Restoring the continuum limit in the time-dependent numerical renormalization group approach

The continuous coupling function in quantum impurity problems is exactly partitioned into a part represented by a finite-size Wilson chain and a part represented by a set of additional reservoirs, each coupled to one Wilson chain site. These additional reservoirs represent high-energy modes of the environment neglected by the numerical renormalization group and are required to restore the continuum limit of the original problem. We present a hybrid time-dependent numerical renormalization group approach which combines an accurate numerical renormalization group treatment of the nonequilibrium dynamics on the finite-size Wilson chain with a Bloch-Redfield formalism to include the effect of these additional reservoirs. Our approach overcomes the intrinsic shortcoming of the time-dependent numerical renormalization group approach induced by the bath discretization with a Wilson parameter $\mathrm{\Lambda }>1$. We analytically prove that for a system with a single chemical potential, the thermal equilibrium reduced density operator is the steady-state solution of the Bloch-Redfield master equation. For the numerical solution of this master equation, a Lanczos method is employed which couples all energy shells of the numerical renormalization group. The presented hybrid approach is applied to the real-time dynamics in correlated fermionic QISs. An analytical solution of the resonant-level model serves as a benchmark for the accuracy of the method which is then applied to nontrivial models, such as the interacting resonant-level model and the single-impurity Anderson model.

Figure 1

The semi-infinite Wilson chain depicted up to the chain link $m$.

Figure 2

The reservoir continuum $m-1$ (a) is recursively replaced by a single chain site $f_m$ coupled to a new reservoir degree of freedom $c_{0m}$ by a chain link matrix element $V_m$, shown in (b), to obtain a continuous fraction representation of the original bath by a (c) finite-size tight-binding chain with the continuous reservoir coupled to the end of the chain as used in DMRG calculations [74].

Figure 3

In the modified recursion for the Wilson chain, we divided the reservoir $m$ analog to Fig. 2 into its high-energy and low-energy ($H$ and $L$, respectively) contributions, which are tailored such that the low-energy part is coupled to $f_m$ with the matrix element $t_m<V_m$.

Figure 4

The original bath is replaced by a Wilson chain where each chain site $m<N$ is coupled to a high-energy reservoir $H_{\mathrm{res}}^H\left(m\right)$ and the last site is connected to the remaining reservoir $H_{\mathrm{res}}\left(N\right)$.

Figure 5

Spectral functions ${\overline{\mathrm{\Gamma }}}_m^{H/L}\left(\omega \right)=\mathrm{Im}{\mathrm{\Delta }}_m^H(\omega -i{0}^{+})/{\omega }_m$ of the high (at the top) and low (at the bottom) energy reservoirs vs ${\overline{\omega }}_m=\omega /{\omega }_m$. A bandwidth of $D=100{\mathrm{\Gamma }}_0$ and a discretization parameter $\mathrm{\Lambda }=2$ have been chosen. The sites of the chain that the particular reservoir is coupled to are counted by $m$.

Figure 6

Diagrammatic representation of the recursion relation for calculating the $\mathbf{A}$ tensor. Each box represents a matrix element $P_{r_{m_2+1},k_{m_2}}\left[{\alpha }_{m+1}\right]$ (upper row) or its complex conjugate (lower row). The state labels $k$ and $k^{\prime }$ are plotted horizontally. The state label ${\alpha }_{m_2+1}$ for the $m+1$ site is plotted vertically. A connected line indicates a summation over the corresponding index in analogy to Fig. 2 in Ref. [51].

Figure 7

Real-time dynamics of the local orbital occupancy $n_{\mathrm{d}}\left(t\right)$ (a) for a short timescale and (b) for long timescales obtained by the open-chain approach and shown as full lines compared to the TD-NRG approach which is added as a dotted line in the same color for the RLM. The exact analytical solution of Ref. [51] has been added as black dotted lines. Data for different discretization parameters $\mathrm{\Lambda }$ are presented for a sudden change in the energy of the level from $E_d^i=-{\mathrm{\Gamma }}_0$ to $E_d^f={\mathrm{\Gamma }}_0$. The Wilson chain length $N$ ($\mathrm{\Lambda }=1.59,N=50;\mathrm{\Lambda }=2.17,N=30;\mathrm{\Lambda }=3.21,N=20$) was adjusted such that the same temperature $T=0.01{\mathrm{\Gamma }}_0$ is reached for all curves; the corresponding values for $\mathrm{\Lambda }$ are stated in the legend. NRG parameters used: $D={10}^3{\mathrm{\Gamma }}_0,N_{\mathrm{S}}={10}^3,N_z=4$.

Figure 8

Real-time dynamics of the local orbital occupancy $n_{\mathrm{d}}\left(t\right)$ for $N=50$ with (green) and without (orange) $z$ averaging [51]. $N_z$ accounts for the number of different values used for the $z$ averaging. All other parameters as in Fig. 7.

Figure 9

Impurity occupancy $n_d\left(t\right)$ obtained by the full open chain hybrid approach, i.e., with a coupling of all NRG iterations (green) by an adjacent approximation for ${\mathrm{\Xi }}_{l_1,l_2}(m_1,m_2)$ including $m_1=m_2$ and $m_1=m_2\pm 1$ (orange) and restricting to $m_1=m_2$, i.e., with no coupling between discarded states of different NRG iterations (blue). NRG parameters as in Fig. 7 for $N=50$.

Figure 10

Impurity occupancy $n_d\left(t\right)$ obtained by the OC hybrid approach but with different approximations. The green curve is obtained by the full algorithm (all discarded states are coupled) and is taken from Fig. 7. For the blue curve, we neglected the coupling between discarded states of different iterations and included the relaxation into the kept states for each iteration instead. The loss of the trace $\mathrm{Tr}\left[\rho \right(t\left)\right]$ of the density matrix as a function of time is shown as a blue dashed line. Normalizing the blue curve by the time dependent trace yields the orange curve. NRG parameters as in Fig. 7 for $N=50$.

Figure 11

The real-time dynamics of $n_d\left(t\right)$ vs time in the IRLM for different values of $U$ obtained by the open-chain hybrid approach (solid lines) for a sudden change in the energy of the level from $E_d^i/{\mathrm{\Gamma }}_{\mathrm{eff}}=-1$ to $E_d^f/{\mathrm{\Gamma }}_{\mathrm{eff}}=1$. The analytical $U=0$ result is added as a guidance (black dashed line). The thermodynamic expectation value $n_d^f$ is added as a black arrow on the r.h.s of the figure for comparison. NRG parameters: $\mathrm{\Lambda }=1.59,N=50,D/{\mathrm{\Gamma }}_{\mathrm{eff}}={10}^3,N_{\mathrm{S}}={10}^3,N_z=4$ so $T/{\mathrm{\Gamma }}_{\mathrm{eff}}=0.01$.

Figure 12

The real-time dynamics of $n_d\left(t\right)$ vs time in the IRLM for a fixed $U/D=16$ but different chain length $N$ and NRG discretization parameter $\mathrm{\Lambda }$ combinations. The NRG parameter $\mathrm{\Lambda }$ ($\mathrm{\Lambda }=1.59,N=50;\mathrm{\Lambda }=2.17,N=30;\mathrm{\Lambda }=3.21,N=20$) was adjusted such that the same temperature $T=0.01{\mathrm{\Gamma }}_{\mathrm{eff}}$ is reached for all curves and $N_S=300$. The analytical $U=0$ curve is added as a dashed line for illustration purposes.

Figure 13

The real-time dynamics of $n_d\left(t\right)$ vs time in the IRLM for different values of $U$ obtained by our OC hybrid approach (solid lines) and by the TD-NRG (dotted line) in the same color as well as a fit to Fermi's golden rule (dashed line). NRG parameters as in Fig. 7.

Figure 14

A comparison of analytical estimates (dotted lines) and the numerical values (stars) for three different IRLM parameters. The coupling strength $U/D$ has been varied.

Figure 15

Impurity occupancy $n_d\left(t\right)$ and spin polarization $S_z\left(t\right)$ vs time after the quench on a logarithmic timescale. The upper and lower right panels show the results of scenario (i) leaving the hybridization strength constant. In the upper and lower left panels, the data after switching the hybridization strength on at $t=0$ are plotted. Parameters: $\mathrm{\Lambda }=1.66,D=20{\mathrm{\Gamma }}_0,N=30,T=0.01{\mathrm{\Gamma }}_0,N_z=4,N_S={10}^3$.

Figure 16

Comparison of $N_S=300$ (dashed lines) to $N_S=1000$ (solid lines) different states for the SIAM regarding the impurity occupation $n_d\left(t\right)$ and the spin polarization $S_z\left(t\right)$.

Figure 17

$S_z\left(t\right)$ data taken from Fig. 15 for ${\mathrm{\Gamma }}_i=0$ at the top and ${\mathrm{\Gamma }}_i={\mathrm{\Gamma }}_0$ at the bottom. The time is scaled by the system temperature $T$ on the left and by the respective Kondo temperature $T_K$ (which depends on $U$) on the right.

Figure 18

Impurity occupation $n_d\left(t\right)$ and spin $S_z\left(t\right)$ of the open chain (OC) compared to the closed chain (CC, dotted) and the analytical solution (a.s., dashed) according to Eq. (92) plotted on a double logarithmic scale.