�
k,l
�GS|akbl|GS� ∂2
∂bl∂ak
A(a)B(b):|b=a (A.9) Re-arragements of that kind are needed to compute commutators and correlators of higher, normal ordered operator products. Such calculations are the analytical backbone of the flow equation formalism and account for a significant part of the effort involved with this technique.
A.2 Fermi distribution functions
Due to normal ordering Fermi functions are present in most of the relevant flow equations.
They take in a significant place in the theory as they incorporate important electronic prop-erties. We assume that they are fully responsible for any dependence on temperature and external voltage which is applied to bring the system into out-of-equilibrium. By the way we
A.2 Fermi distribution functions 129
stress that in our theoretical setup no other quantities (like coupling constants, etc.) should depend on such outer conditions.
Fermi distribution functions describe the occupation of states in a fermionic multi-particle system in terms of a one-particle approximation. At zero temperature and in equilibrium they simply reflect the complete filling up to a maximal energy, the Fermi energy�F. A sharp Fermi edge is formed.
Equilibrium Fermi function
For zero temperature, the Fermi distribution in equilibrium is defined simply by n+(�) =
� 1 for �≤�F
0 for � > �F (A.10)
For convenience, we also define its complement n−(�) = 1−n+(�) and mention the point symmetry of their difference with respect to the Fermi energy
�n+−n−� (�) =
1 for � < �F 0 for �=�F
−1 for � > �F
(A.11)
The regular appearance of this difference in the flow equation formalism makes the Fermi energy a preferred point in energy for symmetry considerations.
Out-of-equilibrium Fermi function
We now consider the case where an external voltage drop is applied across the impurity. Our description uses two different chemical potentials for the left lead (µL) and the right lead (µR) of the setup; they are related by the voltage biasµL−µR = V and measured with respect to the Fermi energy. This puts forth the emergence of two sharp Fermi edges at the chemical potentials. Figuratively speaking, the voltage bias opens a ’window’ in between, furtheron called the transport window. Within this window, a coexistence of occupied and unoccupied electronic states fosters electron transport.
We roughly approximate the energy level occupation inside the transport window by a flat filling factornT between zero and one, depending on the asymmetry of the couplings involved.
The Fermi distribution function aquires the following form:
n+(�) =
1 for � < �L nT for �L≤�≤�U
0 for � > �U
(A.12) To keep the notation intuitively, we denote the limits of the transport window by a lower and an upper energy. For the classical case ofµL> µR, i.e. for a net current from the left to the right lead, this trivially implies�L=�F +µR and �U =�F +µL.
For most of our discussions we refer to the symmetric case and set the Fermi energy zero.
Then the transport window is opened symmetrically around the Fermi energy
[�L, �U] = [−V /2, V /2] (A.13)
130 A Normal ordering and Fermi distribution functions
and the filling factor takes the valuenT = 0.5. The difference of the complementary Fermi functions retains its point symmetry and vanishs within the transport window completely.
�n+−n−� (�) =
1 for � < �L 0 for �L≤�≤�U
−1 for � > �U
(A.14)
Later discussions will heavily refer to these properties.
Combined object of Fermi functions
To simplify notation we introduce another combination of Fermi functions which will regularly appear in later calculations.
Q12�2 = n+(s1) n+(s2)−n+(s1) n+(s2�) + n+(s2�)n−(s2) (A.15) For zero temperature this combination of Fermi functions is always positive. This can be easily seen, for instance, by denoting the values of this function in a table spanned by the different energy regimes�1/2�/2≶0.
We mention that this combination of Fermi functions re-appears in Fermi liquid theory and point out at [29] where such aspects are discussed.
Appendix B
Pre-diagonalization for an impurity with several spin levels
On this page we show what conditions are necessary to reduce a two-level impurity to an effective one-level impurity for the purpose of applying the pre-diagonalization. The trans-formationT described in this context is exemplary for similar approaches (e.g. the treatment of asymmetric hybridization strength).
Decoupling of a channel
The starting point is a general two level system. Two different operators d1 and d2 are introduced to describe the impurity orbitals, which might couple differently to the band electrons. The hybridization splits up into V1 and V2, the energy levels of the orbitals are generally non-degenerateε1�=ε2. Under the usual assumptions, the Hamiltonian aquires the following structure:
H =�
k
εkc†kck+ε1d†1d1+ε2d†2d2+�
k
� V1
�
c†kd1+d†1ck� +V2
�
c†kd2+d†2ck��
(B.1) A linear transformation
T : (A1, A2) → (A+, A−) A+ = 1
√2(A1+A2) A− = 1
√2(A1−A2)
is applied to the pairs of operators (d†1, d†2), (d1, d2) and the hybridization (V1, V2) and produces the transformed Hamiltonian
Htransf ormed = ε1+ε2 2
�d†+d++d†−d−�
+ε1−ε2 2
�d†+d−+d†−d+� + V1+V2
√2
�
c†kd++d†+ck�
+V1−V2
√2
�
c†kd−+d†−ck�
It is easily read off that for an energetically degenerate two-level system (ε1 = ε2) with a uniform coupling of both orbitals to the band (V1 = V2) effectively one channel decouples.
131
132 B Impurity with several spin levels
Such models can therefore be treated like one-level impurities with a modified hybridization V+=√
2V after a transformation T is performed.
This result can be directly applied for degenerate spin levels on the impurity which both couple . Effectively, this only leads to a renormalization of the hybridization strength. We simply write downV as a renormalized constant.
Appendix C
Some commutators and correlators
In this part of the appendix some non-trivial commutators of normal-ordered operator prod-ucts are presented. For their derivation, some helpful general relations have been used. We refer to (10) where a detailed calculation of the [:4:,2]-commutator is presented to make the reader familiar with this kind of calculation.
C.1 General relations
Let A,B,C,D,E,F,G,H be arbitrary operators, i.e. elements of a common operator space. Then the following relations hold:
Decomposition of commutators into commutators
[AB, C] = A[B, C] + [A, C]B
[AB, CD] = AC[B, D] +A[B, C]D+C[A, D]B+ [A, C]DB
Decomposition of commutators into anticommutators
[AB, C] = A{B, C} − {A, C}B
[AB, CD] = −AC{B, D}+A{B, C}D−C{A, D}B+{A, C}DB [ABCD, EF] = ABEC{D, F} −ABE{C, F}D+
ABC{D, E}F−AB{C, E}DF + EA{B, F}CD−E{A, F}BCD+ A{B, E}F CD− {A, E}BF CD
133
134 C Some commutators and correlators