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arXiv:1512.06127 · Physical review. B./Physical review. B

Buildup of the Kondo effect from real-time effective action for the Anderson impurity model

Sebastian Bock, Alexander Liluashvili, Thomas Gasenzer

Published Jul 11, 2016• 19 citations
arXivHTMLPDF

Abstract

The nonequilibrium time evolution of a quantum dot is studied by means of dynamic equations for time-dependent Green's functions derived from a two-particle-irreducible (2PI) effective action for the Anderson impurity model. Coupling the dot between two leads at different voltages, the dynamics of the current through the dot is investigated. We show that the 2PI approach is capable of describing the dynamical buildup of the Kondo effect, which shows up as a sharp resonance in the spectral function, with a width exponentially suppressed in the electron self-coupling on the dot. An external voltage applied to the dot is found to deteriorate the Kondo effect at the hybridization scale. The dynamic equations are evaluated within different nonperturbative resummation schemes, within the direct, particle-particle, and particle-hole channels, as well as their combination, and the results compared with those from other methods.

DOI
10.1103/physrevb.94.045108

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The nonequilibrium time evolution of a quantum dot is studied by means of dynamic equations for time-dependent Green's functions derived from a two-particle-irreducible (2PI) effective action for the Anderson impurity model. Coupling the dot between two leads at different voltages, the dynamics of the current through the dot is investigated. We show that the 2PI approach is capable of describing the dynamical buildup of the Kondo effect, which shows up as a sharp resonance in the spectral function, with a width exponentially suppressed in the electron self-coupling on the dot. An external voltage applied to the dot is found to deteriorate the Kondo effect at the hybridization scale. The dynamic equations are evaluated within different nonperturbative resummation schemes, within the direct, particle-particle, and particle-hole channels, as well as their combination, and the results compared with those from other methods.
In this thesis, we use a functional quantum field theoretical approach to investigate the non-equilibrium time evolution of an Anderson quantum dot with the main focus on the Kondo regime. We employ a real-time Keldysh path integral formulation to find an effective action. From the two-particle irreducible effective action, we derive, from the variational principle, the exact real-time Kadanoff–Baym equations of motion for the full propagator. We study these dynamic equations for the single impurity Anderson model, which decribes a quantum dot coupled to two finite-temperature leads. We take the tunnelling to the leads into account exactly. In order to solve the Kadanoff–Baym equations numerically we have to approximate them. For this purpose, we make a non-perturbative approximation by summing an infinite number of Feynman diagrams in the direct (s)-, particle-particle (t)-, particle-hole (u)-, and stu-channels. The aim of our investigation is to analyse the non-equilibrium realtime evolution of the quantum dot after a hybridisation and interaction quench into its stationary state. The main focus is on the narrowing of the Kondo resonance and the formation of the Hubbard side bands. Following on from this, the main achievement is the transient, as well as the stationary electrical current, through the quantum dot and the investigation of the dependence on temperature and magnetic field. We compare our results with other methods, such as functional renormalisation group and iterative sum of path integrals, and find a very good agreement in most situations.
The nonequilibrium time evolution of an Anderson quantum dot is investigated. The quantum dot is coupled between two leads forming a chemical-potential gradient. We use Kadanoff-Baym dynamic equations within a nonperturbative resummation of the sss-channel bubble chains. The effect of the resummation leads to the introduction of a frequency-dependent four-point vertex. The tunneling to the leads is taken into account exactly. The method allows the determination of the transient as well as stationary transport through the quantum dot, and results are compared with different schemes discussed in the literature (functional renormalization group, iterative real-time summation of the path integral, time-dependent matrix renormalization group, and quantum Monte Carlo methods).
The nonequilibrium spectral properties of the Anderson impurity model with a chemical potential bias are investigated within a numerically exact real-time quantum Monte Carlo formalism. The two-time correlation function is computed in a form suitable for nonequilibrium dynamical mean field calculations. Additionally, the evolution of the model's spectral properties are simulated in an alternative representation, defined by a hypothetical but experimentally realizable weakly coupled auxiliary lead. The voltage splitting of the Kondo peak is confirmed and the dynamics of its formation after a coupling or gate quench are studied. This representation is shown to contain additional information about the dot's population dynamics. Further, we show that the voltage-dependent differential conductance gives a reasonable qualitative estimate of the equilibrium spectral function, but significant qualitative differences are found including incorrect trends and spurious temperature dependent effects.
We investigate the time evolution of the Kondo resonance in response to a quench by applying the time-dependent numerical renormalization group (TDNRG) approach to the Anderson impurity model in the strong correlation limit. For this purpose, we derive within the TDNRG approach a numerically tractable expression for the retarded two-time nonequilibrium Green function $G(t+{t}^{\ensuremath{'}},t)$, and its associated time-dependent spectral function, $A(\ensuremath{\omega},t)$, for times ttt both before and after the quench. Quenches from both mixed valence and Kondo correlated initial states to Kondo correlated final states are considered. For both cases, we find that the Kondo resonance in the zero temperature spectral function, a preformed version of which is evident at very short times $t\ensuremath{\rightarrow}{0}^{+}$, only fully develops at very long times $t\ensuremath{\gtrsim}1/{T}_{K}$, where TK{T}_{K}TK​ is the Kondo temperature of the final state. In contrast, the final state satellite peaks develop on a fast time scale $1/\mathrm{\ensuremath{\Gamma}}$ during the time interval $\ensuremath{-}1/\mathrm{\ensuremath{\Gamma}}\ensuremath{\lesssim}t\ensuremath{\lesssim}+1/\mathrm{\ensuremath{\Gamma}}$, where $\mathrm{\ensuremath{\Gamma}}$ is the hybridization strength. Initial and final state spectral functions are recovered in the limits $t\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$ and $t\ensuremath{\rightarrow}+\ensuremath{\infty}$, respectively. Our formulation of two-time nonequilibrium Green functions within the TDNRG approach provides a first step towards using this method as an impurity solver within nonequilibrium dynamical mean field theory.
The Anderson impurity model is a paradigmatic example in the study of strongly correlated quantum systems and describes an interacting quantum dot coupled to electronic leads. Here we investigate its dynamics following a quantum quench based on matrix product state simulations. We examine the behavior of its impurity magnetization. Its relaxation allows us to extract the predicted scaling of the Kondo temperature as a function of the impurity-lead hybridization and quantum dot repulsion. Additionally, our simulations provide estimates of the currents in the nonequilibrium quasisteady state appearing after the quench. Through their values, we examine the dependence of the conductance on the voltage bias Vb{V}_{b}Vb​ and on the impurity chemical potential Vg{V}_{g}Vg​, which displays a zero-bias Kondo peak. Our results are relevant for transport measurements in Coulomb blockaded devices, and, in particular, in quantum dots induced in nanowires.
We develop a nonperturbative dynamical theory (NDT) to calculate the retarded Green's function under nonequilibrium conditions. The NDT is particularly useful for treating nonequilibrium transport problems in systems with strong correlation. We apply our NDT to the well-known single-impurity Anderson model at equilibrium to determine its feasibility. We then apply it to a nonequilibrium transport problem in a system with Kondo coupling. An Anderson model with two metallic reservoirs is studied to understand the phenomenon of Kondo-peak splitting in a single-electron transistor of mesoscopic size. We calculate the nonequilibrium retarded Green's function by using the NDT and analyze it in the atomic limit, where the novel coherent phenomenon manifested only under nonequilibrium conditions can be described in an analytical manner. We finally construct a self-consistent loop to calculate the retarded Green's function and present the results for spectral density and differential conductance obtained by the self-consistent method. Our results explain all the features of Kondo-peak splitting observed in experiments. One remarkable conclusion is that Kondo-peak splitting is not the splitting of a conventional Kondo peak, but the splitting of a novel coherent peak created under nonequilibrium steady-state conditions.