5)

• Figure 1: Operator entanglement dynamics in the $\mathrm{SU}(2)$-symmetric dissipative quantum many-body system. (a) Sketch of the model. We consider a quantum spin chain with coherent nearest-neighbor coupling $J$ (blue arrows) and local dissipation proportional to the dipole interaction between neighbor sites with strength $\gamma$ (green arrows). (b) MPDO decomposition of the density matrix $\rho$ in terms of local tensors $\Gamma$ (blue squares) and $\lambda$ (orange circles). Here $\alpha_{i-1}$ and $\alpha_{i}$ are the bond indices, while $s_{i}$ denotes the combined physical index of the bra and ket legs at site $i$. The operator entanglement $S_{\mathrm{op}}$ at certain bond can be calculated from the Schmidt vectors $\lambda$. (c) Time evolution of $S_{\mathrm{op}}$ for the product initial state of singlet pairs with different dissipation strength $\gamma=0.05J$, $0.10J$, $0.15J$, $0.20J$, and $0.25J$. The black dashed line indicates the logarithmic growth of operator entanglement at long times (log-scale time axis), i.e., $S_{\mathrm{op}}(t\to\infty)=\eta\log_{2}(tJ)+S_{0}$. (d) Numerical prefactor $\eta$ and offset $S_{0}$ obtained from the local tangent of operator entanglement at time $t_{0}J$. The results are converged for time step $\delta tJ=0.5$ and maximum bond dimension $\chi=50000$.
• Figure 2: Symmetry-resolved operator entanglement in spin sectors. (a) Probabilities $p_{S}$ of the infinite chain with total spin $S$ in the half system at increasingly late times ($50\leq tJ\leq250$ from light to dark) for $\gamma=0.05J$. The lines are fits with the trial function shown in the main text. (b) The corresponding symmetry-resolved operator entanglement $S_{\mathrm{op},S}$ as a function of time. Only the data with probabilities $p_{S}>10^{-4}$ are presented. (c) (d) The classical Shannon entropy $-\sum p_{S}\log_{2}p_{S}$ and the averaged symmetry-resolved operator entanglement $\sum p_{S}S_{\mathrm{op},S}$ as a function of time for $\gamma/J=0.05$, $0.10$, $0.15$, $0.20$, and $0.25$. The black dashed lines indicate the logarithmic growth at late times (log-scale time axis). The inserts present the corresponding prefactors. The results are converged for the time step $\delta tJ=0.5$ and maximum bond dimension $\chi=50000$.
• Figure 3: Symmetry-resolved operator entanglement in magnetization sectors. (a) Probabilities $p_{S_{z}}$ of the infinite chain with magnetization $S_{z}$ in the half system at increasingly late times ($50\leq tJ\leq250$ from light to dark) for $\gamma=0.05J$. Lines are the Gaussian fits. (b) Variance $\delta$ of the Gaussian fits as a function of time for $\gamma/J=0.05$, $0.10$, $0.15$, $0.20$, and $0.25$. The black dashed line indicates $\delta\sim(tJ)^{\alpha}$ at late times (double-log scale). The corresponding exponent $\alpha$ for each $\gamma$ is shown in the insert. (c) Symmetry-resolved operator entanglement difference $\Delta S_{\mathrm{op},S_{z}}$ as a function of time for $\gamma=0.05J$ and $S_{z}=1,2,3,4,5$ (light to dark). The insert shows the data scaled by $1/S_{z}^{2}$, and the black line is the fit $(a+btJ)^{-c}$ with $a=2.4964$, $b=0.2554$, and $c=1.1228$. (d) Symmetry-resolved operator entanglement $S_{\mathrm{op},S_{z}}$ in the magnetization sector $S_{z}=0$ as a function of time for various $\gamma$. The black dashed line indicates the logarithmic growth of $S_{\mathrm{op},S_{z}=0}$ at late times (log-scale time axis), with the corresponding prefactors shown in the insert. Here the results are converged for time step $\delta tJ=0.5$ and maximum bond dimension $\chi=50000$.
• Figure 4: Operator entanglement for quantum dynamics with symmetry being broken to $\mathrm{U}(1)$. We consider the Néel initial state in (a) and the product initial state of triplet pairs in (b). The black dashed lines indicate the logarithmic growth of operator entanglement at late times (log-scale time axis), with the prefactor $\eta\approx 0.24$ for (a) and $\approx0.60$ for (b), respectively. The inserts show the time evolution of symmetry-resolved operator entanglement $S_{\mathrm{op},S_{z}}$ for $S_{z}=0,1,2,3,4$ (from light to dark). Here we set $\gamma=0.5J$. The time step $\delta tJ$ is $0.25$ for (a) and $0.5$ for (b). We choose the maximum bond dimension $\chi$ as $4000$ for (a) and $12000$ for (b).
• Figure 5: Numerical convergence of operator entanglement $S_{\mathrm{op}}$. (a) Convergence in the time step $\delta t$. (b) Convergence in the maximum bond dimension $\chi$. Here we set $\gamma=0.05J$. The maximum bond dimension $\chi$ in (a) is $50000$, and the time step $\delta tJ$ in (b) is $0.5$.