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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeMay 6th 2022

a stub entry, for the moment, just to make links work

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeMay 11th 2022

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.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeMay 11th 2022

Maybe I have found something in this direction:

This old article

• Yuejin Guo, Jean-Marc Langlois, William A. Goddard: Electronic Structure and Valence-Bond Band Structure of Cuprate Superconducting Materials, Science, New Series 239 4842 (1988) 896-899 $[$jstor:1700316$]$

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:

• C. David Sherrill, Henry F. Schaefer, The Configuration Interaction Method: Advances in Highly Correlated Approaches, Advances in Quantum Chemistry 34 (1999) 143-269 $[$doi:https://doi.org/10.1016/S0065-3276(08)60532-8$]$

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)

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeMay 12th 2022

Have forwarded the question to Physics.SE

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeMay 12th 2022

Found this one relevant article here:

• Jingsan Hu, Jianfei Gu, Weiyi Zhang, Bloch’s band structures of a pair of interacting electrons in simple one- and two-dimensional lattices, Physics Letters A 414 (2021) 127634 $[$doi:10.1016/j.physleta.2021.127634$]$

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…