- 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.

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.