nLab Brillouin torus

Idea

Given a crystalline material, its Brillouin torus is the space of distinguishable momenta/wave vectors of its electronic excitations. Due to the periodicity of the crystal, also these momenta/wave vectors are (dually) periodically identified, which makes this space an n-torus.

Mathematically, the discrete space underlying a crystal is a subset of a Euclidean space $S \;\subset\;E^n \simeq \mathbb{R}^n$ (the set of atomic sites) which is preserved (as a subset) by the action of a crystallographic group $\Lambda \rtimes G \,\subset\, Iso(E)$, where $\Lambda$ is a full lattice (and $G$ the corresponding point group). Then its Brillouin torus is the Pontrjagin dual group of this lattice (eg. Freed-Moore 13):

$$\mathbb{T}^n_{Br} \,=\, Hom_{Grp}\big(\Lambda, \, S^1 \big) \,.$$

Related concepts

• Brillouin zone

• Bravais lattice

• Bloch-Floquet theory

• valence bundle, conduction bundle

References

The concept is named after:

• Léon Brillouin, Les électrons libres dans les métaux et le role des réflexions de Bragg, J. Phys. Radium 1 (1930) 377-400 [doi:10.1051/jphysrad:01930001011037700, hal:jpa-00233038, pdf]

Textbooks on solid state physics traditionally speak of the “reciprocal lattice” (e.g. Kittel 1953, p. 27) which is the dual lattice $Hom_{Grp}\big(\Lambda, \, \mathbb{Z} \big)$, e.g.:

• Michael Reed, Barry Simon, p. 311 of: Sec. XIII.16 Schrödinger operators with periodic potentials, of: Methods of Modern Mathematical Physics – IV: Analysis of Operators, Academic Press (1978)

  (ISBN:9780080570457)

• David Tong, §2.2.2 in: Lectures on solid state physics (2017) $[$pdf, webpage$]$

The resulting Brillouin torus $Hom_{Grp}\big(\Lambda, \, \mathbb{R} \big)/Hom_{Grp}\big(\Lambda, \, \mathbb{Z} \big)$ is often left implicit. It is made explicit for instance in:

• Daniel Freed, Gregory Moore, p. 52 of: Twisted equivariant matter, Ann. Henri Poincaré (2013) 14: 1927 [arXiv:1208.5055, doi:10.1007/s00023-013-0236-x]

• Guo Chuan Thiang, §2.1 in: Topological Semimetals, Encyclopedia of Mathematical Physics 2nd ed (2024) [arXiv:2407.12692]

• Kiyonori Gomi, Guo Chuan Thiang, p. 9 in: Crystallographic T-duality, J. Geom. Phys 139 (2019) 50-77 [doi:10.1016/j.geomphys.2019.01.002, arXiv:1806.11385]

Texts that make explicit the choice of spin structure on Brillouin tori:

• Ümit Ertem: Weyl semimetals and $spin^c$ cobordism [arXiv:2003.04082]

• Andy Knoll, Carsten Timm: Irreducible momentum-space spin structure of Weyl semimetals and its signatures in Friedel oscillations, Phys. Rev. B 109 (2024) 035145 [doi:10.1103/PhysRevB.109.035145]