Flavor fluctuations in 3-level quantum dots: Generic SU(3)-Kondo fixed point in equilibrium and non-Kondo fixed points in nonequilibrium
Carsten J. Lindner Fabian B. Kugler Herbert Schoeller Jan von Delft





Institut für Theorie der Statistischen Physik, RWTH Aachen, 
52056 Aachen, Germany and JARA - Fundamentals of Future Information Technology

Physics Department, Arnold Sommerfeld Center for Theoretical Physics, and Center
for NanoScience,Ludwigs-Maximilians-Universität München, Theresienstr. 37, 80333 Munich, Germany

[Email: ]schoeller@physik.rwth-aachen.de
Institut für Theorie der Statistischen Physik, RWTH Aachen, 
52056 Aachen, Germany and JARA - Fundamentals of Future Information Technology

Physics Department, Arnold Sommerfeld Center for Theoretical Physics, and Center
for NanoScience,Ludwigs-Maximilians-Universität München, Theresienstr. 37, 80333 Munich, Germany


We study a 3-level quantum dot in the singly occupied cotunneling regime coupled 
via a generic tunneling matrix to several multi-channel leads in equilibrium or nonequilibrium. 
Denoting the three possible states of the quantum dot by the quark flavors up (u), down (d) 
and strange (s), we derive an effective model where also each reservoir has three flavors labelled by 
u, d and s with an effective density of states polarized w.r.t. an 
eight-dimensional F-spin corresponding to the eight generators of SU(3).
In equilibrium we perform a standard poor man scaling analysis 
and show that tunneling via virtual intermediate states induces flavor fluctuations 
on the dot which become SU(3)-symmetric at a characteristic and exponentially small low-energy 
scale T_K. Close to T_K the system is described by a 
single isotropic Kondo coupling J>0 diverging at T_K. Using the numerical
renormalization group (NRG) we study in detail the linear conductance and 
confirm the SU(3)-symmetric Kondo fixed point with universal conductance 
G=3sin^2(π/3)e^2 h=2.25e^2 h for 
various tunneling setups by tuning the level spacings on the dot. We also identify regions 
of the level positions where the SU(2)-Kondo fixed point is obtained and find a rather complex 
dependence of the various Kondo temperatures as function of the gate voltage and the 
tunneling couplings. In contrast to the equilibrium case, we find in nonequilibrium that
the fixed point model is not SU(3)-symmetric but characterized by rotated F-spins for 
each reservoir with total vanishing sum.
At large voltage we analyse the F-spin magnetization and the current in golden rule as function
of a longitudinal (h_z) and perpendicular (h_⊥) magnetic field for the isospin and the 
level spacing Δ to the strange quark. As a smoking gun to detect the nonequilibrium fixed point
we find that the curve of zero F-spin magnetization in (h_z,h_⊥,Δ)-space is a circle when
projected onto the (h_z,h_⊥) plane. We propose that our findings can be generalized to the
case of quantum dots with an arbitrary number N of levels. 




05.60.Gg, 72.10.Bg, 73.23.-b,73.63.Kv







Flavor fluctuations in 3-level quantum dots: Generic SU(3)-Kondo fixed point
  in equilibrium and non-Kondo fixed points in nonequilibrium
    Jan von Delft
    
==========================================================================================================================================





§ INTRODUCTION



Over the last three decades transport properties of correlated quantum dots have gained an enormous
interest in many experimental and theoretical research activities in condensed matter physics. As
artificial atoms they allow for a controlled study of interesting phenomena playing a central role
in many different fields of applied and fundamental research in nanoelectronics, spintronics, 
quantum information processing, dissipative quantum mechanics, and many-body physics and 
nonequilibrium phenomena in correlated systems, see e.g. Refs. review_qi,andergassen_etal for 
reviews. Of particular interest is the cotunneling or Coulomb blockade regime of quantum dots with strong
charging energy, where the charge is fixed and only the spin and orbital degrees of freedom can 
fluctuate by second-order tunneling processes via virtual intermediate states. In this regime 
effective models can be derived which are equivalent to Kondo models well-known from solid state
physics <cit.>, see e.g. Ref. glazman_pustilnik_05 for a review of the 
Kondo effect in quantum dots. The standard model is the SU(2)-Kondo model, where a local 
spin-1 2 is coupled via an isotropic exchange coupling to the spins of two large reservoirs.
Below a characteristic low energy scale, called the Kondo temperature T_K, the local spin is
completely screened and the remaining potential scattering leads to resonant transport through the
system with universal conductance 2e^2 h. This Kondo effect has been theoretically predicted 
for quantum dots <cit.> and has been experimentally observed <cit.>. 
After this discovery the research for Kondo physics in quantum dots has gained an enormous interest
and further realizations have been proposed and observed, like e.g. the realization of higher spin values
<cit.>, singlett-triplett fluctuations <cit.>, 
non-Fermi liquid behaviour in 2-channel realizations <cit.>, and 
the SU(4)-Kondo effect <cit.>. Recently, also the realization of SU(N)-Kondo physics for
arbitrary N has been proposed in coupled quantum dots <cit.>.

The enormous variety of possible realizations of Kondo physics raises the question what happens in the
generic case when a quantum dot in the regime of fixed charge with N_dot≥ 1 electrons and 
N≥ 2 levels is coupled via a generic tunneling matrix to several multi-channel reservoirs. Even for
the simplest case N_dot=1 and N=2, this issue is nontrivial since the quantum number l=1,2 
labelling the two dot levels is in general a non-conserved quantity in tunneling, like e.g. 
for ferromagnetic leads <cit.>, orbital degrees of freedom <cit.>, 
Aharonov-Bohm geometries <cit.>, and spin-orbit or
Dzyaloshinski-Moriya interactions <cit.>. 
In Ref. kashcheyevs_etal_prb07 it was shown via a singular value decomposition of
the total tunneling matrix (i.e. containing all reservoirs) that all these different cases can 
be mapped onto an effective model which is equivalent to the anisotropic spin-1 2 Kondo model which
flows into the isotropic SU(2)-symmetric fixed point at low energies below the Kondo temperature. 
This explains why in all linear response transport calculations of quantum dot models with 
N_dot=1 and N=2, the Kondo effect with universal conductance is observed provided that
local effective magnetic fields are explicitly cancelled by external ones <cit.>. 
However, this result is only valid in the linear response regime and for proportional 
couplings to all the reservoirs where the linear conductance can be related to
the equilibrium spectral density of the dot <cit.>. To calculate the latter all reservoirs 
can be taken together to a single one and only the total tunneling matrix matters. 
However, when all reservoirs are coupled in a generic way to the dot or when
they are characterized by different temperatures or chemical potentials, the analysis of
Ref. kashcheyevs_etal_prb07 is no longer valid. This fact was emphasized in 
Ref. goettel_reininghaus_schoeller_15, where it was shown that in a generic 
nonequilibrium situation, the proper effective model for N_dot=1 and N=2 is
a spin-valve model, where the spin polarizations of all reservoirs point in different directions, such
that at the low-energy fixed point their sum is equal to zero. This has the consequence that the
fixed point model in nonequilibrium is essentially not SU(2)-symmetric and new interesting
nonequilibrium fixed point models emerge with different non-Kondo like properties in the weak 
as well as in the strong coupling regime. Only in the equilibrium situation when all 
reservoirs are characterized by the same temperature and chemical potential, all reservoirs can be taken
together resulting in an unpolarized reservoir with SU(2)-symmetry at the fixed point. The
nonequilibrium properties at and away from the fixed point model have been studied for
large voltages above the Kondo temperature <cit.> and a 
smoking gun was identified in the nontrivial magnetic field dependence of the magnetization and the
transport current characterizing the fixed point model.

The proposals of new nonequilibrium fixed point models are of particular interest for the constant
effort to generalize well-established analytical and numerical methods for the study of equilibrium properties
of quantum impurity models <cit.> to the nonequilibrium case. Recent developments of 
perturbative renormalization group methods <cit.> 
have shown how the voltage dependence and the physics of cutoff scales by decay rates can be implemented 
<cit.> and how the time evolution into the stationary state can be 
calculated <cit.>. Even in the strong coupling regime 
<cit.> results in agreement with experiments 
<cit.> were obtained, although the used methods are essentially perturbative
and not capable of describing the strong coupling regime in general. Therefore, numerically exact methods
are required for the description of quantum dot systems in nonequilibrium, like e.g. 
the time-dependent numerical renormalization group <cit.>, time-dependent density matrix
renormalization group <cit.>, iterative stochastic path integrals <cit.>, and quantum 
Monte Carlo methods <cit.>. Recently, a promosing thermofield approach has been suggested
by a combination of TD-NRG and TD-DMRG <cit.> showing a good agreement with the
strong coupling results for the nonequilibrium Kondo model of 
Rfs. strong_coupling_RTRG,smirnov_grifoni_03,strong_coupling_exp.

The aim of the present paper is to analyse the generic case N_dot=1 and arbitrary N to
see how the results of Ref. goettel_reininghaus_schoeller_15 can be generalized to the
case N>2. In particular we will study the case N=3 and, starting from a generic tunneling matrix,
will show that an effective tunneling model can be derived where also the reservoirs are 
characterized by three flavors which we will conveniently label by the up (u),
down (d), and strange (s) quark flavors. 
The effective model in the cotunneling regime of a singly occupied
quantum dot can be described by flavor fluctuations and we will show by a poor man scaling analysis
that the low-energy fixed point model is indeed the SU(3)-symmetric Kondo model. This result is 
shown to hold also for arbitrary N within the poor man scaling analysis and will be explicitly 
confirmed for N=3 by a numerically exact NRG analysis for the linear response conductance, 
similiar to Rfs. Moca2012,lopez_etal_13. In addition to these references we will 
study the dependence of the SU(3)-Kondo temperature on the tunneling matrix elements and will show how
the SU(3)-symmetric point is obtained by a proper adjustment of the level spacings of the dot. 
Subsequently, we will analyse the nonequilibrium situation and generalize the spin-valve model of
Ref. goettel_reininghaus_schoeller_15 for N=2 to the case of three levels N=3. In 
this case a fixed point model arises where the reservoirs are characterized by eight-dimensional
F-spins corresponding to the eight generators of the SU(3)-group which cancel when all reservoirs
are taken together. For large voltages and two reservoirs we find that the nonequilibrium 
fixed point model has a characteristic dependence on the dot parameters for zero F-spin magnetization
on the dot providing a smoking gun for the detection of the fixed point. Thus we conclude that
the results of Ref. goettel_reininghaus_schoeller_15 can indeed be generalized to the
case of N>2 levels with a great potential for a variety of new interesting nonequilibrium fixed point
models where the low-energy behaviour in the strong coupling regime is still unknown.  

The paper is organized as follows. In Section <ref> we will derive various effective models.
We will set up effective tunneling models in Section <ref> and the effective model 
in the cotunneling regime in Section <ref>. The fixed point model is obtained  
via a poor man scaling analysis in Section <ref> for arbitrary N. In Section <ref> 
we consider the particular case N=3 and will set up the relation to the representation of the SU(3)-group 
and the physical picture in terms of F-spin interactions. In Section <ref> we will use the 
NRG method to confirm the SU(3)-symmetric fixed point model
in the linear response regime. Finally, in Section <ref> we analyse 
the nonequilibrium properties of the fixed point model in the perturbative regime of large voltage via
a golden rule approach. The general formulas are derived in Section <ref> and the
magnetization and the current are calculated as function of characteristic dot parameters for the
case of two reservoirs in Section <ref> where the smoking gun for the detection of
the fixed point model is derived. We close with a summary of our results in Section <ref>.
We use units e=ħ=1 throughout this paper. 







§ DERIVATION OF EFFECTIVE MODELS



In this section we start from a quantum dot with N levels coupled via a generic tunneling matrix to
N_res multi-channel noninteracting reservoirs in grandcanonical
equilibrium. We show in Subsection <ref> that this model is equivalent 
to an effective one where the number of channels in each reservoir is the same as the number N 
of the quantum dot levels. For the special case of N=3 this sets the basis to use a notation in 
terms of three flavor states for the three channels and to characterize the reservoirs by 
rotated F-spins with a certain isospin and hypercharge polarization. In addition we will set up 
various effective tunneling models and characterize the properties of the central fixed point model
derived in Sections <ref>-<ref> for the cotunneling regime, 
where the number N_dot of particles on the dot is fixed to
N_dot=1, such that only flavor fluctuations 
via virtual intermediate states can occur. In this regime we will derive an effective model 
describing flavor fluctuations and propose the fixed point model from a poor man scaling analysis.




 §.§ Effective tunneling models



The starting point is a quantum dot consisting of N levels characterized by some quantum number 
l=1,2,…,N, together with a Coulomb energy E_N_dot depending only on the total 
particle number operator N_dot=∑_l c^†_l c_l of the dot

    H_dot    = ∑_ll' h_ll' c^†_l c_l' +  E_N_dot  ,
    
    E_N_dot    =  E_C(N_dot-n_x)^2  ,

where c^†_l/c_l are the creation/annihilation operators of the single-particle states of the dot.
The charging energy E_c is assumed to be the 
largest energy scale in the problem such that, for small h_l, the parameter n_x determines the 
occupation of the dot. If n_x=n is integer, the ground state will be dominated 
by N_dot=n, whereas for half-integer n_x=n+1 2, states with
N_dot=n,n+1 are degenerate w.r.t. the Coulomb interaction. For convenience we define the gate voltage by

    V_g = E_c (2n_x - N)   ,

such that V_g=0 (or n_x=N/2) defines the particle-hole symmetric point for h_ll'=0. 
With this definition we can also write the dot Hamiltonian in second quantized form as

    H_dot = ∑_ll'h̃_ll' c^†_l c_l' + U 2∑_ll' c^†_l c^†_l' c_l' c_l   ,

with U=2E_c and h̃_ll' = h_ll' -(V_g + (U/2)(N-1))δ_ll'. 

The quantum dot is coupled via a generic tunneling matrix to several infinitely large reservoirs 
α=1,2,…,N_res kept at grandcanonical equilibrium with temperature T and 
chemical potential μ_α, such that the total Hamiltonian reads 

    H_tot = H_dot + H_res + H_T  ,

with the reservoir Hamiltonian

    H_res = ∑_αν_α k ϵ_αν_α k a^†_αν_α k a_αν_α k  ,

and the tunneling Hamiltonian

    H_T = 1√(ρ^(0)) ∑_αν_α l k {t^α_ν_α l a^†_αν_α k c_l  +  
    (t^α_ν_α l)^* c^†_l a_αν_α k}  .

Here, ν_α=1,2,…,N_α is the channel index for reservoir α (with N_α channels 
in total), ϵ_αν_α k is the band dispersion of reservoir α for 
channel ν_α relative to the chemical potential μ_α and labelled by k 
(which becomes continuous in the thermodynamic limit), and 
t^α_ν_α l is the tunneling matrix between the dot and reservoir α. 
ρ^(0) is some average density of states (d.o.s.) in the reservoirs,
which is set to ρ^(0)=1 in the following
defining the energy units. In vector-matrix notation, the tunneling Hamiltonian can be written in a more
compact form as

    H_T = ∑_α k{a^†_α k t_α c + c^† t^†_α a_α k}  ,
 
where c^†=(c^†_1,…,c^†_N), 
a^†_α k=(a^†_α 1 k,…,a^†_α N_α k), and 
t_α is a N_α× N-matrix with matrix elements
t^α_ν_α l. For convenience we have taken here a tunneling matrix independent of k which is
usually a very good approximation for rather flat reservoir bands on the scale of the low energy scales of
interest. 

Using Keldysh formalism it is straightforward <cit.> to relate
the stationary current I_αν_α in reservoir α and channel ν_α to the 
stationary nonequilibrium greater/lesser Green's functions G^≷_ll'(ω) of the dot via

    I_αν_α    = e h∫ dω Tr Γ_αν_α{(1-f_α(ω))   i G^<(ω) +
       
    + f_α(ω)  i G^>(ω)}  ,

where Tr denotes the trace over the single-particle states of the dot, 
f_α(ω)=(e^β(ω-μ_α)+1)^-1 is the Fermi function of reservoir α, and
the N× N-hybridization matrix Γ_αν_α is defined by

    (Γ_αν_α)_ll' = 
    2πρ_αν_α (t_ν_α l^α)^* t_ν_α l'^α  .

Here, ρ_αν_α=∑_k δ(ω-ϵ_αν_α k) denotes the d.o.s. in 
reservoir α for channel ν_α, which is assumed to be rather flat so that the energy 
dependence can be neglected. The influence of the reservoirs and the tunneling on the Green's functions
is determined by the reservoir part of the lesser/greater self-energy given by

    Σ^<_res(ω)    = i ∑_α f_α(ω)Γ_α  ,
    Σ^>_res(ω)    = -i ∑_α (1-f_α(ω))Γ_α  ,

where

    Γ_α = ∑_ν_αΓ_αν_α = 2π t_α^† ρ_α t_α

is the hybridization matrix for reservoir α including all channels and 
(ρ_α)_ν_αν^'_α=δ_ν_αν^'_αρ_αν_α is the
diagonal matrix for the d.o.s. of reservoir α. As a consequence we see that 
the Green's functions depend on the reservoirs and the tunneling matrix only via the
hybridization matrices Γ_α of all the reservoirs. Thus, two models with
the same hybridization matrices give exactly the same Green's functions. Once the Green's functions
are known, the channel-resolved currents I_αν_α can be calculated from (<ref>), where
the channel-resolved hybridization matrix Γ_αν_α of the concrete model
under consideration has to be inserted. The stationary expectation values of 
single-particle operators of the dot can be directly calculated from the lesser Green's functions via 
⟨ c_l'^† c_l ⟩ = 1 2π i∫ dω G^<_ll'(ω) and thus are exactly 
the same for two models with the same hybridization matrices
Γ_α. 

We note that for the equilibrium case, where all Fermi functions of the reservoirs are the same, the reservoir 
self-energies involve only the total hybridization matrix

    Γ = ∑_α Γ_α  ,

with the result that the equilibrium Green's functions are the same for two models with the same 
Γ. However, the current in linear response can not be related to the single-particle 
Green's functions in equilibrium via (<ref>) since also the Green's functions have to be 
expanded in the voltages. A special case is the one of proportional couplings where it is assumed that
Γ_α = x_αΓ with ∑_α x_α=1.
Using current conservation ∑_α I_α=0, with I_α=∑_ν_α I_αν_α denoting
the total current in reservoir α, we get in this case from (<ref>) the Landauer-Büttiker
type formula <cit.>

    I_α = e h∑_βα∫ dω T_αβ(ω) (f_α-f_β)(ω)  ,

with the transmission probability

    T_αβ(ω) = 2π x_α x_βTr Γ  ρ(ω)  ,

where ρ(ω)=i 2π(G^R-G^A)(ω) is the
spectral density on the dot. From this formula one can see that in linear response, where 
(f_α-f_β)(ω)≈ -f'(ω)(μ_α-μ_β), one needs only the spectral density 
in equilibrium and, with μ_α=-eV_α, the current can be written as

    I_α = ∑_β G_αβ (V_β-V_α)  ,

with the conductance tensor

    G_αβ = -e^2 h ∫ dω T_αβ(ω)   f'(ω)  .


With the knowledge that the hybridization matrices Γ_α are the only input we need
to characterize the reservoirs and the tunneling matrix, we can now proceed to define effective models with
the same hybridization matrices. Since Γ_α is a positive definite hermitian matrix, 
we can diagonalize it with a unitary matrix U_α 

    Γ_α = U_α Γ_α^d U_α^†

where (Γ_α^d)_ll'=δ_ll'Γ_α l is a diagonal matrix with 
positive eigenvalues Γ_α l=2π t_α l^2 > 0. We exclude here the exotic case that one 
of the eigenvalues Γ_α l is zero since this would mean that one of the reservoir channels 
effectively decouples from the system. Following Ref. goettel_reininghaus_schoeller_15, we can 
write the hybridization matrix in two equivalent forms by shifting the whole information either to an
effective tunneling matrix or to an effective d.o.s. of the reservoirs. In the first case we 
introduce an effective tunneling matrix t_α^eff by

    (t_α^eff)_ll' = t_α l (U^†_α)_ll'  ,

and get

    Γ_α = 
    2π (t_α^eff)^† t_α^eff  .

Since t_α^eff is a N× N-matrix this effective model consists of 
reservoirs which have exactly the same number N of channels as we have levels on the dot, i.e. 
the quantum number on the dot is also the quantum number labelling the channels in the effective
reservoirs but this quantum number is in general not conserved by tunneling. Comparing
(<ref>) to (<ref>), we see that the
effective d.o.s. in the reservoirs is unity, i.e. we consider unpolarized reservoirs. 

In the second case, we define an effective d.o.s. ρ^eff_α 
in reservoir α by

    ρ^eff_α = 
    N U_α (Γ^d_α/Γ_α)
    U_α^†  ,

with Γ_α=∑_l Γ_α l. Defining an average tunneling matrix element t_α>0 by
t^2_α=1 N∑_l t_α l^2, we can then write the hybridization matrix as

    Γ_α = 2π t_α^2   ρ^eff_α  .

In this case the effective tunneling matrix is proportional to unity, the tunneling 
conserves the flavor and is flavor-independent. In contrast, the effective d.o.s. 
contains the whole nontrivial information of the hybridization matrix and describes a 
unitary transformation of the diagonal matrix 
N  Γ^d_α/Γ_α. The latter matrix can be decomposed in a 
basis of all diagonal matrices and the coefficients can be interpreted as physical parameters
characterizing the effective reservoirs. Using 
Tr Γ^d_α = Γ_α we get for N=2 

    2  Γ^d_α/Γ_α = 1_2 + p_ασ_z  ,

where σ_z is the Pauli matrix in z-direction and p_α describes the spin polarization
in reservoir α. Since the matrix has only positive diagonal elements we get the 
condition -1< p_α <1. If one orders the eigenvalues according to 
Γ_α 1≥Γ_α 2 one gets 0<p_α<1. 

For N=3 we obtain

    3  Γ^d_α/Γ_α = 1_3 + p_αλ_3
    + q_α√(3)λ_8   ,

where

    λ_3    = ([  1  0  0;  0 -1  0;  0  0  0 ])  ,
    λ_8    = 1√(3)([  1  0  0;  0  1  0;  0  0 -2 ])  ,

are the two diagonal generators of the SU(3) group, 
describing the isospin in z-direction of the up/down quark and the hypercharge operator 
Y=1√(3)λ_8, respectively. Therefore we
interpret p_α as the isospin polarization and q_α as the hypercharge polarization characterizing the
reservoirs in the 3-channel case. The fact that all matrix elements of (<ref>) are
positive leads to the two conditions

    |p_α| <  1 + q_α 3  ,  
    0 < 1+q_α 3 < 3 2  .

If one orders the eigenvalues according to 
Γ_α 1≥Γ_α 2≥Γ_α 3 one gets 0<p_α<q_α<3/2.
 
The unitary transformation U_α describes a
rotation of the direction of the spin-1 2 in the N=2 case, and a rotation of the 
eight-dimensional F-spin with F_i=1 2λ_i 
for N=3, see Fig. <ref> for an illustration. 
Thus, the form (<ref>) allows for a nice physical interpretation in
terms of physical parameters characterizing the reservoirs. For
N=3, we can label the three flavors of the reservoirs and the dot by l=u,d,s
for the up, down and strange quark and describe with the form (<ref>) a system
where the flavor is conserved in tunneling with equal tunneling amplitudes for all flavors. 
However, the polarization p_α of the isospin described by the up and down quark and the
hypercharge polarization q_α can be different for each reservoir, and the F-spins in the reservoirs can all 
be rotated relative to the F-spin of the dot. This naturally generalizes the effective spin-valve model
set up in Ref. goettel_reininghaus_schoeller_15 for N=2 to the N=3 case, which is the
main subject of this paper. 

The form (<ref>) in terms of an effective tunneling matrix allows for
another representation of the hybridization matrix which will turn out to be crucial to interpret
the fixed point model derived in Section <ref> for the cotunneling regime. 
Taking all effective tunneling matrices together in a N· N_res× N-matrix

    t^eff = ([     t_1^eff;           ·;           ·;           ·; t_N_res^eff ])  ,

we can write this matrix via a singular value decomposition as

    t^eff = V ([ γ; 0 ])
    W^†  ,

where V is a unitary N · N_res× N· N_res-matrix, 
γ is a N× N-diagonal matrix containing the positive singular values 
γ_1 ≥γ_2 ≥…≥γ_N > 0, and W is a 
unitary N× N-matrix. We assume here that N singular values exist, excluding exotic 
cases where some channels decouple effectively from the system. As a consequence, we can express
all effective tunneling matrices in terms of the singular value matrix γ 
as follows 

    t_α^eff = V_α γ  W^†  ,

where V_α are the N× N-matrices occuring in the first N columns of 
V, which are in general not unitary. However,
since V is unitary, we note the important property

    ∑_α V_α^† V_α = 1  .

The unitary matrix W can be eliminated by transforming the basis of the
single-partice states of the dot using new field operators 
c'=W^†c, such that the dot Hamiltonian 
(<ref>) and the tunneling Hamiltonian (<ref>) obtain the form

    H_dot    = (c^')^†h^'c^' +  E_N_dot  ,
    
    H_T    = ∑_α k{a^†_α k (t^eff_α)^' c'  + 
    (c^')^† ((t^eff_α)^')^† a_α k}  ,

with h'=W^† h W and

    (t_α^eff)^' = V_α γ  .

For simplicity we will drop the prime in the following and replace h'→h
and (t_α^eff)^'→t_α^eff keeping in 
mind that these matrices result from the matrices of the original model by transforming the dot channels
with the unitary matrix W.

In terms of the effective tunneling matrices (<ref>) the
hybridization matrices (<ref>) obtain the form

    Γ_α =  2π γ  V_α^†  V_α  γ  .

This form is of particular interest since it separates the hybridization matrix in a part
γ which is independent of the reservoirs and a reservoir-dependent part 
V_α^† V_α. Comparing (<ref>) with 
(<ref>) we can interpret γ as an effective 
tunneling matrix  which conserves the flavor index and is the same for all reservoirs. This 
effective tunneling matrix contains the information of the eigenvalues Γ_l=2πγ_l^2 
of the total hybridization matrix since we get from (<ref>) 

    Γ = ∑_α Γ_α = 
    2π  γ^2  .

The reservoir-dependent part V_α^† V_α can be interpreted 
as an effective d.o.s. of the reservoirs. Taking N=3 and decomposing this hermitian matrix in the 
basis of the F-spin generators F_i=1 2λ_i of SU(3) we get

    V^†_α V_α = 
    x_α (1 + ∑_i=1^8 d_α^i F_i)   ,

with real coefficients x_α and d_α^i which, due to (<ref>), fulfil the
property

    ∑_α x_α = 1   ,  ∑_α x_α d_α^i = 0   .

This means that the sum of the F-spins of all reservoirs is zero. A similiar property holds for
arbitrary N. In equilibrium, where all chemical potentials μ_α=μ are the same and all 
reservoirs can be taken together to one big reservoir, this means that an unpolarized reservoir with
SU(3)-symmetry couples to the dot. However, since the effective tunneling matrix elements γ_l
still depend on the flavor index, SU(3)-symmetry does not hold for the total system even in 
equilibrium. 

Most importantly, we will see in Section <ref> by a poor man scaling analysis 
in the cotunneling regime of a singly occupied dot N_dot=1 that a generic fixed point model 
with an isotropic matrix γ=γ1 emerges,
such that the effective tunneling matrix (<ref>) reads

    t_α^eff = γV_α  .

γ>0 can be related to an isotropic Kondo coupling J via 

    γ^2 = 1  4 J D   ,  1 D = 1 2(1 E_0+1 E_2)
      ,

where E_N_dot is given by (<ref>) and J fulfils the poor man scaling equation

    d J  dΛ = -N 2J^2Λ  ,

with Λ denoting the effective band width. In addition, a special potential scattering term
emerges in the original tunneling model at the fixed point which is given by

    V_sc = v_sc∑_kk'∑_αα'
    :a^†_α k V_α  V^†_α' a_α' k':   ,

where

    v_sc = γ^2 D(N-2 N + δ)  ,  δ = E_0-E_2  E_2 + E_0  ,

with E_0 and E_2 from (<ref>), 
and :…: denotes normal-ordering. This potential scattering term vanishes for N=2 
and δ=0 (i.e. n_x=1 where E_0=E_2) and is
such that it cancels the potential scattering term emerging in an effective model for  
the cotunneling regime, see Section <ref>. Due to V_sc
the reservoir part of the self-energy of the dot is more 
complicated than (<ref>-<ref>) and does not only
depend on the hybridization matrix. However, as is shown in Appendix <ref>, the  
effect of V_sc is just that γ is changed to an effective 
γ̃ given by

    γ̃ = γ√(1+π^2 v_sc^2)  ,

such that the self-energies (<ref>-<ref>) from the reservoirs
can be written at the fixed point with effective hybridization matrices which can be either expressed via an
effective tunneling matrix analog to (<ref>)

    Γ_α = 
    2π (t_α^eff)^† t_α^eff  ,  t_α^eff = γ̃V_α  ,

such that the d.o.s. of the reservoirs is unity, or via an effective d.o.s. 
analog to (<ref>)

    Γ_α    = 2πγ̃^2 ρ_α^eff  ,  ρ_α^eff = V_α^†  V_α  ,

with a trivial tunneling matrix given by γ̃1 which is the same for all 
reservoirs and proportional to unity w.r.t the flavor indices. The particular property of the
effective d.o.s. at the fixed point is the condition

    ∑_α ρ_α^eff = 1  ,

following from (<ref>). This means that in contrast to the general case depicted in 
Fig. <ref> for N=3, the particular property of the fixed point model is that the sum 
over all reservoir F-spins is equal to zero and the tunneling matrix 
γ=γ1 is isotropic. 
As a consequence we get overall SU(3)-symmetry in equilibrium,
whereas in nonequilibrium the fixed point model is essentially not SU(3)-symmetric since the 
F-spins of the reservoirs are nonzero.
A similiar statement holds for any number N of dot levels, generalizing the picture found in
Ref. goettel_reininghaus_schoeller_15 for N=2 to a generic multi-level quantum dot. 

We note that for the particular case of two reservoirs N_res=2 with α=L,R, 
we get from (<ref>) that 
V_L^† V_L = 1 - V_R^† V_R,
such that we can find a common unitary matrix U_V which diagonalizes both
V_α^† V_α for α=L,R

    V_α^† V_α = U_V  A_α^d  U^†_V  ,

where A_α^d are diagonal matrices with the property

    ∑_α=L,R A_α^d = 1  .

For N=3, the matrix A_α^d can be decomposed analog to (<ref>) as

    A^d_α =  x_α(1_3 + p_αλ_3
    + q_α√(3)λ_8 )  ,

where, due to the property (<ref>), we get

    1    = x_L + x_R   ,
    
    0    = x_L p_L + x_R p_R   ,
    
    0    = x_L q_L + x_R q_R   ,

together with 0<x_α<1 and (<ref>).
Thus, the hybridization matrices at the fixed point obtain the following form for two reservoirs

    Γ_α =  2πγ̃^2  U_V   A_α^d  U^†_V   .

Omitting the unitary matrix U_V by choosing a different single-particle basis for the 
dot states and redefining the parameters h_ll' (analog to the transformation by the unitary matrix
W, see (<ref>-<ref>)), we get finally the diagonal form

    Γ_α =  2πγ̃^2  A_α^d  ,

which, for N=3, by inserting the decomposition (<ref>), can be written as

    Γ_α = 1 3Γ_α  (1_3 + p_αλ_3
    + q_α√(3)λ_8)   ,

with Γ_α=2πγ̃^2 x_α. This form 
for the hybrization matrices constitutes the central generic fixed point model for N=3 and two 
reservoirs in the cotunneling regime of a singly occupied dot. This will be confirmed
in Section <ref> by NRG in equilibrium and analysed in Section <ref> by
a golden rule approach in nonequilibrium. It generalizes the spin-valve model for 
a 2-level quantum dot with opposite spin polarizations in the two reservoirs analysed in 
Ref. goettel_reininghaus_schoeller_15 to the case of
a 3-level quantum dot, where the isospin and hypercharge polarizations have to be opposite in
the two reservoirs. An analog fixed point model arises for an arbitrary number of dot levels, in this case
one obtains in the two reservoirs opposite parameters corresponding to the N-1 diagonal generators of SU(N). 
Whereas in equilibrium the fixed point model is SU(N)-symmetric (at least if the dot 
parmeters h_ll' are adjusted properly, see Section <ref>) and leads generically to the
SU(N)-Kondo effect, the nonequilibrium fixed point model is not SU(N)-symmetric and generically
non-Kondo physics has to be expected. This will be analysed in Section <ref> in the
perturbative golden rule regime of large voltage, where we will see that zero F-spin 
magnetization on the dot occurs only for particular values of the dot parameters h_ll' providing a
smoking gun for the detection of the fixed point model.  
  



 §.§ Effective model in the cotunneling regime



The effective model in the cotunneling regime where the particle number on the dot is fixed to 
N_dot=1 can easily be obtained by projecting the Hamiltonian matrix on this subspace analog 
to Brillouin-Wigner perturbation theory. Taking only one virtual process into the particle number 
sectors N_dot=0,2 into account we get 

    H_tot^eff = H_res + P_1 H_dot P_1 
     -  :P_1 H_T Q_1 1 H_dot Q_1 H_T P_1:    ,

where P_1 projects onto the 1-particle subspace of the dot and Q_1=1-P_1. We have introduced the
normal-ordering :… : w.r.t. the reservoir field operators since we are not interested in terms
renormalizing the dot Hamiltonian leading to effective parameters h_ll'. For H_res
and H_T we take a model with the effective tunneling matrix (<ref>) and the unity matrix
for the effective d.o.s. in the reservoirs, as has been discussed in Section <ref>. 
Inserting H_dot and H_T from (<ref>) and
(<ref>) and using E_N_dot from (<ref>) we get with 
P_1 c_l^† c_l'P_1 = |l⟩⟨ l'|

    H_tot^eff    = H_res + ∑_ll' h_ll'|l⟩⟨ l'| + E_1
       
    - 1 E_2∑_αα'∑_kk' 
    :P_1 a^†_α k t^eff_α c c^† (t^eff_α')^† a_α' k' P_1:
       
    - 1 E_0∑_αα'∑_kk' 
    :P_1 c^† (t^eff_α')^† a_α' k' a^†_α k t^eff_α cP_1:  .

Using 

    P_1(c c^†)_ll'P_1     =  -|l'⟩⟨ l| + δ_ll'P_1   ,
    
    :(a_α' k' a^†_α k)_l'l:    = -:a^†_α l k  a_α'l'k':  ,

we get after inserting (<ref>) for the tunneling matrix and leaving out the unimportant 
constant E_1

    H_tot^eff = H_res + ∑_ll' h_ll'|l⟩⟨ l'|
     + V_eff  ,

with the effective interaction

    V_eff = ∑_αα'∑_kk':a^†_α k V_α Ĵ V_α'^† a_α' k':  ,

and

    Ĵ_ll' = γ_lγ_l'(2 D|l'⟩⟨ l| 
     - 1 E_2δ_ll'1̂)
      ,

with 2/D=1/E_0+1/E_2, see (<ref>). We note that the hat on Ĵ_ll'
indicates that this object is a dot operator in the 1-particle subspace for each fixed value 
of l and l', i.e. Ĵ represents a N× N-matrix with dot operators in 
each matrix element. By using 1̂=∑_l |l⟩⟨ l|, a straighforward calculation
leads to the decomposition

    Ĵ_ll'    = ξ_ll' |l'⟩⟨ l| (1-δ_ll') + 
       
    + ∑_l_1 lη_ll_1(1 N1̂-|l_1⟩⟨ l_1|)δ_ll' + v_l δ_ll'1̂  ,

with

    ξ_ll'    = 2 Dγ_lγ_l'  ,  η_ll' = 2 Dγ_l^2  ,
    
    v_l    = -1 Dγ_l^2 (N-2 N + δ)   ,

and 2δ = D/E_2-D/E_0, see (<ref>). We note that the bare
parameters η_ll' are independent of l' but obtain a strong dependence on l' under the 
renormalization group flow described below. The decomposition (<ref>) exhibits the
non-diagonal matrix |l'⟩⟨ l| for l l', all traceless diagonal matrices 
1 N1̂-|l_1⟩⟨ l_1| for l_1 l, and the unity matrix 1̂
describing the effective potential scattering. 

We note that the effective interaction (<ref>) can also be written in terms of
reservoir field operators for a single reservoir only

    V_eff = ∑_kk':ã^†_k  Ĵ ã_k':  ,

where 

    ã_k = ∑_α V_α^† a_α k

fulfil commutation relations of field operators for a single effective reservoir with N flavors
due to the property (<ref>). However, this is 
only possible if all the reservoirs can be taken together, i.e. they must have the same temperature
and chemical potential. In nonequilibrium this is not possible. Nevertheless, for the poor man scaling
analysis described in the next section, this form of the Hamiltonian can be applied since the poor man
scaling analysis integrates out only energy scales above the temperatures and chemical potentials of
the reservoirs. 




 §.§ Poor man scaling and fixed point model for N levels



Taking the effective Hamiltonian in the cotunneling regime in the form 
(<ref>) and (<ref>), 
we now proceed to find an effective low-energy theory by
integrating out all energy scales from the high-energy cutoff Λ=D down to some low-energy scale 
Λ_c defined by the largest physical low energy scale in the system set by the parameters h_ll'
of the dot Hamiltonian, the temperature T of the reservoirs, and the chemical potentials μ_α of
the reservoirs 

    Λ_c = max{{|h_ll'|}_ll',T,{μ_α}_α}  .

This can be achieved by a standard poor man scaling analysis leading to the RG equations

    d Ĵ_ll' ds =  -∑_l_1[ Ĵ_ll_1,Ĵ_l_1 l']   ,

where [·,·] denotes the commutator and s=lnDΛ is the flow parameter. 
This RG equation has obviously the two invariants
TrĴ_ll' and ∑_l Ĵ_ll. Defining 

    η_l     = ∑_l' lη_ll'  ,  η = ∑_lη_l   ,   v = ∑_l v_l   ,

we obtain from the decomposition (<ref>) 

    TrĴ_ll'    =  N v_l δ_ll'  ,
    ⟨ l|∑_l'Ĵ_l'l'|l⟩    = 1 Nη - η_l  + v   ,

and get the invariants 

    0    = d dsv_l   ,
    
    0    = d ds(η_l - 1 Nη)   .

The first equation means that there is no renormalization for the potential scattering. The
second equation holds for all l=1,…,N and gives N-1 independent invariants. 

Inserting the decomposition (<ref>) in (<ref>) we find
after some straightforward algebra the RG equations for the parameters ξ_ll' and  
η_ll' characterizing the effective operator-valued matrix Ĵ
at scale Λ in terms of (<ref>) (l l' in all following equations)

    dξ_ll' d s    = 2ξ_ll'η̅_ll' + ∑_l_1 l,l'ξ_ll_1ξ_l_1 l'  ,
    dη_ll' d s    = 2ξ_ll'ξ_l'l + ∑_l_1 l,l'ξ_ll_1ξ_l_1 l  ,

where we defined the symmetric matrix

    η̅_ll' = 1 2(η_ll'+η_l'l)   ,

which fulfils the RG equation

    dη̅_ll' d s    = 2ξ_ll'^2 + 1 2∑_l_1 l,l'(ξ_ll_1^2 + ξ_l_1 l'^2)   ,

since ξ_ll' stays symmetric during the whole RG flow

    ξ_ll' = ξ_l'l  .

These differential equations have to be solved starting from the initial conditions at s=0 
given by (<ref>). 

The RG equation for η_ll' can be solved by the ansatz

    η_ll' = η̅_ll' + r_l - r_l'  ,

where the r_l are determined from the RG equations

    d r_l  ds = 1 2∑_l' lξ_ll'^2   ,

with initial condition r_l=γ_l^2/D. Using the form (<ref>) we can express the 
N-1 independent invariants (<ref>) as

    0 = d ds(r - N r_l  + η̅_l  - 1 Nη̅)  ,

where we have defined in analogy to (<ref>) 

    η̅_l  = ∑_l' lη̅_ll'   ,   η̅ = η = ∑_lη̅_l    ,    r = ∑_l r_l   .

With these invariants all N-1 differences r_l-r_l' can be expressed via the
symmetric matrix η̅_ll' and it is only necessary to consider the RG equations 
(<ref>) and (<ref>) for the symmetric matrices 
ξ_ll' and η̅_ll'. As we will see in Section <ref>, these 
coupling constants can be interpreted as the transverse and longitudinal Kondo couplings
J_⊥ and J_z corresponding to the SU(2)-subgroup formed by the level pair (l,l'). 

As one can see from (<ref>) the parameters η_ll' obtain a significant
dependence on l' not present in the initial condition. Furthermore, all parameters ξ_ll' and
η_ll' stay positive and increase monotonously under the RG flow until they diverge at a 
certain low-energy scale T_K. The fixed point is the one where all parameters are the same and 
proportional to an isotropic Kondo-like coupling J

    ξ_ll' = η_ll' = 1 2 J   ,

where J fulfils the RG equation (<ref>) 

    d J ds = N 2J^2   ⇒   T_K = Λ e^-2 N J = const  .

T_K is the energy scale where all coupling constants diverge and is called the Kondo temperature 
in the following. This scale is exponentially sensitive to the choice of the initial conditions. 
Therefore, one defines a typical initial coupling J_0 via

    4 γ_l^2  D =  y_l J_0  ,  ∑_l y_l = 1   ,

such that y_l∼ O(1) are fixed parameters, and defines formally the scaling limit by 

    J_0 → 0   ,   D → ∞  ,   
    T_K = const  .


Close to the fixed point we can neglect the small
potential scattering term and get from (<ref>) the form

    Ĵ_ll'    = 1 2J |l'⟩⟨ l| (1-δ_ll') + 
       
    + 1 2 J ∑_l_1 l(1 N1̂-|l_1⟩⟨ l_1|)δ_ll'  ,

which can also be written in the more compact form

    Ĵ_ll'    = 1 2J |l'⟩⟨ l|  - 1  2N J 1̂δ_ll'  .

Using this form in the effective interaction (<ref>) we get at the
fixed point in the 1-particle subspace of the dot 

    V_eff    = - 1 2N J ∑_kk' : ã^†_k ã_k': 
        + 1 2J ∑_kk'∑_ll' c^†_l' c_l  :ã^†_lk ã_l'k':   .

At the fixed point the effective interaction is obviously SU(N)-invariant under a common unitary 
transformation of the N flavors of the reservoir and dot field operators. We note that this holds only 
in the case of the single reservoir described by the field operators ã_lk but not for the original
model in nonequilibrium where the reservoirs have different chemical potentials μ_α. In this
case one has to insert (<ref>) in (<ref>) and finds that the effective interaction
is not invariant under a common unitary transformation of all dot field operators c_l and reservoir
field operators a_α l k due to the presence of the matrices V_α.

We finally show that the fixed point Hamiltonian corresponds to a projection of the effective tunneling model 
(<ref>) together with the potential scattering term (<ref>)
on the N=1 subspace of the dot. Comparing (<ref>) with (<ref>) 
we find that we get indeed a unity matrix for γ=γ1 with γ 
given by (<ref>). Furthermore, the potential scattering is absent in the fixed point model 
(<ref>) and, therefore, we have to introduce the potential
scattering term (<ref>) in the effective tunneling model with a coupling constant 
v_sc given by (<ref>) of opposite sign compared to (<ref>) 
(where γ_l is replaced by γ) such that (<ref>) cancels the potential 
scattering generated by projecting the effective tunneling model on the N=1 subspace. 
 



 §.§ Poor man scaling in SU(3)-representation



For the 3-level case N=3, which is the main subject of this paper, it is quite instructive
to write the Hamiltonian and the poor man scaling equations also in the representation of the generators 
of the SU(3)-group. This provides a nice physical picture how the reservoir and dot F-spins are coupled
and how the interaction can be interpreted in terms of the dot and reservoir quark flavors. 

Since each matrix element Ĵ_ll' is an operator in the 3-dimensional dot space we
can decompose it in the F-spin components F̂_i=1 2λ̂_i of the dot as

    Ĵ_ll' = ∑_i=1^8 J^i_ll'F̂_i  + v_lδ_ll'1̂  ,

where the last term contains the potential scattering. Furthermore 
each 3× 3-matrix J^i can again be decomposed in the 
generators λ_j in reservoir space (note that we still consider here only one effective
reservoir due to the form (<ref>) of the effective interaction in the poor man
scaling regime). Comparing (<ref>) with (<ref>) we find after some
straightforward algebra

    J^i    =  J_i λ_i   ,  for  i=1,2,4,5,6,7   ,
    J^3    =  J_3  λ_3  + J_38 λ_8
      + 2 3 c_3 1  ,
    J^8    =  J_8  λ_8  +  J_83 λ_3 
     + 2 3√(3) c_8 1  ,

where the various coupling constants are defined by 

    J_1    = J_2 = ξ_12  ,   K_1 = η̅_12  ,
    
    J_4    = J_5 = ξ_13  ,   K_4 = η̅_13  ,
    
    J_6    = J_7 = ξ_23  ,   K_6 = η̅_23  ,
    
    J_3    = K_1   ,   J_8 = 1 3 (2K_4 + 2K_6 - K_1)  ,
    
    J_38    = J_83 = 1√(3) (K_4-K_6)  ,

together with the two invariants

    c_3 = γ_1^2-γ_2^2 D  ,  
    c_8 = γ_1^2+γ_2^2-2γ_3^2 D  .

c_3 and c_8 must be invariants since

    ∑_l Ĵ_ll = ∑_i=1^8 (TrJ^i) F̂_i  + v1̂

is an invariant such that all coefficients TrJ^i must be invariants for i=1,… 8.
Using (<ref>-<ref>) we see that the trace for 
i=1,2,4,5,6,7 is trivially zero but for i=3,8 we get that TrJ^3=2 c_3 and
TrJ^8=(2/√(3)) c_8 must  be invariants. 

We note that only the 6 coupling constants (J_1,J_4,J_6)=(ξ_12,ξ_13,ξ_23) and
(K_1,K_4,K_6)=(η̅_12,η̅_13,η̅_23) are independent. This is consistent
with our general analysis in Section <ref> where we showed that only the parameters
ξ_ll' and η̅_ll' are needed. 



Since all coupling constants grow under the RG flow and diverge at T_K, the small invariants c_3,
c_8 and v_l can be omitted from the effective interaction V_eff defined
in (<ref>). Inserting 
Ĵ ≈ ∑_i=1^8 J^iF̂_i from (<ref>)
and the decompositions (<ref>-<ref>) we can write V_eff
in the compact form

    V_eff = 2 ∑_i=1^8  J_i f̂_i F̂_i 
     + 2J_38 (f̂_8F̂_3 + f̂_3F̂_8)   ,

where we defined the reservoir f-spin operator as 

    f̂_i = 1 2∑_kk'ã_k^† λ_i ã_k'  .

The form (<ref>) exhibits very clearly how the reservoir f-spin couples
to the dot F-spin. There are three possible isospin pairs
formed by the up/down quark (i=1,2), the up/strange quark (i=4,5), 
or the down/strange quark (i=6,7), corresponding to the flavor pairs l=1,2, l=1,3 and
l=2,3, respectively. For each isospin pair we can define a transverse and
longitudinal coupling, denoted by (J_1,K_1), (J_4,K_4), and (J_6,K_6), respectively, 
analog to the transverse and longitudinal Kondo couplings (J_⊥,J_z) for a single spin 1/2.
The three transverse couplings belong to the 6 independent generators λ_i for i=1,2,4,5,6,7.
Therefore, the effective interaction does not contain any transverse couplings between different
isospins of the reservoir and the dot but only the product f̂_i F̂_i for i=1,2,4,5,6,7.
In contrast, the three longitudinal parts of the isospins are not independent. By convention
one chooses the longitudinal part of the up/down isospin (represented by λ_3) and
the sum over the longitudinal parts of the up/strange and down/strange isospins (represented by
the hypercharge generator √(3)λ_8) as basis for the two independent traceless matrices. 
Therefore, there is not only a longitudinal isospin coupling J_3 and a
hypercharge coupling J_8 but also a mixed coupling J_38 describing an
interaction of the longitudinal reservoir isospin with the hypercharge polarization of the dot and
vice versa. This picture naturally generalizes to arbitrary N providing a physical interpretation
of the coupling constants ξ_ij and η̅_ij in terms of the transverse and 
longitudinal couplings for the isospin formed by the two flavors l=i,j. 

Using (<ref>) and (<ref>) for N=3, we obtain the RG equations

    dJ_1 ds    = 2 J_1 K_1 + J_4 J_6   ,
    dJ_4 ds    = 2 J_4 K_4 + J_1 J_6   ,
    dJ_6 ds    = 2 J_6 K_6 + J_1 J_4   ,
    dK_1 ds    = 2 J_1^2 + 1 2(J_4^2 + J_6^2)   ,
    dK_4 ds    = 2 J_4^2 + 1 2(J_1^2 + J_6^2)   ,
    dK_6 ds    = 2 J_6^2 + 1 2(J_1^2 + J_4^2)   ,
 
with the initial conditions at s=0 given by (<ref>)

    J_1(0)   =2γ_1γ_2 D  ,  
    J_4(0)=2γ_1γ_3 D  ,
    
    J_6(0)   =2γ_2γ_3 D  ,  
    K_1(0)=γ^2_1+γ^2_2 D  ,
    
    K_4(0)   =γ_1^2+γ_3^2 D  ,  
    K_6(0)=γ_2^2+γ^2_3 D  .

A numerical study of these RG equations shows that independent of the initial conditions all couplings become equal
during the RG flow and diverge at some low-energy scale T_K, in agreement with (<ref>). 
Using (<ref>-<ref>) this means that all J_i=J/2 become the same 
for i=1,…,8 and the mixed coupling J_38 scales to zero. Thus, at the fixed point the 
effective interaction can be written in the isotropic and SU(3)-invariant form

    V_eff = J ∑_i=1^8  f̂_i F̂_i   ,

which is identical with (<ref>). Applying the analog scheme to an arbitrary number N 
of dot levels we obtain at the fixed point the same result, one just has to sum in (<ref>) 
over all generators of SU(N). Fig. <ref> shows an example for the RG flow where the
longitudinal and tranverse couplings K_i≈ J_i are initially nearly the same but different for each 
i=1,4,6.




To obtain a feeling for the nature of the strong-coupling ground state, we assume a two-site model 
with Hamiltonian (<ref>). In particular, we consider a tight-binding model for the reservoir 
and the two sites are the dot and the first site of the reservoir (i.e. the one that couples to the dot), 
respectively, while the other reservoir sites are not taken into account. The crucial point about 
determining the ground state lies in choosing the approrpriate representation for the eigenstates of 
the SU(3)-symmetric interaction in (<ref>). The SU(3) group has two fundamental 
representations <cit.>, which we denote by the multiplett notation [3] and [3]. 
We represent the eigenstates of the dot in the representation [3] where the F-spin components are 
F̂_i=1  2λ̂_i. Denoting the states by the quark flavors l=1,2,3=u,d,s, we have

    |u⟩    = |1 2, 1  3⟩  ,
    |d⟩    = |-1 2, 1  3⟩  ,
    |s⟩    = |0, -2  3⟩  ,

where the states on the r.h.s. are the eigenstates of F̂_3 and F̂_8 and the 
first (second) quantum number in the label is the corresponding eigenvalue of 
F̂_3 (2 √(3)F̂_8). Therefore, we refer to these eigenvalues as 
isospin (hypercharge) quantum numbers. Choosing the same representation for the first site in the 
reservoir is not useful since the states of the composite system are part of either the sextet [6] 
or the triplet [3] due to [3] ⊗ [3]= [6] ⊕ [3] <cit.>. 
Such a representation is not suitable since the system has a distinct non-degenerate ground state. 
Instead, we represent the first site of the reservoir with [3] and obtain  
[3] ⊗ [3]= [8] ⊕ [1] where all but one state of the two-site system 
form an octet together with the remaining state being a unique singlet state. [3] is the 
complex conjugate representation of [3] and has  therefore the generators 
f̂_i= - 1  2λ̂^*_i. Consequently, we label the states of the second site 
with the anti-quark flavor l=1,2,3=u,d,s and get

    |u⟩    = |-1 2,-1  3⟩  ,
    |d⟩    = |1 2, -1  3⟩  ,
    |s⟩    = |0, 2  3⟩  .

In this basis, the operators λ̂^*_i have the same matrix representation as the 
Gell-Mann matrices λ̂_i.

Indeed, we will show in appendix <ref> that the singlet state

    |gs⟩ = 1 √(3)( |u u⟩ 
    + |d d⟩ + |s s⟩)

is the ground state with energy E_gs=-4  3 J while the octet states are degenerate
with energy E_8=1  6J. Since |l l⟩=|l⟩⊗|l⟩ 
it is straightforward to define the reduced dot density matrix

    ρ̂ = ∑_l=u,d,s⟨l|( |gs⟩⟨gs|) |l⟩ = 1  31̂  ,

which yields n_l=1  3 in perfect agreement with the NRG analysis in section <ref>.

Together with the SU(3)-symmetric interaction term, the outcome (<ref>) 
motivates the term "quantum fluctuations" for the significant physical processes in the fixed point model. 
The ground state is a symmetric linear combination of bound states with quark-antiquark-flavor. This is 
in accordance with the observation that no free quarks exist, i.e. they always gather to form a particle 
with integer electric charge. The interaction term (<ref>) preserves this since the 
fluctuation terms (i=1,2,4,5,6,7) always annihilate a quark-antiquark-pair while creating a 
different quark-antiquark-bound state simultaneously. Furthermore, we will discuss in 
Appendix <ref> that the eigenstates of (<ref>) 
are identical to those of the quark model for light pseudoscalar mesons<cit.>.

In this context, choosing J_1 ≈ K_1 ≫ J_4 ≈ J_6 and K_4 ≈ K_6
for the initial values reveals a nice physical picture in terms of the isospin of the up and down
quark. Fig. <ref> shows that in the whole regime from weak to intermediate coupling 
the couplings stay approximately degenerate with J_1 ≈ K_1, J_4 ≈ J_6 and K_4 ≈ K_6. 
Here, the model exhibits an approximated SU(2)-symmetry for the isospin with an isotropic isospin 
coupling J_I = 1  2 (J_1 + K_1) ≫| J_1 - K_1 |. Furthermore, the interaction of 
isospin and hypercharge degrees of freedom disentangle in leading order since J_38≪ J_3, J_8. 
In the same way, J_4 ≈ J_6 characterizes transitions between states differing in the hypercharge 
quantum number, compare with (<ref>-<ref>). In total, we find an 
isotropic isospin model where the presence of the third level (strange quark) mainly results in a 
potential scattering (J_8 ∼ J_I) for the isospin with suppressed transitions to states with different 
hypercharge (J_4, J_6 ≪ J_I). However, finally the RG flow approaches the generic SU(3)-symmetric fixed point 
on the Kondo scale T_K also in this case. 




§ NRG ANALYSIS IN EQUILIBRIUM





In Section <ref> we have shown for a 3-level quantum dot in the cotunneling regime 
that the generic fixed point model is an SU(3)-invariant isotropic effective interaction 
(<ref>) between the F-spins of the reservoir and the dot. This holds  
for the equilibrium case where all reservoirs can be taken together to a single reservoir and it 
requires also SU(3)-symmetry of the dot. This means that the dot parameters h_ll' have to be 
adjusted appropriately (including renormalizations arising from the coupling to the reservoir) 
such that the populations of all dot states are the same n_l=⟨ c^†_l c_l⟩=1/3. 
The aim of this section is to confirm that in equilibrium the SU(3)-symmetric fixed point can be
established independent of the tunneling matrix by an adjustment of the dot parameters. To this end we
use the numerically exact NRG method <cit.> and analyse the linear conductance G for N=3
and two reservoirs (α=L,R) for the case of proportional couplings 
Γ_α=x_αΓ where G can be calculated from
(<ref>) and (<ref>)

    g = G/G_0 = -π 2∫ dω Tr Γ ρ(ω) f'(ω)   ,

with the dimensionless conductance g in units of G_0=(e^2/h)/(4x_L x_R).
As explained in Section <ref> the equilibrium spectral density 
ρ(ω) depends only on the total hybridization matrix Γ,
i.e. we can use a unitary transformation of the dot states such that this matrix is diagonal (see
(<ref>)) and the spectral density in this basis depends only on 
the eigenvalues Γ_l=2πγ_l^2. In this case the linear conductance (<ref>)
can be written as

    g = -π 2∫ dω ∑_l  Γ_l  ρ_ll(ω)  f'(ω)   .

In the new dot basis we assume for simplicity that the dot Hamiltonian contains only diagonal elements

    H = ∑_l h_l c^†_l c_l   .

Other cases with nondiagonal elements h_ll' can also be studied but are of no interest
because they just destroy SU(3)-symmetry of the dot and drive the system away from the fixed point model.
Here, we are interested in a systematic study how, for arbitrary tunneling parameters Γ_l, 
SU(3)-symmetry can be restored by tuning the level positions h_l appropriately. In addition we will
also study the dependence of the SU(3)-Kondo temperature T_K^(3) as function of Γ_l and compare
it to the corresponding SU(2)-Kondo temperature T_K^(2), where only two levels contribute to transport.  
This analysis goes beyond the one of Ref. lopez_etal_13 which has concentrated on the 
linear conductance for the SU(3)-symmetric case (i.e. all Γ_l are the same and h_l=0) and
the destruction of SU(3)-symmetry by different Γ_l or finite values for h_ll'. As a 
signature of SU(3)-symmetry we take the Friedel sum rule (used also in Ref. Moca2012,lopez_etal_13)

    g = ∑_l sin^2(π n_l)   ,

which holds exactly at zero temperature and gives the value g=2.25 for equal populations
n_l=1/3 corresponding to the SU(3)-symmetric fixed point. The occupations n_l can be calculated from
the spectral density via n_l=∫ dωρ_ll(ω)f(ω). For the parameters in all figures we use 

    1 2∑_lΓ_l = 1   ,   U = 10   ,   W = 10^4   ,

where 2W denotes the width of a flat d.o.s. of the reservoirs (i.e. |ω|<W for the integral in 
(<ref>)). 



The calculations are performed using the full-density-matrix NRG <cit.>, 
where we exploit either the individual charge conservation or the full SU(N) symmetry by 
means of the QSpace tensor library developed by A. Weichselbaum <cit.>. 
For the final results we employ a discretization parameter of Λ=3, and we keep states up to a 
rescaled energy of E_trunc=9 and maximal number N_keep during the NRG iteration. 
In the calculations without SU(N) symmetry we set N_keep=8000. In the SU(N)-symmetric cases 
we can further increase the precision to very high level and explicitly confirm that results for g are 
converged up to 1% and results for n_l are converged up to 10^-6 with respect to the numerical 
parameters. Note that in many calculations we optimize the level positions to achieve equal occupation of 
certain levels. Since the values of such optimized level positions h_l^* depend on the discretization of 
the bath, we refrain from using z-averaging <cit.>. Finally, we need not broaden the NRG data as 
the computation of both g and n_l requires only discrete spectral weights.

To set the scene we show in Fig. <ref> known curves for the conductance depending on gate 
voltage and temperature 
in the SU(N)-symmetric cases for N=2,3, where all Γ_l are the same and all h_l=0. We find
converged, plateau-like features when decreasing T below the Kondo temperature T_K in the cotunneling 
regime of a singly occupied dot. Note that n_l shows a very weak dependence on temperature in this regime and, 
at T<T_K, the Friedel sum rule (<ref>) is fulfilled. Furthermore, we find
that the Kondo temperatures T_K^(N) are similiar for N=2 and N=3
(recall that ∑_l Γ_l is fixed). In contrast, the p-h-symmetric point 
V_g=0 corresponds to very different physics for the two cases, since for N=3 there are strong 
charge fluctuations due to E_1=E_2, whereas for N=2 spin fluctuations
dominate. Therefore, at V_g=0, the relevant low-energy scale is the hybridization Γ_l for N=3
<cit.> and the Kondo temperature for N=2. 



Next we study the case Γ_1=Γ_2Γ_3 and h_1=h_2=0. 
In this case the different tunneling couplings lead to a different renormalization of h_3 of 
O(Γ_1Γ_3/U) relativ to h_1/2. Therefore, h_1=h_2=h_3=0 is not 
the SU(3)-symmetric point and the level position h_3 has to be adjusted appropriately to recover 
equal populations of the states and conductance g=2.25 at zero temperature. Calling this optimized
value h_3^* we show in Fig. <ref> the conductance as function of |h_3-h_3^*|.
For temperatures T<T_K^(3) we see that the conductance reaches the SU(3)-symmetric value g=2.25 for
|h_3-h_3^*|∼ T_K^(3) as expected. The Kondo temperature T_K^(3) does not depend strongly on the
value of Γ_3 and is nearly the same for Γ_3<Γ_1/2 
(Fig. <ref>(a)) and Γ_3>Γ_1/2 (Fig. <ref>(b)). 
For |h_3-h_3^*|>T_K^(3) and h_3>h_3^* (solid lines in Fig. <ref>) we see 
that the SU(2)-Kondo effect with g=2 appears at low enough 
temperatures T<T_K^(2). Whereas T_K^(2)≈ T_K^(3) for relatively small Γ_3<Γ_1/2, we find 
that T_K^(2)<T_K^(3) for Γ_3>Γ_1/2. The latter can be explained by the fact that 
the two levels l=1,2 form the SU(2)-Kondo effect and therefore T_K^(2) decreases if the 
coupling to these two levels Γ_1, Γ_2 is lowered. In contrast, when all three levels contribute to the
SU(3)-Kondo effect, we have a total coupling of ∑_l Γ_l/2=1 and find that the relative 
distribution of the Γ_l influences T_K^(3) only weakly. Furthermore, in the regime where 
the SU(2)-Kondo effect occurs we see a strong difference when moving over from h_3>h_3^* to h_3<h_3^* 
(dashed lines in Fig. <ref>) since then level 3 forms the ground state and thus the 
Kondo effect is much weaker compared to the case when the two levels l=1,2 are lower in
energy. In the regime of the SU(3)-Kondo effect it is hardly relevant whether level 3 approaches the
other two levels from above or below.  



In Fig. <ref> we show the conductance as function of the gate voltage again for
h_1=h_2=0 and the two cases Γ_1=Γ_2≷Γ_3 as in Fig. <ref> but at
each value of the gate voltage we choose the optimized value h_3=h_3^*(V_g) for which the populations
of the three states are the same at zero temperature. As in Fig. <ref> we confirm 
that T_K^(3) depends only weakly on Γ_3 but the overall tendency is that T_k^(3) decreases when 
increasing |Γ_1/2-Γ_3|. At the p-h symmetric point V_g=0, the situation is completely
different since charge fluctuations dominate for N=3. Therefore, the conductance around V_g=0 
depends strongly on the relative distribution of the Γ_l. In fact, comparing various cases we find 
that the conductance at V_g=0 (where also h_3^*=0) decreases monotonously when increasing the variance 
of the couplings Γ_l. At large variance as in Fig. <ref>(b), g around V_g=0 
is strongly surpressed. In contrast, in the cotunneling regime V_g≈-U/2 the conductance is 
rather insensitive to the distribution of the Γ_l. The combination of these phenomena leads 
to a surprising shape of the curve g(V_g) which exhibits a local minimum at the p-h symmetric point 
for intermediate temperatures.

Finally, we consider in Fig. <ref> three different hybridizations Γ_1<Γ_2<Γ_3 
and tune h_2 and h_3 at fixed h_1=0, V_g=-U/2, and T=10^-10. From the plots of the occupations 
n_l we can easily distinguish three regions where only one level is involved. At the intersections 
of two such regions we observe a two-level Kondo effect with conductance g=2. The widths 
of these intersections in the h_2-h_3 plane define three different Kondo temperatures T_K^(2)
which are ordered according to the size of the corresponding hybridizations 
Γ_1+Γ_2<Γ_1+Γ_3<Γ_2+Γ_3. In the center, where all “one-level sections” 
intersect, we observe a wide region of a three-level Kondo effect with conductance g=2.25.
The corresponding Kondo temperature T_K^(3) is of the same order as the maximum of the three
two-level Kondo temperatures. 

In summary, we find that for any kind of (diagonal) hybridization,
whether with no, two, or three identical elements, we can find carefully optimized
level positions (and low enough temperatures) to observe the behavior known from the 
SU(3)-symmetric quantum dot. For other hybridizations with two identical hybridization elements or, 
again, optimized level positions we can also reproduce the behavior of a two-level 
Kondo effect such that one level is (effectively) excluded. For arbitrary Γ_l and 
h_l (corresponding to most parts of a version of Fig. <ref> zoomed out) 
the typical behavior is that of the single, (effectively) lowest-lying level. 

 





§ NONEQUILIBRIUM FIXED POINT MODEL



The aim of this section is to analyse the nonequilibrium properties of the system for N=3 in the
perturbative regime where the cutoff scale Λ_c defined by (<ref>) is much larger
than the Kondo temperature Λ_c≫ T_K. Most importantly, as already emphasized several times 
in the previous sections, even if the fixed point model (<ref>) is reached at 
scale Λ_c (which will be the case if we take the formal scaling limit defined by 
(<ref>)), it is essentially not SU(3)-invariant if the chemical
potentials of all reservoirs are different. This leads to new interesting nonequilibrium fixed point
models similiar to the ones discussed in Ref. goettel_reininghaus_schoeller_15 for the N=2
case which show a completely different behaviour of physical observables like the magnetization or the current
compared to the SU(N)-symmetric Kondo model. Moreover, in practical situations the initial cutoff
D∼ E_c is fixed leading to deviations from the fixed point model. Therefore, the aim of this section
is to analyse the perturbative effects of the full effective interaction on physical observables and to identify a
smoking gun for the fixed point model together with a parameter measuring the distance from this
fixed point. 



 §.§ Golden rule approach



We start from the effective interaction in the form (<ref>) in terms of the original reservoir 
field operators a_α l k. Inserting (<ref>-<ref>) 
and leaving out all small terms ∼ v_l, c_3, c_8, we obtain

    V_eff = ∑_αα'∑_kk' :a^†_α k  V_α Ĵ V_α'^† a_α'k'  ,

with

    Ĵ    ≈ ∑_i J^i F̂_i 
    J^i    = J_i λ_i
     + J_38(δ_i3 λ_8 + δ_i8 λ_3)  .


The total Hamiltonian is given by H_tot=H_res + H_dot + V_eff, with a unity 
d.o.s. in the reservoirs and the dot Hamiltonian H_dot=∑_ll'h_ll'|l⟩⟨ l'| in 
the 1-particle subspace. To apply golden rule we first diagonalize the dot Hamiltonian by
a unitary transformation Û such that 

    H̃_dot = Û^† H_dotÛ = ∑_l ϵ_l |l⟩⟨ l|   .

The golden rule rate for a transition from l'→ l in the diagonalized basis is then given by 

    Γ_l'→ l    = 
    2π∑_rr'|⟨ l r|Û^† V_effÛ |l'r'⟩|^2 
    ⟨ r'|ρ_res|r'⟩ ·
       · δ(ϵ_l + E_r - ϵ_l' - E_r')   ,

where |r⟩ denote the many-particle states of the reservoirs with energy E_r and
ρ_res=∏_αρ_res^α is the product of the grandcanonical 
distributions of the reservoirs. Inserting the effective interaction (<ref>) we find


    Γ_l'→ l =  2π∑_αα'∫ dω∫ dω' (1-f_α(ω))f_α'(ω')
    δ(ϵ_l-ϵ_l' +ω + μ_α - ω' - μ_α')
    ∑_l_1 l_1^'
    |⟨ l|Û^† (V_α Ĵ V_α'^†)_l_1 l_1^'Û|l'⟩|^2  .


At zero temperature we get

    Γ_l'→ l    =  2π∑_αα' w(ϵ_l-ϵ_l'+μ_α - μ_α') ·
       · ∑_l_1 l_1^'|⟨ l|Û^† (V_α Ĵ V_α'^†)_l_1 l_1^'Û|l'⟩|^2  ,

with w(x)=|x|θ(x). Here, |ϵ_l-ϵ_l'+μ_α - μ_α'|
is just the available energy phase space in the reservoirs for the energy gain 
ϵ_l'-ϵ_l+μ_α' - μ_α > 0. Inserting 
(<ref>) we can write the golden rate in the compact form

    Γ_l'→ l    = 2π∑_αα' w(ϵ_l-ϵ_l'+μ_α - μ_α') ·
       · ∑_ij ⟨ l| Û^†F̂_i Û |l'⟩ ⟨ l'| Û^†F̂_j Û |l⟩ τ_ij^αα'  ,

where 

    τ_ij^αα' = Tr V_α^† V_α J^i
     V_α'^† V_α' J^j   .
 
As expected only the combination V_α^† V_α enters into
this expression which is consistent with our discussion in Section <ref> where it 
was shown that the hybridization matrices Γ_α depend only on this combination, see
(<ref>).  

The stationary probability distribution p_l in the diagonalized basis follows from

    ∑_l' p_l' Γ_l'→ l = 0   ,  ∑_l p_l = 1  .

In an analog way one can calculate the stationary current I_β flowing in reservoir β from
the current rates W^β_ll' in golden rule

    I_β = ∑_ll' p_l' Γ^β_l'→ l  ,

with

    ∑_l Γ^β_l'→ l    = 2π∑_αα' (δ_αβ-δ_α'β)
    w(ϵ_l-ϵ_l'+μ_α - μ_α') ·
       · ∑_ij ⟨ l| Û^†F̂_i Û |l'⟩ ⟨ l'| Û^†F̂_j Û |l⟩ τ_ij^αα'  .


Once the input of the matrices V_α, the coupling constants (J_1,J_4,J_6) and
(K_1,K_4,K_6) (determining the matrices J^i for i=1,…,8), the unitary transformation
Û and the eigenvalues ϵ_l of the dot Hamiltonian are known, the stationary 
probabilities and the current can be calculated in a straightforward way from the above golden
rule expressions. Thereby, we have neglected small renormalizations of the dot parameters
induced by the coupling to the reservoirs which are assumed to be much smaller
than the level spacings in the dot. 





 §.§ F-spin magnetization for two reservoirs



We now calculate the F-spin magnetization of the dot

    m_F = √(∑_i=1^8 (⟨F̂_i ⟩)^2)

for the special case of two reservoirs. We will show that the condition of zero F-spin 
magnetization requires special dot parameters characterizing the deviation from the fixed point model. In the 
basis of the diagonalized dot Hamiltonian the density matrix of the dot is diagonal in golden rule
approximation so that only the two diagonal generators F̂_3 and F̂_8 contribute to m_F

    m_F    = √((⟨F̂_3 ⟩)^2 + ⟨F̂_8 ⟩)^2)
       = 1 2√((p_1-p_2)^2 + 1 3(p_1+p_2-2p_3)^2)  .

Zero F-spin magnetization is then equivalent to an equal population of the three states

    m_F = 0   ⇔   p_1=p_2=p_3   .


As explained in Section <ref> via (<ref>) 
the case of two reservoirs has the advantage
that both matrices V_α^† V_α=
U_V A_α^d U^†_V can be diagonalized by 
a common unitary matrix U_V and the diagonal matrices A_α^d
are parametrized via (<ref>) by the parameters x_α, p_α and q_α,
which fulfil the conditions (<ref>-<ref>) 
and (<ref>). Furthermore it was shown that the special property of the fixed point 
model is that the unitary transformation U_V can be shifted to the dot such that in
the new basis an
effective diagonal tunneling model (<ref>) emerges.
Thus, the particular property of the fixed point model is that the expectation value of the
F-spin magnetization and the current I_α are independent of the unitary matrix 
U_V. In contrast, for the model away from the fixed point this is no longer the case. 

The unitary matrix U_V provides a mean to parametrize the dot Hamiltonian by 
convenient parameters. After transforming the dot Hamiltonian with 
Û_V=∑_ll'(U_V)_ll'|l⟩⟨ l'| we take the form

    Û_V^† H_dotÛ_V  = h_xF̂_1 + h_yF̂_2 + h_zF̂_3 
    + 2√(3)ΔF̂_8   ,

such that h⃗ can be interpreted as an effective magnetic field 
acting on the isospin of the up/down quark, and Δ is the level distance between the strange
quark and the average level position of the up and down quark

    Δ    = 1 2(ϵ_1 + ϵ_2) - ϵ_3   ,

see also Fig. <ref> for an illustration.
The eigenvalues ϵ_l of H_dot and the unitary operator Û can then be expressed 
by the dot parameters h⃗ and Δ by 

    ϵ_1/2    = ±1 2h + 1 3Δ  ,  ϵ_3 = -2 3Δ  ,
    Û    = Û_V Û_h  ,  U_h = ([ x_1  x_2        0;        0        1 ])   ,

where h=√(h_⊥^2 + h_z^2), h_⊥^2=h_x^2+h_y^2, and

    x_1/2 = 1√(2h(h∓ h_z))([ ± (h_x-ih_y);       h∓ h_z ])   .

Inserting Û=Û_VÛ_h and V_α^† V_α=
U_V A_α^d U^†_V in the golden rate
(<ref>) we get

    Γ_l'→ l    = 2π∑_αα' w(ϵ_l-ϵ_l'+μ_α - μ_α') ·
       · ∑_ij ⟨ l| Û^†Û_V^†F̂_i Û_VÛ_h |l'⟩ ⟨ l'| Û^†_hÛ^†_V F̂_j Û_VÛ_h |l⟩ τ_ij^αα'  ,

with 

    τ_ij^αα' = Tr A^d_α(U_V^† J^i U_V)
     A^d_α'(U_V^† J^j U_V)   .
 
For the special case of the fixed point model where J^i=1 2Jλ_i
we can see that the unitary matrix U_V indeed drops out as expected due to the invariant

    ∑_i=1^8 (U_V^† λ_i U_V)
    (Û_V^†F̂_i Û_V)  = ∑_i=1^8  λ_i F̂_i   .

An analog property holds for the current rate (<ref>).

In the following, we consider the strong nonequilibrium regime where
the bias voltage V=μ_L-μ_R>0 is assumed to be larger than all level spacings, i.e.

    V > |h|,|Δ± h/2|   ,

From (<ref>) we see directly that the condition m_F=0 is equivalent to 
⟨F̂_3 ⟩ = ⟨F̂_8 ⟩ =0. Consequently, this are two 
conditions revealing that m_F=m_F(h_z,h_⊥,Δ)=0 generically defines a closed curve 
in (h_z, h_⊥, Δ)-space. Inserting (<ref>) for A_α^d, 
(<ref>) for J^i, (<ref>) for ϵ_l, and (<ref>) 
for Û_h, we evaluate the golden rule rates (<ref>) and (<ref>)
in Appendix <ref> for the special case U_V=1 
from which we can determine the shape of this curve. This gives a generic result for the fixed point model
(where the matrix U_V drops out) whereas for the model away from the fixed point
we consider only the special case of a diagonal tunneling model. 

From the condition m_F(h_z,h_⊥,Δ)=0 or p_1=p_2=p_3=1/3 we obtain in Appendix <ref>
the two equations

    Δ    = x_L q_L V + J_4^2 - J_6^2  J_4^2 + J_6^2 (x_L p_L V -1 2 h_z)   ,
    θ_2^2 x_L^2 p_L^2 V^2    = θ_1^2 h_⊥^2 + θ_2^2 (h_z-x_L p_L V)^2   ,

where 

    θ_1^2    = J_1^2 + J_3^2 + J_38^2 + 1 2(J_4^2 + J_6^2)   ,
    θ_2^2    = 2 J_1^2 + 3 2J_4^2 - 1 2 J_6^2   .

This means that the projection of the curve m_F(h_z,h_⊥,Δ)=0 on the (h_z,h_⊥)-plane is 
an ellipse with the ratio 

    s_1 = θ_1/θ_2

of the two shape parameters. θ_1 is the major-axis (minor-axis) if s_1>1 (s_1<1). We point 
out that this is different to the SU(2)-model (i.e. J_38=J_4=J_6=0) where θ_1 is always the 
major-axis. Furthermore, the derivative of Δ w.r.t. h_z is given by

    s_2 = dΔ d h_z = -1 2J_4^2 - J_6^2  J_4^2 + J_6^2  .

The two parameters s_1/2 provide smoking guns for the detection of the fixed point model since
for J_i = J/2 and J_38=0 we obtain

    s_1 = 1   ,   s_2 = 0   ,

i.e. a circle in the (h_z,h_⊥)-plane as shown in Fig. <ref> and 
no dependence of Δ=q_L V on h_z at the fixed point.


In this sense 1-s_1 and s_2 can both be viewed as parameters measuring the distance from the 
fixed point model. Furthermore we see that the parameters x_L p_L=-x_R p_R and x_L q_L=-x_R q_R 
of the fixed point model can be determined from the two equations

    Δ = x_L q_L V   ,   h_⊥^2 + (h_z - x_L p_L V)^2  = x_L p_L^2 V^2  .

To fix the remaining parameter x_L x_R and the coupling J from a physical quantity 
we have also evaluated the current in Appendix <ref> and obtained at the fixed point and
for m_F=0

    I_L    = -I_R
       = π x_L x_R J^2{-q_L - q_R 6Δ - p_L - p_R  4 h_z  + 
       
    + 1 3(4  - q_L q_R  9 - p_L p_R 3)V }
       = π J^2{1 6x_R q_RΔ + 1 4x_R p_R h_z + 
       1 3(4x_L x_R + 1 9x_R^2 q_R^2 + 1 3 x_R^2 p_L^2) V }  ,

where we used x_L x_R (q_L-q_R)=-x_R q_R and x_L x_R (p_L-p_R)=-x_R p_R in the last equation.
J^2 is just the overall scale of the current and the parameter x_L x_R appears explicitly.
Together with x_L+x_R=1, the two parameters x_L/R can thus be fixed. 

In summary, we have shown in the strong nonequilibrium regime that the condition of vanishing 
F-spin magnetization m_F=0 defines a closed curve in (h_z, h_⊥, Δ)-space that is an 
ellipse in the special case of a diagonal tunneling model. A golden rule calculation has revealed 
that the geometric properties of this ellipse are a measure for the distance to the fixed point 
model where the ellipse turns into a circle being embedded in a plane defined by a constant value 
for Δ. At the fixed point, the parameters of the effective model can experimentally be 
obtained from identifying the position of this circle together with measuring the current at the 
corresponding dot parameters h⃗ and Δ.






§ SUMMARY



The results obtained in this paper show that the area of nonequilibrium low-temperature transport through
generic quantum dot models contains a huge variety of interesting fixed point models not accessible in the
equilibrium case. Previous studies have analysed many generic Kondo scenarios for equilbrium systems and
used the finite voltage V just as a probe for the equilibrium dot spectral density for quantum dots
coupled very asymmetrically to two reservoirs <cit.>. In addition, the finite voltage
together with corresponding decay rates was just expected to act as a cutoff scale for RG flows in the
weak coupling regime <cit.> analog to the temperature, leading
to quantitatively but not qualitatively different physical properties. In contrast, the
analysis performed in this paper shows that, for generic tunneling matrices, the cutoff set 
by the voltage is essentially different from
the temperature since it drives the system towards a fixed point characterized by a 
different symmetry compared to the equilibrium case. Our main result is that if 
an electron on a singly-occupied dot in the cotunneling regime can occupy N levels flavor fluctuations 
lead to a model in the nonequilibrium situation which is essentially not SU(N)-invariant. In
the scaling limit for fixed values of V and T_K, a fixed point model appears at scale V where each 
reservoir is characterized by N effective flavors with N^2-1-dimensional polarizations 
(corresponding to the N^2-1 generators of the
SU(N)-group) pointing in different directions such that the total sum is equal to zero. This leads
to a SU(N)-symmetric equilibrium fixed point where all reservoirs can be taken together,
but to a SU(N)-nonsymmetric nonequilibrium fixed point with qualitatively different 
physical properties. We have
demonstrated this for the special case N=3 and two reservoirs in the weak coupling regime V≫ T_K
and have seen that the condition of equal population of all dot states is realized for special dot 
parameters providing a smoking gun to identify the special symmetry of the nonequilibrium 
fixed point model. 

Strictly speaking the numerical solution of the RG flow shows that even for rather large ratios D/T_K, the
coupling constants become all equal only very close to T_K, where the poor man scaling approach is no
longer valid. This means that the fixed point model can not be reached for voltages V≫ T_K,
except for cases where the initial parameters have already been set close to the fixed point. It is therefore
of high interest for the future to develop numerically exact approaches to describe the strong coupling 
regime in nonequilibrium. In particular for voltages V∼ T_K we expect that the fixed point model has
been reached and the scaling of the conductance and the F-spin magnetization as function of the dot 
parameters will be essentially different from the SU(N)-symmetric case. In agreement with 
Rfs. Moca2012,lopez_etal_13 we have demonstrated in this paper that in equilibrium the 
fixed point model is indeed reached for temperatures below the Kondo temperature T_K providing 
evidence that a similiar result will also hold in the nonequilibrium case when the voltage 
reaches T_K. It will be interesting for the future to test this conjecture and to provide 
signatures of the nonequilibrium fixed point model in the strong coupling regime. 

Finally, it will also be very interesting for the future to study the nonequilibrium fixed points 
in regimes where the particle number of the dot is larger than one N_dot>1. Already
in the equilibrium case it has been demonstrated that not only the Coulomb interaction but also 
other kinds of interactions (e.g. spin-dependent terms) are very important to find the correct ground state,
see e.g. Ref. glazman_pustilnik_05 for a review. Based on this and our results for 
N_dot=1 we expect that even a richer variety of new nonequilibrium fixed point models 
has to be expected for N_dot>1.




§ ACKNOWLEDGMENTS

This work was supported by the Deutsche Forschungsgemeinschaft via RTG 1995 (C.J.L. and H.S.).
We thank A. Weichselbaum and S.-S. B. Lee for 
helpful discussions on the NRG setup. F. B. K. and J. v. D. acknowledge support by the Cluster of 
Excellence Nanosystems Initiative Munich; F. B. K. acknowledges funding from the research school IMPRS-QST.






§ RESERVOIR SELF ENERGY



In this Appendix we calculate the greater/lesser self-energies
Σ^≷_res(ω) of the dot
arising from the tunneling Hamiltonian (<ref>) with an effective tunneling matrix given
by (<ref>) together with the potential scattering term V_sc, 
see (<ref>). The effective d.o.s. of the reservoirs is given by unity since 
the whole nontrivial information of the reservoirs is included in the effective tunneling matrix. 
Using standard Keldysh formalism we get

    Σ^≷_res(ω) = γ^2 ∑_αα'∑_kk' V_α^† G^≷_α k,α' k'(ω) V_α'  ,

where G^≷_α k,α' k'(ω) are the greater/lesser reservoir Green's 
functions arising from the reservoir part of the Hamiltonian including the potential scattering term.
These Green's functions can be calculated from the Dyson equation with V_sc defining the self-energy

    G^≷_α k,α' k'(ω)    = g^≷_α k(ω)δ_αα'δ_kk'
       
    + v_sc∑_α_1 k_1 g^≷_α k(ω) V_α V^†_α_1 G^A_α_1 k_1,α' k'(ω)
       
    + v_sc∑_α_1 k_1 g^R_α k(ω) V_α V^†_α_1 G^≷_α_1 k_1,α' k'(ω)
      ,

where G^A_α_1 k_1,α' k'(ω) denotes the advanced Green's function which follows
from the Dyson equation

    G^A_α k,α' k'(ω)    = g^A_α k(ω)δ_αα'δ_kk'
       
    + v_sc∑_α_1 k_1 g^A_α k(ω) V_α V^†_α_1 G^A_α_1 k_1,α' k'(ω)
      .

g^x_α k (with x=R,A,≷) denote the free Green's functions of reservoir α 
without V_sc given by

    g^R/A_α k(ω)    = 1ω-ϵ_α k1± iη  ,
    g^<_α k(ω)    = -f_α(ω)(g^R-g^A)(ω)  ,
    g^>_α k(ω)    = (1-f_α(ω))(g^R-g^A)(ω)  .

Since the d.o.s. of the reservoirs is unity we get 

    ∑_k g^R/A_α k(ω)    = ∓ iπ1  ,
    ∑_k g^<_α k(ω)    = 2π i f_α(ω)1  ,
    ∑_k g^>_α k(ω)    = -2π i (1-f_α(ω))1  .

Using these properties together with 
∑_α V_α^† V_α=1 and defining

    G̅^x(ω) = ∑_αα'∑_kk' V_α^† G_α k,α' k'^x(ω) V_α'  ,

with x=R,A,≷, we obtain from the Dyson equations (<ref>) and
(<ref>) after a straightforward calculation

    G̅^A    = iπ1 + iπ v_sc G̅^A  ,
    G̅^<(ω)    = - iπ v_sc G̅^<(ω) 
       + 2π i∑_α f_α(ω) V_α^† V_α (1 + v_sc G̅^A)
      ,
    G̅^>(ω)    = - iπ v_sc G̅^>(ω) 
       
    - 2π i∑_α (1-f_α(ω)) V_α^† V_α (1 + v_sc G̅^A)
      .

Solving this set of matrix equations for G̅^≷(ω) and inserting the solution in

    Σ^≷_res(ω) = γ^2 G̅^≷(ω)  ,

we finally get the result (<ref>-<ref>) 
for the self-energies with an effective hybridization matrix given by 
(<ref>).






§ EQUILIBRIUM GROUND STATE OF THE FIXED POINT MODEL



In section <ref>, we have argued why the dot representation has the [3] 
fundamental representation while the first state of the reservoir the complex conjugate of this 
fundamental representation [3]. Representing both sites by [3] (or, equivalently, 
by the complex conjugate of this representation [3]), leads to a decomposition of the 
Hilbert space of the composite system into a sextet and a triplet. Accordingly, a SU(3)-symmetric 
Hamiltonian in this representation has an either threefold or sixfold degenerate ground state which 
is in contrast to the outcome of our analysis. Choosing the complex conjugate representation 
[3] for the reservoir site instead leads to a Hilbert space that decomposes into an 
octet and a singlet. A SU(3)-symmetric Hamiltonian in this representation yields two different 
eigenenergies of which one is non-degenerate and the other eightfold degenerate.

We want to emphasize that this is fundamentally different to the situation in the corresponding 
SU(2) model. Generally, the fundamental representation of the spin 1  2 [2] is equivalent 
to its complex conjugate, i.e. [2]=[2]. This is consistent with the observation that no 
anti-spin 1  2 exists. However, this a special property of the SU(2) group that holds no 
longer for SU(N) with N>2 and we anticipate for an analog SU(N)-model a ground state inspired by 
flavor-antiflavor pairs.

We consider the following set of basis states for the composite system

    |u s⟩    = |u⟩⊗|s⟩  , 
    |d s⟩    = |d⟩⊗|s⟩  , 
    |d u⟩    = |d⟩⊗|u⟩  , 
    |u d⟩    = |u⟩⊗|d⟩  , 
    |s u⟩    = |s⟩⊗|u⟩  , 
    |s d⟩    = |s⟩⊗|d⟩  , 
    |u u⟩    = |u⟩⊗|u⟩  , 
    |d d⟩    = |d⟩⊗|d⟩  , 
    |s s⟩    = |s⟩⊗|s⟩  .

In a quark picture, these states are meaningful since they are all eigenstates of the total charge operator

    q̂_tot = Q̂ + q̂  ,

where Q̂=F̂_3 + 1 √(3)F̂_8 and q̂=f̂_3 + 1 √(3)f̂_8 
are defined as usual in the quark model <cit.>, with an integer eigenvalue. This is analog to the 
observation that no elementary particle with non-integer electrical charge exist in nature.

Let the effective Hamiltonian V_eff (<ref>) act on the states 
(<ref>-<ref>), we find that 
|u s⟩, |d s⟩, |d u⟩, |u d⟩, 
|s u⟩ and |s d⟩ are eigenstates with eigenvalue E_8=1  6 J. 
Instead, the remaining states (<ref>-<ref>) are no eigenstates since

    V_eff|u u⟩    =  - J  3|u u⟩ 
    - J  2(|d d⟩ + |s s⟩)   , 
    
    V_eff|d d⟩    =  - J  3|d d⟩ 
    - J  2(|u u⟩ + |s s⟩)   , 
    
    V_eff|s s⟩    =  - J  3|s s⟩ 
    - J  2(|u u⟩ + |d d⟩)   .

Finding the remaining eigenstates is a trivial diagonalization problem in the 3 × 3 subspace 
of |u u⟩, |d d⟩ and |s s⟩. The first two linear combinations

    |1⟩    = 1 √(2)(|u u⟩ - |d d⟩)   ,
    |2⟩    = 1 √(6)(|u u⟩ + |d d⟩ 
    - 2 |s s⟩)   ,

with eigenvalue E_8 complement the octet. Being orthogonal to |1⟩ and |2⟩, the singlet 
eigenstate is the ground state (<ref>) with eigenvalue 
E_gs=-4  3 J. We note that this set of eigenstates is the same as for pseudoscalar 
mesons in the light quark model<cit.>.




§ EVALUATION OF GOLDEN RULE RATE



In this appendix we evaluate the golden rule rates (<ref>) and (<ref>)
for the special case U_V=1. 
We denote the three states by the quark flavors, i.e. l=1,2,3≡ u,d,s. 
First, we evaluate the matrix elements τ_i j^αα^' from (<ref>) 
by employing the algebra of the Gell-Mann matrices. Writing

    τ_i j^αα^' = x_α x_α'τ̅_i j^αα^'  ,

we obtain for the non-vanishing matrix elements

    τ̅_1 1^αα^'   = τ̅_2 2^αα^' =   
    2 J^2_1 M^-_1,αα^'  , 
    τ̅_1 2^αα^'   =  -τ̅_2 1^αα^' 
    =  2 i J^2_1 M^-_2,αα^'  , 
    τ̅_4 4^αα^'   = τ̅_5 5^αα^' 
    =  J^2_4 M^++_3,αα^'  , 
    τ̅_4 5^αα^'   =  - τ̅_5 4^αα^' 
    =  i J^2_4 M^+-_3,αα^'  , 
    τ̅_6 6^αα^'   = τ̅_7 7^αα^' 
    =  J^2_6 M^-+_3,αα^'  , 
    τ̅_6 7^αα^'   =  - τ̅_7 6^αα^' 
    =  i J^2_6 M^–_3,αα^'  , 
    τ̅^αα^'_i j|_i,j ∈ (3,8)   =  2J_3i J_3j M^+_1,αα^' 
    + 2 J_i 8 J_j 8 M_4,αα^'
       
    + 2 √(3)(J_3 i J_j 8 + J_i 8 J_3 j) M^+_2,αα^'  ,

where J_33=J_3, J_88=J_8, and

    M^σ_1,αα^'   = q_αq_α^' 
    + σ p_α p_α^'  ,
    
    M^σ_2,αα^'   =  p_αq_α^' 
    + σq_α p_α^'  , 
    
    M^σσ^'_3,αα^'   = (q_α  
    + σ p_α) q̃_α^' + σ^'q̃_α(q_α^' 
    + σ p_α^')   , 
    
    M_4,αα^'   =  1+ p_α p_α^' 
    + q_α q_α^' 3 - q_α + q_¸α^' 3  ,

with q_α=1+q_α 3 and q̃_α=1 - 2 q_α 3. 
Introducing the notation

    χ_1/3^αα^'   = π 2(τ_1 1^αα^'±τ_3 3^αα^')   ,
    χ_2^αα^'   = i πτ_1 2^αα^'  ,  χ_s^α = x_αq̃_α  ,
    
    χ_u/d^α   = 2 π x_α[ J^2_+
    (q_α± p_αϕ_z )
    + J^2_- (p_α±q_αϕ_z ) ]   ,

with J^2_±=1  2(J^2_4 ± J^2_6) and ϕ_z= h_z  h, we obtain
by inserting (<ref>) and (<ref>) in (<ref>) after
a straightforward calculation

    Γ_d→ u   = ∑_αα^' w(μ_α - μ_α^' - h) ·
       · [χ_1^αα^' 
    - χ_2^αα^'ϕ_z + χ_3^αα^'ϕ_z^2]    ,
    Γ_u→ d   = ∑_αα^' w(μ_α - μ_α^' + h) ·
       · [χ_1^αα^' 
    + χ_2^αα^'ϕ_z + χ_3^αα^'ϕ_z^2]   ,
    Γ_s→ u   = ∑_αα^' 
    w(μ_α - μ_α^' - Δ - h  2) 
    χ_u^αχ_s^α^'  ,
    Γ_u→ s   = ∑_αα^' 
    w(μ_α - μ_α^' + Δ + h  2)
    χ_s^αχ_u^α^'  ,
    Γ_s→ d   = ∑_αα^' 
    w(μ_α - μ_α^' - Δ + h  2) 
    χ_d^αχ_s^α^'  ,
    Γ_d→ s   = ∑_αα^' 
    w(μ_α - μ_α^' + Δ - h  2) 
    χ_s^αχ_d^α^'  .


In the following we consider the case of two reservoirs in the strong nonequilibrium regime as 
defined in (<ref>). From the properties 
(<ref>-<ref>) 
and the results (<ref>-<ref>) for τ_ij^αα^', we obtain


    Γ_d→ u   = [χ_1^LR - χ_2^LRϕ_z + χ_3^LRϕ_z^2] (V-h)  
    + w(-h) ∑_α[χ_1^αα + χ_3^ααϕ_z^2]   , 
    Γ_u→ d   = Γ_d→ u + 2 χ_2^LRϕ_z V 
    + h (χ_1 + χ_3ϕ_z^2 )   ,  
    Γ_s→ u   = χ_u^L χ_s^R (V - Δ - h  2) 
    + w(-Δ - h  2) ∑_αχ_u^αχ_s^α  , 
    Γ_u→ s   = Γ_s→ u + (χ_s^L χ_u^R 
    - χ_u^L χ_s^R ) V + (Δ + h  2) χ_u χ_s   ,  
    Γ_s→ d   = χ_d^L χ_s^R (V - Δ + h  2)  
    + w(-Δ + h  2) ∑_αχ_d^αχ_s^α  , 
    Γ_d→ s   = Γ_s→ d + (χ_s^L χ_d^R 
    - χ_d^L χ_s^R ) V + (Δ - h  2) χ_d χ_s   .


where we have defined

    χ_1/3    = ∑_αα^'χ_1/3^αα^' = π[J_1^2 ± J_3^2 ±  J_38^2]   , 
    χ_u/d    = ∑_αχ_u/d^α =  2 π (J^2_+ ± J^2_- ϕ_z)   , 
    χ_s     = ∑_αχ_s^α =  1   ,

and note that 

    χ_2^LR   = - 2 π J_1^2 x_L p_L   , 
    χ_s^L χ_u/d^R - χ_u/d^L χ_s^R    =  
    - 2 π[x_L q_L (J^2_+ ± J^2_- ϕ_z)  .
       . + x_L p_L (J^2_- ± J^2_+ ϕ_z) ]   .

The stationary probability distribution p_l follows from inserting 
(<ref>-<ref>) in (<ref>). 
Finally, we can compute m_F from (<ref>).

We note that m_F=0 is equivalent to ⟨F̂_3 ⟩ = ⟨F̂_8 ⟩ =0. 
Therefore, we consider ⟨F̂_3 ⟩ = 1  2 (p_u - p_d) 
and ⟨F̂_8 ⟩ = 1  3 (p_u + p_d - 2 p_s) in the following and analyze under 
which conditions both expectation values become zero in the strong nonequilibrium regime. 
A cumbersome but straightforward analysis yields

    ⟨F̂_3 ⟩    = 1  2 N[ℱ_1 (Γ_s→ u + Γ_s→ d).
       . + ℱ_2 (Γ_s→ u - Γ_s→ d)]   , 
    2√(3)⟨F̂_8 ⟩    = 
    -1  3 N{ℱ_1[2 π J^2_+ (h - 2 x_L p_L ϕ_z V )
    ..
       ..+ 4 π J^2_- (Δ - x_L q_L V) ϕ_z + Γ_s→ u - Γ_s→ d]
    . 
       . + ℱ_2 [2 ℱ_2 
    + 2  (Γ_d→ u + Γ_u→ d)
    + Γ_s→ u + Γ_s→ d]}  .

Here, the factor N follows from the normalization condition (<ref>). 
Furthermore, we have defined the following functions in (h_z,h_⊥,Δ)-space

    ℱ_1    =  - 2 χ_2^LRϕ_z V - h (χ_1 + χ_3ϕ_z^2 ) 
       
    - π[ J^2_+ (h - 2 x_L p_L ϕ_z V ) 
    + 2 J^2_- (Δ - x_L q_L V) ϕ_z ]   , 
    ℱ_2    = π[2 J^2_+ (Δ - x_L q_L V )
    + J^2_- (h_z - 2 p_l V) ]

ℱ_1=ℱ_2=0 fulfills the condition 
⟨F̂_3 ⟩ = ⟨F̂_8 ⟩ =0. Moreover, it defines a curve in 
(h_z,h_⊥,Δ)-space that provides us with a tool to measure the distance to the fixed 
point model. ℱ_2=0 directly yields (<ref>) and defines the plane in 
(h_z,h_⊥,Δ)-space where the curve lies in. The shape of the curve follows from ℱ_1=0. 
To that end, we insert (<ref>) into (<ref>) and 
obtain (<ref>). That is, we project the curve onto the (h_z,h_⊥)-plane.

Finally, we prove (<ref>). To this end, we decompose (<ref>) as

    ⟨ I_β⟩   = ∑_l l^'Γ^β_l^'→ l p_l^'
       =   I^β_0 + I^β_3 ⟨F̂_3 ⟩ 
    + I^β_8 2√(3)⟨F̂_8 ⟩  ,

with

    I^β_0    = 1  3∑_l l^'Γ^β_l^'→ l  , 
    
    I^β_3    = ∑_l (Γ^β_u→ l - Γ^β_d→ l)   , 
    
    I^β_8    = 1  2∑_l (Γ^β_u→ l 
    + Γ^β_d→ l - 2 Γ^β_s→ l)   .

Evaluating (<ref>) for two reservoirs in the strong nonequilibrium regime 
(<ref>), we can express (<ref>-<ref>) in terms of τ̅^αα^'_ij


    I^L_0    = π 3 x_L x_R {[2 τ̅^LR_11 
    + τ̅^LR_33 + 2 (τ̅^LR_44 + τ̅^LR_66) 
    + τ̅^LR_88] V 
       
    + 2 i(τ̅^LR_45 + τ̅^LR_67) Δ 
    + i(2 τ̅^LR_12 + τ̅^LR_45 - τ̅^LR_67) h_z }  , 
    
    I^L_3    = π x_L x_R {[2 i τ̅^LR_12 
    + 2 √(3)τ̅^LR_38 + τ̅^LR_44 
    - τ̅^LR_66 + i (τ̅^LR_45 - τ̅^LR_67) ] ϕ_z V  
       
    + [τ̅^LR_44 - τ̅^LR_66 + i(τ̅^LR_45 
    - τ̅^LR_67)] ϕ_z Δ
       
    + [τ̅^LR_11 + τ̅^LR_33 + (τ̅^LR_11 
    - τ̅^LR_33) ϕ_z^2 + 1  2(τ̅^LR_44 + τ̅^LR_66) 
    + i  2(τ̅^LR_45 + τ̅^LR_67) ] h }  , 
    
    I^L_8    = π 2 x_L x_R {[2 τ̅^LR_11 + τ̅^LR_33 
    - (τ̅^LR_44 + τ̅^LR_66 + τ̅^LR_88) 
    + 3i (τ̅^LR_45 + τ̅^LR_67) ] V 
       
    + [3 (τ̅^LR_44 + τ̅^LR_66) 
    - i (τ̅^LR_45 + τ̅^LR_67) ] Δ
       
    + [2 i τ̅^LR_12 + 3  2(τ̅^LR_44 
    - τ̅^LR_66) - i  2(τ̅^LR_45 
    - τ̅^LR_67) ] h_z }  .

If we consider m_F=0, the current I^β is completely equal to I^β_0. Therefore, 
we can evaluate (<ref>) using (<ref>-<ref>) 
at the fixed point and obtain (<ref>).








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