Brillouin Zones of Integer Lattices and Their Perturbations
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866913476921786368 |
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| author | Edelsbrunner, Herbert Garber, Alexey Ghafari, Mohadese Heiss, Teresa Saghafian, Morteza Wintraecken, Mathijs |
| author_facet | Edelsbrunner, Herbert Garber, Alexey Ghafari, Mohadese Heiss, Teresa Saghafian, Morteza Wintraecken, Mathijs |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_01077 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Brillouin Zones of Integer Lattices and Their Perturbations Edelsbrunner, Herbert Garber, Alexey Ghafari, Mohadese Heiss, Teresa Saghafian, Morteza Wintraecken, Mathijs Combinatorics Computational Geometry Metric Geometry 52C22, 52C35, 05B45 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. |
| title | Brillouin Zones of Integer Lattices and Their Perturbations |
| topic | Combinatorics Computational Geometry Metric Geometry 52C22, 52C35, 05B45 |
| url | https://arxiv.org/abs/2204.01077 |