System of green functions in solid state physics

[siver]

Assume we have a solid state hamiltonian, which is ellectrons plus nuclei plus all possible interactions:

H = $T_{el}$ + $V_{el-el}$ + $T_{n}$ + $V_{n-n}$ + $V_{n-el}$​

Is it wrong to write down a self-consistent system of Green functions for electrons and phonons?

$G_{el}$=g + gS$G_{el}$
$G_{ph}$=g + gS$G_{ph}$​

S operator does the interraction thing.
Looks fair to me, but I've never seen any articles doing this way.

 

[sam_bell]

If by little 'g' you mean free particle propagator, then it's not wrong --for the electrons. The operator 's' is then the self-energy operator. You can look it up. It's not as obvious to me what little 'g' is for phonons. The typical derivation of phonons has already taken into account effects of electrical interactions in the "spring force" between ions. Maybe you could take the 's' in the case of phonons to represent all the deviations from this ideal case.

p.s. you need to use a different symbol 'g' for electrons and phonons, or you'll get confused.

 

[siver]

Thanks for reply!

Sure thing g's (free particle propagators) are different for different particle types.
I'm worried if this "system" of two equations eve exist. Maybe they're just independent equations with totally diferent self-patrs for each quasiparticle. And it's pointless to solve them as a system.

 

[sam_bell]

It's best to use the Born-Oppenheimer approximation. In other words, first consider the locations of the nuclei fixed, and then solve the corresponding electronic problem. This will give you an electronic self-energy that is dependent on the nuclear coordinates. It will also define an effective potential for the nuclear problem according to V(R1, ..., RN) = (total ground state energy when nuclei at coordinates R1, ..., RN). This in turn defines a new problem for the nuclear coordinates. This should be solved in two stages. First stage is harmonic approximation (giving you phonons and little 'g'), and second stage is finding phonon self-energy resulting from anharmonic deviations. Not taking advantage of the Born-Oppenheimer approximation is too ambitious. Only if it fails should you consider chucking it.

 

[siver]

To avoid BO approx is the main perpose, my system likes to dance. =)
Thanks alot.