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The energy-dependent scattering of fermions from a localized orbital at anenergy-dependent rate $\Gamma(\epsilon)\propto |\epsilon|^r$ gives rise toquantum critical points (QCPs) in the pseudogap single-impurity Anderson modelseparating a local moment phase with an unscreened spin moment from astrong-coupling phase which slightly deviates from the screened phase ofstandard Kondo problem. Using the time-dependent numerical renormalizationgroup (TD-NRG) approach we show that local dynamic properties alwaysequilibrate towards a steady-state value even for quenches across the QCP butwith systematic deviations from the thermal equilibrium depending on thedistance to the critical coupling. Local non-equilibrium properties arepresented for interaction quenches and hybridization quenches. We augment ournumerical data by an analytical calculation that becomes exact at short timesand find excellent agreement between the numerics and the analytical theory.For interaction quenches within the screened phase we find a universal functionfor the time-dependent local double occupancy. We trace back the discrepancybetween our results and the data obtained by a time-dependent Gutzwillervariational approach to restrictions of the wave-function ansatz in theGutzwiller theory: while the NRG ground states properly account for theformation of an extended spin moment which decouples from the system in theunscreened phase, the Gutzwiller ansatz only allows the formation of the spinmoment on the local impurity orbital.

Real-time dynamics induced by quenches across the quantum critical points in gapless Fermi systems with a magnetic impurity