Brillouin Zones of Integer Lattices and Their Perturbations

Abstract

For a locally finite set, $A\subseteq {\mathbb{R}}^d$, the $k$-th Brillouin zone of $a\in A$ is the region of points $x\in {\mathbb{R}}^d$ for which $\Vert x-a\Vert$ is the $k$-th smallest among the Euclidean distances between $x$ and the points in $A$. If $A$ is a lattice, the $k$-th Brillouin zones of the points in $A$ are translates of each other, which tile space. Depending on the value of $k$, they express medium- or long-range order in the set. We study fundamental geometric and combinatorial properties of Brillouin zones, focusing on the integer lattice and its perturbations. Our results include the stability of a Brillouin zone under perturbations, a linear upper bound on the number of chambers in a zone for lattices in ${\mathbb{R}}^2$, and the convergence of the maximum volume of a chamber to zero for the integer lattice.