Dissipative Dynamics in Open Fermionic Chains

By merging the Feynman-Vernon's approach with the out-of-equilibrium Keldysh-Schwinger formalism, we construct the reduced generating functional through which all the time-dependent correlation functions of an open fermionic system can be directly derived by applying the appropriate functional derivatives. As a concrete example, we investigate the transverse Ising model, we derive the covariance matrix at the steady state of the system and we investigate its critical behavior.