The statement ͓between Eqs. ͑388͒ and ͑389͔͒ that z 1 and z + can be replaced by 0 and h is not correct because this replacement affects the cutoff scales which determine the broadening of logarithms and absolute values in the following. Therefore, 0 and h have to be replaced by z 1 and z + in Eqs. ͑389͒-͑391͒, respectively. All logarithms and absolute values in the relaxation and dephasing rates and the renormalized magnetic field are then broadened by the difference ⌫ 1 − ⌫ 2 of the relaxation and dephasing rates. Consequently, Eqs. ͑393͒-͑395͒ should readwhere the logarithm L − ͑x͒ and the absolute value ͉x͉ − are defined by ͓cf. Eqs. ͑382͒-͑384͔͒Deriving these quantities with respect to the magnetic field h 0 yieldswhere the broadened ⌰ function ⌰ − ͑x͒ is given by ͓cf. Eq. ͑385͔͒These corrections have an effect on Figs. 6 and 7 and Figs. 15-17. The results which are presented in the other figures are unaffected by this change.

We consider an open quantum system in contact with fermionic metallic reservoirs in a nonequilibrium setup. For the case of spin, orbital or potential fluctuations, we present a systematic formulation of real-time renormalization group at finite temperature, where the complex Fourier variable of an effective Liouvillian is used as flow parameter. We derive a universal set of differential equations free of divergencies written as a systematic power series in terms of the frequency-independent two-point vertex only, and solve it in different truncation orders by using a universal set of boundary conditions. We apply the formalism to the description of the weak to strong coupling crossover of the isotropic spin-1 2 nonequilibrium Kondo model at zero magnetic field. From the temperature and voltage dependence of the conductance in different energy regimes we determine various characteristic low-energy scales and compare their universal ratio to known results. For a fixed finite bias voltage larger than the Kondo temperature, we find that the temperature-dependence of the differential conductance exhibits non-monotonic behavior in the form of a peak structure. We show that the peak position and peak width scale linearly with the applied voltage over many orders of magnitude in units of the Kondo temperature. Finally, we compare our calculations with recent experiments.

We apply the real-time renormalization group (RG) in nonequilibrium to an arbitrary quantum dot in the Coulomb blockade regime. Within one-loop RG-equations, we include self-consistently the kernel governing the dynamics of the reduced density matrix of the dot. As a result, we find that relaxation and dephasing rates generically cut off the RG flow. In addition, we include all other cutoff scales defined by temperature, energy excitations, frequency, and voltage. We apply the formalism to transport through single molecular magnets, realized by the fully anisotropic Kondo model (with three different exchange couplings Jx, Jy, and Jz) in a magnetic field hz. We calculate the differential conductance as function of bias voltage V and discuss a quantum phase transition which can be tuned by changing the sign of JxJyJz via the anisotropy parameters. Finally, we calculate the noise S(Ω) at finite frequency Ω for the isotropic Kondo model and find that the dephasing rate determines the height of the shoulders in dS(Ω)/dΩ near Ω = V .

We apply a recently developed nonequilibrium real-time renormalization group (RG) method in frequency space to describe nonlinear quantum transport through a small fermionic quantum system coupled weakly to several reservoirs via spin and/or orbital fluctuations. Within a weakcoupling two-loop analysis, we derive analytic formulas for the nonlinear conductance and the kernel determining the time evolution of the reduced density matrix. A consistent formalism is presented how the RG flow is cut off by relaxation and dephasing rates. We apply the general formalism to the nonequilibrium anisotropic Kondo model at finite magnetic field. We consider the weakcoupling regime, where the maximum of voltage and bare magnetic field is larger than the Kondo temperature. In this regime, we calculate the nonlinear conductance, the magnetic susceptibility, the renormalized spin relaxation and dephasing rates, and the renormalized g factor. All quantities are considered up to the first logarithmic correction beyond leading order at resonance. Up to a redefinition of the Kondo temperature, we confirm previous results for the conductance and the magnetic susceptibility in the isotropic case. In addition, we present a consistent calculation of the resonant line shapes, including the determination whether the spin relaxation or dephasing rate cuts off the logarithmic divergence. Furthermore, we calculate quantities characterizing the exponential decay of the time evolution of the magnetization. In contrast to the conductance, we find that the derivative of the spin relaxation (dephasing) rate with respect to the magnetic field is logarithmically enhanced (suppressed) for voltages smaller (larger) than the renormalized magnetic field, and that the logarithmic divergence is cut off by the opposite rate. The renormalized g factor is predicted to show a symmetric logarithmic suppression at resonance, which is cut off by the spin relaxation rate. We propose a three-terminal setup to measure the suppression at resonance. For all quantities, we analyze also the anisotropic case and find additional nonequilibrium effects at resonance.