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 \(\vec{S}_{\rm imp}\). We investigate the time-dependent spin-correlation function \(\chi(\vec{r},t) = \langle \vec{S}_\mathrm{imp} \vec{s}(\vec{r}) \rangle (t)\) in the Kondo model with antiferromagnetic and ferromagnetic couplings where \( \vec{s}(\vec{r})\) 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 wave length 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.