We already examined the flow equation for the on-site interaction in a fairly extended way.
We now want consider the flow equation for one-particle energies (�s) and potential scattering amplitudes (Ps�s). This can be done in a joint formalism as both refer to the same operator structures. We will find that the change of these constants is a typical second order effect in U.
5.5.1 Flow equations for the renormalised energies and the potential scat-tering term
Obviously, the band energies �s are subject to renormalization changes, which will be ex-pressed by flow equations for �s(B). Additionally, similar new terms, describing potential scattering, might arise with increasing flux although they were not present in the original Hamiltonian. In order to absorb both into a common formalism, we consider the flow of the constantPs1�s1 appearing in the free part of the Hamiltonian
H0(B) = �
s1�s1
Ps1�s1:b†s1�bs1: (5.39) where Ps1s1 = �s1 with coinciding indices represents the band energies. For initial flow, we have non-vanishing band energies (Ps1s1(B = 0) �= 0) but no scattering terms are present (Ps1�s1(B= 0) = 0).
Evaluation of the commutator terms
As the commutator [η(1), H0] does not contain any operator product of length two, all these products result from the commutator [η(1), Hint(1)] which is quadratic inU. Hence energy reno-malization and potential scattering are second order effects only. The results are degenerated for both spin orientations (↑↑) and (↓↓), so we only consider the first one. Single spin flips are forbidden by spin conservation. Combining four relevant terms by substituting internal indices we finally arrive at the following flow equation.
∂Ps1�˜s˜1(B)
∂B = �
U02 W1˜�˜1
� �
s1s2�s2
W122�2 e−B(�1�˜+�2�−�1−�2)2 e−B(�˜1+�2�−�1−�2)2 2
��1˜�+�˜1
2 + �2�−�1−�2
�
Q12�2 (5.40) We remind the reader of the definition ofQ12�2 in (A.15) where it is introduced as a quadratic combination of Fermi distribution functions at different points in energy. Furthermore we mention that the right hand side is symmetric in the external indicess1˜� and s˜1. The follow-ing discussion of this equation will be done separately for the one-particle energies and the potential scattering amplitudes.
Renormalisation of the one-particle energies
The renormalisation of the one-particle energies is a fundamental effect of Hamiltonian di-agonalization and cannot be avoided. It incorporates a part of the spectral weight which is
5.5Flow equation for the energies and scattering amplitudes 79 flowing from the off-diagonal into the diagonal elements of the Hamiltonian. Nonetheless it is of lower-ranking interest as in all observable results one-particle energies appear next to factors of the interaction. As the energy corrections are of the kind�(B) =�(0) +U2 ∆�they are not visible in second order calculations.
U �(B) = U �(0) +U3 ∆�O(U≈2)U �(0) (5.41) Therefore we neglect all aspects of energy renormalization and treat the one-particle energies in good approximation as constants under the flow.
Potential scattering amplitudes and extended generator
A different perspective should be taken on the issue of the potential scattering term. It is generated under the flow and corrupts the diagonal shape of the Hamiltonian. An exten-tion of the generator has been defined in equaexten-tion (4.11) which fully compensates for these contributions. Using the explicit flow equation for the scattering amplitudes we make this extension explicit:
η(2a)(B) = U02(B)�
s˜1�s˜1
Ws1�˜s˜1
1
�s˜1� −�s˜1
�
s1s2�s2
Ws21s2�s2 Qs1s2�s2
e−B(�s1�˜+�s2�−�s1−�s2)2 e−B(�s˜1+�s2�−�s1−�s2)2 2
��s1�˜ +�s˜1
2 + �s2� −�s1 −�s2
�
:b†s˜1�bs˜1: (5.42) This part of the generator is clearly antihermitian as the coefficients of the operator products are antisymmetric under interchange of the labelss˜1 and s1˜�.
The effect of the extension on the Hamiltonian diagonalization has been the cornerstone of its definition and has improved the diagonalization method. Hence we expect that it –loosely speaking– re-distributes the effects which potential scattering has and inserts additional terms into the flow equations of other observables. We will study this as an aspect of the transfor-mation of observables in the next chapter.
80 5 Diagonalization of the Anderson Hamiltonian
Chapter 6
Transformation of the observables
In chapter 3 we have learned that according to the general framework of the flow equation for-malism the infinitesimal unitary transformations applied to the Hamiltonian simultaneously affect the appearance of all observables. With increasing flow higher order terms of their basis representation are generated and change their behaviour. In this chapter we study the transformation of the creation and of the annihilation operator which we will need to express a one-particle Greens function in the final basis representation.
6.1 Ansatz for the transformation of fundamental operators
Before we progress to the transformtion of a composed observable we study the transforma-tion of a one-particle creatransforma-tion or annihilatransforma-tion operator under the flow equatransforma-tion procedure.
Following the general outline of chapter (3) we choose a truncation scheme for the fundamen-tal operator itself and represent it in terms of generalised coupling constants, then we derive their flow equations. The analysis of these flow equations will show that it is more convenient and sufficient to study a linear superposition of operators.
6.1.1 Creation operator
As a first step we consider the creation operator. As it is an ordinary object of the multi-particle operator space it undergoes the same unitary transformations as any other observable.
Therefore we start with defining its basis representation which is trivial for initial flow but will aquire a more complex structure furtheron. Hence we need to discuss a suitable truncation scheme.
Choice of a truncation scheme
Physical demands impose constraints on the unitary transformations and allow for a reason-able restriction of the trucation scheme. As particle number and spin conservation is assumed the mixing of certain subsets is forbidden. Particle number conservation ensures that only subsets with the same numeral difference of creation and annihilation operators mix, spin conservation excludes some spin parameter regimes within the remaining subsets.
This implies that the number of creation operators minus the number of annihilation operators equals one in any term which is generated under the flow. In principle, arbitrarily high excitations may contribute. In an approximative restriction we only admit those of first
81
82 6 Transformation of the observables
order particle excitations, i.e. operator products up to length three. Thus we start with the following ansatz
The initial conditions are set by physical plausibilities. At initial flow the originial observable has not yet been changed, so
γss5�9�(B = 0) =δss9�5� and Mss5�9�s6�s5(B = 0) = 0.
The newly introduced constants γss5�9�(B) and Mss5�9�s6�s5(B) are subject to a renormalization change under the flow induced by the sequence of infinitesimal transformations.
The obvious next steps are the setup of the flow equations by evaluating various commutators.
We postpone this analysis for a short glimpse at the transformation of annihilation operators.
6.1.2 Annihilation operator
As creation and corresponding annihilation operators are related by hermitian conjugation we can regress to more general aspects of the transformation of hermitian conjugate observables.
Transformation of hermitian conjugate observables
According to the definition of the canonical generator and of its extensionη(2a) the full gen-erator is anti-hermitian (η†(B) =−η(B)). Thus the flow equation of the conjugate operator is simply given by the conjugated flow equation of the operator.
[η(B), b(b)]a.h.= �
b(B)η†(B)−η†(B)b(B)�
= [η(B), b†]† For the flow equation, this means
∂b(B)
Consequentially, the same constants γ and M describe the behaviour of the annihilation operator under the transformation.
All generalised coupling constants are real numbers. We re-label the internal summations to keep a correspondence between primed indices and creation operators
bs9(γss59(B)Mss59s6s5�(B)) = �
s5
γss59(B)bs5 + �
s5�s6s5
Mss59s6s5�(B) :b†s5�bs6bs5: (6.2) Thus we find that it is sufficient to treat the transformation of the creation operator to set up and solve the differential equations for the generalized couplingsγ andM. No other structure will be needed.