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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 6th 2022

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

    v1, current

    • 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 nn electrons in a crystal, for any nn?

    Such that the resulting “nn-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 “NN-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)

    • 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…

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