Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory object of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 6th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexa stub entry, for the moment, just to make links work <a href="https://ncatlab.org/nlab/revision/Bloch%27s+theorem/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Bloch%27s+theorem">current</a>

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

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMay 11th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexI 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.

   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.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 11th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexMaybe 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](https://www.jstor.org/stable/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 $[$<a href="https://doi.org/10.1016/S0065-3276(08)60532-8">doi:https://doi.org/10.1016/S0065-3276(08)60532-8</a>$]$ This looks like it goes in the right direction. For instance, it's fun to observe that [FSV 94](https://ncatlab.org/nlab/show/hypergeometric+KZ-solutions+--+references#FeiginSchechtmanVarchenko94)'s main construction (3.3.1) is a *Slater determinant*! (eg. (3) in the above textbook)

   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)

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeMay 12th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexHave forwarded the question to [Physics.SE](https://physics.stackexchange.com/q/708298/5603)

   Have forwarded the question to Physics.SE

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMay 12th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexFound 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](https://doi.org/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...

   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…

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeSep 12th 2022
   • (edited Sep 12th 2022)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded these pointers: * Joseph Maciejko, Steven Rayan, *Hyperbolic band theory*, Science Advances **7** 36 (2021) &lbrack;[doi:10.1126/sciadv.abe9170](https://doi.org/10.1126/sciadv.abe9170)&rbrack; * Adil Attar, Igor Boettcher, *Selberg trace formula in hyperbolic band theory*, Phys. Rev. E **106** 034114 (2022) &lbrack;[arXiv:2201.06587](https://arxiv.org/abs/2201.06587), [doi:10.1103/PhysRevE.106.034114](https://doi.org/10.1103/PhysRevE.106.034114)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/Bloch%27s+theorem/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/Bloch%27s+theorem/6">v6</a>, <a href="https://ncatlab.org/nlab/show/Bloch%27s+theorem">current</a>

   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]

   diff, v6, current