Origin of Symmetry Breaking in the Grasshopper Model
Abstract
The planar grasshopper problem, originally introduced in (Goulko & Kent 2017 Proc. R. Soc. A 473, 20170494), is a striking example of a model with long-range isotropic interactions whose ground states break rotational symmetry. In this work we analyze and explain the nature of this symmetry breaking with emphasis on the importance of dimensionality. Interestingly, rotational symmetry is recovered in three dimensions for small jumps, which correspond to the non-isotropic cogwheel regime of the two-dimensional problem. We discuss simplified models that reproduce the symmetry properties of the original system in N dimensions. For the full grasshopper model in two dimensions we obtain quantitative predictions for optimal perturbations of the disk. Our analytical results are confirmed by numerical simulations.
Keywords
Cite
@article{arxiv.2311.05023,
title = {Origin of Symmetry Breaking in the Grasshopper Model},
author = {David Llamas and Jaron Kent-Dobias and Kun Chen and Adrian Kent and Olga Goulko},
journal= {arXiv preprint arXiv:2311.05023},
year = {2024}
}
Comments
Ancillary files with animations of 3d shapes included