Many-body systems with both coherent dynamics and dissipation constitute a
rich class of models which are nevertheless much less explored than their
dissipationless counterparts. The advent of numerous experimental platforms
that simulate such dynamics poses an immediate challenge to systematically
understand and classify these models. In particular, nontrivial many-body
states emerge as steady states under non-equilibrium dynamics. While these
states and their phase transitions have been studied extensively with mean
field theory, the validity of the mean field approximation has not been
systematically investigated. In this paper, we employ a field-theoretic
approach based on the Keldysh formalism to study nonequilibrium phases and
phase transitions in a variety of models. In all cases, a complete description
via the Keldysh formalism indicates a partial or complete failure of the mean
field analysis. Furthermore, we find that an effective temperature emerges as a
result of dissipation, and the universal behavior including the dynamics near
the steady state is generically described by a thermodynamic universality
class.
