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Physical review. B./Physical review. B

Real-space numerical renormalization group computation of transport properties in side-coupled geometry

Ana Luiza Ferrari, L. Nunes de Oliveira

Published Aug 16, 2022• 4 citations
Publisher

Abstract

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)].

DOI
10.1103/physrevb.106.075129

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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 Lu0.9\ifmmode†\else\textdagger\fiYb0.1\ifmmode†\else\textdagger\fiRh2\ifmmode†\else\textdagger\fiSi2\ifmmode†\else\textdagger\fi{\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{}}}Lu0.9\ifmmode†\else\textdagger\fi​Yb0.1\ifmmode†\else\textdagger\fi​Rh2\ifmmode†\else\textdagger\fi​Si2\ifmmode†\else\textdagger\fi​, and the remarkably accurate, recent conductance measurements by Xu et al. [Chin. Phys. Lett. 38, 087101 (2021)].
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
The numerical renormalization group (NRG) approach was developed to identify and quantify different equilibrium regimes of quantum impurity systems (QISs) with unprecedented accuracy by a tailored finite size representation. Out of equilibrium, the steady-state density operator is not of the Boltzmannian form but one that is determined by the imposed boundary conditions. We extend the NRG to the nonequilibrium setting by augmenting each Wilson site with a reservoir, whose coupling functions are calculated via a continuous fraction expansion in order to recover the continuum limit exactly. The nonequilibrium parameters such as a finite bias as well as a finite temperature gradient enters through the Bloch-Redfield tensor (BRT), whose zero eigenvector gives the steady-state density operator. We used the resulting open chain full density matrix (OC-FDM) approach with an effective single lead description to investigate the charge and spin transport through a quantum dot (QD) under finite bias and temperature gradient. The influence of lead asymmetry and an external magnetic field on the transport properties are also studied. For completeness, we have also investigated local properties of the QD, such as charge fluctuations and find excellent agreement with real-time quantum Monte Carlo (RT-QMC) data. The OC-FDM approach was able to explore Kondo energy scales as low as TK/D≈10−8T_K/D \approx 10^{-8}TK​/D≈10−8 in the non-equilibrium regime, as well as show convergence with the established equilibrium benchmarks, such as a quantitative agreement with full density matrix numerical renormalization group (FDM-NRG) and Fermi-liquid scaling at small bias and temperature. Owing to the effective single lead description, a single OC-FDM data point takes orders of magnitude less time on a standard laptop, compared to other state-of-the-art numerical methods.
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 TTT and magnetic field BBB of the conductance through a quantum dot described by the impurity Anderson model, for small TTT and BBB. We show that the equation used to extract the dependence on BBB 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 BBB and the disagreement between both approaches increases.
Wilson's Numerical Renormalization Group (NRG) is so far the only nonperturbative technique that can reliably access low-energy properties of quantum impurity systems. We present a recent extension of the method, the DM-NRG, which yields highly accurate results for dynamical quantities at arbitrary frequencies and temperatures. As an application, we determine the spectrum of a quantum dot in an external magnetic field. Furthermore, we discuss magnetic impurities with orbital degeneracy, which have been inferred in recent experiments on quantum dots in an Aharonov-Bohm geometry. It is demonstrated that for spinless electrons, interference between neighbouring levels sets the low-energy scale of the system. Switching on an external field leads to a remarkable crossover into a regime dominated by orbital Kondo screening. We predict that the broadening-induced level splitting should be clearly visible in measurements of the optical absorption power. A more general model including the electron spin is studied within an extended two-band NRG procedure. We observe competition between interference and Kondo screening, similar to the situation in two-impurity models (RKKY).
The numerical renormalization group (NRG) is tailored to describe interacting impurity models in equilibrium, but it faces limitations for steady-state nonequilibrium, arising, e.g., due to an applied bias voltage. We show that these limitations can be overcome by describing the thermal leads using a thermofield approach, integrating out high energy modes using NRG, and then treating the nonequilibrium dynamics at low energies using a quench protocol, implemented using the time-dependent density matrix renormalization group. This yields quantitatively reliable results for the current (with errors ≲3%) down to the exponentially small energy scales characteristic of impurity models. We present results of benchmark quality for the temperature and magnetic field dependence of the zero-bias conductance peak for the single-impurity Anderson model.
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.