Time Evolution of the Kondo Resonance in Response to a Quench

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 $G(t+t^{\prime },t)$, and its associated time-dependent spectral function, $A(\omega ,t)$, 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 $t\to 0^{+}$, only fully develops at very long times $t\gtrsim 1/T_K$, where $T_K$ is the Kondo temperature of the final state. In contrast, the final state satellite peaks develop on a fast time scale $1/\mathrm{\Gamma }$ during the time interval $-1/\mathrm{\Gamma }\lesssim t\lesssim +1/\mathrm{\Gamma }$, where $\mathrm{\Gamma }$ is the hybridization strength. Initial and final state spectral functions are recovered in the limits $t\to -\infty$ and $t\to +\infty$, 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.

Figure 1

(a) Time evolution of the normalized spectral function $\pi \mathrm{\Gamma }A(\omega >0,t)$ for the symmetric Anderson model at positive times, following a quench at $t=0$ specified by ${ϵ}_i=-15\mathrm{\Gamma }$, $U_i=30\mathrm{\Gamma }$, and ${ϵ}_f=-6\mathrm{\Gamma }$, $U_f=12\mathrm{\Gamma }$ with final state Kondo temperature $T_K=2.5\times {10}^{-5}$. A structure on the scale of $T_K$ evolves into the Kondo resonance at long times $t\gtrsim 1/T_K$, while a structure at $\omega ={ϵ}_f+U_f\approx 240T_K$ with negligible time dependence corresponds to the final state satellite peak. Panels (b)–(d) show the spectral function at times $t{T}_K=0.001$, 1, and 1000, respectively. The TDNRG calculations used a discretization parameter $\mathrm{\Lambda }=4$, $z$ averaging [78, 79] with $N_z=32$ and a cutoff energy $E_{\mathrm{cut}}=24$.

Figure 2

Time evolution of the normalized spectral function $\pi \mathrm{\Gamma }A(\omega ,t)$ at positive times, for, (a), negative, and, (b), positive frequencies, for quench (A) from the mixed valence to the symmetric Kondo regime with $T_K=1.0\times {10}^{-4}$. A structure on the scale of $T_K$ evolves into the Kondo resonance at long times $t\gtrsim 1/T_K$, while structures at $\omega =\pm {ϵ}_f\approx \pm 40T_K$, with negligible time-dependence, correspond to the final state satellite peaks. Panels (c)–(e) show the spectral function at times $t{T}_K=0.0001$, 1, and 10000, respectively. TDNRG parameters: $\mathrm{\Lambda }=4$, $z$ averaging with $N_z=64$ and a cutoff energy $E_{\mathrm{cut}}=24$.

Figure 3

$A(\omega ,t)$ vs $t{T}_K$ from, (a), negative, to, (b), positive times for quench (A), and on a linear frequency scale. Dashed lines mark $t\mathrm{\Gamma }=\pm 1$ ($t{T}_K=\pm 2.5\times {10}^{-2}$). Initial state ($\omega =\pm {ϵ}_i=\pm 15\mathrm{\Gamma }\approx \pm 600T_K$) and final state ($\omega =\pm {ϵ}_f=\pm 6\mathrm{\Gamma }\approx \pm 240T_K$) satellite peaks are clearly visible, as are initial and final state Kondo resonances around $\omega =0$. Panels (c)–(f) show the $A(\omega ,t)$ at times $t{T}_K=-1000,-1,-0.001$, and $+1000$, respectively. TDNRG parameters as in Fig. 2.

Figure 4

$A(\omega ,t)$ vs $t{T}_K$ from, (a), negative, to, (b), positive times and on a linear frequency scale for quench (B) (from a mixed valence to a symmetric Kondo regime). Dashed lines mark $t\mathrm{\Gamma }=\pm 1$ ($t{T}_K=\pm {10}^{-1}$). Panels (c)–(f) show the spectral function at times $t{T}_K=-1000,-1,-0.001$, and $+1000$, respectively. TDNRG parameters as in Fig. 2.