The time-dependent numerical renormalization group (TDNRG) method [Anders et al. Phys. Rev. Lett. 95, 196801 (2005)] offers the prospect of investigating in a non-perturbative manner the time-dependence of local observables of interacting quantum impurity models at all time scales following a quantum quench. Here, we present a generalization of this method to arbitrary finite temperature by making use of the full density matrix approach [Weichselbaum et al. Phys. Rev. Lett. 99, 076402 (2007)]. We show that all terms in the projected full density matrix ρ i→ f = ρappearing in the time-evolution of a local observable may be evaluated in closed form at finite temperature, with ρ +− = ρ −+ = 0. The expression for ρ −− is shown to be finite at finite temperature, becoming negligible only in the limit of vanishing temperatures. We prove that this approach recovers the short-time limit for the expectation value of a local observable exactly at arbitrary temperatures. In contrast, the corresponding long-time limit is recovered exactly only for a continuous bath, i.e. when the logarithmic discretization parameter Λ → 1 + . Since the numerical renormalization group approach breaks down in this limit, and calculations have to be carried out at Λ > 1, the long-time behavior following an arbitrary quantum quench has a finite error, which poses an obstacle for the method, e.g., in its application to the scattering states numerical renormalization group method for describing steady state non-equilibrium transport through correlated impurities [Anders, Phys. Rev. Lett. 101, 066804, (2008)]. We suggest a way to overcome this problem by noting that the time-dependence, in general, and the long-time limit, in particular, becomes increasingly more accurate on reducing the size of the quantum quench. This suggests an improved generalized TDNRG approach in which the system is time-evolved between the initial and final states via a sequence of small quantum quenches within a finite time interval instead of by a single large and instantaneous quantum quench. The formalism for this is provided, thus generalizing the TDNRG method to multiple quantum quenches, periodic switching, and general pulses. This formalism, like our finite temperature generalization of the single-quench case, rests on no other approximation than the NRG approximation. The results are illustrated numerically by application to the Anderson impurity model.

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 non-interacting 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.

We investigate the time evolution of the Kondo resonance in response to a quench by applying the timedependent numerical renormalization group (TDNRG) approach to the Anderson impurity model in the strong correlation limit. For this purpose, we derive within TDNRG a numerically tractable expression for the retarded two-time nonequilibrium Green function G(t + t , t), and its associated time-dependent spectral function, A(ω, 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 → 0 + , only fully develops at very long times t 1/T K , where T K is the Kondo temperature of the final state. In contrast, the final state satellite peaks develop on a fast time scale 1/Γ during the time interval −1/Γ t +1/Γ, where Γ is the hybridization strength. Initial and final state spectral functions are recovered in the limits t → −∞ and t → +∞, respectively. Our formulation of two-time nonequilibrium Green functions within TDNRG provides a first step towards using this method as an impurity solver within nonequilibrium dynamical mean field theory.Introduction.-The nonequilibrium properties of strongly correlated quantum impurity models continue to pose a major theoretical challenge. This contrasts with their equilibrium properties, which are largely well understood [1], or can be investigated within a number of highly accurate methods, such as the numerical renormalization group method (NRG) [2][3][4][5], the continuous time quantum Monte Carlo (CTQMC) approach [6], the density matrix renormalization group [7], or the Bethe ansatz method [8,9]. Quantum impurity models far from equilibrium are of direct relevance to several fields of research, including charge transfer effects in lowenergy ion-surface scattering [10][11][12][13][14][15][16][17], transient and steady state effects in molecular and semiconductor quantum dots [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36], and also in the context of dynamical mean field theory (DMFT) of strongly correlated lattice models [37][38][39], as generalized to nonequilibrium [40][41][42]. In the latter, further progress hinges on an accurate non-perturbative solution for the nonequilibrium Green functions of an effective quantum impurity model. Such a solution, beyond allowing timeresolved spectroscopies of correlated lattice systems within DMFT to be addressed [43][44][45][46][47], would also be useful in understanding time-resolved scanning tunnelling microscopy of nanoscale systems [48] and proposed cold atom realizations of Kondo correlated states [49][50][51][52], which could be probed with real-time radio-frequency spectroscopy [53][54][55].In this Letter, we use the time-dependent numerical renormalization group (TDNRG) approach [56][57][58][59][60][61][62] to calculate the retarded two-time ...

We investigate the thermodynamics and transient dynamics of the (unbiased) Ohmic two-state system by exploiting the equivalence of this model to the interacting resonant level model. For the thermodynamics, we show, by using the numerical renormalization group (NRG) method, how the universal specific heat and susceptibility curves evolve with increasing dissipation strength, α, from those of an isolated two-level system at vanishingly small dissipation strength, with the characteristic activated-like behavior in this limit, to those of the isotropic Kondo model in the limit α → 1 − . At any finite α > 0, and for sufficiently low temperature, the behavior of the thermodynamics is that of a gapless renormalized Fermi liquid. Our results compare well with available Bethe ansatz calculations at rational values of α, but go beyond these, since our NRG calculations, via the interacting resonant level model, can be carried out efficiently and accurately for arbitrary dissipation strengths 0 ≤ α < 1 − . We verify the dramatic renormalization of the low-energy thermodynamic scale T0 with increasing α, finding excellent agreement between NRG and density matrix renormalization group (DMRG) approaches. For the zero-temperature transient dynamics of the two-level system, P (t) = σz(t) , with initial-state preparation P (t ≤ 0) = +1, we apply the time-dependent extension of the NRG (TDNRG) to the interacting resonant level model, and compare the results obtained with those from the noninteracting-blip approximation (NIBA), the functional renormalization group (FRG), and the time-dependent density matrix renormalization group (TD-DMRG). We demonstrate excellent agreement on short to intermediate time scales between TDNRG and TD-DMRG for 0 α 0.9 for P (t), and between TDNRG and FRG in the vicinity of α = 1/2. Furthermore, we quantify the error in the NIBA for a range of α, finding significant errors in the latter even for 0.1 ≤ α ≤ 0.4. We also briefly discuss why the long-time errors in the present formulation of the TDNRG prevent an investigation of the crossover between coherent and incoherent dynamics. Our results for P (t) at short to intermediate times could act as useful benchmarks for the development of new techniques to simulate the transient dynamics of spin-boson problems.