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I do want to eventually add content to this page, but I also have a question beyond what should go here:
Is there any discussion of a kind of interacting version of Bloch theory, where one would consider eigenstates of interacting tuples of $n$ electrons in a crystal, for any $n$?
Such that the resulting “$n$-Bloch bundle” would be a Hilbert bundle not over the Brillouin torus, but over the configuration space of points in the Brillouin torus?
Has this been considered at all? I gather if it has, then not under these keywords.
Maybe I have found something in this direction:
This old article
speaks of “$N$-electron band theory” in explicit contrast to the standard “one-electron band” theory. The authors don’t dwell much on the details, but apparently they are referring to the “configuration interaction method” of which a slightly more recent monograph account is in:
This looks like it goes in the right direction.
For instance, it’s fun to observe that FSV 94’s main construction (3.3.1) is a Slater determinant! (eg. (3) in the above textbook)
Have forwarded the question to Physics.SE
Found this one relevant article here:
These authors consider “Bloch theory of 2-electron systems” and compute “bands” which are now functions not on the Brillouin torus, but its product space (i.e. depending not on one but on a pair of Bloch momenta).
That’s the kind of discussion that I am looking to see in the literature.
Next we’d want to argue that the 2-electron quantum states that correspond to these bands form a vector bundle over the configuration space of 2 points in the Brillouin torus. That’s pretty obvious (notice how the config space instead of the product space is necessary here to account for the vanishing of all 2-electron wavefunctions at coinciding momenta, which rules out such a vector bundle over the full product space).
But I am getting the impression that nobody has considered this before…
added these pointers:
Joseph Maciejko, Steven Rayan, Hyperbolic band theory, Science Advances 7 36 (2021) [doi:10.1126/sciadv.abe9170]
Adil Attar, Igor Boettcher, Selberg trace formula in hyperbolic band theory, Phys. Rev. E 106 034114 (2022) [arXiv:2201.06587, doi:10.1103/PhysRevE.106.034114]
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