Brillouin Zones of Integer Lattices and Their Perturbations

TL;DR

This paper explores the geometric and combinatorial properties of Brillouin zones in integer lattices and their perturbations, revealing stability, bounds, and volume convergence behaviors.

Contribution

It introduces new results on the stability under perturbations, bounds on the number of chambers, and volume convergence of Brillouin zones in lattices.

Findings

Brillouin zones are stable under perturbations.

Number of chambers in 2D lattices is linearly bounded.

Maximum chamber volume converges to zero for the integer lattice.

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 $\|x-a\|$ 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.