A coupled quantum dot system in a phonon bath in nonequilibrium was examined. We have explained the close relation of this problem to the previously studied spin-boson model. As in case of the spin-boson model we applied the RTRG to this problem and determined the stationary tunnel current. In this case it was necessary to formulate the RTRG also for observables, which are linear in the bath field operators, and we had to account for additional signs arising from the commutation of fermionic field operators.
By accounting for both the coupling to the leads and the coupling to the environmen-tal phonons nonperturbatively we achieved a reliable solution for the stationary tunnel current in the whole parameter space with moderately large couplings. Especially, we obtained accurate results for arbitrary energy differencebetween the dots and tempera-tureT, which is in contrast to approximations using the NIBA. Furthermore, we included off-diagonal terms of the electron-phonon interaction, which give a contribution in case of finite extensions of the electron densities within the dots. Thereby, for the first time both the elastic and the inelastic current of the experiment could quantitatively be re-produced [35]. Our analysis shows the importance of the finite width σ of the electron densities within one dot, and for the experiment, the dependence ofσonwas calculated.
For the coupling parameters realized in the experiment [30] the neglecting of double and higher-order vertex objects was justified. In contrast, in the next chapter we will study a Kondo model, for which we will have to account for double vertices. We have already mentioned in Section 3.2, that we will use a modified formulation of the RTRG to treat these double vertex objects.
Chapter 5
Two-lead Kondo model
The Kondo model describes a magnetic impurity coupled to a band of conduction elec-trons. Over the past decades this problem has often been studied both experimentally and theoretically (for a review see Ref. [36]). When Kondo first studied the model perturba-tively in 1964 [83], he found low-temperature divergencies. In fact, perturbative studies give only a good description of the problem forT TK, where TK is the Kondo tem-perature. Renormalization-group studies show that the coupling constant increases when reducing the relevant energy scale. Thus, perturbative approaches like Poor Man’s scal-ing break down atTK. In 1975, the problem was solved by Wilson [5] by applying the numerical renormalization-group method to the Kondo model. Later, a solution was also found by using the Bethe ansatz [84,85], an approach which had first been introduced in Ref. [86].
In recent experiments quantum dot systems served as realizations of the Kondo model [87,88,89,77]. Such systems have the advantage that the parameters can be tuned within a wide range. When one applies an external bias voltage to the quantum dot, one deals with a nonequilibrium problem, the two-lead Kondo model (TLKM). It has attracted much theoretical interest and the question has been raised, if an external voltage induces two-channel physics [37] - [43].
We apply the RTRG method to the problem. As the bare Hamiltonian already involves double vertices, the usual formulation would lead to complicated equations because of re-tardation effects (see AppendixC). Therefore, we use a formulation in energy space, by which we avoid the problem of retardation. However, one then has to account for an addi-tional frequency-dependence of the vertices. Furthermore, this method naturally leads to divergencies, if the cutoff is defined only with respect to the frequency of the contraction vertices. We introduce a generalized definition of the cutoff-function, which also depends on the external vertices. This new scheme allows a RG study of the coupling constants of the two-lead Kondo model. We study an effective Hamiltonian for vanishing temperature T. Thus, effects which are only taken into account by an analysis on the Keldysh con-tour, such as rates, are neglected. Thereby we quantitatively find a two-channel behaviour for the running couplings which, on this level, was previously proposed in a qualitative analysis [37]. The influence of rates, which arise for a finite bias, is discussed.
97
V R
L J
J J
LL
RL
LR
RR
S
Figure 5.1: The two-lead Kondo model. The voltageV is applied between two leads (L andR), which are coupled to the impurity spin S. The coupling constants~ JLL andJRR (JLRandJRL) correspond to reflection (transmission) of conduction electrons.
5.1 Model Hamiltonian
In 1964 Kondo studied a model Hamiltonian for a magnetic impurity in a conduction band [83]. There an exchange scattering potential gives rise to an interaction part of the HamiltonianHV, which is parametrized by the energyJ:
HV =X
kσσ0
J c†kσ~σσσ0ckσ0S .~ (5.1) Here,S~is the spin of the impurity and~σis the vector consisting of the Pauli spin matrices.
c†kσ (ckσ) creates (annihilates) an electron in the conduction band with the energyk and the spinσ. The total Hamiltonian can be written as a sum H = HB +HV, where HB denotes the free electron part for the conduction electrons. When two bands are coupled to the impurity, we end up with
HB = X
αkσ
αkc†αkσcαkσ, (5.2)
HV = X
αα0kσσ0
Jαα0c†αkσ~σσσ0cα0kσ0S ,~ (5.3) where the index α = L(R) refers to the left (right) electron reservoir. The coupling constantsJLL and JRR correspond to a reflection of conduction electrons, whereas JLR andJRL give rise to transmission from one reservoir to the other. Between them a finite external voltageV may be applied (see Fig.5.1).
The case of vanishing bias involves a well-known renormalization-group flow [5,36].
Introducing the high-energy cutoffωcas a flow parameter the antiferromagnetic coupling
constants increase, when ωc is successively reduced. This leads to a low-temperature regime where the impurity spin is quenched by the band electrons. The physics is then determined by the only relevant energy scale, the Kondo temperature TK. Regarding the above Hamiltonian we use the density of states of the reservoir α, ρα, to define the dimensionless couplingJ¯αα0 :=ραJαα0. In the following we consider the symmetric case, whereJ¯:= ¯JLL = ¯JRR = ¯JLR = ¯JRL. The Kondo temperature is given by
TK =Dp
2 ¯J e−1/4 ¯J, (5.4)
whereD TK is the bandwidth of the electron reservoirs. Note that, in this chapter we again use~ =kB = e= 1. For the caseV = 0the increase of the coupling constant for TK > T follows from the RG equation [5,36]
dJ¯
dlnωc = −4 ¯J2
⇒ dJ¯−1
dlnωc = 4. (5.5)
For the case of finiteV, Coleman et al. [37] proposed for the running couplings the existence of two different regimes, which are separated by the energy scaleV:
dJ¯
dlnωc = −4 ¯J2 for ωc V > TK > T , (5.6) dJ¯LL
dlnωc = dJ¯RR
dlnωc =−2 ¯J2, dJ¯LR
dlnωc = dJ¯RL dlnωc = 0
for V ωc TK > T . (5.7)
However, since Coleman et al. used a qualitative argumentation within a Poor Man’s scaling approach, these equations have not been derived quantitatively. Furthermore, the above equations do not reflect the effects of rates which arise for a finite voltage.