[drdeca]

Yes. The theory that explains the better understood ones works based on the idea of "cooper pairs", where there are, aiui, entangled pairs of electrons which are in a lower energy state due to how they are entangled, and where in order to get like, bumped around in a way that would produce resistance, they would have to change in energy in a way by an amount that isn't available?

So, very much a quantum mechanical effect.

[cyberax]

Cooper pairs are not entangled. They stick together because the first electron causes atoms to clump a little bit, so that they produce a locally higher positive charge, that in turn attracts the other electron.

[drdeca]

Are you sure? I mean, not the second part. That part matches my impression of things. But them not being entangled. My understanding is that collectively, the pair of electrons acts like a boson, unlike an individual electron, which allows multiple such pairs to be in the same state, and, I would expect that this would require the electrons in each pair to be entangled?

Because reasoning about the real deal sounds like it would be rather difficult for me (and likely too difficult for me without significant assistance, at least, within any reasonable about of time for me to spend on this), I want to make up a toy model now.

Say I have a single-electron Hilbert space and a single electron Hamiltonian on it (where, I would later maybe add terms for interactions between the electrons, and/or a space for how the surrounding material can change in ways tied to the electrons). Below a certain energy level, I want to say that the eigenspace of the single electron Hamiltonian for each lower energy level, is finite-dimensional, and also the spectrum is discrete. (and, I want to consider the electrons only with energies below this threshold.)

For any wavefunction in the single-electron Hilbert space, there is a corresponding creation operator acting on the Fock space.

Now, I imagine that without any interactions, the concept of a cooper pair is probably inapplicable. But, I am imagining that when we add in the interactions, that, at least at low energies, the Fock space obtained from the single-electron Hilbert space, should work as a Hilbert space for the many-particle system with interactions between electrons (and then, I guess take the tensor-product with another Hilbert space representing how nuclei can change, when handling that part).

Everything in the Fock space can be obtained as a linear combination of applications of some number of creation operators to the vacuum state. As such, a state with a single cooper pair can be expressed as a linear combination of states obtained by applying two creation operators to the vacuum state.

Unless this "linear combination" can be done with only a single term, this would be an entangled state, right?

I would think that there should be creation operators for cooper pairs, consisting of a linear combination of products of two creation operators for electrons?

[cyberax]

You don't need them to be entangled to act as a boson.

A Cooper pair is a composite object, and you absolutely need to consider the atomic lattice of the superconductor.

So you end up with your electron being entangled with all the surrounding atoms in the lattice. And all other electrons.

[drdeca]

Ah, so it isn't so much that they don't need to be entangled, so much as, describing them as entangled isn't a good way to frame it, because all the electrons (and the rest of the material) will be all entangled with each-other, so, it isn't a distinguishing feature, and misleading to focus on the two in the pair being entangled?

[cyberax]

Yes, pretty much.