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arXiv:1111.6982 · Physical Review B

Renormalization group approach to time-dependent transport through correlated quantum dots

Dante Marvin Kennes, Severin G. Jakobs, Christoph Karrasch, V. Meden

Published Feb 21, 2012• 77 citations
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Abstract

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.

DOI
10.1103/physrevb.85.085113

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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.
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We study the dynamics in low dimensional correlated quantum systems. Advances in this thesis are three-fold. First, as a methodical point we extend the real-time formulation of the functional renormalization group approach introduced in [1]. Describing the transient time dependence of a given microscopic model prepared in an initial non-equilibrium state poses a formidable challenge in the presence of interaction. We successfully extend the functional renormalization group approach of [1] to finite temperature as well as extended interacting geometries. Furthermore, for both extensions we test the validity of the perturbatively motivated treatment by explicitly benchmarking to complementary methods and find that for not too large interactions the functional renormalization group proves to be quantitatively accurate. Second, we examine the non-equilibrium dynamics of open as well as closed quantum systems in the presence of correlations. We study a prominent prototype model for coherence in small quantum systems coupled to dissipative environments, namely the spin-boson model. We reveal that prevailing literature does not cover the full complexity of the dynamics encountered. The relaxation dynamics formerly classified as coherent and incoherent should be refined according to the results presented in this thesis: we propose a subdivision of the coherent regime into partially and asymptotically coherent. For the former the coherence signature of the dynamics is restricted only to intermediate times, while in the latter it extends in the asymptotic regime. Interestingly, we explicitly show that elevating temperature might be conducive to coherence, which is opposite to the expected behavior. For those closed quantum systems known to fall into the so called Luttinger Liquid universality class in equilibrium, we provide evidence that the equilibrium concept of universality seemingly carries over to the non-equilibrium steady-state. This is rather counter-intuitive, as universality is usually attributed to the low energy degrees of freedom only. Third, we study the thermoelectric transport properties of quantum dots. Quantum dots are promising candidates for highly efficient thermoelectric devices as they are not bounded by the Wiedemann-Franz law which strongly limits the efficiency of ordinary bulk systems. For this we study a charge fluctuating model and a model which additionally allows for spin fluctuations and phononic degrees of freedom. We find that in certain regimes quantum dots show very high efficiency when operated as power engines. [1] D. M. Kennes, ‘A Renormalization Group Approach to the Time Evolution of Correlated Quantum Dots’ Master’s thesis, RWTH Aachen (2011).
Using a real-time renormalization group method we study the minimal model of a quantum dot dominated by charge fluctuations, the two-lead interacting resonant level model, at finite bias voltage. We develop a set of RG equations to treat the case of weak and strong charge fluctuations, together with the determination of power-law exponents up to second order in the Coulomb interaction. We derive analytic expressions for the charge susceptibility, the steady-state current, and the conductance in the situation of arbitrary system parameters, in particular away from the particle-hole symmetric point and for asymmetric Coulomb interactions. In the asymmetric situation we find that power laws can be observed for the current only as a function of the level position (gate voltage) but in general not as a function of the voltage except for extremely large voltages. Furthermore, we study the quench dynamics after sudden switch-on of the level-lead couplings. The time evolution of the dot occupation and current is governed by exponential relaxation accompanied by voltage-dependent oscillations and characteristic algebraic decay.
We study transport properties of quantum impurity systems using the functional renormalization group. The latter is an RG-based diagrammatic tool to treat Coulomb interactions in a fast and flexible way. Prior applications, which employed a simple first-order (Hartree-Fock-like) scheme to truncate the FRG flow equations within the Matsubara formalism, succeeded in accurately describing linear transport of various quantum dot geometries at zero temperature T=0. In a nutshell, advance in this Thesis is three-fold. First, we introduce a frequency-dependent second-order approximation in order to eventually compute finite-energy properties such as the conductance at T>0 (mainly focusing on the single impurity Anderson model). Second, a generalisation of the Hartree-Fock-like approach to Keldysh space allows for addressing the non-equilibrium steady-state dynamics of the interacting resonant level model. Third, we investigate the physics of a quantum dot Josephson junction as well as the charging of a single narrow level using the first-order scheme.
We apply the real-time renormalization group (RG) in nonequilibrium to an arbitrary quantum dot in the Coulomb blockade regime. Within one-loop RG equations, we include self-consistently the kernel governing the dynamics of the reduced density matrix of the dot. As a result, we find that relaxation and dephasing rates generically cut off the RG flow. In addition, we include all other cutoff scales defined by temperature, energy excitations, frequency, and voltage. We apply the formalism to transport through single molecular magnets, realized by the fully anisotropic Kondo model (with three different exchange couplings Jx{J}_{x}Jx​, Jy{J}_{y}Jy​, and Jz{J}_{z}Jz​) in a magnetic field hz{h}_{z}hz​. We calculate the differential conductance as function of bias voltage VVV and discuss a quantum phase transition which can be tuned by changing the sign of JxJyJz{J}_{x}{J}_{y}{J}_{z}Jx​Jy​Jz​ via the anisotropy parameters. Finally, we calculate the noise $S(\ensuremath{\Omega})$ at finite frequency $\ensuremath{\Omega}$ for the isotropic Kondo model and find that the dephasing rate determines the height of the shoulders in $dS(\ensuremath{\Omega})∕d\ensuremath{\Omega}$ near $\ensuremath{\Omega}=V$.
We develop a general approach to the nonequilibrium dynamics of quantum-impurity systems for arbitrary coupling strength. The numerical renormalization group is used to generate a complete basis set necessary for the correct description of the time evolution. We benchmark our method with the exact analytical solution for the resonant-level model. As a first application, we investigate the equilibration of an ultrasmall quantum dot subject to a sudden change of gate voltage and external magnetic field. Two distinct relaxation times are identified for the spin and charge dynamics.