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=2₅ (T). In the antiferromagnetic regime, the long-time decay is governed for T<T₊ by the Kondo time scale 1∕T₊. Here ₅ is the conduction-electron density of states, T₊ is the Kondo temperature, and J (T) 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.