Spin precession and real-time dynamics in the Kondo model: Time-dependent numerical renormalization-group study

A detailed derivation of the recently proposed time-dependent numerical renormalization-group (TD-NRG) approach to nonequilibrium dynamics in quantum-impurity systems is presented. We demonstrate that the method is suitable for fermionic as well as bosonic baths. Comparisons with exact analytical results for the charge relaxation in the resonant-level model and for dephasing in the spin-boson model establish the accuracy of the method. The real-time dynamics of a single spin coupled to each type of bath is investigated. We use the TD-NRG to calculate the spin relaxation and spin precession of a single Kondo impurity. The short- and long-time dynamics are studied as a function of temperature in the ferromagnetic and antiferromagnetic regimes. The short-time dynamics agrees very well with analytical results obtained at second order in the exchange coupling $J$. In the ferromagnetic regime, the transient spin decay is described by the scaling variable $x=2{\rho }_F\mid J\left(T\right)\mid Tt$. In the antiferromagnetic regime, the long-time decay is governed for $T<T_K$ by the Kondo time scale $1∕T_K$. Here ${\rho }_F$ is the conduction-electron density of states, $T_K$ is the Kondo temperature, and $J\left(T\right)$ is the effective exchange coupling at temperature $T$. Results for spin precession are obtained by rotating the external magnetic field from the $x$ axis to the $z$ axis.

Figure 1

(Color online) The full Wilson chain of length $N$ is divided into a subchain of length $m$ and the “environment” $R_{m,N}$. The Hamiltonian ${\mathcal{H}}_m$ can be viewed either as acting only on the subchain of length $m$, or as acting on the full chain of length $N$, but with the hopping matrix elements $t_m,\cdots ,t_{N-1}$ all set to zero. The former picture is the traditional one. In this paper we adopt the latter point of view.

Figure 2

(Color online) (a) Diagrammatic representation of the recursion relation of Eq. (40) for ${\rho }_{s_i,r_i}^{\mathrm{red},0}\left(m\right)$. Each box stands for a matrix element $P_{{l^{\prime }}_i,l_i}\left[{\alpha }_{m+1}\right]$ (upper row) or its complex conjugate (lower row). The state labels $l_i$ and $l_i^{\prime }$ are plotted horizontally, with $l_i$ to the left and $l_i^{\prime }$ to the right. The state label ${\alpha }_{m+1}$ for the $m+1$ site is plotted vertically. A connected line indicates summation over the corresponding index. The reduced density matrix ${\rho }_{s_i,r_i}^{\mathrm{red},0}\left(m\right)$ has two indices $s_i$ and $r_i$, carried by the external legs on the left. Iterating (a) up to $m^{\prime }=N$ relates the reduced density matrix ${\rho }_{s_i,r_i}^{\mathrm{red},0}\left(m\right)$ to the equilibrium NRG density matrix $⟨{k^{\prime }}_i;N\mid {\widehat{\rho }}_0\mid k_i,N⟩$. This process is visualized diagrammatically in (b). The sequence of connected vertical lines amounts to tracing over the environment degrees of freedom ${\left\{{\alpha }_i\right\}}_{i=m+1}^N$.

Figure 3

(Color online) Comparison of the time-dependent occupancy $n_d\left(t\right)$ of the resonant-level model, obtain analytically by Eq. (53) and numerically by the TD-NRG through evaluation of Eq. (25). We consider a sudden change in the energy of the level from $E_d^0=0$ to $E_d^1=-2\Gamma$, without any change in the hybridization strength: ${\Gamma }_0={\Gamma }_1=\Gamma$. Different values of $\Lambda =1.3$, ${1.3}^2$, ${1.3}^3$, ${1.3}^4$ are used. The number of NRG iterations (i.e., the Wilson chain lengths) have been adjusted so that all curves are calculated at approximately the same temperature $T_N$. In going from $\Lambda =1.3$ to $\Lambda ={1.3}^4=2.8561$, $T_N∕\Gamma$ equals 0.00171, 0.00176, 0.00183, 0.00193, corresponding to $N=96$, 48, 36, 24 NRG iterations. All temperatures are sufficiently low to justify comparison to Eq. (53). The remaining NRG parameters are as follows: $N_s=1000$, $N_z=16$, $D∕\Gamma =500$, and ${\alpha }_d=0$ (i.e., no additional damping). A single curve with $\Lambda ={1.3}^4$ and $N_z=1$ is presented for comparison.

Figure 4

(Color online) Spin-boson model: Decoherence of the off-diagonal component ${\rho }_{01}\left(t\right)$ of the reduced density matrix for the spin as a function of time. Solid lines are the results of the TD-NRG, obtained by evaluating the expectation value of $\frac{1}{2}{\sigma }_x\left(t\right)$. Dashed lines with the same color code show the exact analytical result of Eq. (58). Here $T∕{\omega }_c=0.0078$, $\alpha =0.1$, $ϵ=0$, $\Delta (t<0)=100{\omega }_c$, and $\Delta (t>0)=0$. The power $s$ is varied from the super-Ohmic to the sub-Ohmic regime according to the values indicated in the legends. The TD-NRG accurately reproduces the exact analytic result up to $t\sim 1∕T$. NRG parameters: $N_s=150$, $N_z=16$, $\Lambda =\sqrt{2}$, and ${\alpha }_d=0.1$. The total number of NRG iterations is $N=14$, and the number of bosonic states added at each iteration is $N_b=8$ (see Ref. 20).

Figure 5

(Color online) Spin polarization $⟨S_z\left(t\right)⟩$ vs $t$, for an Ohmic bath $(s=1)$ and different coupling strengths $\alpha$. The system is prepared in a state where the spin is fully polarized, achieved by setting $ϵ∕{\omega }_c=100$ and $\Delta =0$ in ${\mathcal{H}}^i$. At time $t=0$, $ϵ$ is switched off and $\Delta ∕{\omega }_c=0.1$ is switched on. The temperature equals $T∕{\omega }_c=9.53\times {10}^{-7}$. NRG parameters: $N_s=100$, $N_z=32$, $\Lambda =2$, and ${\alpha }_d=0$ (i.e., no additional damping). The total number of NRG iterations is $N=14$, and the number of bosonic states added at each iteration is equal to $N_b=8$.

Figure 6

(Color online) Time evolution of the spin expectation value $S_z\left(t\right)$ for the anisotropic Kondo model in the antiferromagnetic regime, for fixed $J_{\perp }∕D=0.15$ and different values of $J_z∕D=-0.1$, 0, 0.05, 0.1, 0.12, and 0.15. Panel (a) shows the short-time behavior and the analytic result of Eq. (72) (magenta curve), along with its truncation to second order in $t$ (dotted line). Panel (b) displays the long-time behavior of $S_z\left(t\right)$ on a logarithmic time scale. The same data are replotted in panel (c) vs $t∕t_K$. The universal curve of Ref. 51 has been added to panel (c), after rescaling their definition of $T_K$ by a factor of $1∕1.67$ to match the definition used here. Initial condition: $J_z^0=J_{\perp }^0=0$ and ${\gamma }_s{H}_z^0∕D=0.1$. NRG parameters: $\Lambda =1.5$, $N_s=1000$, ${\alpha }_d=0.1$, and $N_z=16$. The temperature equals $T∕D=3.7\times {10}^{-12}$, corresponding to $N_{\mathrm{iter}}=130$ NRG iterations.

Figure 7

(Color online) Time evolution of the spin expectation value $S_z\left(t\right)$ for the isotropic Kondo model with $J∕D=0.15$ and different temperatures $T∕T_K=0.075$, 4.3, 33, and 250. Setting ${\gamma }_s{H}_z^0∕D=1$ and $J_z^0=J_{\perp }^0=0$, the impurity spin is initially polarized and decoupled from the band. The Kondo temperature equals $T_K∕D=0.00054$. Panel (b) displays the same data as in panel (a), but replotted as a function of $t\bullet T$. NRG parameters are the same as in Fig. 6.

Figure 8

(Color online) Spin relaxation in the isotropic Kondo model for $J∕D=0.15$ and different strengths of the remaining magnetic field $H_z∕H_K=0.01$, 0.1, 1, 10, and 100. Here $H_K=T_K∕{\gamma }_s$ is the Kondo field. The temperature equals $T∕D=1.2\times {10}^{-8}$ and the initial polarizing field equals ${\gamma }_s{H}_z^0∕D=0.1$. Panel (a) shows the case where $J^0=0$. Panel (b) displays the case where $J^0=J$. The inset to (b) shows the estimated spin-relaxation scale $t_{\text{spin}}$ as defined in the text, both for $J^0=0$ and $J^0=J$. NRG parameters are the same as in Fig. 6.

Figure 9

(Color online) The same data as in panel (a) of Fig. 8, replotted vs $t_{\text{spin}}\left(H_z\right)$ in the form of the reduced spin function $f\left(t\right)$ of Eq. (75). The curves for $H_z∕H_K=0.01$, 0.1, and 1 fall on top of one another. Note that $f\left(t\right)$ turns negative for $t⪢t_{\text{spin}}$.

Figure 10

(Color online) Time evolution of the normalized spin expectation value $S_z\left(t\right)∕S_z\left(0\right)$ for the isotropic Kondo model in the FM regime. Two distinct values of $J∕D=-0.2$ and $-0.1$, and four different temperatures $T∕D=0.13$, 0.018, 0.0023, and $4\times {10}^{-5}$ are shown. An initial polarizing field of magnitude ${\gamma }_s{H}_z^0∕D=0.01$ is switched off at time $t=0$, without altering the Kondo couplings. Panel (a) shows the time evolution of the normalized spin expectation value vs the dimensionless variable $tD$. Panel (b) plots the same data vs the dimensionless scaling variable $x=2{\rho }_F\mid J\left(T\right)\mid Tt$. The resulting curves lose their accuracy for $x>1$, as the TD-NRG cannot access long time scales that exceed $1∕T$.[15] NRG parameters: $\Lambda =1.5$, $N_s=1000$, ${\alpha }_d=0.1$, and $N_z=16$.

Figure 11

(Color online) Spin precession of a Kondo spin, isotropically coupled to the conduction electrons with $J>0$. Two different values of $J=0.2$ and 0.4 are shown. For $t<0$, a magnetic field of magnitude ${\gamma }_{s}H∕D=0.1$ is applied along the $x$ axis: ${\gamma }_s\vec{H}=0.1D{\vec{e}}_x$. At $t=0$, the field is rotated around the $y$ axis such that it points for $t>0$ along the $z$ axis: ${\gamma }_s\vec{H}=0.1D{\vec{e}}_z$. The Kondo couplings are left unchanged, i.e., $J^0=J$. In addition to the individual components of $⟨\vec{S}\left(t\right)⟩$, we plot its magnitude $\mid ⟨\vec{S}⟩\mid$ as a function of time. NRG Parameters used: $N_s=1000$, $N_z=16$, $\Lambda =1.6$, $N_{\mathrm{iter}}=60$, and ${\alpha }_d=0.1$.

Figure 12

(Color online) Relaxation rates for the spin components $S_x$ and $S_z$ (in units of $1∕D$) vs $J$, as obtained from the phenomenological fits $S_x\left(t\right)=(1∕2)\cos \left({\omega }_{x}t\right)\exp (-{\Gamma }_{x}t)$ and $S_z\left(t\right)=S_z\left(\infty \right)[1-\cos \left({\omega }_{z}t\right)\exp (-{\Gamma }_{z}t)]$. Here the impurity spin was initially decoupled from the band [i.e., $J(t<0)=0$]. Both ferromagnetic (FM) and antiferromagnetic (AF) couplings are displayed. NRG Parameters: $N_s=1000$, $N_z=16$, $\Lambda =1.8$, $N_{\mathrm{iter}}=60$, and ${\alpha }_d=0.1$.