I've been reading up about the BCS theory of superconductivity, and the treatments I've seen begin rather mysteriously with a Hamiltonian that (in the language of second quantization) looks something like this:
$$\mathcal{H}=\sum_{\vec k\sigma}\xi_{\vec k}c_{\vec k\sigma}^{\dagger}c_{\vec k\sigma}+\sum_{\vec k\vec l}g_{\vec k\vec l}c_{\vec k\uparrow}^{\dagger}c_{-\vec k\downarrow}^{\dagger}c_{-\vec l\downarrow}c_{\vec l\uparrow},$$
where $\sigma \in \{\uparrow,\downarrow\}$ labels possible spin states of an electron, $c_{\vec k\sigma}^{\dagger}$(respectively $c_{\vec k\sigma}$) creates (respectively annihilates) an electron of momentum $\vec k$, $\xi_{\vec k}\equiv\epsilon_{\vec k}-\mu$ is the kinetic energy of an electron of momentum $\vec k$ measured relative to the chemical potential $\mu$ and $g_{\vec k,\vec l}$ is the coupling strength of a (phonon mediated) interaction between an electron of momentum $\vec k$ and an electron of momentum $\vec l$.
Now the first, "kinetic" term represents the kinetic energy of the electrons after accounting for the band structure, and I think I understand it alright. I have some doubts regarding the second "interaction" term.
- How do we compute $g_{\vec k,\vec l}$? Are there models which allow us to explicitly see how properties of the lattice (e.g. isotope mass) affect the strength of the interaction? Or do we simply extract it from experimental measurements?
- I'm relatively new to the language of second quantization, so could someone explain to me why this sequence of creation and annihilation operators (in this particular order) describes a phonon mediated interaction between an electron of momentum $\vec k$ and an electron of momentum $\vec l$?
- It seems like we're only including terms corresponding to interactions between pairs which have total spin 0. (Am I reading this term wrong somehow?) Why can't we have spin-1 quasiparticles condensing into a charge carrying "superfluid" ground state?
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