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arXiv:1109.5731

The Fermi liquid nature of the ground state of double quantum dots in parallel from a 1/N expansion

Manas Kulkarni, Robert Konik

Published Sep 26, 2011• 1 citation
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Abstract

A large-N diagrammatic approach is used to study coupled quantum dots in a parallel geometry. We show that the Friedel sum rule (FSR) holds at lowest order in a 1/N expansion for this system, thereby suggesting that the ground state is a Fermi liquid. Using the FSR together with the dot occupancy, we compute the dot system's conductance. Our finding that the 1/N expansion indicates the system is a Fermi liquid is in agreement with both prior results based on the Bethe ansatz and slave boson mean field theory.

DOI
10.48550/arxiv.1109.5731

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A large-N diagrammatic approach is used to study coupled quantum dots in a parallel geometry. We show that the Friedel sum rule (FSR) holds at lowest order in a 1/N expansion for this system, thereby suggesting that the ground state is a Fermi liquid. Using the FSR together with the dot occupancy, we compute the dot system's conductance. Our finding that the 1/N expansion indicates the system is a Fermi liquid is in agreement with both prior results based on the Bethe ansatz and slave boson mean field theory.
We argue through a combination of slave-boson mean-field theory and the Bethe ansatz that the ground state of closely spaced double quantum dots in parallel coupled to a single effective channel are Fermi liquids. We do so by studying the dots' conductance, impurity entropy, and spin correlation. In particular, we find that the zero-temperature conductance is characterized by the Friedel sum rule, a hallmark of Fermi-liquid physics, and that the impurity entropy vanishes in the limit of zero temperature, indicating that the ground state is a singlet. This conclusion is in opposition to a number of numerical renormalization-group studies. We suggest a possible reason for the discrepancy.
We resolve the controversy regarding the ground state of the parallel double quantum dot system near half filling. The numerical renormalization group predicts an underscreened Kondo state with residual spin-1/2 magnetic moment, ln2 residual impurity entropy, and unitary conductance, while the Bethe ansatz solution predicts a fully screened impurity, regular Fermi-liquid ground state, and zero conductance. We calculate the impurity entropy of the system as a function of the temperature using the hybridization-expansion continuous-time quantum Monte Carlo technique, which is a numerically exact stochastic method, and find excellent agreement with the numerical renormalization group results. We show that the origin of the unconventional behavior in this model is the odd-symmetry "dark state" on the dots.
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 study correlated two-level quantum dots, coupled in effective one-channel fashion to metallic leads, with electron interactions including on-level and interlevel Coulomb repulsions, as well as the interorbital Hund's rule exchange favoring the spin-1 state in the relevant sector of the free dot. For arbitrary dot occupancy, the underlying phases, quantum phase transitions (QPTs), thermodynamics, single-particle dynamics, and electronic transport properties are considered, and direct comparison is made to conductance experiments on lateral quantum dots. Two distinct phases arise generically, one characterized by a normal Fermi liquid fixed point (FP) and the other by an underscreened (USC) spin-1 FP. Associated QPTs, which occur in general in a mixed valent regime of nonintegral dot charge, are found to consist of continuous lines of Kosterlitz-Thouless transitions, separated by first-order level-crossing transitions at high symmetry points. A ``Friedel-Luttinger sum rule'' is derived and, together with a deduced generalization of Luttinger's theorem to the USC phase (a singular Fermi liquid), is used to obtain a general result for the T=0T=0T=0 zero-bias conductance, expressed solely in terms of the dot occupancy and applicable to both phases. Relatedly, dynamical signatures of the QPT show two broad classes of behavior, corresponding to the collapse of either a Kondo resonance, or antiresonance, as the transition is approached from the Fermi liquid phase, the latter behavior being apparent in experimental differential conductance maps. The problem is studied using the numerical renormalization group method, combined with analytical arguments.
We explore the effects of asymmetry of hopping parameters between double parallel quantum dots and the leads on the conductance and a possibility of local magnetic moment formation in this system using functional renormalization group approach with the counterterm. We demonstrate a possibility of a quantum phase transition to a local moment regime [so-called singular Fermi liquid (SFL) state] for various types of hopping asymmetries and discuss respective gate voltage dependencies of the conductance. We show that, depending on the type of the asymmetry, the system can demonstrate either a first-order quantum phase transition to an SFL state, accompanied by a discontinuous change of the conductance, similarly to the symmetric case, or the second-order quantum phase transition, in which the conductance is continuous and exhibits Fano-type asymmetric resonance near the transition point. A semianalytical explanation of these different types of conductance behavior is presented.
The equations-of-motion (EOM) hierarchy satisfied by the Green functions of a quantum dot embedded in an external mesoscopic network is considered within a high-order decoupling approximation scheme. Exact analytic solutions of the resulting coupled integral equations are presented in several limits. In particular, it is found that at the particle-hole symmetric point the EOM Green function is temperature independent due to a discontinuous change in the imaginary part of the interacting self-energy. However, this imaginary part obeys the Fermi liquid unitarity requirement away from this special point, at zero temperature. Results for the occupation numbers, the density of states, and the local spin susceptibility are compared with exact Fermi liquid relations and the Bethe ansatz solution. The approximation is found to be very accurate far from the Kondo regime. In contrast, the description of the Kondo effect is valid on a qualitative level only. In particular, we find that the Friedel sum-rule is considerably violated, up to 30%, and the spin susceptibility is underestimated. We show that the widely used simplified version of the EOM method, which does not account fully for the correlations on the network, fails to produce the Kondo correlations even qualitatively.