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. For that matter, how do we know that the coupling is attractive? We have a qualitative justification based on how we intuitively understand the formation of cooper pairs (i.e. as the first electron moves though the lattice it leaves in its wake a positively charged disturbance to which the second electron is attracted), but is there a more rigorous treatment of this?
3. 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 $\vec l$?
4. 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?
5. In one of the treatments I came across, they talked about restricting the Hamiltonian further to interactions involving pairs with 0 total linear momentum, i.e. $\vec l=-\vec k$. Does this seem right? Why would you do this?
6. As a more general question about the Hamiltonian framework, why (and when) is it ok to leave out certain terms? For instance, in this case we do not the kinetic term for the phonons themselves (in fact I am not clear how this Hamiltonian works when it doesn't include phonon creation/annihilation operators at all), and we leave out several interactions, some of which are stronger than the Cooper pairing interaction, e.g., the Coulomb repulsion between electrons.

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?

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. For that matter, how do we know that the coupling is attractive? We have a qualitative justification based on how we intuitively understand the formation of cooper pairs (i.e. as the first electron moves though the lattice it leaves in its wake a positively charged disturbance to which the second electron is attracted), but is there a more rigorous treatment of this?
3. 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 $\vec l$?
4. 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?
5. In one of the treatments I came across, they talked about restricting the Hamiltonian further to interactions involving pairs with 0 total linear momentum, i.e. $\vec l=-\vec k$. Does this seem right? Why would you do this?
6. As a more general question about the Hamiltonian framework, why (and when) is it ok to leave out certain terms? For instance, in this case we do not the kinetic term for the phonons themselves (in fact I am not clear how this Hamiltonian works when it doesn't include phonon creation/annihilation operators at all), and we leave out several interactions, some of which are stronger than the Cooper pairing interaction, e.g., the Coulomb repulsion between electrons.

Thanks in advance!

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. For that matter, how do we know that the coupling is attractive? We have a qualitative justification based on how we intuitively understand the formation of cooper pairs (i.e. as the first electron moves though the lattice it leaves in its wake a positively charged disturbance to which the second electron is attracted), but is there a more rigorous treatment of this?
3. 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 $\vec l$?
4. 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?
5. In one of the treatments I came across, they talked about restricting the Hamiltonian further to interactions involving pairs with 0 total linear momentum, i.e. $\vec l=-\vec k$. Does this seem right? Why would you do this?
6. As a more general question about the Hamiltonian framework, why (and when) is it ok to leave out certain terms? For instance, in this case we do not the kinetic term for the phonons themselves (in fact I am not clear how this Hamiltonian works when it doesn't include phonon creation/annihilation operators at all), and we leave out several interactions, some of which are stronger than the Cooper pairing interaction, e.g., the Coulomb repulsion between electrons.