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arXiv:0909.5555 · Physical Review B

Comparison between scattering-states numerical renormalization group and the Kadanoff-Baym-Keldysh approach to quantum transport: Crossover from weak to strong correlations

Sebastian Schmitt, Frithjof B. Anders

Published Apr 9, 2010• 40 citations
arXivHTMLPDF

Abstract

The quantum transport through nanoscale junctions is governed by the charging energy $U$ of the device. We employ the recently developed scattering-states numerical renormalization-group approach to open quantum systems to study nonequilibrium Green's functions and current-voltage characteristics of such junctions for small and intermediate values of $U$. We establish the accuracy of the approach by a comparison with diagrammatic Kadanoff-Baym-Keldysh results which become exact in the weak-coupling limit $U\ensuremath{\rightarrow}0$. We demonstrate the limits of the diagrammatic expansions at intermediate values of the charging energy. While the numerical renormalization-group approach correctly predicts only one single, universal low-energy scale at zero bias voltage, some diagrammatic expansions yield two different low-energy scales for the magnetic and the charge fluctuations. At large voltages, however, the self-consistent second Born as well as the GW approximation reproduce the scattering-states renormalization-group spectral functions for symmetric junctions while for asymmetric junctions the voltage-dependent redistribution of spectral weight differs significantly in the different approaches. The second-order perturbation theory does not capture the correct single-particle dynamics at large bias and violates current conservation for asymmetric junctions.

DOI
10.1103/physrevb.81.165106

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The quantum transport through nanoscale junctions is governed by the charging energy UUU of the device. We employ the recently developed scattering-states numerical renormalization-group approach to open quantum systems to study nonequilibrium Green's functions and current-voltage characteristics of such junctions for small and intermediate values of UUU. We establish the accuracy of the approach by a comparison with diagrammatic Kadanoff-Baym-Keldysh results which become exact in the weak-coupling limit U\ensuremath→0U\ensuremath{\rightarrow}0U\ensuremath→0. We demonstrate the limits of the diagrammatic expansions at intermediate values of the charging energy. While the numerical renormalization-group approach correctly predicts only one single, universal low-energy scale at zero bias voltage, some diagrammatic expansions yield two different low-energy scales for the magnetic and the charge fluctuations. At large voltages, however, the self-consistent second Born as well as the GW approximation reproduce the scattering-states renormalization-group spectral functions for symmetric junctions while for asymmetric junctions the voltage-dependent redistribution of spectral weight differs significantly in the different approaches. The second-order perturbation theory does not capture the correct single-particle dynamics at large bias and violates current conservation for asymmetric junctions.
Scattering states fulfill the correct boundary conditions of a current carrying open quantum system. Discretizing the energy continuum of these states allows for employing Wilson's numerical renormalization group approach without violating the boundary conditions by using a finite size system. We evolve the analytically known steady-state density operator for a non-interacting quantum-system at finite bias to the full interacting problem by the time-dependent numerical renormalization group after switching on the local charging energy. Using a newly developed algorithm for steady-state nonequilibrium Green functions, we can calculate the current I as function of bias voltage V for arbitrary temperature and magnetic field. A comparison with second-order and GW Kadanoff-Baym-Keldysh results shows excellent agreement for weak interaction strength U .
We propose a time-dependent many-body approach to study the short-time dynamics of correlated electrons in quantum transport through nanoscale systems contacted to metallic leads. This approach is based on the time propagation of the Kadanoff-Baym equations for the nonequilibrium many-body Green's function of open and interacting systems out of equilibrium. An important feature of the method is that it takes full account of electronic correlations and embedding effects in the presence of time-dependent external fields, while at the same time satisfying the charge conservation law. The method further extends the Meir-Wingreen formula to the time domain for initially correlated states. We study the electron dynamics of a correlated quantum wire attached to two-dimensional leads exposed to a sudden switch on of a bias voltage using conserving many-body approximations at Hartree-Fock, second Born and GW level. We obtain detailed results for the transient currents, dipole moments, spectral functions, charging times, and the many-body screening of the quantum wire as well as for the time-dependent density pattern in the leads, and we show how the time dependence of these observables provides a wealth of information on the energy level structure of the quantum wire out of equilibrium. For moderate interaction strengths the second Born and GW results are in excellent agreement at all times. We find that many-body effects beyond the Hartree-Fock approximation have a large effect on the qualitative behavior of the system and lead to a bias-dependent gap closing and quasiparticle broadening, shortening of the transient times and washing out of the step features in the current-voltage curves.
We present the real-time renormalization group (RTRG) method as a method to describe the stationary state current through generic multilevel quantum dots in nonequilibrium. The employed approach consists of a very rudimentary approximation for the renormalization group (RG) equations which neglects all vertex corrections while it provides a means to compute the effective dot Liouvillian self-consistently. Being based on a weak-coupling expansion in the tunneling between dot and reservoirs, the RTRG approach turns out to reliably describe charge fluctuations in and out of equilibrium for arbitrary coupling strength, even at zero temperature. We confirm this in the linear response regime with a benchmark against highly accurate numerical renormalization group data in the exemplary case of three-level quantum dots. For small to intermediate bias voltages and weak Coulomb interactions, we find an excellent agreement between RTRG and functional renormalization group data, which can be expected to be accurate in this regime. As a consequence, we advertise the presented RTRG approach as an efficient and versatile tool to describe charge fluctuations in quantum dot systems.
The progress in micro-fabrication technologies makes it possible to address individual, tailored, mesoscopic structures with sizes on a nanometer to micrometer scale. The transport properties of such quasi zero- or one-dimensional systems are considerably influenced by the electron-electron interaction. In theoretical studies this often becomes apparent in the emergence of divergences in a perturbative treatment of the interaction. A successful approach may then be given by renormalization group (RG) methods which are based on the idea of treating energy scales successively from high to low. In the functional RG (fRG) an exact infinite hierarchy of flow equations for multi-particle Green or vertex functions is derived, that allows for systematic approximations on different levels. In numerous studies the fRG has been found to be a powerful tool for the investigation of low dimensional Fermi systems. Usually it is formulated within Matsubara formalism which applies to a system in thermal equilibrium. A central topic of this thesis is the implementation of the fRG within Keldysh formalism, providing the possibility to treat equilibrium and non-equilibrium situations on equal footing. The resulting Keldysh fRG is used in this thesis to explore the transport properties of quantum wires with several barriers and of the single impurity Anderson model. The thesis starts with an introduction to Keldysh formalism and to related concepts. Then the structure of the Green and vertex functions of Keldysh formalism is investigated. This includes studying the consequences of causality, characterizing thermal equilibrium, and formulating corresponding relations for the multi-particle functions. After that, the implementation of the fRG within Keldysh formalism is presented in detail. The flow equations for the vertex functions are derived via their diagrammatic expansion. Two especially useful flow parameters are determined: one of them cuts off the imaginary frequencies that appear as poles of the Fermi functions of the reservoirs. The second flow parameter is an artificially enhanced reservoir hybridization; on one hand it has the advantage to preserve the fluctuation dissipation theorem in equilibrium even in frequency dependent approximations; on the other hand, its use can be more cumbersome. One of the models studied in this thesis is a tight-binding chain with nearest-neighbour interaction. At weak and intermediate interactions it belongs to the class of Luttinger liquids (LLs) whose low energy properties are characterized by power laws. The transport properties of LLs have been studied experimentally for example on carbon nanotubes. However, the presence of transport barriers at the contacts make the interpretation of the experimental data difficult. A theoretical treatment with the fRG permits a rather large flexibility to model the barrier configuration. In this thesis the fRG is first used in its equilibrium Matsubara implementation in order to analyse the temperature dependence of the linear conductance in the presence of two, three or four barriers. It is shown that temperature induces an averaging over quantum phases of propagation, leading to combinations of the power law exponents of the individual barriers. An experimental set-up that allows to verify this behaviour is proposed. In a second step the Keldysh version of the fRG is applied to the problem of transport through a quantum wire at finite bias voltage. For a wire with two high contact barriers it is found that the non-equilibrium particle distribution in the wire is responsible for the appearance of a two-fold power law in the density of states. An additional weakly attached probe lead can be used to make the two different power laws visible in the conductance. The second model which is analysed is the one of the single Anderson impurity, often considered as a paradigm for the investigation of strong local electron correlations. The latter become noticeable in the appearance of the Kondo scale, an energy exponentially small in the interaction strength, that characterizes the low energy physics of the model in the regime of spin fluctuations. On the basis of this model an advanced frequency dependent approximation for the Keldysh fRG is developed in this thesis. Such an approximation can account for subleading effects of the interaction, namely the fact that two-particle interactions can generate decay rates for single-particle states. The derivation of a consistent approximation scheme is presented, that turns out to yield a good description of the model at moderate interaction strength.
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.
It is shown that the lesser and greater self energies and Green functions calculated in a recent work on universal noequilibrium transport in interacting quantum dots [E. Muñoz, C. J. Bolech, and S. Kirchner, Phys. Rev. Lett. 110, 016601 (2013)] are incorrect. The authors start from renormalized second-order perturbation theory in the Coulomb repulsion in the particle-hole symmetric impurity Anderson model. The reported lesser self-energy is compared with the correct one, and with an analytic result valid for small frequency and applied voltage. As a consequence of the mistakes, the conservation of the current is not established and the results are unreliable.