basics
Examples
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):
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)
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 in:
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