Hybrid NRG-DMRG approach to real-time dynamics of quantum impurity systems

A hybrid approach to nonequilibrium dynamics of quantum impurity systems is presented. The numerical renormalization group serves as a means to generate a suitable low-energy Hamiltonian, allowing for an accurate evaluation of the real-time dynamics of the problem up to exponentially long times using primarily the time-adaptive density-matrix renormalization group. In particular, by constructing a suitable hybrid chain, discretization errors are essentially eliminated on all time scales of interest. We extract the decay time of the interaction-enhanced oscillations in the interacting resonant-level model and show their quadratic divergence with the interaction strength $U$. Our numerical analysis is in excellent agreement with analytic predictions based on an expansion in $1/U$.

Figure 1

Time evolution of $n_d\left(t\right)$ on a double Wilson chain, following a sudden change of the level energy from $E_d^i$ to $E_d^f$. Here ${\Gamma }_0/D={10}^{-5}$ and $U=0$. The full red line depicts the exact solution on the double Wilson chain (obtained by exact diagonalization), the dashed green line displays the hybrid NRG-DMRG, and the full black line is the exact analytical continuum-limit solution (Ref. 17) in the wide-band limit. Chain parameters: $M=29$, ${\Lambda }_1=1.8$, ${\Lambda }_2=1.02$, $N_{\mathrm{inter}}=4$, and $N=180$. $N_s=50$ states are retained both in the NRG and in the course of the TD-DMRG. Inset: Exact $n_d\left(t\right)$ on a pure Wilson chain, for $E_d^f=-E_d^i=10{\Gamma }_0$ and different $\Lambda$'s. Here $N=500$ (1000) for $\Lambda \ge 1.1$ ($\Lambda \le 1.05$).

Figure 2

Same as Fig. 1, for $E_d^f=-E_d^i={\Gamma }_{\mathrm{eff}}$ and different values of $U$. ${\Gamma }_0$ was adjusted separately for each value of $U$ so as to maintain ${\Gamma }_{\mathrm{eff}}/D={10}^{-5}$. All other chain, NRG, and TD-DMRG parameters are the same as in Fig. 1.

Figure 3

(a) Interaction-enhanced oscillations (red line) for $E_d^f=-E_d^i=3{\Gamma }_{\mathrm{eff}}$ and $U/D=10$, along with a fit to Eq. (17) (black crosses). (b) Same as (a), for $E_d^f=-E_d^i={\Gamma }_{\mathrm{eff}}$. (c) The fitted frequency $\Omega$ vs $U$ for $E_d^f/{\Gamma }_{\mathrm{eff}}=-E_d^i/{\Gamma }_{\mathrm{eff}}=1$ (black circles), 3 (red circles), and 10 (green circles). Full lines display the analytical strong-coupling expression for $\Omega$, employing the numerical values of $V$. Arrows on the right-hand side mark the asymptotic $U\to \infty$ values of $\Omega$. (d) Same as (c) for the decay time $\tau$. All remaining parameters are the same as in Fig. 2.