transformation but can constantly assume to work with an unchanged non-interacting ground state. Nonetheless we add the remark that this is different if other operators should be considered, e.g. for the current operator.
3.3 Continuous sequence of infinitesimal transformations
At the heart of the flow equation method lies the idea of applying an ordered, continuous sequence of infinitesimal transformations to the operator space to obtain a significant transfor-mation of an observable. This sequence is chosen under the specific constraint that finally the full net transformation diagonalizes the Hamiltonian. This splitting up of an effective (net) transformation into an infinite sequence of steps introduces an order parameter comparable with time and gives way to separate the treatment of different aspects of the transformation process at different values of this order parameter (at different ’times’). The flow equation method makes use of this opportunity in particular by dealing with different energy scales in a controlled and ordered manner.
3.3.1 Parametrizations
Creating such a sequence of transformations is naturally accompanied by an introduction of parametrization. A one-dimensional flow parameter, denoted by B, is introduced and fixed by the following boundary conditions: B = 0 shall refer to the initial step before the trans-formation is applied,B → ∞characterizes the final outcome of the full net transformation.
Both the observables (directly or via the generalized coupling constants) and the generator become dependent on this parameter. Studying these quantities or the procedure itself for various values ofB is commonly referred to as studying them ’under the flow of B’.
As we restrict to unitary transformations, all infinitesimal transformations appearing in this sequence show the same differential structure explained above, but differ from each other by the precise form of their generators. This freedom to change the generator under the flow is a decisive strength of the method and is the source of energy scale separation. In particular, this makes it different from more conventional approaches which restrict to single unitary transformations. A good example may be the re-examination of the Schrieffer- Wolff-transformation with flow equations [17].
Consequently, the first step is to define a continuous parametrization of the generator with respect to a flow parameter B. This will be presented in the next section. We just briefly mention that various possibilities to choose the generators exist.
3.3.2 Representation of the net transformation
After a parametrization has been introduced, the net transformation performed at a certain value ofB can be evaluated by integrating up the infinitesimal transformations. This is done by solving a differential equation forU(B) which directly follows from the definition of the generator in equation (3.3).
η(B) = (∂BU(B))U−1
44 3 Introduction to the flow equation method
It can be re-written as a Schroedinger-like equation for a time evolution operatorU(B) with the flow parameter replacing time.
∂BU(B) =η(B)U(B) (3.22)
The calculation can be found in many textbooks on quantum mechanics and leads to aB -ordered product (denoted by the ordering operatorTB) of the form
U(B) =TBeR0B dB�η(B�) (3.23) Although this is an exact result, the complicated B-ordering involved renders this result unpractical for non-trivial generators. Therefore it is usually avoided to refer to the full net transformationU(B).
3.3.3 Active and passive view on the transformation
Unitary transformations are linear maps which implement a change between basis represen-tations. They can be equivalently approached from two different points of view.
Sometimes, the features of a basis change are highlighted which means that, in particular, the basis elements are changed themselves. This is commonly known under the name of a
’passive’ transformation (figuratively, this is the rotation of the coordinate system under an fixed object). On the other hand, it is sometimes advantageous to think of the transformation as a plain linear map between different objects which are all represented in a fixed operator basis. Then we call it an ’active’ transformation (which corresponds to an rotation of the object in a fixed frame). Both views are equivalent but should not be confused.
For the purpose of diagonalizing the Hamiltonian it is commonly referred to the ’passive’
view. Yet the transformation of any observable in the flow equation framework is most easily understood if continuous unitary transformations in the constantly fixed original operator basis are considered.
Continuous change of basis representation
In many theories the diagonalization of an Hamiltonian is commonly discussed in terms of the
’passive’ view. This could be done in the flow equation framework as well. Then it implies that the basis elements aquire a dependence on the flow parameter. In order to keep the flow equation formalism valid throughout the flow, we have to impose the same canonic (fermionic) anticommutation relations on the fundamental operators at any value ofB. This resembles
’equal time anticommutation relations’ in conventional quantum field theory.
�b†s�(B), bs(B)�
=δss�
� b†s�
1(B), b†s�
2(B)�
={bs1(B), bs2(B)}= 0
Such conditions define the character of the fundamental operators and, in consequence, of all basis elements of a canonical normal ordered multiparticle operator basis at arbitrary B. Comparing operators at different values of the flow parameter is only possible if both are expressed in a common basis representation, i.e. if they are considered at ’equal time’.
Therefore a general procedure is needed to change between different basis representations; it is given by the transformation of the basis elements (observables).
3.3Continuous sequence of infinitesimal transformations 45
Fixed operator basis
For obvious reasons, such a transformation is best studied in the active picture. Exemplified for a creation operator we see that in the permanently changing frame the operator maintains a simple formal representationFB�
b†(B = 0)�
=b†(B). Here we denoted the effective unitary transformation byFB.
If we, instead, express the transformed operator in terms of the original basis, the action of the unitary transformation is made more explicit:
FB
�
b†(B = 0)�
=�
m,n
�
s�1...s�m,s1...sn
gs�
1...s�m,s1...sn(B):b†s�
1(0). . . b†s�
m(0)bs1(0). . . bsn(0): (3.24) Under the flow, new contributions to the basis expansion emerge. The behaviour of the generalized coupling constants of such observables can be again studied using the flow equation formalism.
In the following part of this work we will only refer to the active view. It has the major advantage that allB-dependence is attributed to the generalized coupling constants and can be easily evaluated by means of real-valued differential equations.
3.3.4 Discussion of initial and final picture
As the flow equation method is a diagonalization procedure it singles out two particular choices of a basis: An initial one is used to define the Hamiltonian; it corresponds toB = 0.
The final basis is defined as the (approximate) eigenbasis of the Hamiltonian. It is reached at the end of the flow procedure (B → ∞). We briefly review the physical aspects of both descriptions.
Initial basis representation
In the initial representation the diagonal part of the Hamiltonian describes a conduction band of particles. Due to the pre-diagonalization applied to the Anderson impurity model these particles are no physical band electrons of the metallic leads any more. Yet they are quasi-particles which already incorporate the hybridization with a local impurity. To keep notation easy we call them particles nonetheless. The nature of the free Hamiltonian (H0) defines the character of the creation and annihilation operators which create and destroy one such particle in a specified state. This can be easily expressed by the commutator
[H0(B = 0), b†s(B = 0)] =�s(B= 0)b†s(B = 0) (3.25) In this physically accessible basis the two-particle interaction on the impurity is defined in a straightforward manner, similarly the (approximate) non-interacting ground state of the system (the Fermi sea) and, consequently, the one-point correlator are known.
Yet we study a (strongly) interacting system which has to be described by an unknown and potentially complex interacting ground state. This is why we perform a change into the final basis representation.
46 3 Introduction to the flow equation method
Final basis representation
When the flow parameter approaches infinity the diagonal basis of the Hamiltonian is reached.
Now the full Hamiltonian defines the physical character of the fundamental operators.
[H(B =∞), b†s(B =∞)] =�s(B =∞)b†s(B=∞) (3.26) It describes the creation and annihilation of quasiparticles which additionally include the two-particle interaction on the impurity. They are the proper excitations of the fully interacting system. In this eigenbasis of the system we study its density of states to gain insight into the effects caused by the interactions in and out of equilibrium. A more detailed understanding of the nature of the quasiparticles is provided by the (inverse) transformation of the observable:
It shows that expressing the ’quasiparticle’ (i.e. B → ∞) creation and annihilation operators in terms of the ’particle’ (B = 0) basis makes a sequence of multiparticle contributions explicit.
Their quantitative weight increases with the progression of the flow parameter.
3.3.5 Flow of generalized coupling constants
Such and similar quantitative studies of the behaviour of observables under the flow are an interesting aspect of the flow equation method. Again we refer to the expansion of the observable in terms of generalized coupling constants. The opportunity to focus on the individual behaviour of flowing coupling constants is a major strength of the method.
After parametrization of the generalized coupling constants and of the generator by a flow parameter, the flow equations derived in (3.16) hold for all values ofB, i.e. they describe the behaviour of the generalized coupling constants under the change ofB throughout the flow.
Solving this system of differential equations for all values ofB makes the continuous change of the observables, in particular the stepwise diagonalization of the Hamiltonian, accessible in a quantitative way. This reminds of scaling and renormalization group ideas which we will briefly discuss in the following section.