Spin Precession and Real Time Dynamics in the Kondo Model: A 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. A comparison with exact analytical results for the charge relaxation in the resonant-level model and for dephasing in the spin-boson model establishes the accuracy of the method. The real-time dynamics of a single spin coupled to both types of baths 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 is 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 long-time spin decay is described by the scaling variable $x = 2\rho_F J(T) T t$. In the antiferromagnetic regime it is governed for $T < T_K$ by the Kondo time scale $1/T_K$. Here $\rho_F$ is the conduction-electron density of states and $T_K$ is the Kondo temperature. Results for spin precession are obtained by rotating the external magnetic field from the x axis to the z axis.