Signatures of quantum phases in a dissipative system

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.