Fermi liquid approach to the quantum RC circuit: Renormalization group analysis of the Anderson and Coulomb blockade models

We formulate a general approach for studying the low-frequency response of an interacting quantum dot connected to leads in the presence of oscillating gate voltages. The energy dissipated is characterized by the charge relaxation resistance, which under the loose assumption of Fermi liquid behavior at low energy, is shown to depend only on static charge susceptibilities. The predictions of the scattering theory are recovered in the noninteracting limit while the effect of interactions is simply to replace densities of states by charge susceptibilities in formulas. In order to substantiate the Fermi liquid picture in the case of a quantum RC geometry, we apply a renormalization group analysis and derive the low-energy Hamiltonian for two specific models: the Anderson and the Coulomb blockade models. The Anderson model is shown, using a field theoretical approach based on Barnes slave bosons, to map onto the Kondo model. We recover the well-known expression of the Kondo temperature for the asymmetric Anderson model and compute the charge susceptibility. The Barnes slave bosons are extended to the Coulomb blockade model where the renormalization-group analysis can be carried out perturbatively up to zero energy. All calculations agree with the Fermi liquid nature of the low-energy fixed point and satisfy the Friedel sum rule.