Discussion of Advanced Flow (1/ √

B �∆) at zero Temperature In this regime, the Gaussian cut-off function determines the effective range of contributions to the dx-integral which we discuss in a first step. Inside the cut-off window the first order Taylor expansion of the hybridization is a sufficient approximation. The most relevant aspects for an evaluation of the integrations are introduced by the Fermi function. According to the general philosophy of our approach we will express its effects by explicit restrictions on the integration domain. For this procedure we again apply symmetry considerations and use graphical motivations.

Our further discussion will follow the following line: Noticing that either even or odd functions are present in the argument of thedx-integration, we separately consider these cases to make use of obvious symmetries and calculate the value of the integral. This will lead to a functional expression which still depends on y. In a second step we apply the dy-integration. Then symmetry considerations come into operation again.

Even functions in x

Even functions in the integrand can be assumed to be of the form x^2ke^−2Bx2 for arbitrary integer numbersk. They are axial symmetric with respect tox= 0 and show an exponential decay.

Their axial symmetry combined with the point symmetry of the Fermi function leads to the opening of a cancellation window for large absolute values ofxas contributions in the tails of the function ]− ∞,−x_C] and [xC,∞[ compensate for each other. Depending on the position of the transport window,x_C =x_U (for y >0) orx_C =−x_L (fory≤0).

The effective contribution depends in a similar way on the asymmetry of the position of the transport window with respect to the symmetry of the Gaussian curve. It might open a contribution window of length 2|y|, either [−x_U, x_L] (fory >0) or [x_U,−x_L] (fory <0). The

74 5 Diagonalization of the Anderson Hamiltonian

effectivedx-integration is performed over this window only.

� _∞ Integrations over even arguments inx andy vanish:

An important observation shows that for fixed external voltage bias the result must be point symmetric in y. The relative position of the symmetry centres (x= 0 for the even functions and x=y for the Fermi function) decides on the position of the contribution window which vanishes completely in the case of their coincidence. Therefore the contribution window is symmetrically opened on both sides for corresponding values of y, but the Fermi function induces different signs.

This implies that the dy-integration vanishes whenever the dy integrand consists of an even weight iny and the result of the dx-integration. Hence terms of the form

�

dy y^2l

�

dx N_y(x) x^2k e^−2Bx2 = 0 (5.29) do not contribute to the flow of the interaction for any integer valuesl, k.

Odd functions in x

We start with a similar analysis for a class of odd functions inxcharacterized by the argument x^2k+1e^−2Bx2 for arbitrary integer values ofk. The interesting situation for these terms arises with|y|> V^b. In this case a cancellation window of length 2(|y| −V^b) is opened around the zero point. Now the contribution window is split into two parts containing the tails of the function. We consider both casesy > 0 and y <0 explicitely which differ by the position of the windows only. Integrations over odd arguments inx produce axial symmetric result in y:

We observe that the contributions are axial symmetric for corresponding values ofy, i.e.

� _∞

−∞

dx N(y;x)x^2k+1e^−2Bx2 =� _∞

−∞

dx N(−y;x)x^2k+1e^−2Bx2 (5.31) Integrations over odd arguments inx and y vanish:

Consequentially, thedy-integration vanishes if an odd weight iny is applied, i.e. for arbitrary integer valuesk and l

�

dyy^2l+1

�

dxx^2k+1 N_y(x) e^−2Bx2 = 0 (5.32)

5.4Flow of the interaction for the asymmetric model 75 Restriction of integration for arguments of the formy^2l x^2k+1:

Moreover, we can simplify all further calculations which involve odd functions inxand even weights in y by restricting the dy-integration to half of the axis and doubling the obtained result.

for arbitrary integer valuesk andl.

Discussion of the terms in the flow equation

Now we apply our analysis of the integrals to the different terms in the right hand side of (5.26) which originated from the Taylor expansion of the hybridization. They show different behaviour depending on their parity inx and y. The linear term vanishes because of sym-metries already discussed: y x e^−2Bx2 is an odd term in x and y and does not contribute according to (5.32), for the even termx² e^−2Bx2 we can apply the result (5.29).

� _∞

Only the quadratic terms∼(y²−x²) x e^−2Bx2) produce nonvanishing contributions. Again we make use of the symmetries and restrict the range of integration according to (5.33) to valuesy <0. We calculate both terms independently using standard Gaussian integrals which can be found in the appendix.

Calculation of the term ∼y²xe^−2Bx2

76 5 Diagonalization of the Anderson Hamiltonian

The two underbraced terms differ in a sign only and are treated in a similar way. We introduce new variablesz=y±V, substitute and separately calculate the integrals.

= 1

Using the symmetry of the Gaussian,�₋V

−∞dz e^−2Bz2 =�_∞

V dz e^−2Bz2 the integrals can be combined to full Gaussian integrals and evaluated. In total one obtains the following result:

� _∞

Calculation of the term ∼x³e^−2Bx2 : Using similar techniques we calculate the second nonvanishing contribution to the flow equation. As a detailed presentation of the calculation would not shed new light on the topic we just state the result:

� _∞

We remark that this result is independent of the voltage bias. Therefore we can expect specific out-of-equilibrium changes to the flow of the interaction only by the first contribution.

Flow equation in the advanced flow regime

Now we insert the results (5.34), (5.35) and (5.36) into (5.26). Hence the flow equation for the interactionU(B) is given by the difference of both terms calculated above:

dUF(B)

In order to approximately evaluate the flow equation we assume that the solutions for initial and advanced flow coincide atB = 1/∆² which delivers an initial valueU_∆=U(B = 1/∆²) (cf. 5.15). Now the differential equation can be integrated easily and produces the following result:

5.4Flow of the interaction for the asymmetric model 77 Here we notice that forB → ∞, i.e. in the case of diagonal Hamiltonian, there is a renormal-ization of the interaction which depends on the voltage bias applied to the system. The deriva-tive of the hybridization function at the Fermi level amounts to^∂W_∂�²|�_F = 2πρ0�d

∆ W²(�_F)<2ρ0�d

and we know from the result (5.15) thatU_∆< U(B = 0). ∆

Now a more detailed discussion of this result for various regimes is possible. Depending on the parameters there is a configuration such that with increasing outer voltage the on-site interaction is reduced. This may correspond to results obtained in other models² where out of equilibrium conditions provoke a suppression of correlation effects. Further inquiries could be of interest.