We investigate several definitions of the time-dependent spectral function $A(\ensuremath{\omega},t)$ of the Anderson impurity model following a quench and within the time-dependent numerical renormalization group (TDNRG) method. In terms of the single-particle two-time retarded Green function
Gr(t1,t2), the definitions we consider differ in the choice of the time variable
t with respect to
t1 and/or
t2 (which we refer to as the time reference). In a previous study [H. T. M. Nghiem et al., Phys. Rev. Lett. 119, 156601 (2017)], we investigated the spectral function $A(\ensuremath{\omega},t)$, obtained from the Fourier transform of
Im[Gr(t1,t2)] with respect to the time difference ${t}^{\ensuremath{'}}={t}_{1}\ensuremath{-}{t}_{2}$, with time reference
t=t2. Here, we complement this work by deriving expressions for the retarded Green function for the choices
t=t1 and the average, or Wigner, time
t=(t1+t2)/2, within the TDNRG approach. We compare and contrast the resulting $A(\ensuremath{\omega},t)$ for the different choices of time reference. While the choice
t=t1 results in a spectral function with no time dependence before the quench (
t<0) (being identical to the equilibrium initial-state spectral function for
t<0), the choices
t=(t1+t2)/2 and
t=t2 exhibit nontrivial time evolution both before and after the quench. Expressions for the lesser, greater, and advanced Green functions are also derived within TDNRG for all choices of time reference. The average-time lesser Green function ${G}^{<}(\ensuremath{\omega},t)$ is particularly interesting, as it determines the time-dependent occupied density of states $N(\ensuremath{\omega},t)={G}^{<}(\ensuremath{\omega},t)/(2\ensuremath{\pi}i)$, a quantity that determines the photoemission current in the context of time-resolved pump-probe photoemission spectroscopy. We present calculations for $N(\ensuremath{\omega},t)$ for the Anderson model following a quench, and discuss the resulting time evolution of the spectral features, such as the Kondo resonance and high-energy satellite peaks. We also discuss the issue of thermalization at long times for $N(\ensuremath{\omega},t)$. Finally, we use the results for $N(\ensuremath{\omega},t)$ to calculate the time-resolved photoemission current for the Anderson model following a quench (acting as the pump) and study the different behaviors that can be observed for different resolution times of a Gaussian probe pulse.
We develop a method for calculating the self-energy of a quantum impurity coupled to a continuous bath by stochastically generating a distribution of finite Anderson models that are solved by exact diagonalization, using the noninteracting local spectral function as a probability distribution for the sampling. The method enables calculation of the full analytic self-energy and single-particle Green's function in the complex frequency plane, without analytic continuation, and can be used for both finite and zero temperature at arbitrary fillings. Results are in good agreement with imaginary frequency data from continuous-time quantum Monte Carlo calculations for the single-impurity Anderson model and the two-orbital Hubbard model within dynamical mean-field theory (DMFT) as well as real frequency data for self-energy of the single-band Hubbard model within DMFT using the numerical renormalization group. The method should be applicable to a wide range of quantum impurity models and particularly useful when high-precision real frequency results are sought.
We introduce a method to obtain the specific heat of quantum impurity models via a direct calculation of the impurity internal energy requiring only the evaluation of local quantities within a single numerical renormalization group (NRG) calculation for the total system. For the Anderson impurity model we show that the impurity internal energy can be expressed as a sum of purely local static correlation functions and a term that involves also the impurity Green function. The temperature dependence of the latter can be neglected in many cases, thereby allowing the impurity specific heat
Cimp to be calculated accurately from local static correlation functions; specifically via ${C}_{\mathrm{imp}}=\frac{\ensuremath{\partial}{E}_{\mathrm{ionic}}}{\ensuremath{\partial}T}+\frac{1}{2}\frac{\ensuremath{\partial}{E}_{\mathrm{hyb}}}{\ensuremath{\partial}T}$, where
Eionic and
Ehyb are the energies of the (embedded) impurity and the hybridization energy, respectively. The term involving the Green function can also be evaluated in cases where its temperature dependence is non-negligible, adding an extra term to
Cimp. For the nondegenerate Anderson impurity model, we show by comparison with exact Bethe ansatz calculations that the results recover accurately both the Kondo induced peak in the specific heat at low temperatures as well as the high-temperature peak due to the resonant level. The approach applies to multiorbital and multichannel Anderson impurity models with arbitrary local Coulomb interactions. An application to the Ohmic two-state system and the anisotropic Kondo model is also given, with comparisons to Bethe ansatz calculations. The approach could also be of interest within other impurity solvers, for example, within quantum Monte Carlo techniques.
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 impurity Green function and dynamical susceptibilities for the two-channel Anderson impurity model are calculated. An exact expression for the self-energy of the impurity Green function as ratio of two correlation functions is given. The imaginary part of the self-energy scales as $\sqrt{\ensuremath{\mid}\ensuremath{\omega}∕{T}_{K}\ensuremath{\mid}}$ for $T\ensuremath{\rightarrow}0$, serving as a hallmark for non-Fermi behavior. The many-body resonance is pinned to a universal value of $1∕(2\ensuremath{\pi}\ensuremath{\Delta})$ at $\ensuremath{\omega}=0$ for arbitrary local occupation in contrary to the single-channel Anderson impurity model. Its shape becomes increasingly more symmetric for the Kondo regimes of the model. The dynamical spin and channel susceptibilities are governed by two energy scales
TK and
Th, and their ${\ensuremath{\chi}}^{\ensuremath{''}}(\ensuremath{\omega})$ approach a constant value for $\ensuremath{\omega}\ensuremath{\rightarrow}0$.
The nonequilibrium spectral properties of the Anderson impurity model with a chemical potential bias are investigated within a numerically exact real-time quantum Monte Carlo formalism. The two-time correlation function is computed in a form suitable for nonequilibrium dynamical mean field calculations. Additionally, the evolution of the model's spectral properties are simulated in an alternative representation, defined by a hypothetical but experimentally realizable weakly coupled auxiliary lead. The voltage splitting of the Kondo peak is confirmed and the dynamics of its formation after a coupling or gate quench are studied. This representation is shown to contain additional information about the dot's population dynamics. Further, we show that the voltage-dependent differential conductance gives a reasonable qualitative estimate of the equilibrium spectral function, but significant qualitative differences are found including incorrect trends and spurious temperature dependent effects.