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

TL;DR

This paper develops a Fermi liquid framework for analyzing the low-frequency response of quantum dots in RC circuits, using renormalization group techniques on Anderson and Coulomb blockade models to confirm Fermi liquid behavior.

Contribution

It introduces a renormalization group approach to derive low-energy Hamiltonians for quantum RC circuits, extending Fermi liquid theory to interacting models like Anderson and Coulomb blockade.

Findings

Charge relaxation resistance depends only on static charge susceptibilities.

The Anderson model maps onto the Kondo model with known T_K expression.

All models confirm Fermi liquid behavior at low energy.

Abstract

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.