Brillouin Zones of Integer Lattices and Their Perturbations

Authors

Herbert Edelsbrunner
Alexey Garber, The University of Texas Rio Grande ValleyFollow
Mohadese Ghafari
Teresa Heiss
Morteza Saghafian
Mathijs Wintraecken

Document Type

Article

Publication Date

2024

Abstract

For a locally finite set, 𝐴⊆ℝ𝑑 , the 𝑘 th Brillouin zone of 𝑎∈𝐴 is the region of points 𝑥∈ℝ𝑑 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, and together they 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 ℝ2 , and the convergence of the maximum volume of a chamber to zero for the integer lattice.

Recommended Citation

Edelsbrunner, Herbert, Alexey Garber, Mohadese Ghafari, Teresa Heiss, Morteza Saghafian, and Mathijs Wintraecken. "Brillouin zones of integer lattices and their perturbations." SIAM Journal on Discrete Mathematics 38, no. 2 (2024): 1784-1807. https://doi.org/10.1137/22M1489071

Publication Title

SIAM Journal on Discrete Mathematics

DOI

https://doi.org/10.1137/22M1489071