Signatures of a quantum stabilized fluctuating phase and critical dynamics in a kinetically-constrained open many-body system with two absorbing states

We introduce and investigate an open many-body quantum system in which kinetically constrained coherent and dissipative processes compete. The form of the incoherent dissipative dynamics is inspired by that of epidemic spreading or cellular-automaton-based computation related to the density-classification problem. It features two non-fluctuating absorbing states as well as a $\mathcal{Z}_2$-symmetric point in parameter space. The coherent evolution is governed by a kinetically constrained $\mathcal{Z}_2$-symmetric many-body Hamiltonian which is related to the quantum XOR-Fredrickson-Andersen model. We show that the quantum coherent dynamics can stabilize a fluctuating state and we characterize the transition between this active phase and the absorbing states. We also identify a rather peculiar behavior at the $\mathcal{Z}_2$-symmetric point. Here the system approaches the absorbing-state manifold with a dynamics that follows a power-law whose exponent continuously varies with the relative strength of the coherent dynamics. Our work shows how the interplay between coherent and dissipative processes as well as symmetry constraints may lead to a highly intricate non-equilibrium evolution and may stabilize phases that are absent in related classical problems.