In this paper, we review the technique of numerical renormalization group (NRG) originally introduced to study the properties of correlated systems. Besides the revision stage of this study, we apply the to NRG calculus of the conductance quantum dots considering two con gurations, depending on the disposition towards the electrodes.In the rst the dot is located between two metal electrodes and in the second the quantum dot is laterally connected to a conductor wire. Theoretical models have been introduced to describe the process of electron transport through quantum dots, as well as allowed to establish basic theoretical relationships between the conductance and the parameters of the model Hamiltonian. With the numerical diagonalization by renormalization group, we calculate the conductance through the eigenvalues and eigenvectors. We also managed to analytically calculate the conductance for the two arrangements of quantum dots considered and con rm the numerical results obtained. Finally, this study allowed physically comprise the conduction of electrons through quantum dots as well as attest the e ciency of the technique of renormalization
In this paper, we review the technique of numerical renormalization group (NRG) originally introduced to study the properties of correlated systems. Besides the revision stage of this study, we apply the to NRG calculus of the conductance quantum dots considering two con gurations, depending on the disposition towards the electrodes.In the rst the dot is located between two metal electrodes and in the second the quantum dot is laterally connected to a conductor wire. Theoretical models have been introduced to describe the process of electron transport through quantum dots, as well as allowed to establish basic theoretical relationships between the conductance and the parameters of the model Hamiltonian. With the numerical diagonalization by renormalization group, we calculate the conductance through the eigenvalues and eigenvectors. We also managed to analytically calculate the conductance for the two arrangements of quantum dots considered and con rm the numerical results obtained. Finally, this study allowed physically comprise the conduction of electrons through quantum dots as well as attest the e ciency of the technique of renormalization
Quantum dots are perfect systems for observing quantum phenomena. This thesis is dedicated to the investigation of conductance through a QD, more specifically with two energy levels and both direct Coulomb interaction and Hund's spin exchange interaction. Basic theory of conductance in interacting quantum systems is presented and Kondo physics are discussed. \nWe then present the fairly recent Functional Renormalization Group approach to calculate full Green's functions of interacting systems. Our approach is based on the one by Karrasch et al. We apply it to the aforementioned QD system and recover the key characteristics of the transport properties with very little numerical effort. We present results on the conductance in the vicinity of the so-called singlet-triplet transition and compare our results both with previous numerical renormalization group results and with predictions of the perturbative renormalization group. Most features from the NRG results are reproduced qualitatively.
A real-space formulation of the numerical renormalization group (NRG) procedure is introduced. The real-space construction, dubbed eNRG, is more straightforward than the NRG discretization, and allows a faithful description of the coupling between quantum dots and conduction states even if the design of the couplings is intricate. General features of the two procedures are discussed comparatively. A more specific comparison is then developed, based on computations of the zero-bias transport properties for an Anderson-model description of a quantum wire side-coupled to a single quantum dot. An eNRG computation is shown to reproduce accurately the temperature-dependent electrical conductance for the uncorrelated model in the continuum limit, while significant deviations mark the corresponding NRG computation for thermal energies comparable to the conduction bandwidth. A combination of analytical and numerical results for the transport properties of the correlated model then provides a more exacting check on the accuracy of the eNRG procedure. A recent NRG analysis mapping the transport properties of a single-electron transistor onto universal functions of the temperature scaled by the Kondo temperature is extended to the side-coupled device, on the basis of eNRG reasoning. Numerical results for the electrical conductance, thermopower, and thermal conductance in side-coupled geometry are then shown to agree very well with the mappings. The numerical results are also checked against the thermal dependence of the thermopower measured by K\"ohler et al. [Phys. Rev. B 77, 104412 (2008)] in ${\mathrm{Lu}}_{0.9}^{\phantom{\ifmmode\dagger\else\textdagger\fi{}}}{\mathrm{Yb}}_{0.1}^{\phantom{\ifmmode\dagger\else\textdagger\fi{}}}{\mathrm{Rh}}_{2}^{\phantom{\ifmmode\dagger\else\textdagger\fi{}}}{\mathrm{Si}}_{2}^{\phantom{\ifmmode\dagger\else\textdagger\fi{}}}$, and the remarkably accurate, recent conductance measurements by Xu et al. [Chin. Phys. Lett. 38, 087101 (2021)].
The ground-state properties of quartet quantum-dot arrays are studied using the numerical renormalization group (NRG) method with a four-site Hubbard model connected to two noninteracting leads. Specifically, we calculate the conductance and local charge in the dots from the many-body phase shifts, which can be deduced from the fixed-point eigenvalues of NRG. As a function of the on-site energy ϵd which corresponds to the gate voltage, the conductance shows alternatively wide peaks and valleys. Simultaneously, the total number of electrons Nel in the four dots shows a quantized staircase behavior due to a large Coulomb interaction U. The conductance plateaus of the unitary limit emerging for odd Nel are caused by the Kondo effect. The valleys of the conductance emerge for even Nel, and their width becomes substantially large at half-filling. It can be regarded as a kind of the Mott-Hubbard insulating behavior manifesting in a small system. These structures of the plateaus and valleys become weak for large values of the hybridization strength $\ensuremath{\Gamma}$ between the chain and leads. We also discuss the parallel conductance for the array connected to four leads.
The equilibrium transport properties of an elementary nanostructured device with sidecoupled geometry are computed and related to universal functions.The computation relies on a real-space formulation of the numerical renormalization-group (NRG) procedure.The real-space construction, dubbed eNRG, is more straightforward than the NRG discretization and allows more faithful description of the coupling between quantum dots and conduction states.The procedure is applied to an Anderson-model description of a quantum wire side-coupled to a single quantum dot.A gate potential controls the dot occupation.In the Kondo regime, the electrical conductance through this device is known to map linearly onto a universal function of the temperature scaled by the Kondo temperature.Here, the energy moments from which the Seebeck coefficient and the thermal conductance can be computed are shown to map linearly onto universal functions also.The moments and transport properties computed by the eNRG procedure are shown to agree very well with these analytical developments.Algorithms facilitating comparison with experimental results are discussed.As an illustration, one of the algorithms is applied to thermal dependence of the thermopower measured by Köhler in Lu 0.9 Yb 0.1 Rh 2 Si 2 , and to resistivity measurements in CeRh 6 Ge 4 .1,2
We describe a formulation to deduce the phase shifts, which determine the ground-state properties of interacting quantum-dot systems with the inversion symmetry, from the fixed-point eigenvalues of the numerical renormalization group (NRG). Our approach does not assume the specific form of the Hamiltonian nor the electron-hole symmetry, and it is applicable to a wide class of quantum impurities connected to noninteracting leads. We apply the method to a triple dot which is described by a three-site Hubbard chain connected to two noninteracting leads, and calculate the dc conductance away from half-filling. The conductance shows the typical Kondo plateaus of Unitary limit in some regions of the gate voltages, at which the total number of electrons N_el in the three dots is odd, i.e., N_el =1, 3 and 5. In contrast, the conductance shows a wide minimum in the gate voltages corresponding to even number of electrons, N_el = 2 and 4. We also discuss the parallel conductance of the triple dot connected transversely to four leads, and show that it can be deduced from the two phase shifts defined in the two-lead case.
In a recent work [Merker, Kirchner, Mu\~noz, and Costi, Phys. Rev. B 87, 165132 (2013)], the authors compared the results of numerical renormalization group (NRG) and a perturbative approach for the dependence on temperature T and magnetic field B of the conductance through a quantum dot described by the impurity Anderson model, for small T and B. We show that the equation used to extract the dependence on B from NRG results is incorrect out of the particle-hole symmetric case. As a consequence, in the Kondo regime, the correct NRG results have a weaker dependence on B and the disagreement between both approaches increases.