  Lindbladian formalism, as tuned to dissipative and open systems, has been
all-pervasive to interpret non-equilibrium steady states of quantum many-body
systems. We study the fate of free fermionic and superconducting phases in a
dissipative one-dimensional Kitaev model - where the bath acts both as a source
and a sink of fermionic particles with different coupling rates. As a function
of these two couplings, we investigate the steady state, its entanglement
content, and its approach from varying initial states. Interestingly, we find
that the steady state phase diagram retains decipherable signatures of ground
state critical physics. We also show that early-time fidelity is a useful
marker to find a subclass of phase transitions in such situations. Moreover, we
show that the survival of critical signatures at late-times, strongly depend on
the thermal nature of the steady state. This connection hints at a
correspondence between quantum observables and classical magnetism in the
steady state of such systems. Our work uncovers interesting connections between
dissipative quantum many-body systems, thermalization of a classical spin and
many-body quantum critical phenomena.

Strongly Correlated Electrons;Statistical Mechanics;Quantum Physics