dω C(ω) = 0 (7.24)
We note that the argument of the integration is not the exact variation of the spectral density any more but an approximative expression in second order of U. This might introduce an error ∆Cwhich in the density of states scales withU2. Thus we expect that the approximate analytical solution of the set of differential equation does not extend easily into the regime of strong coupling. We will study this error numerically.
7.3 Conservation of spectral weight
The impurity spectral density is a normalised function which describes the relative distribution of spectral weight with respect to energy. Thus we expect that its normalisation should not change under the influence of interactions. We will prove the conservation of spectral weight both for the system of differential flow equations (6.11 and 6.13) and for their approximate analytical solution (6.14 and 6.25) up to second order inU in a mathematically precise way.
We start with the general examination of the set of differential flow equations for the creation operator and study the retarded Greens function at a zero time.
7.3.1 Spectral integration and zero time retarded Greens function
Calculating the normalisation of the spectral function clearly means integrating over its energy dependence. This integral can be related to the retarded Greens function if it is evaluated at zero time. Using the relation (7.13) we write
�
dω ρd(ω) = −1 π
�
dω �(GRd(ω)) = GRd(t= 0+, B) (7.25) So we can check the conservation of spectral weight throughout the full flow equation trans-formation (i.e. for allB) by studying the change of the retarded Greens function at zero time under the flow.
7.3Conservation of spectral weight 101
7.3.2 Stability of Greens function at zero time
The following calculation will show that the retarded Greens function at zero time does not change under the flow.
In the case of final flow we have expressed the correlator in terms of the generalized coupling constants of an transformed observable. We assume that this expression holds in good approx-imation generically for arbitrary values of the flow parameter. This assumption corresponds to the neglection of any change of the ground state under the flow which we consider to be –in equilibrium– a Fermi sea. Hence the change of the operators becomes the only relevant effect caused by the unitary transformations; it is described by the flowing coupling constants γ and M. Note that the time dependent Greens function is more complicated for arbitrary values of time and of the flow parameter. This is due to the fact that for arbitrary values of B the Hamiltonian is not diagonal and the time evolution of the Heisenberg operators is nontrivial. We make use of the flow equations for the generalized coupling constants as expressed in (6.11) and (6.13).
Inserting the flow equations and a suitable relabelling of the various terms involved shows that all terms which originated from the canonical generator cancel each other directly. Moreover, even the contribution of the extended generator to the flow equation ofγ vanishes due to a symmetry argument. In the following excursion we briefly present the calculation. Inserting the flow equations leads to Relabelling the internal indices in the second summand according to the scheme (5,1,2,2�)→ (1,5,6,5�) clearly shows the cancellation of the second and third term such that only the con-tribution induced by the extension of the generator (the first term) remains. In chapter (5.5.1) we have observed that the flow equation of the scattering amplitudesPs1�s1 is symmetric in its indices. This implies that in total the argument of the summation in the potential scattering term is asymmetric under index permutations5↔s1. Consequently, the potential scattering contribution also vanishes and we read off
i ∂
∂BGRd(t= 0+, B) = 0
102 7 Impurity spectral density
Note that this result holds generically for all values of outer parameters, i.e. (in the usual ground state approximation) even under out-of-equilibrium conditions. It is a feature of the set of differential flow equations and does not refer to any assumptions made for an approximate analytic evaluation. Therefore we expect it to hold in principle even for a more detailed, e.g. numerical solution of the set of flow equations.
Confirmation of unitarity
We add the remark that the analysis is completely the same for a single ground state correlator of the anticommutator between an annihilation and a creation operator (i.e. if no matrix elements of the pre-diagonalising transformation would be present in the Greens function).
The only difference then is in the meaning of the generalized coupling constants. Such an approach would show more clearly that the canonical anticommutator itself does not change under the flow equation procedure. Hence we confirm that up to second order inU a unitary transformation has been implemented (cf. 2.2.2).
7.3.3 Conservation of spectral weight by the flow equations
The invariance of the time zero retarded Green’s function under the flow reflects the con-servation of spectral weight by the flow equations in the chosen trucation scheme and up to second order in the perturbative parameterU. Thus we can write
∂
∂B
�
dωρd(ω) = 0 (7.28)
Hence a calculation which provides an exact solution of the set of differential flow equations which we have set up in chapter 6 conserves the invariance of the spectral density under the flow.
7.3.4 Conservation of spectral weight by the approximate analytical solu-tion
In a similar calculation we can prove the invariance of the spectral weight under the flow even for the approximate analytical solution. We show that the right hand side of equation (7.27) is zero if we insert the explicit solutions (6.14 and 6.25) and their explicit derivatives with respect to the flow parameter.
0=? �
s5
γ5∂γ5
∂B + �
s5�s5s6
M565�↑↓↓Q55�6
∂M565�↑↓↓
∂B (7.29)
We note that the derivative ∂γ∂Bs(B) ∼ O(U2). For a consistent calculation in second order of U we only need to consider the constant term ofγ5(B) =γ5(0)+O(U2) in the first summand.
In the second summand bothM and its derivative ∂M(B)∂B contribute in first order of U and cannot be neglected.
Basic idea of the proof and comments
A striking feature of the following proof is that only symmetry arguments are used to show that equation (7.29) does hold. Therefore it is independent of a correct treatment of the pole
7.3Conservation of spectral weight 103 structure which we have discussed in (6.3.4). Thus we can refer to this result for a meaningful implementation (or correction) of the emerging, formaly diverging integrals.
Moreover we directly see that the symmetry holds for all values of the flow parameter and the conservation of spectral weight cannot be violated for any finite value ofB, i.e. anywhere within the transformation process. The fact that the approximate analytical solutions respects this important sum rule suggests that it is a reliable approach to study the out-of-equilibrium behaviour of the Anderson impurity model within its perturbatively defined validity.
For the interested reader we briefly present the explicit form of the terms involved.
Aspects of the calculation∗
Again we make use of the short-hand notation introduced in (6.3.3) and explicitly denote the analytical expressions and their derivatives. In order to maintain the full symmetry of the terms we refer to their crude form, including all summations.
γ˜5�(B) = γ˜5�(B = 0) + We insert these expressions into (7.29) and make the following replacements of index labels in the second term:
(˜5�˜6�˜512) −→ (1,2,2�xy)
�6˜� −�˜5+�5˜� −→ −a
This relabelling is nothing else but a change of notation.
We observe that the terms ∼ e−B(y+a)2 cancel each other directly and the remaining term
∼e−B[(x+a)2+(y+a)2] can be written in the following form We note that the argument of these summations is asymmetric under interchange ofx andy.
Thus the right hand side vanishes for all values of the flow parameter if unlimited bandwidth is assumed.
104 7 Impurity spectral density