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arXiv:1304.4454 · Physical Review B

Admittance of the SU(2) and SU(4) Anderson quantum RC circuits

Michele Filippone, Karyn Le Hur, Christophe Mora

Published Jul 3, 2013• 15 citations
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Abstract

We study the Anderson model as a description of the quantum $RC$ circuit for spin-$1/2$ electrons and a single level connected to a single lead. Our analysis relies on the Fermi liquid nature of the ground state, which fixes the form of the low-energy effective model. The constants of this effective model are extracted from a numerical solution of the Bethe ansatz equations for the Anderson model. They allow us to compute the charge relaxation resistance ${R}_{q}$ in different parameter regimes. In the Kondo region, the peak in ${R}_{q}$ as a function of the magnetic field is recovered and proven to be in quantitative agreement with previous numerical renormalization group results. In the valence-fluctuation region, the peak in ${R}_{q}$ is shown to persist, with a maximum value of $h/2{e}^{2}$, and an analytical expression is obtained using perturbation theory. We extend our analysis to the SU(4) Anderson model where we also derive the existence of a giant peak in the charge relaxation resistance.

DOI
10.1103/physrevb.88.045302

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We study the Anderson model as a description of the quantum RCRCRC circuit for spin-1/21/21/2 electrons and a single level connected to a single lead. Our analysis relies on the Fermi liquid nature of the ground state, which fixes the form of the low-energy effective model. The constants of this effective model are extracted from a numerical solution of the Bethe ansatz equations for the Anderson model. They allow us to compute the charge relaxation resistance Rq{R}_{q}Rq​ in different parameter regimes. In the Kondo region, the peak in Rq{R}_{q}Rq​ as a function of the magnetic field is recovered and proven to be in quantitative agreement with previous numerical renormalization group results. In the valence-fluctuation region, the peak in Rq{R}_{q}Rq​ is shown to persist, with a maximum value of h/2e2h/2{e}^{2}h/2e2, and an analytical expression is obtained using perturbation theory. We extend our analysis to the SU(4) Anderson model where we also derive the existence of a giant peak in the charge relaxation resistance.
We investigate the dynamical charge response of the Anderson model viewed as a quantum RC circuit. Applying a low-energy effective Fermi liquid theory, a generalized Korringa-Shiba formula is derived at zero temperature, and the charge relaxation resistance is expressed solely in terms of static susceptibilities which are accessible by Bethe ansatz. We identify a giant charge relaxation resistance at intermediate magnetic fields related to the destruction of the Kondo singlet. The scaling properties of this peak are computed analytically in the Kondo regime. We also show that the resistance peak fades away at the particle-hole symmetric point.
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 RCRCRC 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.
We generalize Nozi\`eres' Fermi-liquid theory for the low-energy behavior of the Kondo model to that of the single-impurity Anderson model. In addition to the electrons' phase shift at the Fermi energy, the low-energy Fermi-liquid theory is characterized by four Fermi-liquid parameters: the two given by Nozi\`eres that enter to first order in the excitation energy, and two additional ones that enter to second order and are needed away from particle-hole symmetry. We express all four parameters in terms of zero-temperature physical observables, namely the local charge and spin susceptibilities and their derivatives with respect to the local level position. We determine these in terms of the bare parameters of the Anderson model using Bethe ansatz and numerical renormalization group (NRG) calculations. Our low-energy Fermi-liquid theory applies throughout the crossover from the strong-coupling Kondo regime via the mixed-valence regime to the empty-orbital regime. From the Fermi-liquid theory, we determine the conductance through a quantum dot symmetrically coupled to two leads in the regime of small magnetic field, low temperature, and small bias voltage, and compute the coefficients of the $\ensuremath{\sim}{B}^{2}, \ensuremath{\sim}{T}^{2}$, and $\ensuremath{\sim}{V}^{2}$ terms exactly in terms of the Fermi-liquid parameters. The coefficients of T2,V2{T}^{2}, {V}^{2}T2,V2, and B2{B}^{2}B2 are found to change sign during the Kondo to empty-orbital crossover. The crossover becomes universal in the limit that the local interaction is much larger than the level width. For completeness, we also compute the shot noise and discuss the resulting Fano factor.
This paper is a corrected version of Phys. Rev. B 95, 165404 (2017), which we have retracted because it contained a trivial but fatal sign error that lead to incorrect conclusions. We extend a recently developed Fermi liquid (FL) theory for the asymmetric single-impurity Anderson model [C. Mora et al., Phys. Rev. B 92, 075120 (2015)] to the case of an arbitrary local magnetic field. To describe the system's low-lying quasiparticle excitations for arbitrary values of the bare Hamiltonian's model parameters, we construct an effective low-energy FL Hamiltonian whose FL parameters are expressed in terms of the local level's spin-dependent ground-state occupations and their derivatives with respect to level energy and local magnetic field. These quantities are calculable with excellent accuracy from the Bethe ansatz solution of the Anderson model. Applying this effective model to a quantum dot in a nonequilibrium setting, we obtain exact results for the curvature of the spectral function, cA{c}_{A}cA​, describing its leading $\ensuremath{\sim}{\ensuremath{\varepsilon}}^{2}$ term, and the transport coefficients cV{c}_{V}cV​ and cT{c}_{T}cT​, describing the leading $\ensuremath{\sim}{V}^{2}$ and $\ensuremath{\sim}{T}^{2}$ terms in the nonlinear differential conductance. A sign change in cA{c}_{A}cA​ or cV{c}_{V}cV​ is indicative of a change from a local maximum to a local minimum in the spectral function or nonlinear conductance, respectively, as is expected to occur when an increasing magnetic field causes the Kondo resonance to split into two subpeaks. We find that the fields BA{B}_{A}BA​, BT{B}_{T}BT​, and BV{B}_{V}BV​ at which cA{c}_{A}cA​, cT{c}_{T}cT​, and cV{c}_{V}cV​ change sign, respectively, are all of order TK{T}_{K}TK​, as expected, with BA=BT=BV=0.75073TK{B}_{A}={B}_{T}={B}_{V}=0.75073{T}_{K}BA​=BT​=BV​=0.75073TK​ in the Kondo limit.
Using different techniques, and Fermi-liquid relationships, we calculate the variation with the applied magnetic field (up to second order) of the zero-temperature equilibrium conductance through a quantum dot described by the impurity Anderson model. We focus on the strong-coupling limit $U\ensuremath{\gg}\mathrm{\ensuremath{\Delta}}$, where UUU is the Coulomb repulsion and $\mathrm{\ensuremath{\Delta}}$ is half the resonant-level width, and consider several values of the dot level energy Ed{E}_{d}Ed​, ranging from the Kondo regime ${\ensuremath{\epsilon}}_{F}\ensuremath{-}{E}_{d}\ensuremath{\gg}\mathrm{\ensuremath{\Delta}}$ to the intermediate-valence regime ${\ensuremath{\epsilon}}_{F}\ensuremath{-}{E}_{d}\ensuremath{\sim}\mathrm{\ensuremath{\Delta}}$, where ${\ensuremath{\epsilon}}_{F}$ is the Fermi energy. We have mainly used the density-matrix renormalization group (DMRG) and the numerical renormalization group (NRG) combined with renormalized perturbation theory (RPT). Results for the dot occupancy and magnetic susceptibility from the DMRG and NRG+RPT\mathrm{NRG}+\mathrm{RPT}NRG+RPT are compared with the corresponding Bethe ansatz results for $U\ensuremath{\rightarrow}\ensuremath{\infty}$, showing an excellent agreement once Ed{E}_{d}Ed​ is renormalized by a constant Haldane shift. For $U<3\mathrm{\ensuremath{\Delta}}$ a simple perturbative approach in UUU agrees very well with the other methods. The conductance decreases with the applied magnetic field for dot occupancies ${n}_{d}\ensuremath{\sim}1$ and increases for ${n}_{d}\ensuremath{\sim}0.5$ or ${n}_{d}\ensuremath{\sim}1.5$ regardless of the value of UUU. We also relate the energy scale for the magnetic-field dependence of the conductance with the width of the low-energy peak in the spectral density of the dot.
We compute exactly the low-energy single-electron Green's function, the impurity and electron self-energies, and the resistivity for the two-channel Anderson impurity model. These results are obtained by exploiting the boundary conformal field theory identified from the Bethe ansatz solution of the model. Using that solution we can make contact with the parameters of the original Hamiltonian and provide the detailed crossover between the two integer valence limits. Our results generalize those obtained previously in the context of the two-channel Kondo model.