Influence of a bosonic environment onto the non-equilibrium dynamics of local electronic states in a quantum impurity system close to a quantum phase transition

We investigate the influence of an additional bosonic bath onto the real-time dynamics of a localized orbital coupled to conduction band with an energy-dependent coupling function $\Gamma(\varepsilon) \propto |\varepsilon|^r$. Recently, a rich phase diagram has been found in this Bose-Fermi Anderson model, where the transitions between competing ground states are governed by quantum critical points. In addition to a transition between a Kondo singlet and a local moment, a localized phase has been established once the coupling to a sub-ohmic bosonic bath exceeds a critical value. Using the time-dependent numerical renormalization group approach, we show that the non-equilibrium dynamics with F-type of bath exponents can be fully understand within an effective single-impurity Anderson model using a renormalized local Coulomb interaction $U_{\rm ren}$. For regimes with B-type of bath exponents, the nature of the bosonic bath and the coupling strength has a profound impact on the electron dynamics which can only partially be understood using an appropriate $U_{\rm ren}$. The local expectation values always reach a steady state at very long times. By a scaling analysis for $\Lambda \to 1^+$, we find thermalization of the system only in the strong coupling regime. In the local moment and in the localized phase significant deviations between the steady-state value and the thermal equilibrium value are found that are related to the distance to the quantum critical point.