The key idea of the flow equation approach consists in performing a continuous sequence of infinitesimal unitary transformations on a given Hamiltonian

With an anti-Hermitian generator the solution of equation (2.2) describes a family of unitarily equivalent Hamiltonians parameterized by the

The concrete realization of this approach for the Kondo model was discussed in Ref. (16). The starting point is the bosonized form(17) of the Hamiltonian (2.1). Since we will be mainly interested in describing the basic ideas of our approach, we restrict ourselves to a linear dispersion relation. Notice, however, that the flow equation approach can also be used for a nontrivial conduction band density of states as it does not rely on the integrability of the model.(18) With a linear dispersion relation the Kondo problem becomes effectively one-dimensional, the charge density excitations in (2.1) decouple, and we only need to look at the spin density part

with . Here are the bosonic spin density modes with the bosonic spin density field defined by . For simplicity we have set the Fermi velocity . is proportional to the inverse conduction band width. All our latter results will be expressed as universal functions of the low-energy Kondo scale , and we can consider (2.3) to be equivalent to our original Kondo Hamiltonian if .

Eq. (2.3) was used as the starting point
of the flow equation approach in Ref. (16). Away
from the Toulouse point the unitary equivalence of the flow holds only
approximately, but this approximation can be controlled by a small
parameter(15) and yields very accurate
results. During the flow the Hamiltonian can be parameterized as

Here , and denote normalized vertex operators with scaling dimension in momentum space,

that obey and . For the special case they fulfill fermionic anticommutation relations and can therefore be interpreted as creation and annihilation operators for fermions.

In Ref. (16) the following flow equations for the
parameters in (2.4) have been derived

and a differential equation for the flow of the scaling dimension

It can be shown(16) that one always finds in the strong-coupling phase of the Kondo model, i.e. in the low-energy limit the vertex operators in (2.4) become fermions. In the following we will use an improved version of the above flow equations by taking into account that all approximations should be performed with respect to the interacting ground state: It turns out that the only necessary modification in (2.5), (2.6) and (2.7) is that the exponent in gets replaced by , i.e. it is not a running exponent anymore.(19)