Functional RG in nonequilibrium

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During the first period two new nonequilibrium RG methods have been derived. One is based on kinetic equations in Liouville space and expands in the coupling between system and reservoir, the so-called real-time RG in frequency space (RTRG-FS). The other is based on the Keldysh formalism and the Wetterich RG scheme and expands in the Coulomb interaction on the local system, the so-called functional RGmethod in nonequilibrium (FRG-NE). In the next period it is planned to further develop these methods and to adress many open questions of fundamental interest in the Kondo model, the single-impurity Anderson model (SIAM), the interacting resonant level model (IRLM), and Luttinger liquids. An important open question concerns the strong coupling limit of the Kondo model (or, equivalently, the SIAM), where all energy scales are of the same order as the Kondo temperature, so that spin fluctuations can no longer be treated perturbatively. Based on some preliminary and promising considerations within RTRG-FS, we will try to include the Z-factor and 2-loop terms, and we will use a fully selfconsistent determination of the decay rates. For the SIAMwe found previously that the FRG-NE scheme works in the strong coupling regime for moderate Coulomb interactions. To treat strong Coulomb interactions we will try to include certain static parts of the 3-particle vertex and study their influence on the flow of the two-particle vertex.Another challenging topic will be the generalization of RTRG-FS to the case when explicitly time-dependent and harmonic fields are applied to quantum dots. We will develop systematic schemes for small amplitudes (linear response), small frequencies (adiabatic quantum pumping), and large frequencies (Floquet theory, expansion in Fourier components). The method will be used to analyse the dynamic response of the interacting resonant level model with one or two levels.To understand the fundamental interplay of charge and spin fluctuations in nonequilibrium, we will apply the RTRG-FS scheme to the SIAM in the well-controlled regime of weak spin fluctuations. Arbitrary Coulomb interaction, temperature, magnetic field, gate and bias voltage will be considered, and all stationary properties (level renormalization, resonant line shapes) together with the time evolution into the stationary state will be calculated.Concerning quantum wires, a fundamental unsolved issue is the description of energy relaxation and dephasing induced by the Coulomb interaction in a nonequilibrium situation. Based on our experience in quantum dots, we plan to study this problem by considering the frequency-dependence of the 2-particle vertex. We will include the flow of the Z-factor and the frequency-dependent part of the imaginary part of the self-energy. The aim is to calculate the nonequilibrium one-particle distribution function and the line shape of the splitting of the zero-bias anomaly, and to understand the competition between dephasing and phase-averaging in the presence of more than two barriers.