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arXiv:1012.4293 · Physical Review B

Real-time effective-action approach to the Anderson quantum dot

Dénes Sexty, Thomas Gasenzer, Jan Martin Pawlowski

Published Apr 22, 2011• 7 citations
arXivHTMLPDF

Abstract

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 $s$-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).

DOI
10.1103/physrevb.83.165315

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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).
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 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.
We have developed a numerical approach to compute real-time path integral expressions for quantum transport problems out of equilibrium. The scheme is based on a deterministic iterative summation of the path integral for the generating function of the nonequilibrium current. Self-energies due to the leads, being nonlocal in time, are fully taken into account within a finite memory time, thereby including non-Markovian effects, and numerical results are extrapolated both to vanishing (Trotter) time discretization and to infinite memory time. This extrapolation scheme converges except at very low temperatures, and the results are then numerically exact. The method is applied to nonequilibrium transport through an Anderson dot.
We develop an iterative, numerically exact approach for the treatment of nonequilibrium quantum transport and dissipation problems that avoids the real-time sign problem associated with standard Monte Carlo techniques. The method requires a well-defined decorrelation time of the nonlocal influence functional for proper convergence to the exact limit. Since finite decorrelation times may arise either from temperature or from a voltage drop at zero temperature, the approach is well suited for the description of the real-time dynamics of single-molecule devices and quantum dots driven to a steady state via interaction with two or more electron leads. We numerically investigate two nontrivial models: the evolution of the nonequilibrium population of a two-level system coupled to two electronic reservoirs, and quantum transport in the nonequilibrium Anderson model. For the latter case, two distinct formulations are described. Results are compared to those obtained by other techniques.
We construct a real time current-conserving functional renormalization group (RG) scheme on the Keldysh contour to study frequency-dependent transport and noise through a quantum dot in the local moment regime. We find that the current vertex develops a nontrivial nonlocal structure in time that is governed by a new set of RG equations. Solving these RG equations, we compute the complete frequency and temperature dependence of the noise spectrum. For voltages that are large compared to the Kondo temperature (i.e., $\mathit{eV}\ensuremath{\gg}{k}_{B}{T}_{K}$), two sharp antiresonances are found in the noise spectrum at frequencies $\ensuremath{\hbar}\ensuremath{\omega}=\ifmmode\pm\else\textpm\fi{}eV$ and, correspondingly, two Kondo-assisted peaks appear in the ac conductance through the dot.
We introduce a real time version of the functional renormalization group, which allows us to study correlation effects on nonequilibrium transport through quantum dots. Our method is equally capable to address (i) the relaxation out of a nonequilibrium initial state into a (potentially) steady state driven by a bias voltage and (ii) the dynamics governed by an explicitly time-dependent Hamiltonian. All time regimes from transient to asymptotic can be tackled; the only approximation is the consistent truncation of the flow equations at a given order. As an application we investigate the relaxation dynamics of the interacting resonant level model, which describes a fermionic quantum dot dominated by charge fluctuations. Moreover, we study incoherence and relaxation phenomena within the ohmic spin-boson model by mapping the latter to the interacting resonant level model.