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The study of quantum many-body physics in Liouvillian open quantum systems becomes increasingly important with the recent progress in experimental control on dissipative systems and their exploitation for technological purposes. A central question in open quantum systems concerns the fate of quantum correlations, and the possibility of controlling them by engineering the competition between the Hamiltonian dynamics and the coupling of the system to a bath. Such a question is very challenging from a theoretical point of view, as numerical methods faithfully accounting for quantum correlations are either relying on exact diagonalization, drastically limiting the sizes that can be treated numerically, or on approximations on the range or strength of quantum correlations, associated with the choice of a specific ansatz for the density matrix. In this work we propose a new method to treat open quantum spin lattices, based on stochastic quantum trajectories for the solution of the open-system dynamics. Along each trajectory, the hierarchy of equations of motion for many-point spin-spin correlators is truncated to a given finite order, assuming that multivariate kth-order cumulants vanish for k exceeding a cutoff k_{c}. This scheme allows one to track the evolution of quantum spin-spin correlations up to order k_{c} for all length scales. We validate this approach in the paradigmatic case of the dissipative phase transitions of the two-dimensional XYZ lattice subject to spontaneous decay. We convincingly assess the existence of steady-state phase transitions from paramagnetic to ferromagnetic, and back to paramagnetic, upon increasing one of the Hamiltonian spin-spin couplings, as well as the classical Ising nature of such transitions. Moreover, the approach allows us to show the presence of significant quantum correlations in the vicinity of the dissipative critical point, and to unveil the presence of spin squeezing, which can be proven to be a tight lower bound to the quantum Fisher information. |
