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arXiv:1912.08474 · Physical review. B./Physical review. B

Time-dependent spectral functions of the Anderson impurity model in response to a quench with application to time-resolved photoemission spectroscopy

Hoa T. M. Nghiem, H. T. Dang, Theo A. Costi

Published Mar 11, 2020• 8 citations
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Abstract

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 ${G}^{r}({t}_{1},{t}_{2})$, the definitions we consider differ in the choice of the time variable $t$ with respect to ${t}_{1}$ and/or ${t}_{2}$ (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 $\mathrm{Im}[{G}^{r}({t}_{1},{t}_{2})]$ with respect to the time difference ${t}^{\ensuremath{'}}={t}_{1}\ensuremath{-}{t}_{2}$, with time reference $t={t}_{2}$. Here, we complement this work by deriving expressions for the retarded Green function for the choices $t={t}_{1}$ and the average, or Wigner, time $t=({t}_{1}+{t}_{2})/2$, within the TDNRG approach. We compare and contrast the resulting $A(\ensuremath{\omega},t)$ for the different choices of time reference. While the choice $t={t}_{1}$ 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 $t=({t}_{1}+{t}_{2})/2$ and $t={t}_{2}$ 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.

DOI
10.1103/physrevb.101.115117

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We investigate several definitions of the time-dependent spectral function A(\ensuremathω,t)A(\ensuremath{\omega},t)A(\ensuremathω,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){G}^{r}({t}_{1},{t}_{2})Gr(t1​,t2​), the definitions we consider differ in the choice of the time variable ttt with respect to t1{t}_{1}t1​ and/or t2{t}_{2}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(\ensuremathω,t)A(\ensuremath{\omega},t)A(\ensuremathω,t), obtained from the Fourier transform of Im[Gr(t1,t2)]\mathrm{Im}[{G}^{r}({t}_{1},{t}_{2})]Im[Gr(t1​,t2​)] with respect to the time difference t\ensuremath′=t1\ensuremath−t2{t}^{\ensuremath{'}}={t}_{1}\ensuremath{-}{t}_{2}t\ensuremath′=t1​\ensuremath−t2​, with time reference t=t2t={t}_{2}t=t2​. Here, we complement this work by deriving expressions for the retarded Green function for the choices t=t1t={t}_{1}t=t1​ and the average, or Wigner, time t=(t1+t2)/2t=({t}_{1}+{t}_{2})/2t=(t1​+t2​)/2, within the TDNRG approach. We compare and contrast the resulting A(\ensuremathω,t)A(\ensuremath{\omega},t)A(\ensuremathω,t) for the different choices of time reference. While the choice t=t1t={t}_{1}t=t1​ results in a spectral function with no time dependence before the quench (t<0t<0t<0) (being identical to the equilibrium initial-state spectral function for t<0t<0t<0), the choices t=(t1+t2)/2t=({t}_{1}+{t}_{2})/2t=(t1​+t2​)/2 and t=t2t={t}_{2}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<(\ensuremathω,t){G}^{<}(\ensuremath{\omega},t)G<(\ensuremathω,t) is particularly interesting, as it determines the time-dependent occupied density of states N(\ensuremathω,t)=G<(\ensuremathω,t)/(2\ensuremathπi)N(\ensuremath{\omega},t)={G}^{<}(\ensuremath{\omega},t)/(2\ensuremath{\pi}i)N(\ensuremathω,t)=G<(\ensuremathω,t)/(2\ensuremathπi), a quantity that determines the photoemission current in the context of time-resolved pump-probe photoemission spectroscopy. We present calculations for N(\ensuremathω,t)N(\ensuremath{\omega},t)N(\ensuremathω,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(\ensuremathω,t)N(\ensuremath{\omega},t)N(\ensuremathω,t). Finally, we use the results for N(\ensuremathω,t)N(\ensuremath{\omega},t)N(\ensuremathω,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 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 ttt 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{T}_{K}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 self-energy method for quantum impurity models expresses the correlation part of the self-energy in terms of the ratio of two Green's functions and allows for a more accurate calculation of equilibrium spectral functions than is possible directly from the one-particle Green's function [Bulla et al., J. Phys.: Condens. Matter 10, 8365 (1998)], for example, within the numerical renormalization group method. In addition, the self-energy itself is a central quantity required in the dynamical mean field theory of strongly correlated lattice models. Here, we show how to generalize the self-energy method to the time-dependent situation for the prototype model of strong correlations, the Anderson impurity model. We use the equation-of-motion method to obtain closed expressions for the local Green's function in terms of a time-dependent correlation self-energy, with the latter being given as a ratio of a one-particle time-dependent Green's function and a higher-order correlation function. We benchmark this self-energy approach to time-dependent spectral functions against the direct approach within the time-dependent numerical renormalization group method. The self-energy approach improves the accuracy of time-dependent spectral function calculations, and the closed-form expressions for the Green's function allow for a clear picture of the time evolution of spectral features at the different characteristic time scales. The self-energy approach is of potential interest also for other quantum impurity solvers for real-time evolution, including time-dependent density matrix renormalization group and continuous-time quantum Monte Carlo techniques.
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.
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 use the numerical renormalization group to calculate the auxiliary spectral functions of the $U=\ensuremath{\infty}$ Anderson impurity model. The slave-boson and pseudo-fermion spectral functions diverge at the threshold with exponents ${\ensuremath{\alpha}}_{b}$ and ${\ensuremath{\alpha}}_{f}$ given by the x-ray photoemission and the x-ray absorption exponents, respectively. In contrast to the noncrossing approximation, the exact exponents obtained here depend on the impurity occupation number. In general, vertex corrections in the convolution formulas for physical Green's functions are singular at the threshold and may not be neglected in the Fermi liquid regime.
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.