Spatial and temporal propagation of Kondo correlations

Abstract

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 $\chi (\vec{r},t)=⟨{\vec{S}}_{\mathrm{imp}}\vec{s}(\vec{r})⟩(t)$ 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.

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