  We discuss a notion of quantum critical exponents in open quantum many-body
systems driven by quantum noise. We show that in translationally invariant
quantum lattice models undergoing quasi-local Markovian dissipative processes,
mixed states emerge as stationary points that show scaling laws for the
divergence of correlation lengths giving rise to well-defined critical
exponents. The main new technical tool developed here is a complete description
of steady states of free bosonic or fermionic translationally invariant systems
driven by quantum noise: This approach allows to express all correlation
properties in terms of a symbol, paralleling the Fisher-Hartwig theory used for
ground state properties of free models. We discuss critical exponents arising
in bosonic and fermionic models. Finally, we relate the findings to recent work
on dissipative preparation of pure dark and matrix product states by Markovian
noisy processes and sketch further perspectives.
