  We study the effects of dissipative boundaries in many-body systems at
continuous quantum transitions, when the parameters of the Hamiltonian driving
the unitary dynamics are close to their critical values. As paradigmatic
models, we consider fermionic wires subject to dissipative interactions at the
boundaries, associated with pumping or loss of particles. They are induced by
couplings with a Markovian baths, so that the evolution of the system density
matrix can be described by a Lindblad master equation. We study the quantum
evolution arising from variations of the Hamiltonian and dissipation
parameters, starting at t=0 from the ground state of the critical Hamiltonian.
Two different dynamic regimes emerge: (i) an early-time regime for times t ~ L,
where the competition between coherent and incoherent drivings develops a
dynamic finite-size scaling, obtained by extending the scaling framework
describing the coherent critical dynamics of the closed system, to allow for
the boundary dissipation; (ii) a large-time regime for t ~ L^3 whose dynamic
scaling describes the late quantum evolution leading to the t->infty stationary
states.
