Dimension reduction techniques for the minimization of theta functions on lattices

Abstract

We consider the minimization of theta functions $\theta \_\Lambda (\alpha )=\sum \_p\in \Lambda e^{-\pi \alpha |p{|}^2}$ amongst lattices $\Lambda \subset {\mathbb{R}}^d$, by reducing the dimension of the problem, following as a motivation the case $d=3$, where minimizers are supposed to be either the BCC or the FCC lattices. A first way to reduce dimension is by considering layered lattices, and minimize either among competitors presenting different sequences of repetitions of the layers, or among competitors presenting different shifts of the layers with respect to each other. The second case presents the problem of minimizing theta functions also on translated lattices, namely minimizing $(L,u)\mapsto \theta \_L+u(\alpha )$. Another way to reduce dimension is by considering lattices with a product structure or by successively minimizing over concentric layers. The first direction leads to the question of minimization amongst orthorhombic lattices, whereas the second is relevant for asymptotics questions, which we study in detail in two dimensions.

45 pages. 7 figures

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• On minima of sum of theta functions and Mueller-Ho Conjecture
• On energy ground states among crystal lattice structures with prescribed bonds
• An Extremal Property of the Hexagonal Lattice
• Effect of periodic arrays of defects on lattice energy minimizers
• Local variational study of 2d lattice energies and application to Lennard-Jones type interactions
• Euclidean lattices, theta invariants, and thermodynamic formalism
• A note on energy minimization in dimension 2