Operator entanglement in SU(2)-symmetric dissipative quantum many-body dynamics

The presence of symmetries can lead to nontrivial dynamics of operator entanglement in open quantum many-body systems, which characterizes the cost of a matrix product density operator representation of the density matrix in tensor-network methods and provides insights into the corresponding classical simulability. One example is 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 SU (2) symmetry and dissipations beyond dephasing. We show that after the initial rise and fall, the operator entanglement increases again in a logarithmic manner at late times in the SU (2) -symmetric case. We find that this behavior can be fully understood from the corresponding U (1) subsymmetry by considering the symmetry-resolved operator entanglement. Especially, the probability distribution of different U (1) sectors also follows the Gaussian distribution observed in the U (1) -symmetric case with dephasing, with the variance growing as a power law. But unlike the latter, both the classical Shannon entropy associated with the probabilities for the half system to be in different symmetry sectors and the corresponding symmetry-resolved operator entanglement now make nontrivial contributions to the late-time logarithmic growth of operator entanglement in our SU (2) -symmetric case. 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 U (1) as the symmetry or subsymmetry and for broader dissipations beyond dephasing, although more analytical and numerical proof from future studies is still required for this conjecture. By breaking the SU (2) symmetry of our quantum many-body dynamics to U (1), we also show that the latter property---the logarithmic growth behavior of operator entanglement for broader dissipations beyond dephasing---is valid even for open quantum systems with only U (1) symmetry.