Questions concerning BCS theory (particularly the "pairing Hamiltonian")

Question

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.

1. 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?
2. 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 $\vec l$?
3. 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?

Answer 1

1. The BCS Hamiltonian is derived from a Hamiltonian describing, in particular, electron-phonon interaction. I read the (rather cumbersome) derivation in A.S. Davydov's "Quantum Mechanics", but I am sure it can be found in many other places.

Answer 2

It might make more sense to address your questions in non-numerical order.

To answer your second question, this isn't an interaction between an electron with momentum $k$ and another with momentum $l$. It describes the scattering of two electrons with opposite spin and momenta $\pm l$ to momenta $\pm k$. You destroy an up electron with momentum $l$ and a down electron with momentum $-l$ and create an up electron with momentum $k$ and a down electron with momentum $-k$.

Addressing your third question, this model is based upon the assumption that the electrons pair with opposite momentum and opposite spin. That is, that the Cooper pair has a net spin of zero and a net momentum of zero. This does not necesarily have to be the case, but was assumed to be the case in BCS theory. There are a few justifications for making this initial assumption:

1. Typical phonon energies are much smaller than the Fermi energy. This means that an iteraction with a phonon will only scatter electrons within a thin shell around the Fermi-surface. If you imagine two rings. Their overlap is greatest when they are centred on each other and rapidly decreases as they are seperated. The cross-section for interaction with phonons is large if the momentum is zero.

2. If you are looking for a lowest energy configuration, the zero-momentum and zero-spin case seems like a sensible start.

As for the form of $g_{kl}$, there isn't an all-encompasing theory in the same way that we can calculate band structures very accurately from atomic positions. There are, of course, theories for calculating general coupling strengths. Eliashberg theory, in brief, takes the electronic and phononic densities and computes a coupling strength based upon the probabilites of scattering from one electronic state to another via a phonon.

You need to look further into triplet pairing and p-wave superconductivity to hear more about spin-1 pairs. Its a hugely active area of research and I think it is fair to say that the jury is out on whether it is actually realised in the candidate materials.