  We study the steady states of translation-invariant open quantum many-body
systems governed by Lindblad master equations, where the Hamiltonian is
quadratic in the ladder operators, and the Lindblad operators are either linear
or quadratic and Hermitian. These systems are called quasifree and quadratic,
respectively. We find that steady states of one-dimensional systems with
finite-range interactions necessarily have exponentially decaying Green's
functions. For the quasifree case without quadratic Lindblad operators, we show
that fermionic systems with finite-range interactions are noncritical for any
number of spatial dimensions and provide bounds on the correlation lengths.
Quasifree bosonic systems can be critical in $D>1$ dimensions. Last, we address
the question of phase transitions in quadratic systems and find that, without
symmetry constraints beyond invariance under single-particle basis and
particle-hole transformations, all gapped Liouvillians belong to the same
phase.
