A What to read for proving that the self-consistent Born approximation is conserving

Summary
How to prove from non-equilibrium field-theory that the retarded, advanced, <, and > self-energies calculated using the self-consistent Born approximation give currents which conserve mass and conserve energy?
I am wondering if there is a book with solved example problems I can follow so that I can carry out the following calculation:

I have a set of unperturbed electron Green functions $g^{r0},g^{a0},g^{<0},g^{>0}$ and phonon Green functions $d^{r0},d^{a0},d^{<0},d^{>0}$ which are for a central region $C$ adjoined by a left and right reservoir $L,R$ each at a temperature $T_{L,R}$ and potential $\mu_{L,R}$. I use the self-consistent Born approximation to calculate $g^r,g^a,g^<,g^>$ and $d^r,d^a,d^<,d^>$ due to an electron/phonon interaction from three diagrams: one Hartree-like diagram and one Fock-like diagram for an electron colliding with a phonon and one polarization-bubble-like diagram for a phonon exciting a charge-neutral electron/hole pair. I get band-densities of number currents $N_{L,R}(\omega)$ and energy currents $E_{L,R}(\omega)$ for charge-carriers of energy $\hbar\omega$ from the equations $i\hbar(dN_{L,R}/dt)=[N_{L,R},H]$ and likewise for $E$.

I am able to prove that $N_L=N_R$ in the absence of interactions. However, I am not sure how to prove this in general.

Note that Annals of Physics 236, 1-42 (1994), Eq. (54) through Eq. (56) does treat this problem except for zero temperature. Finite temperature is critical for what I am doing.

(At this zero temperature case treated in Annals of Physics 236, 1-42 (1994), there are delta functions, and conservation of matter results from a fortuitous cancellation on integrating the band-density $N_{L,R}(\omega)$ to give the number-current $N_{L,R}$).

Van Leuwenn's text seems to give the material I need to solve this problem, but the authors seem to use a GW method. This is different from Hyldgaard et al, whose starting point seem to be Langreth. An earlier paper, Phys Rev B 46 11, published by roughly the same research group give their Eq. (76) and Fig. 1 as justification for current-conservation, which I am having trouble understanding.
 
Dear bjnartowt,

Very interesting question! There are several approaches to show if a considered many-body approximation is conserving. Personally, I find the easiest-to-understand method is to consider the time-dependent case, and specialize to the steady-state case afterwards. In this language, particle conservation means that the continuity equation is satisfied locally at every given time, and in steady-state with two terminals means ##I_L = I_R##. Leeuwen's approach is the one of Baym, and considers generating functionals for the self-energy, ##\Phi##-functionals. In the electronic case, ##\Phi = \Phi[g]## is a functional of the interacting electronic Green's function, but in your case, the functionals will be ##\Phi = \Phi[g,d]##. The electronic self-energy is then ##\Sigma(12) = \frac{\delta \Phi[g,d]}{\delta g(21)}##. Finite temperature is then included as a Matsubara track.

The conservation laws will follow from symmetries of the ##\Phi##-functional. Again, taking particle-number conservation as an example, this will follow from the symmetry (this is taken from memory, could be slightly off) ##\Phi[e^{\Lambda} g e^{-\Lambda},d] = \Phi[g,d]##. ##\Lambda## is a small quantity, and this symmetry (which is fulfilled in your case because all your electron loops are closed) gives the particle conservation (for the electrons!) you want, according to Baym's original argument.

The electron-phonon situation is very similar to the ##GW## approximation if you view the phonon as a screened interaction (electronic self-energy is gd). As such, the ideas from the electronic case applies also to the electron-phonon case.

I hope this helps a bit.
 

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