Spatial and temporal propagation of Kondo correlations

We address the fundamental question how the spatial Kondo correlations are building up in time assuming an initially decoupled impurity spin ${\vec{S}}_{\mathrm{imp}}$. We investigate the time-dependent spin-correlation function $\chi (\vec{r},t)=\langle {\vec{S}}_{\mathrm{imp}}\vec{s}\left(\vec{r}\right)\rangle \left(t\right)$ in the Kondo model with antiferromagnetic and ferromagnetic couplings, where $\vec{s}\left(\vec{r}\right)$ denotes the spin density of the conduction electrons after switching on the Kondo coupling at time $t=0$. We present data obtained from a time-dependent numerical renormalisation group (TD-NRG) calculation. We gauge the accuracy of our two-band NRG by the spatial sum rules of the equilibrium correlation functions and the reproduction of the analytically exactly known spin-correlation function of the decoupled Fermi sea. We find a remarkable building up of Kondo correlation outside of the light cone defined by the Fermi velocity of the host metal. By employing a perturbative approach exact in second-order of the Kondo coupling, we connect these surprising correlations to the intrinsic spin-density entanglement of the Fermi sea. The thermal wavelength supplies a cutoff scale at finite temperatures beyond which correlations are exponentially suppressed. We present data for the frequency dependent retarded spin-spin susceptibility and use the results to calculate the real-time response of a weak perturbation in linear response: within the spatial resolution no response outside of the light cone is found.

Figure 1

Normalization constants ${\overline{N}}_{e\left(o\right)}$ for different dimensions $D$ vs the dimensionless distance $x=k_{\mathrm{F}}R/\pi$. For $R\to \infty$ ${\overline{N}}_e$ is equal to ${\overline{N}}_o$.

Figure 2

The spin-correlation function (a) $R{\chi }_{\infty }\left(R\right)=R\langle {\vec{S}}_{\mathrm{imp}}\vec{s}\left(\vec{r}\right)\rangle$ as a function of the dimensionless distance $x=k_{\mathrm{F}}R/\pi$ in 1D for small Kondo couplings ${\rho }_{0}J=0.05,0.075,0.1$ and $R/{\xi }_{\mathrm{K}}\ll 1$. We added RKKY interaction between the impurity spin and a probe spin in distance $R$ for comparison. (b) $R^2{\chi }_{\infty }\left(R\right)$ as a function of the dimensionless distance $x$ in 1D for larger Kondo couplings ${\rho }_{0}J=0.5,0.6,0.7$. The inset shows the value of the correlation function at the origin ${\chi }_{\infty }\left(0\right)$ vs $\rho J$. For large $J$, the value $-3/4$ is reached.

Figure 3

The spin-correlation function $R^2{\chi }_{\infty }\left(R\right)$ as a function of the dimensionless distance $x=k_{\mathrm{F}}R/\pi$ in 2D. In higher dimensions, the envelope of the RKKY and ${\chi }_{\infty }\left(R\right)$ has a more complicated structure. Every second maximum has a lower amplitude. NRG parameters are $\Lambda =5$ and $N_s=3000$.

Figure 4

The spin-correlation function $R{\chi }_{\infty }\left(R\right)$ as a function of the dimensionless distance $x=k_{\mathrm{F}}R/\pi$ in 1D for two different ferromagnetic Kondo couplings ${\rho }_{0}J=-0.1$ and $-0.5$ for $T\to 0$. The inset shows ${\chi }_{\infty }\left(0\right)$ vs $\rho J$. For large ferromagnetic couplings $0.25$ is reached. NRG parameters are $\Lambda =3$ and $N_s=1400$.

Figure 5

The 1D time and spatial dependent spin-correlation function $\chi (R,t)$ vs $x=k_{\mathrm{F}}R/\pi$ and $\tau =tD$ for ${\rho }_{0}J=0.3$ as a color contour plot. Its color map is depicted on the right site. The correlation propagates with $v_{\mathrm{F}}$, which is added as white line as guide to the eye. NRG parameters are $\Lambda =3$, $N_s=1400$, and $N_z=4$.

Figure 6

Time dependent correlation function $\chi (R,t)$ for a fixed distance $k_{\mathrm{F}}R/\pi =0.51$ vs time for different number of $z$ averages $N_z$ and a fixed ${\rho }_{0}J=0.3$. NRG parameters are $\Lambda =3\text{and}{N}_s=1600$.

Figure 7

(a) Time-dependent correlation function $\chi (R,t)$ vs $t^{\prime }=t-R/v_{\mathrm{F}}$ for four different distances $k_{\mathrm{F}}R=0,0.51,1.01,1.51$ for ${\rho }_{0}J=0.3$ in 1D. (b) Rescaled time-dependent correlation function $\chi (R,t)/{\chi }_{\max }$ vs $t-t_{\max }$ for $k_{\mathrm{F}}R/\pi =2.01$ and four different couplings ${\rho }_{0}J=0.05,0.10.15,0.3$. $t_{\max }$ is the position and ${\chi }_{\max }$ is the amplitude of the ferromagnetic peak. NRG parameters are $\Lambda =3,N_s=1200,\text{and}{N}_z=32$.

Figure 8

The time-dependent correlation $\langle {\vec{S}}_{\mathrm{imp}}\vec{s}\left(R\right)\rangle \left(t\right)$ for different distances as a color contour plot. Its color map is depicted on the right site. (a) The oscillation between ferromagnetic and antiferromagnatic correlations for long times is only observed for short distances $R/{\xi }_{\mathrm{K}}\ll 1$. (b) For long times, oscillations only between zero and antiferromagnetic correlations are observed. The ferromagnetic propagation vanishes at around $R/{\xi }_{\mathrm{K}}\approx 1$. Both long-time behaviors are in agreement with the NRG equilibrium results. NRG parameters are $\Lambda =3,N_s=1400,\text{and}{N}_z=32$.

Figure 9

The analytical $\chi (R,t)$ evaluated numerically for ${\rho }_{0}J=0.3$ up to second order in $J$ as a color contour plot. Its color map is depicted on the right site. The light cone $R=v_{\mathrm{F}}t$ has been added as a white line.

Figure 10

The intrinsic spin-spin correlations of the Fermi sea between the spin densities $\vec{s}\left(0\right)$ and $\overrightarrow{s\left(r\right)}$ in 1D. Via the mapping to the even and odd conduction bands, we are able to measure bath properties at distance $R$ and get a perfect agreement between theory and NRG results. NRG parameters are $\Lambda =3$ and $N_s=1200$.

Figure 11

First- and second-order of $\chi (R,t)$ for small times $t$ and ${\rho }_{0}J=0.3$ in 1D. Even for short times, we observe correlations for large distances. A comparison with (a) shows that the positions of these correlations coincide with the positions of the intrinsic correlations of the Fermi sea. So the correlations outside the light cone originate from these intrinsic correlations.

Figure 12

(a) The time-dependent spin correlation function $\chi (R,t)$ vs $x=k_{\mathrm{F}}R/\pi$ and $\tau =tD$ for a ferromagnetic Kondo coupling ${\rho }_{0}J=-0.1$ in 1D as color contour plot. Its color map is depicted on the right site. (b) The analytic spin correlation function $\chi (R,t)$ for the same parameter calculated in second-order perturbation theory in ${\rho }_{0}J$. NRG parameters are $\Lambda =3$, $N_s=1400$, and $N_z=32$.

Figure 13

The correlation function $\chi (R,t)$ vs $x=R/{\xi }_{\mathrm{K}}$ and $\tau =tD$ for the medium coupling ${\rho }_{0}J=0.3$ and different temperatures in 1D. A comparison between the correlation function for (a) zero temperature and (b) $T=2T_{\mathrm{K}}$ reveal that correlations in and outside of the light cone vanish at around $x=R/{\xi }_{\mathrm{K}}=0.5$ for $T=2T_{\mathrm{K}}$. NRG parameters are $\Lambda =3$, $N_s=1400$, and $N_z=4$.

Figure 14

The correlation function $\chi (R,tD=70)$ vs $x=R/{\xi }_{\mathrm{K}}$ for the medium coupling ${\rho }_{0}J=0.3$ and the constant time $tD=70$ in 1D. For $T=0$, the RKKY-like oscillations are observed while for $T=2T_{\mathrm{K}}$ the correlations are cut off. The inset shows the envelope function of $\chi (R,tD=70)$.

Figure 15

(a) The spectral function of retarded host spin-density susceptibility ${\chi }_{\mathrm{c}-\mathrm{c}}^r(R,t)$ in the absence of the impurity for four different distances $k_{\mathrm{F}}R/\pi =0.01,0.51,1.51,2.01$. The solid line shows the exact analytic result and the dashed line the numerical NRG result. (b) The conduction electron spin-density $\langle s^z(R,t)\rangle$ vs time as a response to the local magnetic field $B\left(t\right)=B\theta \left(t\right)$ applied to the spin-density $s^z\left(0\right)$ at the origin. The Fourier transformation of the spectral information depicted in (a) has been used in combination with Eq. (27) where ${\chi }_{\mathrm{imp}-\mathrm{c}}^r(R,t)$ has been replaced by ${\chi }_{\mathrm{c}-\mathrm{c}}^r(R,t)$. The arrows indicate the time $\tau =R/v_{\mathrm{F}}$. Note that the $\langle s^z(R,t)\rangle$ has been normalised to the Zeeman energy ${\Delta }_0=g{\mu }_{\mathrm{B}}B$. NRG parameters are $\Lambda =2.25$, $N_s=3000$, and $N_z=16$.

Figure 16

(a) The spectral function of the retarded host-spin susceptibility ${\rho }_{\mathrm{imp}-\mathrm{c}}^r(R,\omega )$ for four different distances $R=0.01,0.51,1.51,2.01$. (b) The time-dependent spin-polarization $\langle s^z(R,t)\rangle$ vs the dimensionless time $t{T}_{\mathrm{K}}$ after switching on the local magnetic Zeeman splitting $\Delta \left(t\right)=g{\mu }_{\mathrm{B}}B\theta \left(t\right)$ on the impurity spin calculated using Eq. (27) for three distances $0.51,1.51,2.01$. The inset shows the short time behavior in more detail. The horizontal lines on the right indicate the equilibrium value $\langle s^z(R,\infty )\rangle /\left(g{\mu }_{\mathrm{B}}B\right)$ obtained by an equilibrium NRG calculation with $N_z=1$ for a very small magnetic field $g{\mu }_{\mathrm{B}}B/D={10}^{-8}$ acting on the impurity spin. NRG parameters are $\Lambda =2.25$, $N_s=3000$, and $N_z=16$.

Figure 17

The second-order Feynman diagram generating the lowest-order contribution to the RKKY interaction between to localized spins mediated by a spin excitation propagating through the metallic host.