Real-Time Dynamics in Quantum-Impurity Systems: A Time-Dependent Numerical Renormalization-Group Approach

We develop a general approach to the nonequilibrium dynamics of quantum-impurity systems for arbitrary coupling strength. The numerical renormalization group is used to generate a complete basis set necessary for the correct description of the time evolution. We benchmark our method with the exact analytical solution for the resonant-level model. As a first application, we investigate the equilibration of an ultrasmall quantum dot subject to a sudden change of gate voltage and external magnetic field. Two distinct relaxation times are identified for the spin and charge dynamics.

Figure 1

The full NRG chain of length $N$ is divided into a subchain of length $m$ and the environment $R_{m,N}$.

Figure 2

Time-dependent occupancy of the RLM for (a) $E_d^0=0$, $T\to 0$ and three different values of $E_d^1/\Gamma =-1,-2,-3$, and (b) $E_d^0=0$, $E_d^1=-\Gamma$ and four different temperatures $T/\Gamma =0.1,0.5,1,5$. Solid lines are the results of the TD-NRG with $\Lambda =1.3$, $D/\Gamma =500$, $N_z=32$, $\alpha =0$ (no damping factor) and $N_s=1000$ states retained. Dashed lines are the exact solution in the wideband limit. Right arrows mark the saturated NRG occupancies for $\alpha >0$. Inset to (a): Progressive calculation of $n_d(0)$ as a function of the number of backward iterations $N-m$, for $E_d^1=-2\Gamma$.

Figure 3

Time-dependent occupancy $n_d(t)$, (a),(b) and magnetization $S_z(t)$, (c),(d) of the SIAM with $T\to 0$, $D/{\Gamma }_1=20$, $H_0=2E_d^0={\Gamma }_1$, $H_1=0$, $E_d^1=-U/2$, and seven different values of $U/{\Gamma }_1=2,4,6,8,10,12,18$. Here ${\Gamma }_0={\Gamma }_1$ in (a) and (c), whereas ${\Gamma }_0=0$ in (b) and (d). NRG parameters: $\Lambda =1.5$, $N_s=1000$, $N_z=16$, and $\alpha =0.1$.

Figure 4

The impurity magnetization curves of Fig. 3, replotted versus $t/t_K$ with $t_K=1/T_K$. Here ${\Gamma }_0={\Gamma }_1$ in (a), while ${\Gamma }_0=0$ in (b).