We develop an alternative time-dependent numerical renormalization group (TDNRG) formalism for multiple quenches and implement it to study the response of a quantum impurity system to a general pulse. Within this approach, we reduce the contribution of the NRG approximation to numerical errors in the time evolution of observables by a formulation that avoids the use of the generalized overlap matrix elements in our previous multiple-quench TDNRG formalism [Nghiem et al., Phys. Rev. B 89, 075118 (2014); Phys. Rev. B 90, 035129 (2014)]. We demonstrate that the formalism yields a smaller cumulative error in the trace of the projected density matrix as a function of time and a smaller discontinuity of local observables between quenches than in our previous approach. Moreover, by increasing the switch-on time, the time between the first and last quench of the discretized pulse, the long-time limit of observables systematically converges to its expected value in the final state, i.e., the more adiabatic the switching, the more accurately is the long-time limit recovered. The present formalism can be straightforwardly extended to infinite switch-on times. We show that this yields highly accurate results for the long-time limit of both thermodynamic observables and spectral functions, and overcomes the significant errors within the single quench formalism [Anders et al., Phys. Rev. Lett. 95, 196801 (2005); Nghiem et al., Phys. Rev. Lett. 119, 156601 (2017)]. This improvement provides a first step towards an accurate description of nonequilibrium steady states of quantum impurity systems, e.g., within the scattering states NRG approach [Anders, Phys. Rev. Lett. 101, 066804 (2008)].
We investigate the time evolution of the Kondo resonance in response to a quench by applying the time-dependent numerical renormalization group (TDNRG) approach to the Anderson impurity model in the strong correlation limit. For this purpose, we derive within the TDNRG approach a numerically tractable expression for the retarded two-time nonequilibrium Green function $G(t+{t}^{\ensuremath{'}},t)$, and its associated time-dependent spectral function, $A(\ensuremath{\omega},t)$, for times
t both before and after the quench. Quenches from both mixed valence and Kondo correlated initial states to Kondo correlated final states are considered. For both cases, we find that the Kondo resonance in the zero temperature spectral function, a preformed version of which is evident at very short times $t\ensuremath{\rightarrow}{0}^{+}$, only fully develops at very long times $t\ensuremath{\gtrsim}1/{T}_{K}$, where
TK is the Kondo temperature of the final state. In contrast, the final state satellite peaks develop on a fast time scale $1/\mathrm{\ensuremath{\Gamma}}$ during the time interval $\ensuremath{-}1/\mathrm{\ensuremath{\Gamma}}\ensuremath{\lesssim}t\ensuremath{\lesssim}+1/\mathrm{\ensuremath{\Gamma}}$, where $\mathrm{\ensuremath{\Gamma}}$ is the hybridization strength. Initial and final state spectral functions are recovered in the limits $t\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$ and $t\ensuremath{\rightarrow}+\ensuremath{\infty}$, respectively. Our formulation of two-time nonequilibrium Green functions within the TDNRG approach provides a first step towards using this method as an impurity solver within nonequilibrium dynamical mean field theory.
The time-dependent numerical renormalization group method (TDNRG) [Anders et al., Phys. Rev. Lett. 95, 196801 (2005)] was recently generalized to multiple quenches and arbitrary finite temperatures [Nghiem et al., Phys. Rev. B 89, 075118 (2014)] by using the full density matrix approach [Weichselbaum et al., Phys. Rev. Lett. 99, 076402 (2007)]. The formalism rests solely on the numerical renormalization group (NRG) approximation. In this paper, we numerically implement this formalism to study the response of a quantum impurity system to a general pulse and to periodic driving, in which a smooth pulse or a periodic train of pulses is approximated by a sufficient number of quenches. We show how the NRG approximation affects the trace of the projected density matrices and the continuity of the time evolution of a local observable. We also investigate the long-time limit of a local observable upon switching from a given initial state to a given final state as a function of both the pulse shape and the switch-on time, finding that this limit is improved for smoother pulse shapes and longer switch-on times. This lends support to our earlier suggestion that the long-time limit of observables, following a quench between a given initial state and a given final state, can be improved by replacing a sudden large and instantaneous quench by a sequence of smaller ones acting over a finite time interval: longer switch-on times and smoother pulses, i.e., increased adiabaticity, favor relaxation of the system to its correct thermodynamic long-time limit. For the case of periodic driving, we compare the TDNRG results to the exact analytic ones for the noninteracting resonant level model, finding better agreement at short to intermediate time scales in the case of smoother driving fields. Finally, we demonstrate the validity of the multiple-quench TDNRG formalism for arbitrary temperatures by studying the time evolution of the occupation number in the interacting Anderson impurity model in response to a periodic switching of the local level from the mixed valence to the Kondo regime at low, intermediate, and high temperatures.